A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems

Research ArticleA Numerical Iterative Method for Solving Systems ofFirst-Order Periodic Boundary Value Problems

Mohammed AL-Smadi1 Omar Abu Arqub2 and Ahmad El-Ajou2

1 Applied Science Department Ajloun College Al-Balqa Applied University Ajloun 26816 Jordan2Department of Mathematics Al-Balqa Applied University Salt 19117 Jordan

Correspondence should be addressed to Omar Abu Arqub oabuarqubbauedujo

Received 12 September 2013 Accepted 11 February 2014 Published 25 March 2014

Academic Editor Hak-Keung Lam

Copyright copy 2014 Mohammed AL-Smadi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Theobjective of this paper is to present a numerical iterativemethod for solving systems of first-order ordinary differential equationssubject to periodic boundary conditionsThis iterative technique is based on the use of the reproducing kernelHilbert spacemethodin which every function satisfies the periodic boundary conditions The present method is accurate needs less effort to achieve theresults and is especially developed for nonlinear case Furthermore the present method enables us to approximate the solutionsand their derivatives at every point of the range of integration Indeed three numerical examples are provided to illustrate theeffectiveness of the present method Results obtained show that the numerical scheme is very effective and convenient for solvingsystems of first-order ordinary differential equations with periodic boundary conditions

1 Introduction

Systems of ordinary differential equations with periodicboundary value conditions the so-called periodic boundaryvalue problems (BVPs) are well known for their applicationsin sciences and engineering [1ndash5] In this paper we focuson finding approximate solutions to systems of first-orderperiodic BVPs which are a combination of systems of first-order ordinary differential equations and periodic boundaryconditions In fact accurate and fast numerical solutions ofsystems of first-order periodic BVPs are of great importancedue to their wide applications in scientific and engineeringresearch

Numericalmethods are becomingmore andmore impor-tant in mathematical and engineering applications simplynot only because of the difficulties encountered in findingexact analytical solutions but also because of the ease withwhich numerical techniques can be used in conjunctionwith modern high-speed digital computers A numericalprocedure for solving systems of first-order periodic BVPs

based on the use of reproducing kernel Hilbert space (RKHS)method is discussed in this work

Among a substantial number of works dealing withsystems of first-order periodic BVPs we mention [6ndash10]The existence of solutions to systems of first-order periodicBVPs has been discussed as described in [6] In [7] theauthors have discussed some existence anduniqueness resultsof periodic solutions for first-order periodic differentialsystems Also in [8] the authors have provided the existencemultiplicity and nonexistence of positive periodic solutionsfor systems of first-order periodic BVPs Furthermore theexistence of periodic solutions for the coupled first-orderdifferential systems of Hamiltonian type is carried out in [9]Recently the existence of positive solutions for systems offirst-order periodic BVPs is proposed in [10] Formore resultson the solvability analysis of solutions for systems of first-order periodic BVPs we refer the reader to [11ndash15] and fornumerical solvability of different categories of BVPs one canconsult [16ndash19]

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 135465 10 pageshttpdxdoiorg1011552014135465

2 Journal of Applied Mathematics

Investigation about systems of first-order periodic BVPsnumerically is scarce In this paper we utilize a methodicalway to solve these types of differential systems In factwe provide criteria for finding the approximate and exactsolutions to the following system

1199061015840

1(119909) = 119865

1(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

1199061015840

2(119909) = 119865

2(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

1199061015840

119899(119909) = 119865

119899(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

(1)

subject to the periodic boundary conditions

1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

119906119899(0) = 119906

119899(1)

(2)

where 119909 isin [0 1] 119906119904isin 1198822

2[0 1] are unknown functions to

be determined 119865119904(119909 V1 V1 V

119899) are continuous terms in

1198821

2[0 1] as V

119904= V119904(119909) isin 119882

2

2[0 1] 0 le 119909 le 1 minusinfin lt V

119904lt

infin in which 119904 = 1 2 119899 and 1198821

2[0 1] 1198822

2[0 1] are two

reproducing kernel spaces Here we assume that (1) subjectto the periodic boundary conditions (2) has a unique solutionon [0 1]

Reproducing kernel theory has important applications innumerical analysis differential equations integral equationsprobability and statistics and so forth [20ndash22] In the lastyears extensive work has been done using RKHS methodwhich provides numerical approximations for linear andnonlinear equations This method has been implemented inseveral operator differential integral and integrodifferentialequations side by side with their theories The reader iskindly requested to go through [23ndash35] in order to knowmore details about RKHS method including its history itsmodification for use its applications and its characteristics

The rest of the paper is organized as follows In the nextsection two reproducing kernel spaces are described in orderto formulate the reproducing kernel functions In Section 3some essential results are introduced and a method for theexistence of solutions for (1) and (2) is described In Section 4we give an iterative method to solve (1) and (2) numericallyNumerical examples are presented in Section 5 Section 6ends this paper with brief conclusions

2 Construct of Reproducing Kernel Functions

In this section two reproducing kernels needed are con-structed in order to solve (1) and (2) using RKHS methodBefore the construction we utilize the reproducing kernelconcept Throughout this paper C is the set of complexnumbers 1198712[119886 119887] = 119906 | int

119887

1198861199062(119909)119889119909 lt infin and 119897

2= 119860 |

suminfin

119894=1(119860119894)2lt infin

Definition 1 (see [23]) Let 119864 be a nonempty abstract set Afunction119877 119864times119864 rarr C is a reproducing kernel of theHilbertspace119867 if

(1) for each 119909 isin 119864 119877(sdot 119909) isin 119867(2) for each 119909 isin 119864 and 120593 isin 119867 ⟨120593(sdot) 119877(sdot 119909)⟩ = 120593(119909)

Remark 2 Condition (2) in Definition 1 is called ldquothe repro-ducing propertyrdquo which means that the value of the function120593 at the point 119909 is reproducing by the inner product of 120593(sdot)with 119877(sdot 119909) A Hilbert space which possesses a reproducingkernel is called a RKHS

To solve (1) and (2) using RKHS method we first defineand construct a reproducing kernel space 119882

2

2[0 1] in which

every function satisfies the periodic boundary condition119906(0) = 119906(1) After that we utilize the reproducing kernelspace 119882

1

2[0 1]

Definition 3 The inner product space 1198822

2[0 1] is defined as

1198822

2[0 1] = 119906(119909) | 119906 119906

1015840 are absolutely continuous real-valuedfunctions on [0 1] 119906 1199061015840 11990610158401015840 isin 119871

2[0 1] and 119906(0) = 119906(1) On

the other hand the inner product and the norm in 1198822

2[0 1]

are defined respectively by

⟨119906 V⟩1198822

2

=

1

sum

119894=0

119906(119894)

(0) V(119894) (0) + int

1

0

11990610158401015840

(119905) V10158401015840 (119905) 119889119905 (3)

and 1199061198822

2

= radic⟨119906 119906⟩1198822

2

where 119906 V isin 1198822

2[0 1]

It is easy to see that ⟨119906 V⟩1198822

2

satisfies all the require-ments for the inner product First ⟨119906 119906⟩

1198822

2

ge 0 Second⟨119906 V⟩1198822

2

= ⟨V 119906⟩1198822

2

Third ⟨120574119906 V⟩1198822

2

= 120574⟨119906 V⟩1198822

2

Fourth⟨119906 + 119908 V⟩

1198822

2

= ⟨119906 V⟩1198822

2

+ ⟨119908 V⟩1198822

2

where 119906 V 119908 isin 1198822

2[0 1]

It therefore remains only to prove that ⟨119906 119906⟩1198822

2

= 0 if andonly if 119906 = 0 In fact it is obvious that when 119906 = 0 then⟨119906 119906⟩

1198822

2

= 0 On the other hand if ⟨119906 119906⟩1198822

2

= 0 then by

(3) we have ⟨119906 119906⟩1198822

2

= sum1

119894=0(119906(119894)(0))

2

+ int

1

0(11990610158401015840(119905))

2

119889119905 = 0therefore 119906(0) = 119906

1015840(0) = 0 and 119906

10158401015840(119905) = 0 Then we can

obtain 119906 = 0

Definition 4 (see [23]) The Hilbert space 1198822

2[0 1] is called

a reproducing kernel if for each fixed 119909 isin [0 1]there exist 119877(119909 119910) isin 119882

2

2[0 1] (simply 119877

119909(119910)) such that

⟨119906(119910) 119877119909(119910)⟩1198822

2

= 119906(119909) for any 119906(119910) isin 1198822

2[0 1] and 119910 isin

[0 1]

An important subset of the RKHSs is the RKHSs asso-ciated with continuous kernel functions These spaces havewide applications including complex analysis harmonicanalysis quantum mechanics statistics and machine learn-ing

Theorem 5 The Hilbert space 1198822

2[0 1] is a complete repro-

ducing kernel and its reproducing kernel function 119877119909(119910) can be

written as119877119909(119910)

=

1199011(119909) + 119901

2(119909) 119910 + 119901

3(119909) 1199102+ 1199014(119909) 1199103 119910 le 119909

1199021(119909) + 119902

2(119909) 119910 + 119902

3(119909) 1199102+ 1199024(119909) 1199103 119910 gt 119909

(4)

Journal of Applied Mathematics 3

where 119901119894(119909) and 119902

119894(119909) 119894 = 1 2 3 4 are unknown coefficients

of 119877119909(119910) and will be given in the following proof

Proof The proof of the completeness and reproducingproperty of 119882

2

2[0 1] is similar to the proof in

[24] Now let us find out the expression form ofthe reproing kernel function 119877

119909(119910) in the space

1198822

2[0 1] Through several integration by parts we have

int

1

011990610158401015840(119910)1205973

119910119877119909(119910)119889119910 = sum

1

119894=0(minus1)1minus119894

119906(119894)(119910)1205973minus119894

119910119877119909(119910)|119910=1

119910=0+

int

1

0119906(119910)120597

4

119910119877119909(119910)119889119910 Thus from (3) we can write

⟨119906(119910) 119877119909(119910)⟩1198822

2

= sum1

119894=0119906(119894)(0)[120597119894

119910119877119909(0) + (minus1)

1198941205973minus119894

119910119877119909(0)]

+ sum1

119894=0(minus1)1minus119894

119906(119894)(1)1205973minus119894

119910119877119909(1) + int

1

0119906(119910)120597

4

119910119877119909(119910)119889119910 Since

119877119909(119910) isin 119882

2

2[0 1] it follows that 119877

119909(0) = 119877

119909(1) also since

119906(119909) isin 1198822

2[0 1] it follows that 119906(0) = 119906(1) Then

⟨119906 (119910) 119877119909(119910)⟩1198822

2

=

1

sum

119894=0

119906(119894)

(0) [120597119894

119910119877119909(0) + (minus1)

1198941205973minus119894

119910119877119909(0)]

+

1

sum

119894=0

(minus1)119894+1

119906(119894)

(1) 1205973minus119894

119910119877119909(1)

+ int

1

0

119906 (119910) 1205974

119910119877119909(119910) 119889119910 + 119888

1(119906 (0) minus 119906 (1))

(5)

But on the other aspect as well if 1205972

119910119877119909(1) = 0

119877119909(0) + 120597

3

119910119877119909(0) + 119888

1= 0 1205971

119910119877119909(0) minus 120597

2

119910119877119909(0) = 0 and

1205973

119910119877119909(1) + 119888

1= 0 then (5) implies that ⟨119906(119910) 119877

119909(119910)⟩1198822

2

=

int

1

0119906(119910)120597

4

119910119877119909(119910)119889119910 Now for any 119909 isin [0 1] if 119877

119909(119910) satisfies

1205974

119910119877119909(119910) = minus120575 (119909 minus 119910) 120575 dirac-delta function (6)

then ⟨119906(119910) 119877119909(119910)⟩1198822

2

= 119906(119909) Obviously 119877119909(119910) is the

reproducing kernel function of the space 1198822

2[0 1] Next we

give the expression form of the reproducing kernel function119877119909(119910) The characteristic formula of (6) is given by 120582

4= 0

Then the characteristic values are 120582 = 0 with multiplicity4 So let the expression form of the reproducing kernelfunction 119877

119909(119910) be as defined in (4) On the other hand for

(6) let 119877119909(119910) satisfy the equation 120597

119898

119910119877119909(119909+0) = 120597

119898

119910119877119909(119909minus0)

119898 = 0 1 2 Integrating 1205976

119910119877119909(119910) = minus120575(119909minus119910) from 119909minus120576 to 119909+120576

with respect to 119910 and letting 120576 rarr 0 we have the jump degreeof 1205975119910119877119909(119910) at 119910 = 119909 given by 120597

3

119910119877119909(119909 + 0) minus 120597

3

119910119877119909(119909 minus 0) = minus1

Through the last descriptions the unknown coefficients of(4) can be obtained However by using MAPLE 13 softwarepackage the representation form of the reproducing kernelfunction 119877

119909(119910) is provided by

119877119909(119910) =

1

48

(1199093119910 (6 + 3119910 minus 119910

2) + 3119909

2119910 (minus6 minus 3119910119910

2) + 6119909119910 (2 + 119910 + 119910

2) minus 8 (minus6 + 119910

3)) 119910 le 119909

1

48

(48 + 6119909119910 (2 minus 3119910 + 1199102) + 3119909

2119910 (2 minus 3119910 + 119910

2) minus 1199093(8 minus 6119910 minus 3119910

2+ 1199103)) 119910 gt 119909

(7)

This completes the proof

Definition 6 (see [25]) The inner product space 1198821

2[0 1] is

defined as11988212[0 1] = 119906(119909) | 119906 is absolutely continuous real-

valued function on [0 1] and 1199061015840

isin 1198712[0 1] On the other

hand the inner product and the norm in1198821

2[0 1] are defined

respectively by ⟨119906(119909) V(119909)⟩1198821

2

= 119906(0)V(0) + int

1

01199061015840(119909)V1015840(119909)119889119909

and 1199061198821

2

= radic⟨119906 119906⟩1198821

2

where 119906 V isin 1198821

2[0 1]

Theorem7 (see [25]) TheHilbert space11988212[0 1] is a complete

reproducing kernel and its reproducing kernel function 119866119909(119910)

can be written as

119866x (119910) =

1 + 119910 119910 le 119909

1 + 119909 119910 gt 119909

(8)

Reproducing kernel functions possess some importantproperties such as being symmetric unique and nonnega-tive The reader is asked to refer to [23ndash35] in order to knowmore details about reproducing kernel functions includingtheir mathematical and geometrical properties their typesand kinds and their applications andmethod of calculations

3 Formulation of Linear Operator

In this section the formulation of a differential linear oper-ator and the implementation method are presented in thereproducing kernel space 119882

2

2[0 1] After that we construct

an orthogonal function system of the space 1198822

2[0 1] based

on the use of the Gram-Schmidt orthogonalization processin order to obtain the exact and approximate solutions of (1)and (2) using RKHS method

First as in [23ndash35] we transform the problem into adifferential operator To do this we define a differentialoperator 119871 as 119871 119882

2

2[0 1] rarr 119882

1

2[0 1] such that 119871119906(119909) =

1199061015840(119909) As a result (1) and (2) can be converted into the

equivalent form as follows

119871119906119904(119909) = 119865

119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

119906119904(0) minus 119906

119904(1) = 0

(9)

where 0 le 119909 le 1 and 119904 = 1 2 119899 in which 119906119904(119909) isin 119882

2

2[0 1]

and 119865119904(119909 V1 V1 V

119899) isin 119882

1

2[0 1] for V

119904= V119904(119909) isin 119882

2

2[0 1]

minusinfin lt V119904lt infin and 0 le 119909 le 1 It is easy to show that 119871 is


a bounded linear operator from the space 1198822

2[0 1] into the

space1198821

2[0 1]

