Research ArticlePositive Solutions for Nonlinear 119902-Fractional DifferenceEigenvalue Problem with Nonlocal Conditions
Wafa Shammakh and Maryam Al-Yami
Sciences Faculty for Girls King Abdulaziz University Jeddah Saudi Arabia
Correspondence should be addressed to Maryam Al-Yami malyamikauedusa
Received 16 August 2015 Revised 5 November 2015 Accepted 26 November 2015
Academic Editor Svatoslav Stanek
Copyright copy 2015 W Shammakh and M Al-Yami This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
The problem of positive solutions for nonlinear 119902-fractional difference eigenvalue problem with nonlocal boundary conditions isinvestigated Based on the fixed point index theory in cones sufficient existence of positive solutions conditions is derived for theproblem
1 Introduction
The fractional 119902-calculus is the 119902-extension of ordinaryfractional calculus It has been used by many researchers toadequately describe the evolution of a variety of engineeringeconomical physical and biological processes
We consider a nonlinear 119902-fractional difference eigen-value problem with nonlocal boundary conditions given by119862119863120572
where 119862119863120572119902denote the fractional 119902-derivative of the Caputo
type 119899 minus 1 lt 120572 le 119899 119899 gt 2 120582 gt 0 is a parameter and 120579[119906] isgiven by a Riemann-Stieltjes integral 120579[119906] = int1
0119906(119905)119889119902119860(119905)
This type of BC includes as particular cases multipointproblems when 120579[119906] = sum119898minus2
119894=1120572119894119906(120577119894) (see [1]) and a contin-
uously distributed case when 120579[119906] = int10120572(119904)119906(119904)119889
119902119904 (see [2ndash
4])More recently many people pay attention to BVPs involv-
ing nonlinear 119902-difference equations [5ndash12]
In [13] Yuan and Yang dealt with some existence anduniqueness results for nonlinear boundary value problemsfor delayed 119902-fractional difference systems based on a con-traction mapping principle and Krasnoselskiirsquos fixed-pointtheorem
In [14] Yang investigated the sufficient conditions forthe existence and nonexistence positive solutions for BVPinvolving nonlinear 119902-fractional difference equations
Ferreira [4] studied the existence of positive solutions tothe nonlinear 119902-fractional BVPs by means of Krasnoselskiirsquosfixed point theorem in cones
In this paper we obtain the results on the existence ofone and two positive solutions by utilizing the results ofWebb and Lan [15] involving comparison with the principlecharacteristic value of a related linear problem to the 119902-fractional case We then use the theory worked out by Webband Infante in [16ndash19] to study the general nonlocal BCs
2 Preliminaries
In this section we will present some definitions and lemmasthat will be used in the proof of our main results
Let 119902 isin (0 1) defined by [20]
[119886]119902 =119902119886minus 1
119902 minus 1= 119902119886minus1+ sdot sdot sdot + 1 119886 isin R (3)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 759378 10 pageshttpdxdoiorg1011552015759378
2 Abstract and Applied Analysis
The 119902-analogue of the power function (119886 minus 119887)119899 with 119899 isin N is
(119886 minus 119887)0= 1
(119886 minus 119887)(119899)=
119899minus1
prod
119896=0
(119886 minus 119887119902119896) 119886 119887 isin R 119899 isin N
(4)
More generally if 120572 isin R then
(119886 minus 119887)(120572)= 119886120572
infin
prod
119894=0
(119886 minus 119887119902119894)
(119886 minus 119887119902120572+119894) (5)
Note that if 119887 = 0 then 119886(120572) = 119886120572The 119902-gamma function isdefined by
Γ119902 (119909) =
(1 minus 119902)(119909minus1)
(1 minus 119902)119909minus1
119909 isin R 0 minus1 minus2 0 lt 119902 lt 1
(6)
and satisfies Γ119902(119909 + 1) = [119909]
119902Γ119902(119909)
The 119902-derivative of a function 119891(119909) is here defined by
119863119902119891 (119909) =