Initially we construct an orthogonal function system of1198822

2[0 1] To do so put 120593

119894(119909) = 119866

119909119894

(119909) and 120595119894(119909) = 119871

lowast120593119894(119909)

where 119909119894infin

119894=1is dense on [0 1] and 119871

lowast is the adjoint operatorof119871 In terms of the properties of reproducing kernel function119866119909(119910) one obtains ⟨119906

119904(119909) 120595

119894(119909)⟩1198822

2

= ⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

=

⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) 119894 = 1 2 119904 = 1 2 119899

For the orthonormal function system 120595119894(119909)infin

119894=1of the

space 1198822

2[0 1] it can be derived from the Gram-Schmidt

orthogonalization process of 120595119894(119909)infin

119894=1as follows

120595119894(119909) =

119894

sum

119896=1

120573119894119896120595119896(119909) (10)

where 120573119894119896are orthogonalization coefficients and are given as

120573119894119895=

1

10038171003817100381710038171205951

1003817100381710038171003817

for 119894 = 119895 = 1

120573119894119895=

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2

minus sum119894minus1

119896=1(⟨120595119894 120595119896⟩1198822

2

)

2

for 119894 = 119895 = 1

120573119894119895= minus

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2


119896=1(119888119894119896)2

119894minus1

sum

119896=119895

⟨120595119894 120595119896⟩1198822

2

120573119896119895

for 119894 gt 119895

(11)

Clearly 120595119894(119909) = 119871

lowast120593119894(119909) = ⟨119871

lowast120593119894(119909) 119877

119909(119910)⟩1198822

2

=

⟨120593119894(119909) 119871

119910119877119909(119910)⟩1198821

2

= 119871119910119877119909(119910)|119910=119909119894

isin 1198822

2[0 1] Thus 120595

119894(119909)

can be written in the form 120595119894(119909) = 119871

119910119877119909(119910)|119910=119909119894

where 119871119910

indicates that the operator 119871 applies to the function of 119910

Theorem 8 If 119909119894infin

119894=1is dense on [0 1] then 120595

119894(119909)infin

119894=1is a

complete function system of the space 11988222[0 1]

Proof For each fixed 119906119904(119909) isin 119882

2

2[0 1] let ⟨119906

119904(119909) 120595

119894(119909)⟩1198822

2

=

0 In other words one can write ⟨119906119904(119909) 120595

119894(119909)⟩1198822

2

=

⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) = 0 Note

that 119909119894infin

119894=1is dense on [0 1] therefore 119871119906

119904(119909) = 0 It follows

that 119906119904(119909) = 0 119904 = 1 2 119899 from the existence of 119871minus1 So

the proof of the theorem is complete

Lemma 9 If 119906119904(119909) isin 119882

2

2[0 1] then there exist positive

constants 119872119904 such that 119906(119894)119904(119909)119862

le 119872119904119906119904(119909)1198822

2

119894 = 0 1119904 = 1 2 119899 where 119906

119904(119909)119862= max

0le119909le1|119906119904(119909)|

Proof For any 119909 119910 isin [0 1] we have 119906(119894)

119904(119909) =

⟨119906119904(119910) 120597119894

119909119877119909(119910)⟩1198822

2

By the expression form of the kernelfunction 119877

119909(119910) it follows that 120597

119894

119909119877119909(119910)1198822

2

le 119872119904

119894 Thus

|119906(119894)

119904(119909)| = |⟨119906

119904(119909) 120597119894

119909119877119909(119909)⟩1198822

2

| le 120597119894

119909119877119909(119909)1198822

2

119906119904(119909)1198822

2

le

119872119904

119894119906119904(119909)1198822

2

Hence 119906(119894)119904(119909)119862le max

119894=01119872119904

119894119906119904(119909)1198822

2

119894 = 0 1 119904 = 1 2 119899

The internal structure of the following theorem is asfollows firstly we will give the representation form of theexact solutions of (1) and (2) in the form of an infiniteseries in the space 119882

2

2[0 1] After that the convergence of

approximate solutions 119906119904119898

(119909) to the exact solutions 119906119904(119909)

119904 = 1 2 119899 will be proved

Theorem 10 For each 119906119904 119904 = 1 2 119899 in the space1198822

2[0 1]

the series suminfin119894=1

⟨119906119904(119909) 120595

119894(119909)⟩120595

119894(119909) is convergent in the sense of

the norm of 11988222[0 1] On the other hand if 119909

119894infin

119894=1is dense on

[0 1] then the following hold

(i) the exact solutions of (9) could be represented by

119906119904(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(12)

(ii) the approximate solutions of (9)

119906119904119898

(119909)

=

119898

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(13)

and 119906(119894)

119904119898(119909) 119894 = 0 1 are converging uniformly to the

exact solutions 119906119904(119909) and their derivatives as119898 rarr infin

respectively

Proof For the first part let 119906119904(119909) be solutions of

(9) in the space 1198822

2[0 1] Since 119906

119904(119909) isin 119882

2

2[0 1]

suminfin

119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595

119894(119909) is the Fourier series expansion

about normal orthogonal system 120595119894(119909)infin

119894=1 and 119882

2

2[0 1] is

the Hilbert space then the series suminfin

119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595

119894(119909) is

convergent in the sense of sdot 1198822

2

On the other hand using(10) it easy to see that

119906119904(119909) =

infin

sum

119894=1

⟨119906119904(119909) 120595

119894(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 120595

119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 119871

lowast120593119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119871119906119904(119909) 120593

119896(119909)⟩1198821

2

120595119894(119909)


=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119865119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

120593119896(119909)⟩1198821

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896)

119906119899(119909119896)) 120595119894(119909)

(14)

Therefore the form of (12) is the exact solutions of (9) Forthe second part it is easy to see that by Lemma 9 for any 119909 isin

[0 1]10038161003816100381610038161003816119906(119894)

119904119898(119909) minus 119906

(119894)

119904(119909)

10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

⟨119906119904119898

(119909) minus 119906119904(119909) 119877

(119894)

119909(119909)⟩1198822

2

1003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817120597119894

119909119877119909(119909)

100381710038171003817100381710038171198822

2

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

le 119872119904

119894

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

(15)

where 119894 = 0 1 and 119872119904

119894are positive constants Hence if

119906119904119898

(119909) minus 119906119904(119909)1198822

2

rarr 0 as 119898 rarr infin the approximatesolutions 119906

119904119898(119909) and 119906

(119894)

119904119898(119909) 119894 = 0 1 119904 = 1 2 119899

are converged uniformly to the exact solutions 119906119904(119909) and

their derivatives respectively So the proof of the theorem iscomplete

We mention here that the approximate solutions 119906119904119898

(119909)

in (13) can be obtained directly by taking finitely many termsin the series representation for 119906

119904(119909) of (12)

4 Construction of Iterative Method

In this section an iterative method of obtaining the solutionsof (1) and (2) is represented in the reproducing kernelspace 119882

2

2[0 1] for linear and nonlinear cases Initially we

will mention the following remark about the exact andapproximate solutions of (1) and (2)

In order to apply the RKHS technique to solve (1) and(2) we have the following two cases based on the algebraicstructure of the function 119865

119904 119904 = 1 2 119899

Case 1 If (1) is linear then the exact and approximate solu-tions can be obtained directly from (12) and (13) respectively

Case 2 If (1) is nonlinear then in this case the exact andapproximate solutions can be obtained by using the followingiterative algorithm

Algorithm 11 According to (12) the representation form ofthe solutions of (1) can be denoted by

119906119904(119909) =

infin

sum

119894=1

119861119904

119894120595119894(119909) 119904 = 1 2 119899 (16)

where 119861119904

119894= sum

119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896)) In fact 119861

119904

119894in (16) are unknown one

will approximate 119861119904

119894using known 119860

119904

119894 For numerical

computations one defines the initial functions 1199061199040

(1199091) = 0

put 1199061199040

(1199091) = 119906119904(1199091) and define the 119898-term approximations

to 119906119904(119909) by

119906119904119898

(119909) =

119898

sum

119894=1

119860119904

119894120595119894(119909) 119904 = 1 2 119899 (17)

where the coefficients 119860119904

119894of 120595119894(119909) 119894 = 1 2 119899 119904 =

1 2 119899 are given as

119860119904

1= 12057311119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091))

1199061199041

(119909) = 119860119904

11205951(119909)

119860119904

2=

2

sum

119896=1

1205732119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

1199061199042

(119909) =

2

sum

119894=1

119860119904

119894120595119894(119909)

119860119904

119899=

119898

sum

119896=1

120573119898119896

119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

119906119904119898

(119909) =

119898minus1

sum

119894=1

119860119904

119894120595119894(119909)

(18)

Here we note that in the iterative process of (17) we canguarantee that the approximations119906

119904119898(119909) satisfy the periodic

boundary conditions (2) Now the approximate solutions119906119872

119904119898(119909) can be obtained by taking finitely many terms in the

series representation of 119906119904119898

(119909) and

119906119872

119904119898(119909)

=

119872

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061119898minus1

(119909119896) 1199062119898minus1

(119909119896)

119906119899119898minus1

(119909119896)) 120595119894(119909)

119904 = 1 2 119899

(19)

Now we will proof that 119906119904119898

(119909) in the iterative formula(17) are converged to the exact solutions 119906

119904(119909) of (1) In

fact this result is a fundamental in the RKHS theory and itsapplications The next two lemmas are collected in order toprove the prerecent theorem

Lemma 12 If 119906119904119898

(119909) minus 119906119904(119909)1198822

2

rarr 0 119909119898

rarr 119910 as119898 rarr infin and 119865

119904(119909 V1 V2 V

119899) is continuous in [0 1]


with respect to 119909 V119894 for 119909 isin [0 1] and V

119894isin (minusinfininfin)

then 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as 119898 rarr infin

Proof Firstly we will prove that 119906119904119898minus1

(119909119898) rarr 119906

119904(119910) in the

sense of sdot 1198822

2

Since

1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904(119910)

1003816100381610038161003816

=1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910) + 119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

le1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910)1003816100381610038161003816+1003816100381610038161003816119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

(20)

By reproducing property of 119877119909(119910) we have 119906

119904119898minus1(119909119898) =

⟨119906119904119898minus1

(119909) 119877119909119898

(119909)⟩ and 119906119904119898minus1

(119910) = ⟨119906119904119898minus1

(119909) 119877119910(119909)⟩ Thus

|119906119904119898minus1

(119909119898) minus 119906s119898minus1(119910)| = |⟨119906

119904119898minus1(119909) 119877

119909119898

(119909) minus 119877119910(119909)⟩1198822

2

| le

119906119904119898minus1

(119909)1198822

2

119877119909119898

(119909) minus 119877119910(119909)1198822

2

From the symmetryof 119877119909(119910) it follows that 119877

119909119898

(119909) minus 119877119910(119909)1198822

2

rarr 0 as119898 rarr infin Hence |119906

119904119898minus1(119909119898) minus 119906119904119898minus1

(119910)| rarr 0 as soonas 119909119898

rarr 119910 On the other hand by Theorem 10 part (ii)for any 119910 isin [0 1] it holds that |119906

119904119898minus1(119910) minus 119906

119904(119910)| rarr 0

as 119898 rarr infin Therefore 119906119904119898minus1

(119909119898) rarr 119906

119904(119910) in the


2

as 119909119898

rarr 119910 and 119898 rarr infin Thusby means of the continuation of 119865

119904 it is obtained

that 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as119898 rarr infin

Lemma 13 For 119895 le 119898 one has 119871119906119904119898

(119909119895) = 119871119906

119904(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899

Proof The proof of 119871119906119904119898

(119909119895) = 119865

119904(119909119895 1199061119895minus1

(119909119895)

1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) will be obtained by induction

as follows if 119895 le 119898 then 119871119906119904119898

(119909119895) = sum

119898

119894=1119860119904

119894119871120595119894(119909119895) =

sum119898

119894=1119860119904

119894⟨119871120595119894(119909) 120593119895(119909)⟩1198821

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 119871lowast

119895120593(119909)⟩

1198822

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2

Using the orthogonality of120595119894(119909)infin

119894=1 it yields that

119895

sum

119897=1

120573119895119897119871119906119904119898

(119909119897)

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909)

119895

sum

119897=1

120573119895119897120595119897(119909)⟩

1198822

2

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2

= 119860119904

119895

=

119895

sum

119897=1

120573119895119897119865119904(119909119897 1199061119897minus1

(119909119897) 1199062119897minus1

(119909119897) 119906

119899119897minus1(119909119897))

(21)

Now if 119895 = 1 then 119871119906119904119898

(1199091) = 119865

119904(1199091 11990610

(1199091) 11990620

(1199091)

1199061198990

(1199091)) Again if 119895 = 2 then 120573

21119871119906119904119898

(1199091) +

12057322119871119906119904119898

(1199092) = 120573

21119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091)) +

12057322119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906

1198991(1199092)) Thus 119871119906

119904119898(1199092) =

119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906

1198991(1199092)) Indeed it is easy to

see by using mathematical induction that 119871119906119904119898

(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899

But on the other hand from Theorem 10 119906119904119898

(119909) convergeuniformly to 119906

119904(119909) It follows that on taking limits in (17)

119906119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) Therefore 119906

119904119898(119909) = 119875

119898119906119904(119909) where

119875119898is an orthogonal projector from the space1198822

2[0 1] to Span

1205951 1205952 120595

119898 Thus

119871119906119904119898

(119909119895)

= ⟨119871119906119904119898

(119909) 120593119895(119909)⟩1198821

2

= ⟨119906119904119898

(119909) 119871lowast

119895120593 (119909)⟩

1198822

2

= ⟨119875119898119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119906119904(119909) 119875

119898120595119895(119909)⟩1198822

2

= ⟨119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593

119895(119909)⟩1198821

2

= 119871119906119904(119909119895)

(22)

as 119895 le 119898 and 119904 = 1 2 119899

Theorem 14 If 119906119904119898

1198822

2

is bounded and 119909119894infin

119894=1is dense on

[0 1] then the 119898-term approximate solutions 119906119904119898

(119909) in theiterative formula (17) converge to the exact solutions 119906

119904(119909) of


2[0 1] and 119906

119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) 119904 =

1 2 119899 where 119860119904119894is given by (18)

Proof The proof consists of the following three stepsFirstly we will prove that the sequence 119906

119904119898infin

119898=1in (17) is

monotone increasing in the sense of sdot 1198822

2

By Theorem 8120595119894infin

119894=1is the complete orthonormal system in the space

1198822

2[0 1] Hence we have 119906

1199041198982

1198822

2

= ⟨119906119904119898

(119909) 119906119904119898

(119909)⟩1198822

2

=

⟨sum119898

119894=1119860119904

119894120595119894(119909) sum

119898

119894=1119860119904

119894120595119894(119909)⟩1198822

2

= sum119898

119894=1(119860119904

119894)

2

Therefore119906119904119898

1198822

2

119904 = 1 2 119899 is monotone increasing Sec-ondly we will prove the convergence of 119906

119904119898(119909) From (17)

we have 119906119904119898+1

(119909) = 119906119904119898

(119909) + 119860119904

119898+1120595119898+1

(119909) From theorthogonality of 120595

119894(119909)infin

119894=1 it follows that 119906

119904119898+12

1198822

2

=

119906119904119898

2

1198822

2

+ (119860119904

119898+1)2

= 119906119904119898minus1

2

1198822

2

+ (119860119904

119898)2+ (119860119904

119898+1)2

=

sdot sdot sdot = 1199061199040

2

1198822

2

+ sum119898+1

119894=1(119860119904

119894)2 Since the sequence 119906

119904119898infin

119898=1

is monotone increasing in the sense of sdot 1198822

2

Due tothe condition that 119906

1199041198981198822

2

is bounded 119906119904119898

1198822

2

is con-vergent as 119898 rarr infin Then there exist constants 119888

119904

such that suminfin

119894=1(119860119904

119894)2

= 119888119904 It implies that 119860

119904

119894=

sum119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896) 119906

119899119896minus1(119909119896)) isin 119897

2 119894 =

1 2 On the other hand since (119906119904119898

minus 119906119904119898minus1

) perp (119906119904119898minus1

minus

119906119904119898minus2

) perp sdot sdot sdot perp (119906119904119898+1

minus 119906119904119898

) it follows for 119897 gt 119898 that

1003817100381710038171003817119906119904119897(119909) minus 119906

119904119898(119909)

1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909) + 119906119904119897minus1

(119909) minus sdot sdot sdot

+119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909)1003817100381710038171003817

2

1198822

2

+ sdot sdot sdot

+1003817100381710038171003817119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

(23)


Furthermore 119906119904119897(119909) minus 119906

119904119897minus1(119909)2

1198822

2

= (119860119904

119897)2 Conse-

quently as 119897 119898 rarr infin we have 119906119904119897(119909) minus 119906

119904119898(119909)2

1198822

2

=

sum119897

119894=119898+1(119860119904

119894)2

rarr 0 Considering the completeness of1198822

2[0 1] there exists 119906

119904(119909) isin 119882

2

2[0 1] such that 119906

119904119897(119909) rarr

119906119904(119909) 119904 = 1 2 119899 as 119897 rarr infin in the sense of

sdot 1198822

2

Thirdly we will prove that 119906119904(119909) are the solutions

of (9) Since 119909119894infin

119894=1is dense on [0 1] for any 119909 isin [0 1]

there exists subsequence 119909119898119895

infin

119895=1

such that 119909119898119895

rarr 119909 as119895 rarr infin From Lemma 13 it is clear that 119871119906

119904(119909119898119895

) =

119865119904(119909119898119895

1199061119898119895minus1(119909119896) 1199062119898119895minus1(119909119896) 119906

119899119898119895minus1(119909119896)) Hence let

119895 rarr infin by Lemma 12 and the continuity of 119865119904 we have

119871119906119904(119909) = 119865

119904(119909 1199061(119909) 1199062(119909) 119906

119899(119909)) That is 119906

119904(119909) satisfies

(1) Also since 120595119894(119909) isin 119882

2

2[0 1] clearly 119906

119904(119909) satisfies the

periodic boundary conditions (2) In other words 119906119904(119909) are

the solutions of (1) and (2) where 119906119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) and

119860s119894are given by (18) The proof is complete

According to the internal structure of the presentmethodit is obvious that if we let 119906

119904(119909) denote the exact solutions

of (9) 119906119904119898

(119909) denote the approximate solutions obtained bythe RKHS method as given by (17) and 119903

119904

119898(119909) denote the

difference between 119906119904119898

(119909) and 119906119904(119909) where 119909 isin [0 1] and

119904 = 1 2 119899 then 119903119904

119898(119909)

2

1198822

2

= 119906119904(119909) minus 119906

119904119898(119909)2

1198822

2

=

suminfin

119894=119898+1119860119904

119894120595119894(119909)

2

1198822

2

= suminfin

119894=119898+1(119860119904

119894)

2

and 119903119904

119898minus1(119909)

2

1198822

2

=

suminfin

119894=119898(119860119904

119894)

2

or 119903119904119898

(119909)1198822

2

le 119903119904

119898minus1(119909)1198822

2

Consequently thisshows the following theorem

Theorem 15 The difference 119903119904

119898(119909) 119904 = 1 2 119899 is mono-

tone decreasing in the sense of the norm of 11988222[0 1]

5 Numerical Examples

In this section the theoretical results of the previous sectionsare illustrated bymeans of some numerical examples in orderto illustrate the performance of the RKHSmethod for solvingsystems of first-order periodic BVPs and justify the accuracyand efficiency of the method To do so we consider thefollowing three nonlinear examples These examples havebeen solved by the presented method with different valuesof 119898 and 119872 Results obtained by the method are comparedwith the exact solution of each example by computing theabsolute and relative errors and are found to be in goodagreement with each other In the process of computation allexperiments were performed inMAPLE 13 software package

Example 1 Consider the following first-order nonlinear dif-ferential system

1199061015840

1(119909) minus 119906

1(119909) + (119906

2(119909))3

= 1198911(119909)

1199061015840

2(119909) minus sinh (119906

1(119909)) 119906

2(119909) = 119891

2(119909)

1198911(119909) = (119909 minus 1) (cos119909 minus sin119909) + sin119909 + 119890

3119909(119909minus1)

1198912(119909) = (sinh (sin (119909) (1 minus 119909)) + 2119909 minus 1) 119890

119909(119909minus1)

(24)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(25)

The exact solutions are 1199061(119909) = (119909 minus 1) sin(119909) and 119906

2(119909) =

119890119909(119909minus1)

Using RKHS method take 119909119894

= (119894 minus 1)(119872 minus 1) 119894 =

1 2 119872 on [0 1] The numerical results at some selectedgrid points for 119872 = 101 and 119898 = 3 are given in Tables 1 and2 for the dependent variables 119906

1(119909) and 119906

2(119909) respectively

The present method enables us to approximate the solu-tions and their derivatives at every point of the range ofintegrationHence it is possible to pick any point in [0 1] andas well the approximate solutions and their derivatives will beapplicable Next new numerical results for Example 1 whichinclude the absolute error at some selected gird points in [0 1]

for approximating 11990610158401(119909) and 119906

1015840

2(119909) where 119909

119894= (119894minus1)(119872minus1)

119894 = 1 2 119872119872 = 101 and119898 = 3 are given in Table 3

Example 2 Consider the following first-order nonlineardifferential system

1199061015840

1(119909) + radic119906

1(119909) + 1119906

2(119909) = 119891

1(119909)

V10158402(119909) minus 119906

1(119909) (119906

2(119909))2

+ (1199062(119909))2

= 1198912(119909)

1198911(119909) = (119909

4minus 21199093+ 1199092+ 1)

minus12

+ 41199093minus 61199092+ 2119909

1198912(119909) = minus

1199094+ 21199093minus 51199092+ 2119909 minus 1

(1199094minus 21199093+ 1199092+ 1)2

(26)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(27)

The exact solutions are 1199061(119909) = (119909(119909 minus 1))

2 and 1199062(119909) =

1((119909(119909 minus 1))2+ 1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906



1199061015840

1(119909) + 119906

3(119909) 1198901199061(119909)

+ (1199062(119909))2

= 1198911(119909)

1199061015840

2(119909) minus 119906

2(119909) 119890minus1199061(119909)

+ (1199063(119909))2

= 1198912(119909)

1199061015840

3(119909) minus 119906

1(119909) 1199062(119909) 1199063(119909) = 119891

3(119909)


Table 1 Numerical results of 1199061(119909) for Example 1

119909 Exact solution Approximate solution Absolute error Relative error016 minus0133827 minus013382630119666272 992359 times 10

minus7741522 times 10

minus6

032 minus0213905 minus021390423277976867 102844 times 10minus6

480792 times 10minus6










119909 Exact solution Approximate solution Absolute error Relative error016 0874240 08742398572490666 441286 times 10


minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7


Table 3 Absolute error of approximating 1199061015840

1(119909) and 119906

1015840

2(119909) for Example 1

Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7


1198911(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1)) (119909 (119909 minus 1) + 1)

+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2

minus 119890minus119909(119909minus1)

(2119909 minus 1) minus

119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)

minus (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

times ln (119909 (119909 minus 1) + 1) 119890minus119909(119909minus1)

+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)

The exact solutions are 1199061(119909) = ln(119909(119909 minus 1) + 1) 119906

2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =

1 2 119872 on [0 1] The numerical results at some selectedgrid points for 119872 = 101 and 119898 = 3 are given in Tables 67 and 8 for the dependent variables 119906

1(119909) 119906

2(119909) and 119906

3(119909)

respectivelyFrom the previous tables it can be seen that the RKHS

method provides us with the accurate approximate solutionsOn the other aspect as well it is clear that the accuracyobtained using the mentioned method is advanced by usingonly a few tens of iterations

6 Conclusions

Here we use the RKHS method to solve systems of first-order periodic BVPs The solutions were calculated in theform of a convergent series in the space 119882

2

2[0 1] with

easily computable components In the proposed methodthe 119898-term approximations are obtained and proved toconverge to the exact solutions Meanwhile the error of theapproximate solutions is monotone decreasing in the senseof the norm of 119882

2

2[0 1] It is worthy to note that in our

work the approximate solutions and their derivatives con-verge uniformly to the exact solutions and their derivativesrespectively On the other aspect as well the present methodenables us to approximate the solutions and their derivativesat every point of the range of integration The results showthat the present method is an accurate and reliable analyticaltechnique for solving systems of first-order periodic BVPs

Conflict of Interests

The authors declare that there is no conflict of interests

Acknowledgment

The authors would like to express their thanks to unknownreferees for their careful reading and helpful comments

References

[1] E Coddington andN LevinsonTheory of Ordinary DifferentialEquations McGraw-Hill New York NY USA 1955

[2] H I Freedman and J H Wu ldquoPeriodic solutions of single-species models with periodic delayrdquo SIAM Journal on Mathe-matical Analysis vol 23 no 3 pp 689ndash701 1992

[3] J Mawhin and J R Ward ldquoNonuniform nonresonance con-ditions at the two first eigenvalues for periodic solutions of


forced Lienard and Duffing equationsrdquo RockyMountain Journalof Mathematics vol 12 no 4 pp 643ndash654 1982

[4] A C Lazer ldquoApplication of a lemma on bilinear forms to aproblem in nonlinear oscillationsrdquo Proceedings of the AmericanMathematical Society vol 33 pp 89ndash94 1972

[5] K Abd-Ellateef R Ahmed and Z Drici ldquoGeneralized quasi-linearization for systems of nonlinear differential equationswith periodic boundary conditionsrdquo Dynamics of ContinuousDiscrete amp Impulsive Systems A vol 12 no 1 pp 77ndash85 2005

[6] C C Tisdell ldquoExistence of solutions to first-order periodicboundary value problemsrdquo Journal of Mathematical Analysisand Applications vol 323 no 2 pp 1325ndash1332 2006

[7] R P Agarwal and J Chen ldquoPeriodic solutions for first orderdifferential systemsrdquo Applied Mathematics Letters vol 23 no3 pp 337ndash341 2010

[8] R Chen R Ma and Z He ldquoPositive periodic solutions of first-order singular systemsrdquoAppliedMathematics and Computationvol 218 no 23 pp 11421ndash11428 2012

[9] C P Gupta ldquoPeriodic solutions for coupled first order nonlineardifferential systems of Hamiltonian typerdquo Nonlinear AnalysisTheory Methods amp Applications vol 8 no 11 pp 1271ndash12851984

[10] Q Kong andMWang ldquoPositive solutions of even order systemperiodic boundary value problemsrdquoNonlinear AnalysisTheoryMethods amp Applications vol 72 no 3-4 pp 1778ndash1791 2010

[11] A Boucherif and N Merabet ldquoBoundary value problems forfirst order multivalued differential systemsrdquo Archivum Mathe-maticum vol 41 no 2 pp 187ndash195 2005

[12] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[13] WG Li ldquoSolving the periodic boundary value problemwith theinitial value problemmethodrdquo Journal ofMathematical Analysisand Applications vol 226 no 1 pp 259ndash270 1998

[14] J Mawhin Topological Degree Methods in Nonlinear BoundaryValue Problems vol 40 of CBMS Regional Conference Series inMathematics American Mathematical Society Providence RIUSA 1979

[15] I T Kiguradze ldquoOn periodic solutions of 119899th order ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 40 no 1ndash8 pp 309ndash321 2000

[16] O abuArqub A el-Ajou Z al Zhour and SMomani ldquoMultiplesolutions of nonlinear boundary value problems of fractionalorder a new analytic iterative techniquerdquo Entropy vol 16 no 1pp 471ndash493 2014

[17] Z abo-Hammour O abuArqub SMomani andN ShawagfehldquoOptimization solution of Troeschrsquos and Bratursquos problems ofordinary type using novel continuous genetic algorithmrdquo Dis-crete Dynamics in Nature and Society vol 2014 Article ID401696 15 pages 2014

[18] O abu Arqub Z abo-Hammour and S Momani ldquoApplica-tion of continuous genetic algorithm for nonlinear system ofsecond-order boundary value problemsrdquo Applied Mathematicsand Information Sciences vol 8 pp 235ndash248 2014

[19] O abuArqub Z abo-Hammour SMomani andN ShawagfehldquoSolving singular two-point boundary value problems usingcontinuous genetic algorithmrdquo Abstract and Applied Analysisvol 2012 Article ID 205391 25 pages 2012

[20] A Berlinet and C Thomas-Agnan Reproducing Kernel HilbertSpaces in Probability and Statistics Kluwer Academic BostonMass USA 2004

[21] M Cui and Y Lin Nonlinear Numerical Analysis in the Repro-ducing Kernel Space Nova Science New York NY USA 2008

[22] ADanielReproducingKernel Spaces andApplications SpringerBasel Switzerland 2003

[23] F Geng ldquoSolving singular second order three-point bound-ary value problems using reproducing kernel Hilbert spacemethodrdquo Applied Mathematics and Computation vol 215 no6 pp 2095ndash2102 2009

[24] L-H Yang andY Lin ldquoReproducing kernelmethods for solvinglinear initial-boundary-value problemsrdquo Electronic Journal ofDifferential Equations vol 2008 pp 1ndash11 2008

[25] Y Z LinMG Cui and LH Yang ldquoRepresentation of the exactsolution for a kind of nonlinear partial differential equationrdquoApplied Mathematics Letters vol 19 no 8 pp 808ndash813 2006

[26] W Wang M Cui and B Han ldquoA new method for solving aclass of singular two-point boundary value problemsrdquo AppliedMathematics and Computation vol 206 no 2 pp 721ndash7272008

[27] W Jiang and Z Chen ldquoSolving a system of linear Volterraintegral equations using the new reproducing kernel methodrdquoApplied Mathematics and Computation vol 219 no 20 pp10225ndash10230 2013

[28] F Geng and M Cui ldquoA reproducing kernel method for solvingnonlocal fractional boundary value problemsrdquo Applied Mathe-matics Letters vol 25 no 5 pp 818ndash823 2012

[29] F Z Geng and S P Qian ldquoReproducing kernel methodfor singularly perturbed turning point problems having twinboundary layersrdquo Applied Mathematics Letters vol 26 no 10pp 998ndash1004 2013

[30] W Jiang and Z Chen ldquoA collocation method based on repro-ducing kernel for amodified anomalous subdiffusion equationrdquoNumericalMethods for Partial Differential Equations vol 30 no1 pp 289ndash300 2014

[31] F Z Geng S P Qian and S Li ldquoA numerical method forsingularly perturbed turning point problems with an interiorlayerrdquo Journal of Computational and Applied Mathematics vol255 pp 97ndash105 2014

[32] N Shawagfeh O abu Arqub and S Momani ldquoAnalyticalsolution of nonlinear second-order periodic boundary valueproblem using reproducing kernel methodrdquo Journal of Compu-tational Analysis and Applications vol 16 pp 750ndash762 2014

[33] M al-Smadi O abu Arqub and S Momani ldquoA computationalmethod for two-point boundary value problems of fourth-ordermixed integrodifferential equationsrdquoMathematical Problems inEngineering vol 2013 Article ID 832074 10 pages 2013

[34] O abu Arqub M al-Smadi and N Shawagfeh ldquoSolving Fred-holm integro-differential equations using reproducing kernelHilbert space methodrdquo Applied Mathematics and Computationvol 219 no 17 pp 8938ndash8948 2013