119889119902119891 (119909)
119889119902119909=119891 (119902119909) minus 119891 (119909)
(119902 minus 1) 119909 (7)
and 119902-derivatives of higher order are defined by
119863119899
119902119891 (119909) =
119891 (119909) if 119899 = 0
119863119902119863119899minus1
119902119891 (119909) if 119899 isin N
(8)
The 119902-integral of a function 119891 defined in the interval [0 119887] isgiven by
int
119909
0
119891 (119905) 119889119902119905 = 119909 (1 minus 119902)
Theorem 7 (see [23]) Let 119909 gt 0 and 120572 isin R+ NThen thefollowing equality holds
(119868120572
119902 119862119863120572
119902119891) (119909) = 119891 (119909) minus
[120572]minus1
sum
119896=0
119909119896
Γ119902 (119896 + 1)
(119863119896
119902119891) (0) (19)
Lemma 8 (see [24]) Suppose 119879 119870 rarr 119870 is a completelycontinuous operator and has no fixed points on 120597119870
120588cap119870 Then
the following are true(i) If 119879119906 le 119906 for all 119906 isin 120597119870
120588cap 119870 then 119894(119879 119870
120588cap
119870119870) = 1 where 119894 is the fixed point index on 119870(ii) If 119879119906 ge 119906 for all 119906 isin 120597119870
120588cap 119870 then 119894(119879 119870
120588cap
119870119870) = 0
Lemma 9 (see [24]) Let 119870 be a cone in Banach space 119864Suppose that119879 119870
120588rarr 119870 is a completely continuous operator
There exists 1199060isin 1198700 such that 119906minus119879119906 = 120583119906
0for any 119906 isin 120597119870
119903
and 120583 ge 0 119894(119879 119870120588 119870) = 0
Lemma 10 (see [24]) Let 119870 be a cone in Banach space 119864Suppose that119879 119870
120588rarr 119870 is a completely continuous operator
If 119879119906 = 120583119906 for any 119906 isin 120597119870119903and 120583 ge 1 then 119894(119879 119870
120588 119870) = 1
Lemma 11 Let 119910 isin 119862[0 1] be a given function and 119899 minus 1 lt120572 le 119899 then 119906 is a solution of BVP (1)-(2) if and only if 119906 is asolution of the integral equation
119906 (119905) = 120574 (119905) 120579 [119906] + int
(H5) 119891 [0 1] times [0infin) rarr [0infin) satisfies Caratheodoryconditions that is 119891(sdot 119906) is measurable for each fixed 119906 isin[0infin) and 119891(119905 sdot) is continuous for almost every 119905 isin [0 1]and for each 119903 gt 0 there exists 120601
119903isin 119871infin[0 1] such that 0 le
119891(119905 119906) le 120601119903for all 119906 isin [0 119903] and almost all 119905 isin [0 1]
(H6) One has the following 120574 isin 119862[0 1] 120574(119905) ge 0 0 le120579[120574] lt 1
Lemma 14 If 1198660satisfies (H1) (H2) then 119866 satisfies (H1)
(H2) for a function Φ the same interval [119886 119887] and the sameconstant 119888 where Φ satisfies (H4) and 119888 = min1198880(119905) 119905 isin[119886 119887]
Proof We have
119866 (119905 119902119904) =120574 (119905)
[1 minus 120579 [120574]]G119860(119902119904) + 119866
Similar to the proofs of Lemma 15 and Theorem 16 119871119906(119905) iscompact and maps 119875 into119870
We will use the Krein-Rutman theorem We recall that 120582is an eigenvalue of 119871 with corresponding eigenfunction 120601 if120601 = 0 and 120582120601 = 119871120601 The reciprocals of eigenvalues are calledcharacteristic values of 119871 The radius of the spectrum of 119871denoted by 119903(119871) is given by the well-known spectral radiusformula 119903(119871) = lim
119899rarrinfin1198711198991119899
Theorem 17 (see [15]) Let 119870 be a total cone in a real Banachspace 119864 and let 119864 rarr 119864 be a compact linear operator with(119870) sube 119870 If 119903() gt 0 then there is 120601
1isin 119870 0 such that
1206011= 119903()120601
1
Thus 1205821 fl 119903() is an eigenvalue of the largest possiblereal eigenvalue and 1205831 = 1120582
1is the smallest positive
characteristic value
Lemma 18 (see [15]) Assume that (H1)ndash(H3) hold and let 119871be as defined in (39) Then 119903(119871) gt 0