[35] O abu Arqub M al-Smadi and S Momani ldquoApplication ofreproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equationsrdquo Abstract and AppliedAnalysis vol 2012 Article ID 839836 16 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Journal of



Volume 2014

Advances in

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Complex AnalysisJournal of



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Volume 2014

International Journal of

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Advances in



Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society


Advances in

DecisionSciences


Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


Investigation about systems of first-order periodic BVPsnumerically is scarce In this paper we utilize a methodicalway to solve these types of differential systems In factwe provide criteria for finding the approximate and exactsolutions to the following system

1199061015840

1(119909) = 119865

1(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

1199061015840

2(119909) = 119865

2(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

1199061015840

119899(119909) = 119865

119899(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

(1)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

119906119899(0) = 119906

119899(1)

(2)

where 119909 isin [0 1] 119906119904isin 1198822

2[0 1] are unknown functions to

be determined 119865119904(119909 V1 V1 V

119899) are continuous terms in

1198821

2[0 1] as V

119904= V119904(119909) isin 119882

2

2[0 1] 0 le 119909 le 1 minusinfin lt V

119904lt

infin in which 119904 = 1 2 119899 and 1198821

2[0 1] 1198822

2[0 1] are two

reproducing kernel spaces Here we assume that (1) subjectto the periodic boundary conditions (2) has a unique solutionon [0 1]

Reproducing kernel theory has important applications innumerical analysis differential equations integral equationsprobability and statistics and so forth [20ndash22] In the lastyears extensive work has been done using RKHS methodwhich provides numerical approximations for linear andnonlinear equations This method has been implemented inseveral operator differential integral and integrodifferentialequations side by side with their theories The reader iskindly requested to go through [23ndash35] in order to knowmore details about RKHS method including its history itsmodification for use its applications and its characteristics

The rest of the paper is organized as follows In the nextsection two reproducing kernel spaces are described in orderto formulate the reproducing kernel functions In Section 3some essential results are introduced and a method for theexistence of solutions for (1) and (2) is described In Section 4we give an iterative method to solve (1) and (2) numericallyNumerical examples are presented in Section 5 Section 6ends this paper with brief conclusions

2 Construct of Reproducing Kernel Functions

In this section two reproducing kernels needed are con-structed in order to solve (1) and (2) using RKHS methodBefore the construction we utilize the reproducing kernelconcept Throughout this paper C is the set of complexnumbers 1198712[119886 119887] = 119906 | int

119887

1198861199062(119909)119889119909 lt infin and 119897

2= 119860 |

suminfin

119894=1(119860119894)2lt infin

Definition 1 (see [23]) Let 119864 be a nonempty abstract set Afunction119877 119864times119864 rarr C is a reproducing kernel of theHilbertspace119867 if

(1) for each 119909 isin 119864 119877(sdot 119909) isin 119867(2) for each 119909 isin 119864 and 120593 isin 119867 ⟨120593(sdot) 119877(sdot 119909)⟩ = 120593(119909)

Remark 2 Condition (2) in Definition 1 is called ldquothe repro-ducing propertyrdquo which means that the value of the function120593 at the point 119909 is reproducing by the inner product of 120593(sdot)with 119877(sdot 119909) A Hilbert space which possesses a reproducingkernel is called a RKHS

To solve (1) and (2) using RKHS method we first defineand construct a reproducing kernel space 119882

2

2[0 1] in which

every function satisfies the periodic boundary condition119906(0) = 119906(1) After that we utilize the reproducing kernelspace 119882

1

2[0 1]

Definition 3 The inner product space 1198822

2[0 1] is defined as

1198822

2[0 1] = 119906(119909) | 119906 119906

1015840 are absolutely continuous real-valuedfunctions on [0 1] 119906 1199061015840 11990610158401015840 isin 119871

2[0 1] and 119906(0) = 119906(1) On

the other hand the inner product and the norm in 1198822

2[0 1]

are defined respectively by

⟨119906 V⟩1198822

2

=

1

sum

119894=0

119906(119894)

(0) V(119894) (0) + int

1

0

11990610158401015840

(119905) V10158401015840 (119905) 119889119905 (3)

and 1199061198822

2

= radic⟨119906 119906⟩1198822

2

where 119906 V isin 1198822

2[0 1]

It is easy to see that ⟨119906 V⟩1198822

2

satisfies all the require-ments for the inner product First ⟨119906 119906⟩

1198822

2

ge 0 Second⟨119906 V⟩1198822

2

= ⟨V 119906⟩1198822

2

Third ⟨120574119906 V⟩1198822

2

= 120574⟨119906 V⟩1198822

2

Fourth⟨119906 + 119908 V⟩

1198822

2

= ⟨119906 V⟩1198822

2

+ ⟨119908 V⟩1198822

2

where 119906 V 119908 isin 1198822

2[0 1]

It therefore remains only to prove that ⟨119906 119906⟩1198822

2

= 0 if andonly if 119906 = 0 In fact it is obvious that when 119906 = 0 then⟨119906 119906⟩

1198822

2

= 0 On the other hand if ⟨119906 119906⟩1198822

2

= 0 then by

(3) we have ⟨119906 119906⟩1198822

2

= sum1

119894=0(119906(119894)(0))

2

+ int

1

0(11990610158401015840(119905))

2

119889119905 = 0therefore 119906(0) = 119906

1015840(0) = 0 and 119906

10158401015840(119905) = 0 Then we can

obtain 119906 = 0

Definition 4 (see [23]) The Hilbert space 1198822

2[0 1] is called

a reproducing kernel if for each fixed 119909 isin [0 1]there exist 119877(119909 119910) isin 119882

2

2[0 1] (simply 119877

119909(119910)) such that

⟨119906(119910) 119877119909(119910)⟩1198822

2

= 119906(119909) for any 119906(119910) isin 1198822

2[0 1] and 119910 isin

[0 1]

An important subset of the RKHSs is the RKHSs asso-ciated with continuous kernel functions These spaces havewide applications including complex analysis harmonicanalysis quantum mechanics statistics and machine learn-ing

Theorem 5 The Hilbert space 1198822

2[0 1] is a complete repro-

ducing kernel and its reproducing kernel function 119877119909(119910) can be

written as119877119909(119910)

=

1199011(119909) + 119901

2(119909) 119910 + 119901

3(119909) 1199102+ 1199014(119909) 1199103 119910 le 119909

1199021(119909) + 119902

2(119909) 119910 + 119902

3(119909) 1199102+ 1199024(119909) 1199103 119910 gt 119909

(4)


where 119901119894(119909) and 119902




2



119909(119910) in the space

1198822


int

1

011990610158401015840(119910)1205973

119910119877119909(119910)119889119910 = sum

1

119894=0(minus1)1minus119894

119906(119894)(119910)1205973minus119894

119910119877119909(119910)|119910=1

119910=0+

int

1

0119906(119910)120597

4


⟨119906(119910) 119877119909(119910)⟩1198822

2

= sum1

119894=0119906(119894)(0)[120597119894

119910119877119909(0) + (minus1)

1198941205973minus119894

119910119877119909(0)]

+ sum1

119894=0(minus1)1minus119894

119906(119894)(1)1205973minus119894

119910119877119909(1) + int

1

0119906(119910)120597

4

119910119877119909(119910)119889119910 Since

119877119909(119910) isin 119882

2


119909(0) = 119877


119906(119909) isin 1198822


⟨119906 (119910) 119877119909(119910)⟩1198822

2

=

1

sum

119894=0

119906(119894)

(0) [120597119894

119910119877119909(0) + (minus1)

1198941205973minus119894

119910119877119909(0)]

+

1

sum

119894=0

(minus1)119894+1

119906(119894)

(1) 1205973minus119894

119910119877119909(1)

+ int

1

0

119906 (119910) 1205974

119910119877119909(119910) 119889119910 + 119888

1(119906 (0) minus 119906 (1))

(5)


119910119877119909(1) = 0

119877119909(0) + 120597

3

119910119877119909(0) + 119888

1= 0 1205971

119910119877119909(0) minus 120597

2

119910119877119909(0) = 0 and

1205973

119910119877119909(1) + 119888


119909(119910)⟩1198822

2

=

int

1

0119906(119910)120597

4


119909(119910) satisfies

1205974


then ⟨119906(119910) 119877119909(119910)⟩1198822

2

= 119906(119909) Obviously 119877119909(119910) is the


2[0 1] Next we


4= 0




119898

119910119877119909(119909+0) = 120597

119898

119910119877119909(119909minus0)

119898 = 0 1 2 Integrating 1205976



3

119910119877119909(119909 + 0) minus 120597

3

119910119877119909(119909 minus 0) = minus1



119877119909(119910) =

1

48

(1199093119910 (6 + 3119910 minus 119910

2) + 3119909

2119910 (minus6 minus 3119910119910

2) + 6119909119910 (2 + 119910 + 119910

2) minus 8 (minus6 + 119910

3)) 119910 le 119909

1

48

(48 + 6119909119910 (2 minus 3119910 + 1199102) + 3119909

2119910 (2 minus 3119910 + 119910


2+ 1199103)) 119910 gt 119909

(7)



2[0 1] is





2[0 1] are defined


2

= 119906(0)V(0) + int

1

01199061015840(119909)V1015840(119909)119889119909

and 1199061198821

2

= radic⟨119906 119906⟩1198821

2

where 119906 V isin 1198821

2[0 1]



can be written as

119866x (119910) =

1 + 119910 119910 le 119909

1 + 119909 119910 gt 119909

(8)




2



2[0 1] based



2

2[0 1] rarr 119882

1

2[0 1] such that 119871119906(119909) =



119871119906119904(119909) = 119865

119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

119906119904(0) minus 119906

119904(1) = 0

(9)


2

2[0 1]

and 119865119904(119909 V1 V1 V

119899) isin 119882

1

2[0 1] for V

119904= V119904(119909) isin 119882

2

2[0 1]




2[0 1] into the

space1198821

2[0 1]


2[0 1] To do so put 120593

119894(119909) = 119866

119909119894

(119909) and 120595119894(119909) = 119871

lowast120593119894(119909)


119894=1is dense on [0 1] and 119871


119904(119909) 120595

119894(119909)⟩1198822

2

= ⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

=

⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) 119894 = 1 2 119904 = 1 2 119899


119894=1of the

space 1198822



119894=1as follows

120595119894(119909) =

119894

sum

119896=1

120573119894119896120595119896(119909) (10)


120573119894119895=

1

10038171003817100381710038171205951

1003817100381710038171003817

for 119894 = 119895 = 1

120573119894119895=

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2


119896=1(⟨120595119894 120595119896⟩1198822

2

)

2

for 119894 = 119895 = 1

120573119894119895= minus

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2


119896=1(119888119894119896)2

119894minus1

sum

119896=119895

⟨120595119894 120595119896⟩1198822

2

120573119896119895

for 119894 gt 119895

(11)

Clearly 120595119894(119909) = 119871

lowast120593119894(119909) = ⟨119871

lowast120593119894(119909) 119877

119909(119910)⟩1198822

2

=

⟨120593119894(119909) 119871

119910119877119909(119910)⟩1198821

2

= 119871119910119877119909(119910)|119910=119909119894

isin 1198822

2[0 1] Thus 120595

119894(119909)


119910119877119909(119910)|119910=119909119894

where 119871119910



119894=1is dense on [0 1] then 120595

119894(119909)infin

119894=1is a



2

2[0 1] let ⟨119906

119904(119909) 120595

119894(119909)⟩1198822

2

=


119894(119909)⟩1198822

2

=

⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) = 0 Note



119904(119909) = 0 It follows



Lemma 9 If 119906119904(119909) isin 119882

2



le 119872119904119906119904(119909)1198822

2

119894 = 0 1119904 = 1 2 119899 where 119906

119904(119909)119862= max

0le119909le1|119906119904(119909)|


119904(119909) =

⟨119906119904(119910) 120597119894

119909119877119909(119910)⟩1198822

2


119909(119910) it follows that 120597

119894

119909119877119909(119910)1198822

2

le 119872119904

119894 Thus

|119906(119894)

119904(119909)| = |⟨119906

119904(119909) 120597119894

119909119877119909(119909)⟩1198822

2

| le 120597119894

119909119877119909(119909)1198822

2

119906119904(119909)1198822

2

le

119872119904

119894119906119904(119909)1198822

2

Hence 119906(119894)119904(119909)119862le max

119894=01119872119904

119894119906119904(119909)1198822

2

119894 = 0 1 119904 = 1 2 119899


2




119904 = 1 2 119899 will be proved


2[0 1]


⟨119906119904(119909) 120595

119894(119909)⟩120595



119894infin

119894=1is dense on



119906119904(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(12)


119906119904119898

(119909)

=

119898

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(13)

and 119906(119894)



respectively



2[0 1] Since 119906

119904(119909) isin 119882

2

2[0 1]

suminfin

119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595



119894=1 and 119882

2

2[0 1] is


119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595

119894(119909) is


2


119906119904(119909) =

infin

sum

119894=1

⟨119906119904(119909) 120595

119894(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 120595

119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 119871

lowast120593119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119871119906119904(119909) 120593

119896(119909)⟩1198821

2

120595119894(119909)


=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119865119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

120593119896(119909)⟩1198821

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896)

119906119899(119909119896)) 120595119894(119909)

(14)


[0 1]10038161003816100381610038161003816119906(119894)

119904119898(119909) minus 119906

(119894)

119904(119909)

10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

⟨119906119904119898

(119909) minus 119906119904(119909) 119877

(119894)

119909(119909)⟩1198822

2

1003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817120597119894

119909119877119909(119909)

100381710038171003817100381710038171198822

2

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

le 119872119904

119894

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

(15)

where 119894 = 0 1 and 119872119904


119906119904119898

(119909) minus 119906119904(119909)1198822

2


119904119898(119909) and 119906

(119894)

119904119898(119909) 119894 = 0 1 119904 = 1 2 119899




(119909)


119904(119909) of (12)



2




119904 119904 = 1 2 119899




119906119904(119909) =

infin

sum

119894=1

119861119904

119894120595119894(119909) 119904 = 1 2 119899 (16)

where 119861119904

119894= sum

119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896)) In fact 119861

119904




119904



(1199091) = 0

put 1199061199040


to 119906119904(119909) by

119906119904119898

(119909) =

119898

sum

119894=1

119860119904

119894120595119894(119909) 119904 = 1 2 119899 (17)


119894of 120595119894(119909) 119894 = 1 2 119899 119904 =


119860119904

1= 12057311119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091))

1199061199041

(119909) = 119860119904

11205951(119909)

119860119904

2=

2

sum

119896=1

1205732119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

1199061199042

(119909) =

2

sum

119894=1

119860119904

119894120595119894(119909)

119860119904

119899=

119898

sum

119896=1

120573119898119896

119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

119906119904119898

(119909) =

119898minus1

sum

119894=1

119860119904

119894120595119894(119909)

(18)






(119909) and

119906119872

119904119898(119909)

=

119872

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061119898minus1

(119909119896) 1199062119898minus1

(119909119896)

119906119899119898minus1

(119909119896)) 120595119894(119909)

119904 = 1 2 119899

(19)



119904(119909) of (1) In


Lemma 12 If 119906119904119898

(119909) minus 119906119904(119909)1198822

2

rarr 0 119909119898


119904(119909 V1 V2 V





then 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as 119898 rarr infin


(119909119898) rarr 119906

119904(119910) in the


2

Since

1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904(119910)

1003816100381610038161003816

=1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910) + 119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

le1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910)1003816100381610038161003816+1003816100381610038161003816119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

(20)


119904119898minus1(119909119898) =

⟨119906119904119898minus1

(119909) 119877119909119898

(119909)⟩ and 119906119904119898minus1

(119910) = ⟨119906119904119898minus1

(119909) 119877119910(119909)⟩ Thus

|119906119904119898minus1

(119909119898) minus 119906s119898minus1(119910)| = |⟨119906

119904119898minus1(119909) 119877

119909119898

(119909) minus 119877119910(119909)⟩1198822

2

| le

119906119904119898minus1

(119909)1198822

2

119877119909119898

(119909) minus 119877119910(119909)1198822

2


119909119898

(119909) minus 119877119910(119909)1198822

2



(119910)| rarr 0 as soonas 119909119898


119904119898minus1(119910) minus 119906

119904(119910)| rarr 0


(119909119898) rarr 119906

119904(119910) in the


2

as 119909119898



that 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as119898 rarr infin


(119909119895) = 119871119906

119904(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899


(119909119895) = 119865

119904(119909119895 1199061119895minus1

(119909119895)

1199062119895minus1

(119909119895) 119906



(119909119895) = sum

119898

119894=1119860119904

119894119871120595119894(119909119895) =

sum119898

119894=1119860119904

119894⟨119871120595119894(119909) 120593119895(119909)⟩1198821

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 119871lowast

119895120593(119909)⟩

1198822

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2



119895

sum

119897=1

120573119895119897119871119906119904119898

(119909119897)

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909)

119895

sum

119897=1

120573119895119897120595119897(119909)⟩

1198822

2

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2

= 119860119904

119895

=

119895

sum

119897=1

120573119895119897119865119904(119909119897 1199061119897minus1

(119909119897) 1199062119897minus1

(119909119897) 119906

119899119897minus1(119909119897))

(21)

Now if 119895 = 1 then 119871119906119904119898

(1199091) = 119865

119904(1199091 11990610

(1199091) 11990620

(1199091)

1199061198990

(1199091)) Again if 119895 = 2 then 120573

21119871119906119904119898

(1199091) +

12057322119871119906119904119898

(1199092) = 120573

21119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091)) +

12057322119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906

1198991(1199092)) Thus 119871119906

119904119898(1199092) =

119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906



(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899




119906119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) Therefore 119906

119904119898(119909) = 119875

119898119906119904(119909) where


2[0 1] to Span

1205951 1205952 120595

119898 Thus

119871119906119904119898

(119909119895)

= ⟨119871119906119904119898

(119909) 120593119895(119909)⟩1198821

2

= ⟨119906119904119898

(119909) 119871lowast

119895120593 (119909)⟩

1198822

2

= ⟨119875119898119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119906119904(119909) 119875

119898120595119895(119909)⟩1198822

2

= ⟨119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593

119895(119909)⟩1198821

2

= 119871119906119904(119909119895)

(22)

as 119895 le 119898 and 119904 = 1 2 119899

Theorem 14 If 119906119904119898

1198822

2


119894=1is dense on



119904(119909) of


2[0 1] and 119906

119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) 119904 =

1 2 119899 where 119860119904119894is given by (18)


119904119898infin

119898=1in (17) is


2



1198822


1199041198982

1198822

2

= ⟨119906119904119898

(119909) 119906119904119898

(119909)⟩1198822

2

=

⟨sum119898

119894=1119860119904

119894120595119894(119909) sum

119898

119894=1119860119904

119894120595119894(119909)⟩1198822

2

= sum119898

119894=1(119860119904

119894)

2

Therefore119906119904119898

1198822

2


119904119898(119909) From (17)

we have 119906119904119898+1

(119909) = 119906119904119898

(119909) + 119860119904

119898+1120595119898+1


119894(119909)infin


119904119898+12

1198822

2

=

119906119904119898

2

1198822

2

+ (119860119904

119898+1)2

= 119906119904119898minus1

2

1198822

2

+ (119860119904

119898)2+ (119860119904

119898+1)2

=


2

1198822

2

+ sum119898+1

119894=1(119860119904


119904119898infin

119898=1


2


1199041198981198822

2

is bounded 119906119904119898

1198822

2


119904

such that suminfin

119894=1(119860119904

119894)2

= 119888119904 It implies that 119860

119904

119894=

sum119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896) 119906

119899119896minus1(119909119896)) isin 119897

2 119894 =


minus 119906119904119898minus1

) perp (119906119904119898minus1

minus

119906119904119898minus2


minus 119906119904119898


1003817100381710038171003817119906119904119897(119909) minus 119906

119904119898(119909)

1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909) + 119906119904119897minus1


+119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909)1003817100381710038171003817

2

1198822

2

+ sdot sdot sdot

+1003817100381710038171003817119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

(23)



119904119897minus1(119909)2

1198822

2

= (119860119904

119897)2 Conse-


119904119898(119909)2

1198822

2

=

sum119897

119894=119898+1(119860119904

119894)2



119904(119909) isin 119882

2

2[0 1] such that 119906

119904119897(119909) rarr


sdot 1198822

2





infin

119895=1

such that 119909119898119895


119904(119909119898119895

) =

119865119904(119909119898119895

1199061119898119895minus1(119909119896) 1199062119898119895minus1(119909119896) 119906

119899119898119895minus1(119909119896)) Hence let


119871119906119904(119909) = 119865

119904(119909 1199061(119909) 1199062(119909) 119906

119899(119909)) That is 119906

119904(119909) satisfies

(1) Also since 120595119894(119909) isin 119882

2

2[0 1] clearly 119906




infin

119894=1119860119904

119894120595119894(119909) and




of (9) 119906119904119898


119904

119898(119909) denote the



119904 = 1 2 119899 then 119903119904

119898(119909)

2

1198822

2

= 119906119904(119909) minus 119906

119904119898(119909)2

1198822

2

=

suminfin

119894=119898+1119860119904

119894120595119894(119909)

2

1198822

2

= suminfin

119894=119898+1(119860119904

119894)

2

and 119903119904

119898minus1(119909)

2

1198822

2

=

suminfin

119894=119898(119860119904

119894)

2

or 119903119904119898

(119909)1198822

2

le 119903119904

119898minus1(119909)1198822

2



119898(119909) 119904 = 1 2 119899 is mono-





1199061015840

1(119909) minus 119906

1(119909) + (119906

2(119909))3

= 1198911(119909)

1199061015840

2(119909) minus sinh (119906

1(119909)) 119906

2(119909) = 119891

2(119909)


3119909(119909minus1)


119909(119909minus1)

(24)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(25)


2(119909) =

119890119909(119909minus1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906




1015840

2(119909) where 119909

119894= (119894minus1)(119872minus1)



1199061015840

1(119909) + radic119906

1(119909) + 1119906

2(119909) = 119891

1(119909)

V10158402(119909) minus 119906

1(119909) (119906

2(119909))2

+ (1199062(119909))2

= 1198912(119909)

1198911(119909) = (119909

4minus 21199093+ 1199092+ 1)

minus12

+ 41199093minus 61199092+ 2119909

1198912(119909) = minus

1199094+ 21199093minus 51199092+ 2119909 minus 1

(1199094minus 21199093+ 1199092+ 1)2

(26)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(27)


2 and 1199062(119909) =

1((119909(119909 minus 1))2+ 1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906



1199061015840

1(119909) + 119906

3(119909) 1198901199061(119909)

+ (1199062(119909))2

= 1198911(119909)

1199061015840

2(119909) minus 119906

2(119909) 119890minus1199061(119909)

+ (1199063(119909))2

= 1198912(119909)

1199061015840

3(119909) minus 119906

1(119909) 1199062(119909) 1199063(119909) = 119891

3(119909)





minus6














minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7



1(119909) and 119906

1015840


Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014




where 119901119894(119909) and 119902




2



119909(119910) in the space

1198822


int

1

011990610158401015840(119910)1205973

119910119877119909(119910)119889119910 = sum

1

119894=0(minus1)1minus119894

119906(119894)(119910)1205973minus119894

119910119877119909(119910)|119910=1

119910=0+

int

1

0119906(119910)120597

4


⟨119906(119910) 119877119909(119910)⟩1198822

2

= sum1

119894=0119906(119894)(0)[120597119894

119910119877119909(0) + (minus1)

1198941205973minus119894

119910119877119909(0)]

+ sum1

119894=0(minus1)1minus119894

119906(119894)(1)1205973minus119894

119910119877119909(1) + int

1

0119906(119910)120597

4

119910119877119909(119910)119889119910 Since

119877119909(119910) isin 119882

2


119909(0) = 119877


119906(119909) isin 1198822


⟨119906 (119910) 119877119909(119910)⟩1198822

2

=

1

sum

119894=0

119906(119894)

(0) [120597119894

119910119877119909(0) + (minus1)

1198941205973minus119894

119910119877119909(0)]

+

1

sum

119894=0

(minus1)119894+1

119906(119894)

(1) 1205973minus119894

119910119877119909(1)

+ int

1

0

119906 (119910) 1205974

119910119877119909(119910) 119889119910 + 119888

1(119906 (0) minus 119906 (1))

(5)


119910119877119909(1) = 0

119877119909(0) + 120597

3

119910119877119909(0) + 119888

1= 0 1205971

119910119877119909(0) minus 120597

2

119910119877119909(0) = 0 and

1205973

119910119877119909(1) + 119888


119909(119910)⟩1198822

2

=

int

1

0119906(119910)120597

4


119909(119910) satisfies

1205974


then ⟨119906(119910) 119877119909(119910)⟩1198822

2

= 119906(119909) Obviously 119877119909(119910) is the


2[0 1] Next we


4= 0




119898

119910119877119909(119909+0) = 120597

119898

119910119877119909(119909minus0)

119898 = 0 1 2 Integrating 1205976



3

119910119877119909(119909 + 0) minus 120597

3

119910119877119909(119909 minus 0) = minus1



119877119909(119910) =

1

48

(1199093119910 (6 + 3119910 minus 119910

2) + 3119909

2119910 (minus6 minus 3119910119910

2) + 6119909119910 (2 + 119910 + 119910

2) minus 8 (minus6 + 119910

3)) 119910 le 119909

1

48

(48 + 6119909119910 (2 minus 3119910 + 1199102) + 3119909

2119910 (2 minus 3119910 + 119910


2+ 1199103)) 119910 gt 119909

(7)



2[0 1] is





2[0 1] are defined


2

= 119906(0)V(0) + int

1

01199061015840(119909)V1015840(119909)119889119909

and 1199061198821

2

= radic⟨119906 119906⟩1198821

2

where 119906 V isin 1198821

2[0 1]



can be written as

119866x (119910) =

1 + 119910 119910 le 119909

1 + 119909 119910 gt 119909

(8)




2



2[0 1] based



2

2[0 1] rarr 119882

1

2[0 1] such that 119871119906(119909) =



119871119906119904(119909) = 119865

119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

119906119904(0) minus 119906

119904(1) = 0

(9)


2

2[0 1]

and 119865119904(119909 V1 V1 V

119899) isin 119882

1

2[0 1] for V

119904= V119904(119909) isin 119882

2

2[0 1]




2[0 1] into the

space1198821

2[0 1]


2[0 1] To do so put 120593

119894(119909) = 119866

119909119894

(119909) and 120595119894(119909) = 119871

lowast120593119894(119909)


119894=1is dense on [0 1] and 119871


119904(119909) 120595

119894(119909)⟩1198822

2

= ⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

=

⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) 119894 = 1 2 119904 = 1 2 119899


119894=1of the

space 1198822



119894=1as follows

120595119894(119909) =

119894

sum

119896=1

120573119894119896120595119896(119909) (10)


120573119894119895=

1

10038171003817100381710038171205951

1003817100381710038171003817

for 119894 = 119895 = 1

120573119894119895=

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2


119896=1(⟨120595119894 120595119896⟩1198822

2

)

2

for 119894 = 119895 = 1

120573119894119895= minus

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2


119896=1(119888119894119896)2

119894minus1

sum

119896=119895

⟨120595119894 120595119896⟩1198822

2

120573119896119895

for 119894 gt 119895

(11)

Clearly 120595119894(119909) = 119871

lowast120593119894(119909) = ⟨119871

lowast120593119894(119909) 119877

119909(119910)⟩1198822

2

=

⟨120593119894(119909) 119871

119910119877119909(119910)⟩1198821

2

= 119871119910119877119909(119910)|119910=119909119894

isin 1198822

2[0 1] Thus 120595

119894(119909)


119910119877119909(119910)|119910=119909119894

where 119871119910



119894=1is dense on [0 1] then 120595

119894(119909)infin

119894=1is a



2

2[0 1] let ⟨119906

119904(119909) 120595

119894(119909)⟩1198822

2

=


119894(119909)⟩1198822

2

=

⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) = 0 Note



119904(119909) = 0 It follows



Lemma 9 If 119906119904(119909) isin 119882

2



le 119872119904119906119904(119909)1198822

2

119894 = 0 1119904 = 1 2 119899 where 119906

119904(119909)119862= max

0le119909le1|119906119904(119909)|


119904(119909) =

⟨119906119904(119910) 120597119894

119909119877119909(119910)⟩1198822

2


119909(119910) it follows that 120597

119894

119909119877119909(119910)1198822

2

le 119872119904

119894 Thus

|119906(119894)

119904(119909)| = |⟨119906

119904(119909) 120597119894

119909119877119909(119909)⟩1198822

2

| le 120597119894

119909119877119909(119909)1198822

2

119906119904(119909)1198822

2

le

119872119904

119894119906119904(119909)1198822

2

Hence 119906(119894)119904(119909)119862le max

119894=01119872119904

119894119906119904(119909)1198822

2

119894 = 0 1 119904 = 1 2 119899


2




119904 = 1 2 119899 will be proved


2[0 1]


⟨119906119904(119909) 120595

119894(119909)⟩120595



119894infin

119894=1is dense on



119906119904(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(12)


119906119904119898

(119909)

=

119898

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(13)

and 119906(119894)



respectively



2[0 1] Since 119906

119904(119909) isin 119882

2

2[0 1]

suminfin

119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595



119894=1 and 119882

2

2[0 1] is


119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595

119894(119909) is


2


119906119904(119909) =

infin

sum

119894=1

⟨119906119904(119909) 120595

119894(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 120595

119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 119871

lowast120593119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119871119906119904(119909) 120593

119896(119909)⟩1198821

2

120595119894(119909)


=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119865119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

120593119896(119909)⟩1198821

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896)

119906119899(119909119896)) 120595119894(119909)

(14)


[0 1]10038161003816100381610038161003816119906(119894)

119904119898(119909) minus 119906

(119894)

119904(119909)

10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

⟨119906119904119898

(119909) minus 119906119904(119909) 119877

(119894)

119909(119909)⟩1198822

2

1003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817120597119894

119909119877119909(119909)

100381710038171003817100381710038171198822

2

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

le 119872119904

119894

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

(15)

where 119894 = 0 1 and 119872119904


119906119904119898

(119909) minus 119906119904(119909)1198822

2


119904119898(119909) and 119906

(119894)

119904119898(119909) 119894 = 0 1 119904 = 1 2 119899




(119909)


119904(119909) of (12)



2




119904 119904 = 1 2 119899




119906119904(119909) =

infin

sum

119894=1

119861119904

119894120595119894(119909) 119904 = 1 2 119899 (16)

where 119861119904

119894= sum

119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896)) In fact 119861

119904




119904



(1199091) = 0

put 1199061199040


to 119906119904(119909) by

119906119904119898

(119909) =

119898

sum

119894=1

119860119904

119894120595119894(119909) 119904 = 1 2 119899 (17)


119894of 120595119894(119909) 119894 = 1 2 119899 119904 =


119860119904

1= 12057311119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091))

1199061199041

(119909) = 119860119904

11205951(119909)

119860119904

2=

2

sum

119896=1

1205732119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

1199061199042

(119909) =

2

sum

119894=1

119860119904

119894120595119894(119909)

119860119904

119899=

119898

sum

119896=1

120573119898119896

119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

119906119904119898

(119909) =

119898minus1

sum

119894=1

119860119904

119894120595119894(119909)

(18)






(119909) and

119906119872

119904119898(119909)