Theorem 19 (see [15]) When (H1)ndash(H3) hold 119903(119871) is aneigenvalue of 119871 with eigenfunction 120601
1in 119870
Theorem 20 (see [15]) Let 1205831= 1119903(119871) and 120601
1(119905) be a
corresponding eigenfunction in119875 of norm 1Then119898 le 1205831le 119872
If 119892(119905) gt 0 for 119905 isin [0 1] and 119866(119905 119902119904) gt 0 for 119905 119904 isin [0 1] thefirst inequality is strict unless 1206011(119905) is constant for 119905 isin [0 1] If119892(119905)120601(119905) gt 0 for 119905 isin [119886 119887] the second inequality is strict unless1206011(119905) is constant for 119905 isin [119886 119887]
Proof (for the local BVP (1)-(2) if 119892(119905) equiv 1) We now computethe constant 119898 and the optimal value of119872(119886 119887) that is wedetermine 119886 119887 so that119872(119886 119887) is minimal
For 119902119904 le 119905 we have by direct integration
int
119905
0
1198660 (119905 119902119904) 119889119902119904
= int
119905
0
[
[
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
minus (119905 minus 119902119904)(120572minus1)
Γ119902 (120572)
]
]
119889119902119904
=119905 minus 119905 (1 minus 119905)
(120572minus1)
Γ119902 (120572)
minus119905120572
[120572]119902 Γ119902 (120572)
(44)
For 119902119904 ge 119905
int
1
119905
1198660(119905 119902119904) 119889
119902119904 = int
1
119905
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
Theorem 7 (see [23]) Let 119909 gt 0 and 120572 isin R+ NThen thefollowing equality holds
(119868120572
119902 119862119863120572
119902119891) (119909) = 119891 (119909) minus
[120572]minus1
sum
119896=0
119909119896
Γ119902 (119896 + 1)
(119863119896
119902119891) (0) (19)
Lemma 8 (see [24]) Suppose 119879 119870 rarr 119870 is a completelycontinuous operator and has no fixed points on 120597119870
120588cap119870 Then
the following are true(i) If 119879119906 le 119906 for all 119906 isin 120597119870
120588cap 119870 then 119894(119879 119870
120588cap
119870119870) = 1 where 119894 is the fixed point index on 119870(ii) If 119879119906 ge 119906 for all 119906 isin 120597119870
120588cap 119870 then 119894(119879 119870
120588cap
119870119870) = 0
Lemma 9 (see [24]) Let 119870 be a cone in Banach space 119864Suppose that119879 119870
120588rarr 119870 is a completely continuous operator
There exists 1199060isin 1198700 such that 119906minus119879119906 = 120583119906
0for any 119906 isin 120597119870
119903
and 120583 ge 0 119894(119879 119870120588 119870) = 0
Lemma 10 (see [24]) Let 119870 be a cone in Banach space 119864Suppose that119879 119870
120588rarr 119870 is a completely continuous operator
If 119879119906 = 120583119906 for any 119906 isin 120597119870119903and 120583 ge 1 then 119894(119879 119870
120588 119870) = 1
Lemma 11 Let 119910 isin 119862[0 1] be a given function and 119899 minus 1 lt120572 le 119899 then 119906 is a solution of BVP (1)-(2) if and only if 119906 is asolution of the integral equation
119906 (119905) = 120574 (119905) 120579 [119906] + int
(H5) 119891 [0 1] times [0infin) rarr [0infin) satisfies Caratheodoryconditions that is 119891(sdot 119906) is measurable for each fixed 119906 isin[0infin) and 119891(119905 sdot) is continuous for almost every 119905 isin [0 1]and for each 119903 gt 0 there exists 120601
119903isin 119871infin[0 1] such that 0 le
119891(119905 119906) le 120601119903for all 119906 isin [0 119903] and almost all 119905 isin [0 1]
(H6) One has the following 120574 isin 119862[0 1] 120574(119905) ge 0 0 le120579[120574] lt 1
Lemma 14 If 1198660satisfies (H1) (H2) then 119866 satisfies (H1)
(H2) for a function Φ the same interval [119886 119887] and the sameconstant 119888 where Φ satisfies (H4) and 119888 = min1198880(119905) 119905 isin[119886 119887]
Proof We have
119866 (119905 119902119904) =120574 (119905)
[1 minus 120579 [120574]]G119860(119902119904) + 119866
Similar to the proofs of Lemma 15 and Theorem 16 119871119906(119905) iscompact and maps 119875 into119870