=

119872

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061119898minus1

(119909119896) 1199062119898minus1

(119909119896)

119906119899119898minus1

(119909119896)) 120595119894(119909)

119904 = 1 2 119899

(19)



119904(119909) of (1) In


Lemma 12 If 119906119904119898

(119909) minus 119906119904(119909)1198822

2

rarr 0 119909119898


119904(119909 V1 V2 V





then 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as 119898 rarr infin


(119909119898) rarr 119906

119904(119910) in the


2

Since

1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904(119910)

1003816100381610038161003816

=1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910) + 119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

le1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910)1003816100381610038161003816+1003816100381610038161003816119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

(20)


119904119898minus1(119909119898) =

⟨119906119904119898minus1

(119909) 119877119909119898

(119909)⟩ and 119906119904119898minus1

(119910) = ⟨119906119904119898minus1

(119909) 119877119910(119909)⟩ Thus

|119906119904119898minus1

(119909119898) minus 119906s119898minus1(119910)| = |⟨119906

119904119898minus1(119909) 119877

119909119898

(119909) minus 119877119910(119909)⟩1198822

2

| le

119906119904119898minus1

(119909)1198822

2

119877119909119898

(119909) minus 119877119910(119909)1198822

2


119909119898

(119909) minus 119877119910(119909)1198822

2



(119910)| rarr 0 as soonas 119909119898


119904119898minus1(119910) minus 119906

119904(119910)| rarr 0


(119909119898) rarr 119906

119904(119910) in the


2

as 119909119898



that 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as119898 rarr infin


(119909119895) = 119871119906

119904(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899


(119909119895) = 119865

119904(119909119895 1199061119895minus1

(119909119895)

1199062119895minus1

(119909119895) 119906



(119909119895) = sum

119898

119894=1119860119904

119894119871120595119894(119909119895) =

sum119898

119894=1119860119904

119894⟨119871120595119894(119909) 120593119895(119909)⟩1198821

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 119871lowast

119895120593(119909)⟩

1198822

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2



119895

sum

119897=1

120573119895119897119871119906119904119898

(119909119897)

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909)

119895

sum

119897=1

120573119895119897120595119897(119909)⟩

1198822

2

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2

= 119860119904

119895

=

119895

sum

119897=1

120573119895119897119865119904(119909119897 1199061119897minus1

(119909119897) 1199062119897minus1

(119909119897) 119906

119899119897minus1(119909119897))

(21)

Now if 119895 = 1 then 119871119906119904119898

(1199091) = 119865

119904(1199091 11990610

(1199091) 11990620

(1199091)

1199061198990

(1199091)) Again if 119895 = 2 then 120573

21119871119906119904119898

(1199091) +

12057322119871119906119904119898

(1199092) = 120573

21119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091)) +

12057322119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906

1198991(1199092)) Thus 119871119906

119904119898(1199092) =

119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906



(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899




119906119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) Therefore 119906

119904119898(119909) = 119875

119898119906119904(119909) where


2[0 1] to Span

1205951 1205952 120595

119898 Thus

119871119906119904119898

(119909119895)

= ⟨119871119906119904119898

(119909) 120593119895(119909)⟩1198821

2

= ⟨119906119904119898

(119909) 119871lowast

119895120593 (119909)⟩

1198822

2

= ⟨119875119898119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119906119904(119909) 119875

119898120595119895(119909)⟩1198822

2

= ⟨119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593

119895(119909)⟩1198821

2

= 119871119906119904(119909119895)

(22)

as 119895 le 119898 and 119904 = 1 2 119899

Theorem 14 If 119906119904119898

1198822

2


119894=1is dense on



119904(119909) of


2[0 1] and 119906

119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) 119904 =

1 2 119899 where 119860119904119894is given by (18)


119904119898infin

119898=1in (17) is


2



1198822


1199041198982

1198822

2

= ⟨119906119904119898

(119909) 119906119904119898

(119909)⟩1198822

2

=

⟨sum119898

119894=1119860119904

119894120595119894(119909) sum

119898

119894=1119860119904

119894120595119894(119909)⟩1198822

2

= sum119898

119894=1(119860119904

119894)

2

Therefore119906119904119898

1198822

2


119904119898(119909) From (17)

we have 119906119904119898+1

(119909) = 119906119904119898

(119909) + 119860119904

119898+1120595119898+1


119894(119909)infin


119904119898+12

1198822

2

=

119906119904119898

2

1198822

2

+ (119860119904

119898+1)2

= 119906119904119898minus1

2

1198822

2

+ (119860119904

119898)2+ (119860119904

119898+1)2

=


2

1198822

2

+ sum119898+1

119894=1(119860119904


119904119898infin

119898=1


2


1199041198981198822

2

is bounded 119906119904119898

1198822

2


119904

such that suminfin

119894=1(119860119904

119894)2

= 119888119904 It implies that 119860

119904

119894=

sum119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896) 119906

119899119896minus1(119909119896)) isin 119897

2 119894 =


minus 119906119904119898minus1

) perp (119906119904119898minus1

minus

119906119904119898minus2


minus 119906119904119898


1003817100381710038171003817119906119904119897(119909) minus 119906

119904119898(119909)

1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909) + 119906119904119897minus1


+119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909)1003817100381710038171003817

2

1198822

2

+ sdot sdot sdot

+1003817100381710038171003817119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

(23)



119904119897minus1(119909)2

1198822

2

= (119860119904

119897)2 Conse-


119904119898(119909)2

1198822

2

=

sum119897

119894=119898+1(119860119904

119894)2



119904(119909) isin 119882

2

2[0 1] such that 119906

119904119897(119909) rarr


sdot 1198822

2





infin

119895=1

such that 119909119898119895


119904(119909119898119895

) =

119865119904(119909119898119895

1199061119898119895minus1(119909119896) 1199062119898119895minus1(119909119896) 119906

119899119898119895minus1(119909119896)) Hence let


119871119906119904(119909) = 119865

119904(119909 1199061(119909) 1199062(119909) 119906

119899(119909)) That is 119906

119904(119909) satisfies

(1) Also since 120595119894(119909) isin 119882

2

2[0 1] clearly 119906




infin

119894=1119860119904

119894120595119894(119909) and




of (9) 119906119904119898


119904

119898(119909) denote the



119904 = 1 2 119899 then 119903119904

119898(119909)

2

1198822

2

= 119906119904(119909) minus 119906

119904119898(119909)2

1198822

2

=

suminfin

119894=119898+1119860119904

119894120595119894(119909)

2

1198822

2

= suminfin

119894=119898+1(119860119904

119894)

2

and 119903119904

119898minus1(119909)

2

1198822

2

=

suminfin

119894=119898(119860119904

119894)

2

or 119903119904119898

(119909)1198822

2

le 119903119904

119898minus1(119909)1198822

2



119898(119909) 119904 = 1 2 119899 is mono-





1199061015840

1(119909) minus 119906

1(119909) + (119906

2(119909))3

= 1198911(119909)

1199061015840

2(119909) minus sinh (119906

1(119909)) 119906

2(119909) = 119891

2(119909)


3119909(119909minus1)


119909(119909minus1)

(24)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(25)


2(119909) =

119890119909(119909minus1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906




1015840

2(119909) where 119909

119894= (119894minus1)(119872minus1)



1199061015840

1(119909) + radic119906

1(119909) + 1119906

2(119909) = 119891

1(119909)

V10158402(119909) minus 119906

1(119909) (119906

2(119909))2

+ (1199062(119909))2

= 1198912(119909)

1198911(119909) = (119909

4minus 21199093+ 1199092+ 1)

minus12

+ 41199093minus 61199092+ 2119909

1198912(119909) = minus

1199094+ 21199093minus 51199092+ 2119909 minus 1

(1199094minus 21199093+ 1199092+ 1)2

(26)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(27)


2 and 1199062(119909) =

1((119909(119909 minus 1))2+ 1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906



1199061015840

1(119909) + 119906

3(119909) 1198901199061(119909)

+ (1199062(119909))2

= 1198911(119909)

1199061015840

2(119909) minus 119906

2(119909) 119890minus1199061(119909)

+ (1199063(119909))2

= 1198912(119909)

1199061015840

3(119909) minus 119906

1(119909) 1199062(119909) 1199063(119909) = 119891

3(119909)





minus6














minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7



1(119909) and 119906

1015840


Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014





2[0 1] into the

space1198821

2[0 1]


2[0 1] To do so put 120593

119894(119909) = 119866

119909119894

(119909) and 120595119894(119909) = 119871

lowast120593119894(119909)


119894=1is dense on [0 1] and 119871


119904(119909) 120595

119894(119909)⟩1198822

2

= ⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

=

⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) 119894 = 1 2 119904 = 1 2 119899


119894=1of the

space 1198822



119894=1as follows

120595119894(119909) =

119894

sum

119896=1

120573119894119896120595119896(119909) (10)


120573119894119895=

1

10038171003817100381710038171205951

1003817100381710038171003817

for 119894 = 119895 = 1

120573119894119895=

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2


119896=1(⟨120595119894 120595119896⟩1198822

2

)

2

for 119894 = 119895 = 1

120573119894119895= minus

1

radic1003817100381710038171003817120595119894

1003817100381710038171003817

2


119896=1(119888119894119896)2

119894minus1

sum

119896=119895

⟨120595119894 120595119896⟩1198822

2

120573119896119895

for 119894 gt 119895

(11)

Clearly 120595119894(119909) = 119871

lowast120593119894(119909) = ⟨119871

lowast120593119894(119909) 119877

119909(119910)⟩1198822

2

=

⟨120593119894(119909) 119871

119910119877119909(119910)⟩1198821

2

= 119871119910119877119909(119910)|119910=119909119894

isin 1198822

2[0 1] Thus 120595

119894(119909)


119910119877119909(119910)|119910=119909119894

where 119871119910



119894=1is dense on [0 1] then 120595

119894(119909)infin

119894=1is a



2

2[0 1] let ⟨119906

119904(119909) 120595

119894(119909)⟩1198822

2

=


119894(119909)⟩1198822

2

=

⟨119906119904(119909) 119871lowast120593119894(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593119894(119909)⟩1198821

2

= 119871119906119904(119909119894) = 0 Note



119904(119909) = 0 It follows



Lemma 9 If 119906119904(119909) isin 119882

2



le 119872119904119906119904(119909)1198822

2

119894 = 0 1119904 = 1 2 119899 where 119906

119904(119909)119862= max

0le119909le1|119906119904(119909)|


119904(119909) =

⟨119906119904(119910) 120597119894

119909119877119909(119910)⟩1198822

2


119909(119910) it follows that 120597

119894

119909119877119909(119910)1198822

2

le 119872119904

119894 Thus

|119906(119894)

119904(119909)| = |⟨119906

119904(119909) 120597119894

119909119877119909(119909)⟩1198822

2

| le 120597119894

119909119877119909(119909)1198822

2

119906119904(119909)1198822

2

le

119872119904

119894119906119904(119909)1198822

2

Hence 119906(119894)119904(119909)119862le max

119894=01119872119904

119894119906119904(119909)1198822

2

119894 = 0 1 119904 = 1 2 119899


2




119904 = 1 2 119899 will be proved


2[0 1]


⟨119906119904(119909) 120595

119894(119909)⟩120595



119894infin

119894=1is dense on



119906119904(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(12)


119906119904119898

(119909)

=

119898

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896) 119906

119899(119909119896)) 120595119894(119909)

(13)

and 119906(119894)



respectively



2[0 1] Since 119906

119904(119909) isin 119882

2

2[0 1]

suminfin

119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595



119894=1 and 119882

2

2[0 1] is


119894=1⟨119906119904(119909) 120595

119894(119909)⟩120595

119894(119909) is


2


119906119904(119909) =

infin

sum

119894=1

⟨119906119904(119909) 120595

119894(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 120595

119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119906119904(119909) 119871

lowast120593119896(119909)⟩1198822

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119871119906119904(119909) 120593

119896(119909)⟩1198821

2

120595119894(119909)


=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119865119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

120593119896(119909)⟩1198821

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896)

119906119899(119909119896)) 120595119894(119909)

(14)


[0 1]10038161003816100381610038161003816119906(119894)

119904119898(119909) minus 119906

(119894)

119904(119909)

10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

⟨119906119904119898

(119909) minus 119906119904(119909) 119877

(119894)

119909(119909)⟩1198822

2

1003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817120597119894

119909119877119909(119909)

100381710038171003817100381710038171198822

2

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

le 119872119904

119894

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

(15)

where 119894 = 0 1 and 119872119904


119906119904119898

(119909) minus 119906119904(119909)1198822

2


119904119898(119909) and 119906

(119894)

119904119898(119909) 119894 = 0 1 119904 = 1 2 119899




(119909)


119904(119909) of (12)



2




119904 119904 = 1 2 119899




119906119904(119909) =

infin

sum

119894=1

119861119904

119894120595119894(119909) 119904 = 1 2 119899 (16)

where 119861119904

119894= sum

119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896)) In fact 119861

119904




119904



(1199091) = 0

put 1199061199040


to 119906119904(119909) by

119906119904119898

(119909) =

119898

sum

119894=1

119860119904

119894120595119894(119909) 119904 = 1 2 119899 (17)


119894of 120595119894(119909) 119894 = 1 2 119899 119904 =


119860119904

1= 12057311119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091))

1199061199041

(119909) = 119860119904

11205951(119909)

119860119904

2=

2

sum

119896=1

1205732119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

1199061199042

(119909) =

2

sum

119894=1

119860119904

119894120595119894(119909)

119860119904

119899=

119898

sum

119896=1

120573119898119896

119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

119906119904119898

(119909) =

119898minus1

sum

119894=1

119860119904

119894120595119894(119909)

(18)






(119909) and

119906119872

119904119898(119909)

=

119872

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061119898minus1

(119909119896) 1199062119898minus1

(119909119896)

119906119899119898minus1

(119909119896)) 120595119894(119909)

119904 = 1 2 119899

(19)



119904(119909) of (1) In


Lemma 12 If 119906119904119898

(119909) minus 119906119904(119909)1198822

2

rarr 0 119909119898


119904(119909 V1 V2 V





then 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as 119898 rarr infin


(119909119898) rarr 119906

119904(119910) in the


2

Since

1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904(119910)

1003816100381610038161003816

=1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910) + 119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

le1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910)1003816100381610038161003816+1003816100381610038161003816119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

(20)


119904119898minus1(119909119898) =

⟨119906119904119898minus1

(119909) 119877119909119898

(119909)⟩ and 119906119904119898minus1

(119910) = ⟨119906119904119898minus1

(119909) 119877119910(119909)⟩ Thus

|119906119904119898minus1

(119909119898) minus 119906s119898minus1(119910)| = |⟨119906

119904119898minus1(119909) 119877

119909119898

(119909) minus 119877119910(119909)⟩1198822

2

| le

119906119904119898minus1

(119909)1198822

2

119877119909119898

(119909) minus 119877119910(119909)1198822

2


119909119898

(119909) minus 119877119910(119909)1198822

2



(119910)| rarr 0 as soonas 119909119898


119904119898minus1(119910) minus 119906

119904(119910)| rarr 0


(119909119898) rarr 119906

119904(119910) in the


2

as 119909119898



that 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as119898 rarr infin


(119909119895) = 119871119906

119904(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899


(119909119895) = 119865

119904(119909119895 1199061119895minus1

(119909119895)

1199062119895minus1

(119909119895) 119906



(119909119895) = sum

119898

119894=1119860119904

119894119871120595119894(119909119895) =

sum119898

119894=1119860119904

119894⟨119871120595119894(119909) 120593119895(119909)⟩1198821

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 119871lowast

119895120593(119909)⟩

1198822

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2



119895

sum

119897=1

120573119895119897119871119906119904119898

(119909119897)

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909)