We will use the Krein-Rutman theorem We recall that 120582is an eigenvalue of 119871 with corresponding eigenfunction 120601 if120601 = 0 and 120582120601 = 119871120601 The reciprocals of eigenvalues are calledcharacteristic values of 119871 The radius of the spectrum of 119871denoted by 119903(119871) is given by the well-known spectral radiusformula 119903(119871) = lim
119899rarrinfin1198711198991119899
Theorem 17 (see [15]) Let 119870 be a total cone in a real Banachspace 119864 and let 119864 rarr 119864 be a compact linear operator with(119870) sube 119870 If 119903() gt 0 then there is 120601
1isin 119870 0 such that
1206011= 119903()120601
1
Thus 1205821 fl 119903() is an eigenvalue of the largest possiblereal eigenvalue and 1205831 = 1120582
1is the smallest positive
characteristic value
Lemma 18 (see [15]) Assume that (H1)ndash(H3) hold and let 119871be as defined in (39) Then 119903(119871) gt 0
Theorem 19 (see [15]) When (H1)ndash(H3) hold 119903(119871) is aneigenvalue of 119871 with eigenfunction 120601
1in 119870
Theorem 20 (see [15]) Let 1205831= 1119903(119871) and 120601
1(119905) be a
corresponding eigenfunction in119875 of norm 1Then119898 le 1205831le 119872
If 119892(119905) gt 0 for 119905 isin [0 1] and 119866(119905 119902119904) gt 0 for 119905 119904 isin [0 1] thefirst inequality is strict unless 1206011(119905) is constant for 119905 isin [0 1] If119892(119905)120601(119905) gt 0 for 119905 isin [119886 119887] the second inequality is strict unless1206011(119905) is constant for 119905 isin [119886 119887]
Proof (for the local BVP (1)-(2) if 119892(119905) equiv 1) We now computethe constant 119898 and the optimal value of119872(119886 119887) that is wedetermine 119886 119887 so that119872(119886 119887) is minimal
For 119902119904 le 119905 we have by direct integration
int
119905
0
1198660 (119905 119902119904) 119889119902119904
= int
119905
0
[
[
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
minus (119905 minus 119902119904)(120572minus1)
Γ119902 (120572)
]
]
119889119902119904
=119905 minus 119905 (1 minus 119905)
(120572minus1)
Γ119902 (120572)
minus119905120572
[120572]119902 Γ119902 (120572)
(44)
For 119902119904 ge 119905
int
1
119905
1198660(119905 119902119904) 119889
119902119904 = int
1
119905
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
Theorem 7 (see [23]) Let 119909 gt 0 and 120572 isin R+ NThen thefollowing equality holds
(119868120572
119902 119862119863120572
119902119891) (119909) = 119891 (119909) minus
[120572]minus1
sum
119896=0
119909119896
Γ119902 (119896 + 1)
(119863119896
119902119891) (0) (19)
Lemma 8 (see [24]) Suppose 119879 119870 rarr 119870 is a completelycontinuous operator and has no fixed points on 120597119870
120588cap119870 Then
the following are true(i) If 119879119906 le 119906 for all 119906 isin 120597119870
120588cap 119870 then 119894(119879 119870
120588cap
119870119870) = 1 where 119894 is the fixed point index on 119870(ii) If 119879119906 ge 119906 for all 119906 isin 120597119870
120588cap 119870 then 119894(119879 119870
120588cap
119870119870) = 0
Lemma 9 (see [24]) Let 119870 be a cone in Banach space 119864Suppose that119879 119870
120588rarr 119870 is a completely continuous operator
There exists 1199060isin 1198700 such that 119906minus119879119906 = 120583119906
0for any 119906 isin 120597119870
119903
and 120583 ge 0 119894(119879 119870120588 119870) = 0
Lemma 10 (see [24]) Let 119870 be a cone in Banach space 119864Suppose that119879 119870
120588rarr 119870 is a completely continuous operator
If 119879119906 = 120583119906 for any 119906 isin 120597119870119903and 120583 ge 1 then 119894(119879 119870
120588 119870) = 1
Lemma 11 Let 119910 isin 119862[0 1] be a given function and 119899 minus 1 lt120572 le 119899 then 119906 is a solution of BVP (1)-(2) if and only if 119906 is asolution of the integral equation
119906 (119905) = 120574 (119905) 120579 [119906] + int