119895

sum

119897=1

120573119895119897120595119897(119909)⟩

1198822

2

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2

= 119860119904

119895

=

119895

sum

119897=1

120573119895119897119865119904(119909119897 1199061119897minus1

(119909119897) 1199062119897minus1

(119909119897) 119906

119899119897minus1(119909119897))

(21)

Now if 119895 = 1 then 119871119906119904119898

(1199091) = 119865

119904(1199091 11990610

(1199091) 11990620

(1199091)

1199061198990

(1199091)) Again if 119895 = 2 then 120573

21119871119906119904119898

(1199091) +

12057322119871119906119904119898

(1199092) = 120573

21119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091)) +

12057322119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906

1198991(1199092)) Thus 119871119906

119904119898(1199092) =

119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906



(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899




119906119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) Therefore 119906

119904119898(119909) = 119875

119898119906119904(119909) where


2[0 1] to Span

1205951 1205952 120595

119898 Thus

119871119906119904119898

(119909119895)

= ⟨119871119906119904119898

(119909) 120593119895(119909)⟩1198821

2

= ⟨119906119904119898

(119909) 119871lowast

119895120593 (119909)⟩

1198822

2

= ⟨119875119898119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119906119904(119909) 119875

119898120595119895(119909)⟩1198822

2

= ⟨119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593

119895(119909)⟩1198821

2

= 119871119906119904(119909119895)

(22)

as 119895 le 119898 and 119904 = 1 2 119899

Theorem 14 If 119906119904119898

1198822

2


119894=1is dense on



119904(119909) of


2[0 1] and 119906

119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) 119904 =

1 2 119899 where 119860119904119894is given by (18)


119904119898infin

119898=1in (17) is


2



1198822


1199041198982

1198822

2

= ⟨119906119904119898

(119909) 119906119904119898

(119909)⟩1198822

2

=

⟨sum119898

119894=1119860119904

119894120595119894(119909) sum

119898

119894=1119860119904

119894120595119894(119909)⟩1198822

2

= sum119898

119894=1(119860119904

119894)

2

Therefore119906119904119898

1198822

2


119904119898(119909) From (17)

we have 119906119904119898+1

(119909) = 119906119904119898

(119909) + 119860119904

119898+1120595119898+1


119894(119909)infin


119904119898+12

1198822

2

=

119906119904119898

2

1198822

2

+ (119860119904

119898+1)2

= 119906119904119898minus1

2

1198822

2

+ (119860119904

119898)2+ (119860119904

119898+1)2

=


2

1198822

2

+ sum119898+1

119894=1(119860119904


119904119898infin

119898=1


2


1199041198981198822

2

is bounded 119906119904119898

1198822

2


119904

such that suminfin

119894=1(119860119904

119894)2

= 119888119904 It implies that 119860

119904

119894=

sum119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896) 119906

119899119896minus1(119909119896)) isin 119897

2 119894 =


minus 119906119904119898minus1

) perp (119906119904119898minus1

minus

119906119904119898minus2


minus 119906119904119898


1003817100381710038171003817119906119904119897(119909) minus 119906

119904119898(119909)

1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909) + 119906119904119897minus1


+119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909)1003817100381710038171003817

2

1198822

2

+ sdot sdot sdot

+1003817100381710038171003817119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

(23)



119904119897minus1(119909)2

1198822

2

= (119860119904

119897)2 Conse-


119904119898(119909)2

1198822

2

=

sum119897

119894=119898+1(119860119904

119894)2



119904(119909) isin 119882

2

2[0 1] such that 119906

119904119897(119909) rarr


sdot 1198822

2





infin

119895=1

such that 119909119898119895


119904(119909119898119895

) =

119865119904(119909119898119895

1199061119898119895minus1(119909119896) 1199062119898119895minus1(119909119896) 119906

119899119898119895minus1(119909119896)) Hence let


119871119906119904(119909) = 119865

119904(119909 1199061(119909) 1199062(119909) 119906

119899(119909)) That is 119906

119904(119909) satisfies

(1) Also since 120595119894(119909) isin 119882

2

2[0 1] clearly 119906




infin

119894=1119860119904

119894120595119894(119909) and




of (9) 119906119904119898


119904

119898(119909) denote the



119904 = 1 2 119899 then 119903119904

119898(119909)

2

1198822

2

= 119906119904(119909) minus 119906

119904119898(119909)2

1198822

2

=

suminfin

119894=119898+1119860119904

119894120595119894(119909)

2

1198822

2

= suminfin

119894=119898+1(119860119904

119894)

2

and 119903119904

119898minus1(119909)

2

1198822

2

=

suminfin

119894=119898(119860119904

119894)

2

or 119903119904119898

(119909)1198822

2

le 119903119904

119898minus1(119909)1198822

2



119898(119909) 119904 = 1 2 119899 is mono-





1199061015840

1(119909) minus 119906

1(119909) + (119906

2(119909))3

= 1198911(119909)

1199061015840

2(119909) minus sinh (119906

1(119909)) 119906

2(119909) = 119891

2(119909)


3119909(119909minus1)


119909(119909minus1)

(24)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(25)


2(119909) =

119890119909(119909minus1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906




1015840

2(119909) where 119909

119894= (119894minus1)(119872minus1)



1199061015840

1(119909) + radic119906

1(119909) + 1119906

2(119909) = 119891

1(119909)

V10158402(119909) minus 119906

1(119909) (119906

2(119909))2

+ (1199062(119909))2

= 1198912(119909)

1198911(119909) = (119909

4minus 21199093+ 1199092+ 1)

minus12

+ 41199093minus 61199092+ 2119909

1198912(119909) = minus

1199094+ 21199093minus 51199092+ 2119909 minus 1

(1199094minus 21199093+ 1199092+ 1)2

(26)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(27)


2 and 1199062(119909) =

1((119909(119909 minus 1))2+ 1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906



1199061015840

1(119909) + 119906

3(119909) 1198901199061(119909)

+ (1199062(119909))2

= 1198911(119909)

1199061015840

2(119909) minus 119906

2(119909) 119890minus1199061(119909)

+ (1199063(119909))2

= 1198912(119909)

1199061015840

3(119909) minus 119906

1(119909) 1199062(119909) 1199063(119909) = 119891

3(119909)





minus6














minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7



1(119909) and 119906

1015840


Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014




=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896⟨119865119904(119909 1199061(119909) 119906

2(119909) 119906

119899(119909))

120593119896(119909)⟩1198821

2

120595119894(119909)

=

infin

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061(119909119896) 1199062(119909119896)

119906119899(119909119896)) 120595119894(119909)

(14)


[0 1]10038161003816100381610038161003816119906(119894)

119904119898(119909) minus 119906

(119894)

119904(119909)

10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

⟨119906119904119898

(119909) minus 119906119904(119909) 119877

(119894)

119909(119909)⟩1198822

2

1003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817120597119894

119909119877119909(119909)

100381710038171003817100381710038171198822

2

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

le 119872119904

119894

1003817100381710038171003817119906119904119898

(119909) minus 119906119904(119909)

10038171003817100381710038171198822

2

(15)

where 119894 = 0 1 and 119872119904


119906119904119898

(119909) minus 119906119904(119909)1198822

2


119904119898(119909) and 119906

(119894)

119904119898(119909) 119894 = 0 1 119904 = 1 2 119899




(119909)


119904(119909) of (12)



2




119904 119904 = 1 2 119899




119906119904(119909) =

infin

sum

119894=1

119861119904

119894120595119894(119909) 119904 = 1 2 119899 (16)

where 119861119904

119894= sum

119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896)) In fact 119861

119904




119904



(1199091) = 0

put 1199061199040


to 119906119904(119909) by

119906119904119898

(119909) =

119898

sum

119894=1

119860119904

119894120595119894(119909) 119904 = 1 2 119899 (17)


119894of 120595119894(119909) 119894 = 1 2 119899 119904 =


119860119904

1= 12057311119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091))

1199061199041

(119909) = 119860119904

11205951(119909)

119860119904

2=

2

sum

119896=1

1205732119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

1199061199042

(119909) =

2

sum

119894=1

119860119904

119894120595119894(119909)

119860119904

119899=

119898

sum

119896=1

120573119898119896

119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896)

119906119899119896minus1

(119909119896))

119906119904119898

(119909) =

119898minus1

sum

119894=1

119860119904

119894120595119894(119909)

(18)






(119909) and

119906119872

119904119898(119909)

=

119872

sum

119894=1

119894

sum

119896=1

120573119894119896119865119904(119909119896 1199061119898minus1

(119909119896) 1199062119898minus1

(119909119896)

119906119899119898minus1

(119909119896)) 120595119894(119909)

119904 = 1 2 119899

(19)



119904(119909) of (1) In


Lemma 12 If 119906119904119898

(119909) minus 119906119904(119909)1198822

2

rarr 0 119909119898


119904(119909 V1 V2 V





then 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as 119898 rarr infin


(119909119898) rarr 119906

119904(119910) in the


2

Since

1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904(119910)

1003816100381610038161003816

=1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910) + 119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

le1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910)1003816100381610038161003816+1003816100381610038161003816119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

(20)


119904119898minus1(119909119898) =

⟨119906119904119898minus1

(119909) 119877119909119898

(119909)⟩ and 119906119904119898minus1

(119910) = ⟨119906119904119898minus1

(119909) 119877119910(119909)⟩ Thus

|119906119904119898minus1

(119909119898) minus 119906s119898minus1(119910)| = |⟨119906

119904119898minus1(119909) 119877

119909119898

(119909) minus 119877119910(119909)⟩1198822

2

| le

119906119904119898minus1

(119909)1198822

2

119877119909119898

(119909) minus 119877119910(119909)1198822

2


119909119898

(119909) minus 119877119910(119909)1198822

2



(119910)| rarr 0 as soonas 119909119898


119904119898minus1(119910) minus 119906

119904(119910)| rarr 0


(119909119898) rarr 119906

119904(119910) in the


2

as 119909119898



that 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as119898 rarr infin


(119909119895) = 119871119906

119904(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899


(119909119895) = 119865

119904(119909119895 1199061119895minus1

(119909119895)

1199062119895minus1

(119909119895) 119906



(119909119895) = sum

119898

119894=1119860119904

119894119871120595119894(119909119895) =

sum119898

119894=1119860119904

119894⟨119871120595119894(119909) 120593119895(119909)⟩1198821

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 119871lowast

119895120593(119909)⟩

1198822

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2



119895

sum

119897=1

120573119895119897119871119906119904119898

(119909119897)

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909)

119895

sum

119897=1

120573119895119897120595119897(119909)⟩

1198822

2

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2

= 119860119904

119895

=

119895

sum

119897=1

120573119895119897119865119904(119909119897 1199061119897minus1

(119909119897) 1199062119897minus1

(119909119897) 119906

119899119897minus1(119909119897))

(21)

Now if 119895 = 1 then 119871119906119904119898

(1199091) = 119865

119904(1199091 11990610

(1199091) 11990620

(1199091)

1199061198990

(1199091)) Again if 119895 = 2 then 120573

21119871119906119904119898

(1199091) +

12057322119871119906119904119898

(1199092) = 120573

21119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091)) +

12057322119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906

1198991(1199092)) Thus 119871119906

119904119898(1199092) =

119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906



(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899




119906119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) Therefore 119906

119904119898(119909) = 119875

119898119906119904(119909) where


2[0 1] to Span

1205951 1205952 120595

119898 Thus

119871119906119904119898

(119909119895)

= ⟨119871119906119904119898

(119909) 120593119895(119909)⟩1198821

2

= ⟨119906119904119898

(119909) 119871lowast

119895120593 (119909)⟩

1198822

2

= ⟨119875119898119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119906119904(119909) 119875

119898120595119895(119909)⟩1198822

2

= ⟨119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593

119895(119909)⟩1198821

2

= 119871119906119904(119909119895)

(22)

as 119895 le 119898 and 119904 = 1 2 119899

Theorem 14 If 119906119904119898

1198822

2


119894=1is dense on



119904(119909) of


2[0 1] and 119906

119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) 119904 =

1 2 119899 where 119860119904119894is given by (18)


119904119898infin

119898=1in (17) is


2



1198822


1199041198982

1198822

2

= ⟨119906119904119898

(119909) 119906119904119898

(119909)⟩1198822

2

=

⟨sum119898

119894=1119860119904

119894120595119894(119909) sum

119898

119894=1119860119904

119894120595119894(119909)⟩1198822

2

= sum119898

119894=1(119860119904

119894)

2

Therefore119906119904119898

1198822

2


119904119898(119909) From (17)

we have 119906119904119898+1

(119909) = 119906119904119898

(119909) + 119860119904

119898+1120595119898+1


119894(119909)infin


119904119898+12

1198822

2

=

119906119904119898

2

1198822

2

+ (119860119904

119898+1)2

= 119906119904119898minus1

2

1198822

2

+ (119860119904

119898)2+ (119860119904

119898+1)2

=


2

1198822

2

+ sum119898+1

119894=1(119860119904


119904119898infin

119898=1


2


1199041198981198822

2

is bounded 119906119904119898

1198822

2


119904

such that suminfin

119894=1(119860119904

119894)2

= 119888119904 It implies that 119860

119904

119894=

sum119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896) 119906

119899119896minus1(119909119896)) isin 119897

2 119894 =


minus 119906119904119898minus1

) perp (119906119904119898minus1

minus

119906119904119898minus2


minus 119906119904119898


1003817100381710038171003817119906119904119897(119909) minus 119906

119904119898(119909)

1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909) + 119906119904119897minus1


+119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909)1003817100381710038171003817

2

1198822

2

+ sdot sdot sdot

+1003817100381710038171003817119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

(23)



119904119897minus1(119909)2

1198822

2

= (119860119904

119897)2 Conse-


119904119898(119909)2

1198822

2

=

sum119897

119894=119898+1(119860119904

119894)2



119904(119909) isin 119882

2

2[0 1] such that 119906

119904119897(119909) rarr


sdot 1198822

2





infin

119895=1

such that 119909119898119895


119904(119909119898119895

) =

119865119904(119909119898119895

1199061119898119895minus1(119909119896) 1199062119898119895minus1(119909119896) 119906

119899119898119895minus1(119909119896)) Hence let


119871119906119904(119909) = 119865

119904(119909 1199061(119909) 1199062(119909) 119906

119899(119909)) That is 119906

119904(119909) satisfies

(1) Also since 120595119894(119909) isin 119882

2

2[0 1] clearly 119906




infin

119894=1119860119904

119894120595119894(119909) and




of (9) 119906119904119898


119904

119898(119909) denote the



119904 = 1 2 119899 then 119903119904

119898(119909)

2

1198822

2

= 119906119904(119909) minus 119906

119904119898(119909)2

1198822

2

=

suminfin

119894=119898+1119860119904

119894120595119894(119909)

2

1198822

2

= suminfin

119894=119898+1(119860119904

119894)

2

and 119903119904

119898minus1(119909)

2

1198822

2

=

suminfin

119894=119898(119860119904

119894)

2

or 119903119904119898

(119909)1198822

2

le 119903119904

119898minus1(119909)1198822

2



119898(119909) 119904 = 1 2 119899 is mono-





1199061015840

1(119909) minus 119906

1(119909) + (119906

2(119909))3

= 1198911(119909)

1199061015840

2(119909) minus sinh (119906

1(119909)) 119906

2(119909) = 119891

2(119909)


3119909(119909minus1)


119909(119909minus1)

(24)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(25)


2(119909) =

119890119909(119909minus1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906




1015840

2(119909) where 119909

119894= (119894minus1)(119872minus1)



1199061015840

1(119909) + radic119906

1(119909) + 1119906

2(119909) = 119891

1(119909)

V10158402(119909) minus 119906

1(119909) (119906

2(119909))2

+ (1199062(119909))2

= 1198912(119909)