(H5) 119891 [0 1] times [0infin) rarr [0infin) satisfies Caratheodoryconditions that is 119891(sdot 119906) is measurable for each fixed 119906 isin[0infin) and 119891(119905 sdot) is continuous for almost every 119905 isin [0 1]and for each 119903 gt 0 there exists 120601
119903isin 119871infin[0 1] such that 0 le
119891(119905 119906) le 120601119903for all 119906 isin [0 119903] and almost all 119905 isin [0 1]
(H6) One has the following 120574 isin 119862[0 1] 120574(119905) ge 0 0 le120579[120574] lt 1
Lemma 14 If 1198660satisfies (H1) (H2) then 119866 satisfies (H1)
(H2) for a function Φ the same interval [119886 119887] and the sameconstant 119888 where Φ satisfies (H4) and 119888 = min1198880(119905) 119905 isin[119886 119887]
Proof We have
119866 (119905 119902119904) =120574 (119905)
[1 minus 120579 [120574]]G119860(119902119904) + 119866
Similar to the proofs of Lemma 15 and Theorem 16 119871119906(119905) iscompact and maps 119875 into119870
We will use the Krein-Rutman theorem We recall that 120582is an eigenvalue of 119871 with corresponding eigenfunction 120601 if120601 = 0 and 120582120601 = 119871120601 The reciprocals of eigenvalues are calledcharacteristic values of 119871 The radius of the spectrum of 119871denoted by 119903(119871) is given by the well-known spectral radiusformula 119903(119871) = lim
119899rarrinfin1198711198991119899
Theorem 17 (see [15]) Let 119870 be a total cone in a real Banachspace 119864 and let 119864 rarr 119864 be a compact linear operator with(119870) sube 119870 If 119903() gt 0 then there is 120601
1isin 119870 0 such that
1206011= 119903()120601
1
Thus 1205821 fl 119903() is an eigenvalue of the largest possiblereal eigenvalue and 1205831 = 1120582
1is the smallest positive
characteristic value
Lemma 18 (see [15]) Assume that (H1)ndash(H3) hold and let 119871be as defined in (39) Then 119903(119871) gt 0
Theorem 19 (see [15]) When (H1)ndash(H3) hold 119903(119871) is aneigenvalue of 119871 with eigenfunction 120601
1in 119870
Theorem 20 (see [15]) Let 1205831= 1119903(119871) and 120601
1(119905) be a
corresponding eigenfunction in119875 of norm 1Then119898 le 1205831le 119872
If 119892(119905) gt 0 for 119905 isin [0 1] and 119866(119905 119902119904) gt 0 for 119905 119904 isin [0 1] thefirst inequality is strict unless 1206011(119905) is constant for 119905 isin [0 1] If119892(119905)120601(119905) gt 0 for 119905 isin [119886 119887] the second inequality is strict unless1206011(119905) is constant for 119905 isin [119886 119887]
Proof (for the local BVP (1)-(2) if 119892(119905) equiv 1) We now computethe constant 119898 and the optimal value of119872(119886 119887) that is wedetermine 119886 119887 so that119872(119886 119887) is minimal
For 119902119904 le 119905 we have by direct integration
int
119905
0
1198660 (119905 119902119904) 119889119902119904
= int
119905
0
[
[
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
minus (119905 minus 119902119904)(120572minus1)
Γ119902 (120572)
]
]
119889119902119904
=119905 minus 119905 (1 minus 119905)
(120572minus1)
Γ119902 (120572)
minus119905120572
[120572]119902 Γ119902 (120572)
(44)
For 119902119904 ge 119905
int
1
119905
1198660(119905 119902119904) 119889
119902119904 = int
1
119905
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
(H5) 119891 [0 1] times [0infin) rarr [0infin) satisfies Caratheodoryconditions that is 119891(sdot 119906) is measurable for each fixed 119906 isin[0infin) and 119891(119905 sdot) is continuous for almost every 119905 isin [0 1]and for each 119903 gt 0 there exists 120601
119903isin 119871infin[0 1] such that 0 le
119891(119905 119906) le 120601119903for all 119906 isin [0 119903] and almost all 119905 isin [0 1]
(H6) One has the following 120574 isin 119862[0 1] 120574(119905) ge 0 0 le120579[120574] lt 1
Lemma 14 If 1198660satisfies (H1) (H2) then 119866 satisfies (H1)
(H2) for a function Φ the same interval [119886 119887] and the sameconstant 119888 where Φ satisfies (H4) and 119888 = min1198880(119905) 119905 isin[119886 119887]
Proof We have
119866 (119905 119902119904) =120574 (119905)
[1 minus 120579 [120574]]G119860(119902119904) + 119866
Similar to the proofs of Lemma 15 and Theorem 16 119871119906(119905) iscompact and maps 119875 into119870
We will use the Krein-Rutman theorem We recall that 120582is an eigenvalue of 119871 with corresponding eigenfunction 120601 if120601 = 0 and 120582120601 = 119871120601 The reciprocals of eigenvalues are calledcharacteristic values of 119871 The radius of the spectrum of 119871denoted by 119903(119871) is given by the well-known spectral radiusformula 119903(119871) = lim
119899rarrinfin1198711198991119899
Theorem 17 (see [15]) Let 119870 be a total cone in a real Banachspace 119864 and let 119864 rarr 119864 be a compact linear operator with(119870) sube 119870 If 119903() gt 0 then there is 120601
1isin 119870 0 such that
1206011= 119903()120601
1
Thus 1205821 fl 119903() is an eigenvalue of the largest possiblereal eigenvalue and 1205831 = 1120582
1is the smallest positive
characteristic value
Lemma 18 (see [15]) Assume that (H1)ndash(H3) hold and let 119871be as defined in (39) Then 119903(119871) gt 0
Theorem 19 (see [15]) When (H1)ndash(H3) hold 119903(119871) is aneigenvalue of 119871 with eigenfunction 120601
1in 119870
Theorem 20 (see [15]) Let 1205831= 1119903(119871) and 120601
1(119905) be a
corresponding eigenfunction in119875 of norm 1Then119898 le 1205831le 119872
If 119892(119905) gt 0 for 119905 isin [0 1] and 119866(119905 119902119904) gt 0 for 119905 119904 isin [0 1] thefirst inequality is strict unless 1206011(119905) is constant for 119905 isin [0 1] If119892(119905)120601(119905) gt 0 for 119905 isin [119886 119887] the second inequality is strict unless1206011(119905) is constant for 119905 isin [119886 119887]
Proof (for the local BVP (1)-(2) if 119892(119905) equiv 1) We now computethe constant 119898 and the optimal value of119872(119886 119887) that is wedetermine 119886 119887 so that119872(119886 119887) is minimal
For 119902119904 le 119905 we have by direct integration
int
119905
0
1198660 (119905 119902119904) 119889119902119904
= int
119905
0
[
[
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
minus (119905 minus 119902119904)(120572minus1)
Γ119902 (120572)
]
]
119889119902119904
=119905 minus 119905 (1 minus 119905)
(120572minus1)
Γ119902 (120572)
minus119905120572
[120572]119902 Γ119902 (120572)
(44)
For 119902119904 ge 119905
int
1
119905
1198660(119905 119902119904) 119889
119902119904 = int
1
119905
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
Similar to the proofs of Lemma 15 and Theorem 16 119871119906(119905) iscompact and maps 119875 into119870
We will use the Krein-Rutman theorem We recall that 120582is an eigenvalue of 119871 with corresponding eigenfunction 120601 if120601 = 0 and 120582120601 = 119871120601 The reciprocals of eigenvalues are calledcharacteristic values of 119871 The radius of the spectrum of 119871denoted by 119903(119871) is given by the well-known spectral radiusformula 119903(119871) = lim
119899rarrinfin1198711198991119899
Theorem 17 (see [15]) Let 119870 be a total cone in a real Banachspace 119864 and let 119864 rarr 119864 be a compact linear operator with(119870) sube 119870 If 119903() gt 0 then there is 120601
1isin 119870 0 such that
1206011= 119903()120601
1
Thus 1205821 fl 119903() is an eigenvalue of the largest possiblereal eigenvalue and 1205831 = 1120582
1is the smallest positive
characteristic value
Lemma 18 (see [15]) Assume that (H1)ndash(H3) hold and let 119871be as defined in (39) Then 119903(119871) gt 0
Theorem 19 (see [15]) When (H1)ndash(H3) hold 119903(119871) is aneigenvalue of 119871 with eigenfunction 120601
1in 119870
Theorem 20 (see [15]) Let 1205831= 1119903(119871) and 120601
1(119905) be a
corresponding eigenfunction in119875 of norm 1Then119898 le 1205831le 119872