1198911(119909) = (119909

4minus 21199093+ 1199092+ 1)

minus12

+ 41199093minus 61199092+ 2119909

1198912(119909) = minus

1199094+ 21199093minus 51199092+ 2119909 minus 1

(1199094minus 21199093+ 1199092+ 1)2

(26)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(27)


2 and 1199062(119909) =

1((119909(119909 minus 1))2+ 1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906



1199061015840

1(119909) + 119906

3(119909) 1198901199061(119909)

+ (1199062(119909))2

= 1198911(119909)

1199061015840

2(119909) minus 119906

2(119909) 119890minus1199061(119909)

+ (1199063(119909))2

= 1198912(119909)

1199061015840

3(119909) minus 119906

1(119909) 1199062(119909) 1199063(119909) = 119891

3(119909)





minus6














minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7



1(119909) and 119906

1015840


Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014






then 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as 119898 rarr infin


(119909119898) rarr 119906

119904(119910) in the


2

Since

1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904(119910)

1003816100381610038161003816

=1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910) + 119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

le1003816100381610038161003816119906119904119898minus1

(119909119898) minus 119906119904119898minus1

(119910)1003816100381610038161003816+1003816100381610038161003816119906119904119898minus1

(119910) minus 119906119904(119910)

1003816100381610038161003816

(20)


119904119898minus1(119909119898) =

⟨119906119904119898minus1

(119909) 119877119909119898

(119909)⟩ and 119906119904119898minus1

(119910) = ⟨119906119904119898minus1

(119909) 119877119910(119909)⟩ Thus

|119906119904119898minus1

(119909119898) minus 119906s119898minus1(119910)| = |⟨119906

119904119898minus1(119909) 119877

119909119898

(119909) minus 119877119910(119909)⟩1198822

2

| le

119906119904119898minus1

(119909)1198822

2

119877119909119898

(119909) minus 119877119910(119909)1198822

2


119909119898

(119909) minus 119877119910(119909)1198822

2



(119910)| rarr 0 as soonas 119909119898


119904119898minus1(119910) minus 119906

119904(119910)| rarr 0


(119909119898) rarr 119906

119904(119910) in the


2

as 119909119898



that 119865119904(119909119898 1199061119898minus1

(119909119898) 1199062119898minus1

(119909119898) 119906

119899119898minus1(119909119898)) rarr

119865119904(119910 1199061(119910) 1199062(119910) 119906

119899(119910)) 119904 = 1 2 119899 as119898 rarr infin


(119909119895) = 119871119906

119904(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899


(119909119895) = 119865

119904(119909119895 1199061119895minus1

(119909119895)

1199062119895minus1

(119909119895) 119906



(119909119895) = sum

119898

119894=1119860119904

119894119871120595119894(119909119895) =

sum119898

119894=1119860119904

119894⟨119871120595119894(119909) 120593119895(119909)⟩1198821

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 119871lowast

119895120593(119909)⟩

1198822

2

= sum119898

119894=1119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2



119895

sum

119897=1

120573119895119897119871119906119904119898

(119909119897)

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909)

119895

sum

119897=1

120573119895119897120595119897(119909)⟩

1198822

2

=

119898

sum

119894=1

119860119904

119894⟨120595119894(119909) 120595

119895(119909)⟩1198822

2

= 119860119904

119895

=

119895

sum

119897=1

120573119895119897119865119904(119909119897 1199061119897minus1

(119909119897) 1199062119897minus1

(119909119897) 119906

119899119897minus1(119909119897))

(21)

Now if 119895 = 1 then 119871119906119904119898

(1199091) = 119865

119904(1199091 11990610

(1199091) 11990620

(1199091)

1199061198990

(1199091)) Again if 119895 = 2 then 120573

21119871119906119904119898

(1199091) +

12057322119871119906119904119898

(1199092) = 120573

21119865119904(1199091 11990610

(1199091) 11990620

(1199091) 119906

1198990(1199091)) +

12057322119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906

1198991(1199092)) Thus 119871119906

119904119898(1199092) =

119865119904(1199092 11990611

(1199092) 11990621

(1199092) 119906



(119909119895) =

119865119904(119909119895 1199061119895minus1

(119909119895) 1199062119895minus1

(119909119895) 119906

119899119895minus1(119909119895)) 119904 = 1 2 119899




119906119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) Therefore 119906

119904119898(119909) = 119875

119898119906119904(119909) where


2[0 1] to Span

1205951 1205952 120595

119898 Thus

119871119906119904119898

(119909119895)

= ⟨119871119906119904119898

(119909) 120593119895(119909)⟩1198821

2

= ⟨119906119904119898

(119909) 119871lowast

119895120593 (119909)⟩

1198822

2

= ⟨119875119898119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119906119904(119909) 119875

119898120595119895(119909)⟩1198822

2

= ⟨119906119904(119909) 120595

119895(119909)⟩1198822

2

= ⟨119871119906119904(119909) 120593

119895(119909)⟩1198821

2

= 119871119906119904(119909119895)

(22)

as 119895 le 119898 and 119904 = 1 2 119899

Theorem 14 If 119906119904119898

1198822

2


119894=1is dense on



119904(119909) of


2[0 1] and 119906

119904(119909) = sum

infin

119894=1119860119904

119894120595119894(119909) 119904 =

1 2 119899 where 119860119904119894is given by (18)


119904119898infin

119898=1in (17) is


2



1198822


1199041198982

1198822

2

= ⟨119906119904119898

(119909) 119906119904119898

(119909)⟩1198822

2

=

⟨sum119898

119894=1119860119904

119894120595119894(119909) sum

119898

119894=1119860119904

119894120595119894(119909)⟩1198822

2

= sum119898

119894=1(119860119904

119894)

2

Therefore119906119904119898

1198822

2


119904119898(119909) From (17)

we have 119906119904119898+1

(119909) = 119906119904119898

(119909) + 119860119904

119898+1120595119898+1


119894(119909)infin


119904119898+12

1198822

2

=

119906119904119898

2

1198822

2

+ (119860119904

119898+1)2

= 119906119904119898minus1

2

1198822

2

+ (119860119904

119898)2+ (119860119904

119898+1)2

=


2

1198822

2

+ sum119898+1

119894=1(119860119904


119904119898infin

119898=1


2


1199041198981198822

2

is bounded 119906119904119898

1198822

2


119904

such that suminfin

119894=1(119860119904

119894)2

= 119888119904 It implies that 119860

119904

119894=

sum119894

119896=1120573119894119896119865119904(119909119896 1199061119896minus1

(119909119896) 1199062119896minus1

(119909119896) 119906

119899119896minus1(119909119896)) isin 119897

2 119894 =


minus 119906119904119898minus1

) perp (119906119904119898minus1

minus

119906119904119898minus2


minus 119906119904119898


1003817100381710038171003817119906119904119897(119909) minus 119906

119904119898(119909)

1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909) + 119906119904119897minus1


+119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

=1003817100381710038171003817119906119904119897

(119909) minus 119906119904119897minus1

(119909)1003817100381710038171003817

2

1198822

2

+ sdot sdot sdot

+1003817100381710038171003817119906119904119897+1

(119909) minus 119906119904119898

(119909)1003817100381710038171003817

2

1198822

2

(23)



119904119897minus1(119909)2

1198822

2

= (119860119904

119897)2 Conse-


119904119898(119909)2

1198822

2

=

sum119897

119894=119898+1(119860119904

119894)2



119904(119909) isin 119882

2

2[0 1] such that 119906

119904119897(119909) rarr


sdot 1198822

2





infin

119895=1

such that 119909119898119895


119904(119909119898119895

) =

119865119904(119909119898119895

1199061119898119895minus1(119909119896) 1199062119898119895minus1(119909119896) 119906

119899119898119895minus1(119909119896)) Hence let


119871119906119904(119909) = 119865

119904(119909 1199061(119909) 1199062(119909) 119906

119899(119909)) That is 119906

119904(119909) satisfies

(1) Also since 120595119894(119909) isin 119882

2

2[0 1] clearly 119906




infin

119894=1119860119904

119894120595119894(119909) and




of (9) 119906119904119898


119904

119898(119909) denote the



119904 = 1 2 119899 then 119903119904

119898(119909)

2

1198822

2

= 119906119904(119909) minus 119906

119904119898(119909)2

1198822

2

=

suminfin

119894=119898+1119860119904

119894120595119894(119909)

2

1198822

2

= suminfin

119894=119898+1(119860119904

119894)

2

and 119903119904

119898minus1(119909)

2

1198822

2

=

suminfin

119894=119898(119860119904

119894)

2

or 119903119904119898

(119909)1198822

2

le 119903119904

119898minus1(119909)1198822

2



119898(119909) 119904 = 1 2 119899 is mono-





1199061015840

1(119909) minus 119906

1(119909) + (119906

2(119909))3

= 1198911(119909)

1199061015840

2(119909) minus sinh (119906

1(119909)) 119906

2(119909) = 119891

2(119909)


3119909(119909minus1)


119909(119909minus1)

(24)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(25)


2(119909) =

119890119909(119909minus1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906




1015840

2(119909) where 119909

119894= (119894minus1)(119872minus1)



1199061015840

1(119909) + radic119906

1(119909) + 1119906

2(119909) = 119891

1(119909)

V10158402(119909) minus 119906

1(119909) (119906

2(119909))2

+ (1199062(119909))2

= 1198912(119909)

1198911(119909) = (119909

4minus 21199093+ 1199092+ 1)

minus12

+ 41199093minus 61199092+ 2119909

1198912(119909) = minus

1199094+ 21199093minus 51199092+ 2119909 minus 1

(1199094minus 21199093+ 1199092+ 1)2

(26)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(27)


2 and 1199062(119909) =

1((119909(119909 minus 1))2+ 1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906



1199061015840

1(119909) + 119906

3(119909) 1198901199061(119909)

+ (1199062(119909))2

= 1198911(119909)

1199061015840

2(119909) minus 119906

2(119909) 119890minus1199061(119909)

+ (1199063(119909))2

= 1198912(119909)

1199061015840

3(119909) minus 119906

1(119909) 1199062(119909) 1199063(119909) = 119891

3(119909)





minus6














minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7



1(119909) and 119906

1015840


Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014





119904119897minus1(119909)2

1198822

2

= (119860119904

119897)2 Conse-


119904119898(119909)2

1198822

2

=

sum119897

119894=119898+1(119860119904

119894)2



119904(119909) isin 119882

2

2[0 1] such that 119906

119904119897(119909) rarr


sdot 1198822

2





infin

119895=1

such that 119909119898119895


119904(119909119898119895

) =

119865119904(119909119898119895

1199061119898119895minus1(119909119896) 1199062119898119895minus1(119909119896) 119906

119899119898119895minus1(119909119896)) Hence let


119871119906119904(119909) = 119865

119904(119909 1199061(119909) 1199062(119909) 119906

119899(119909)) That is 119906

119904(119909) satisfies

(1) Also since 120595119894(119909) isin 119882

2

2[0 1] clearly 119906




infin

119894=1119860119904

119894120595119894(119909) and




of (9) 119906119904119898


119904

119898(119909) denote the



119904 = 1 2 119899 then 119903119904

119898(119909)

2

1198822

2

= 119906119904(119909) minus 119906

119904119898(119909)2

1198822

2

=

suminfin

119894=119898+1119860119904

119894120595119894(119909)

2

1198822

2

= suminfin

119894=119898+1(119860119904

119894)

2

and 119903119904

119898minus1(119909)

2

1198822

2

=

suminfin

119894=119898(119860119904

119894)

2

or 119903119904119898

(119909)1198822

2

le 119903119904

119898minus1(119909)1198822

2



119898(119909) 119904 = 1 2 119899 is mono-





1199061015840

1(119909) minus 119906

1(119909) + (119906

2(119909))3

= 1198911(119909)

1199061015840

2(119909) minus sinh (119906

1(119909)) 119906

2(119909) = 119891

2(119909)


3119909(119909minus1)


119909(119909minus1)

(24)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(25)


2(119909) =

119890119909(119909minus1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906




1015840

2(119909) where 119909

119894= (119894minus1)(119872minus1)



1199061015840

1(119909) + radic119906

1(119909) + 1119906

2(119909) = 119891

1(119909)

V10158402(119909) minus 119906

1(119909) (119906

2(119909))2

+ (1199062(119909))2

= 1198912(119909)

1198911(119909) = (119909

4minus 21199093+ 1199092+ 1)

minus12

+ 41199093minus 61199092+ 2119909

1198912(119909) = minus

1199094+ 21199093minus 51199092+ 2119909 minus 1

(1199094minus 21199093+ 1199092+ 1)2

(26)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

(27)


2 and 1199062(119909) =

1((119909(119909 minus 1))2+ 1)


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) and 119906



1199061015840

1(119909) + 119906

3(119909) 1198901199061(119909)

+ (1199062(119909))2

= 1198911(119909)

1199061015840

2(119909) minus 119906

2(119909) 119890minus1199061(119909)

+ (1199063(119909))2

= 1198912(119909)

1199061015840

3(119909) minus 119906

1(119909) 1199062(119909) 1199063(119909) = 119891

3(119909)





minus6














minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7



1(119909) and 119906

1015840


Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014







minus6














minus7

032 0804447 08044464859485744 670233 times 10minus7


048 0779112 07791116154224935 750275 times 10minus7


064 0794216 07942151498560056 702761 times 10minus7


080 0852144 08521432738935479 515073 times 10minus7


096 0962328 09623277968729329 135849 times 10minus7



1(119909) and 119906

1015840


Derivative 119909 = 016 119909 = 048 119909 = 064 119909 = 096

1199061015840

1(119909) 396943 times 10




minus6

1199061015840

2(119909) 888178 times 10




minus7




minus5

032 00473498 004734840000000003 135999 times 10minus6


048 00623002 006229859999999997 156000 times 10minus6


064 00530842 005308272000000007 143999 times 10minus6


080 00256000 002559900000000005 999999 times 10minus7


096 00014746 000147432000000012 239999 times 10minus7





minus7

032 0954791 0954792235061675 135412 times 10minus6


048 0941354 0941355026122341 150133 times 10minus6


064 0949592 0949593136843461 141570 times 10minus6


080 0975039 0975040039211652 103765 times 10minus6


096 0998528 0998527858662958 247537 times 10minus7





minus6















minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014







minus7

032 124309 1243088505342329 122692 times 10minus6


048 128351 1283510460180459 144905 times 10minus6


064 125910 1259102236286934 131435 times 10minus6


080 117351 1173510014918347 856073 times 10minus7


096 103915 1039146624444131 184037 times 10minus7





minus7

032 0806168 08061672349997103 120922 times 10minus6


048 0781712 07817107460286419 139143 times 10minus6


064 0796259 07962584197762563 128185 times 10minus6


080 0852827 08528264441003238 885879 times 10minus7


096 0962337 09623371578286029 212773 times 10minus7



+ 119890minus2119909(119909minus1)

+

2119909 minus 1

119909 (119909 minus 1) + 1

1198912(119909) = (cosh (119909 (119909 minus 1)) + 119909 (119909 minus 1))

2



119890minus119909(119909minus1)

119909 (119909 minus 1) + 1

1198913(119909) = sinh (119909 (119909 minus 1)) (2119909 minus 1)



+ 2119909 minus 1

(28)


1199061(0) = 119906

1(1)

1199062(0) = 119906

2(1)

1199063(0) = 119906

3(1)

(29)


2(119909) =

119890119909(1minus119909) and 119906

3(119909) = 119909(119909 minus 1) + cosh(119909(119909 minus 1))


= (119894 minus 1)(119872 minus 1) 119894 =


1(119909) 119906

2(119909) and 119906

3(119909)



6 Conclusions


2

2[0 1] with


2





Acknowledgment


References













































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014












































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014










Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014



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A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems

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