If 119892(119905) gt 0 for 119905 isin [0 1] and 119866(119905 119902119904) gt 0 for 119905 119904 isin [0 1] thefirst inequality is strict unless 1206011(119905) is constant for 119905 isin [0 1] If119892(119905)120601(119905) gt 0 for 119905 isin [119886 119887] the second inequality is strict unless1206011(119905) is constant for 119905 isin [119886 119887]
Proof (for the local BVP (1)-(2) if 119892(119905) equiv 1) We now computethe constant 119898 and the optimal value of119872(119886 119887) that is wedetermine 119886 119887 so that119872(119886 119887) is minimal
For 119902119904 le 119905 we have by direct integration
int
119905
0
1198660 (119905 119902119904) 119889119902119904
= int
119905
0
[
[
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
minus (119905 minus 119902119904)(120572minus1)
Γ119902 (120572)
]
]
119889119902119904
=119905 minus 119905 (1 minus 119905)
(120572minus1)
Γ119902 (120572)
minus119905120572
[120572]119902 Γ119902 (120572)
(44)
For 119902119904 ge 119905
int
1
119905
1198660(119905 119902119904) 119889
119902119904 = int
1
119905
[120572 minus 1]119902 119905 (1 minus 119902119904)(120572minus2)
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
so that 119879119906 le 1205881015840 = 119906 for all 119906 isin 1205971198811205881015840 Now Lemma 8 yields
119894119896(119879 1198811205881015840) = 1 (88)
On the other hand in view of (A2) we may take 120588lowast gt 1205881015840so that (69) holds (see the proof of Theorem 21) From (A3)we may take 119877
1isin (0 120588
1015840) so that (77) holds (see the proof of
Theorem 22)Combining (88) (69) and (77) we arrive at
119894119896 (119879119870120588lowast 1198811205881015840) = 0 minus 1 = minus1
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
On the other hand in view of (A1) we may take 120588 isin (0 1205881015840)so that (64) holds (see the proof of Theorem 21) In additionfrom (A4) we may take 119877 gt 1205881015840 so that (85) holds (see theproof of Theorem 22)
Combining (91) (64) and (85) we arrive at
119894119896(119879119870119877 1198811205881015840) = 1 minus 0 = 1
119894119896(119879 1198811205881015840 119870120588) = 0 minus 1 = minus1
(92)
Hence 119879 has at least two fixed points with one on 1198811205881015840 119870120588
and the other on 119870119877 1198811205881015840 Therefore (1)-(2) had at least two
positive solutions
We illustrate the applicability of these results with someexamples
It is readily shown that 1198910 = 1198910= 3 119891infin = 21 119891
infin= 7
Also 3119906 le 119891(119906) le 21119906 for 119906 ge 0 By calculation wefind119898 = 019722 and the smallest119872 calculated is119872(119886 119887) asymp119872(0484405 1) asymp 074665 We find 120583
1asymp 030366 Hence
by Theorem 21 there is at least one positive solution if 3120582 lt1205831and 7120582 gt 120583
1 that is there is a positive solution if 120582 isin
(047047 109773)
ByTheorem 22 there does not exist a positive solution ifeither 3120582 gt 120583
1or 21120582 lt 120583
1 that is if 120582 lt 109773 or 120582 gt
015682 no positive solution exists
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] N Pongarm S Asawasamrit J Tariboon and S K NtouyasldquoMulti-strip fractional q-integral boundary value problemsfor nonlinear fractional q-difference equationsrdquo Advances inDifference Equations vol 2014 no 1 article 193 2014
[2] R Almeida and N Martins ldquoExistence results for fractional q-difference equations of order120572 isin ]2 3[with three-point bound-ary conditionsrdquo Communications in Nonlinear Science andNumerical Simulation vol 19 no 6 pp 1675ndash1685 2014
[3] Y ZhaoHChen andQZhang ldquoExistence results for fractionalq-difference equations with nonlocal q-integral boundary con-ditionsrdquo Advances in Difference Equations vol 2013 article 482013
[4] R A Ferreira ldquoPositive solutions of a nonlinear q-fractionaldifference equation with integral boundary conditionsrdquo Inter-national Journal of Difference Equations vol 9 no 2 pp 135ndash145 2014
[5] B Ahmad and S K Ntouyas ldquoFractional 119902-difference hybridequations and inclusions with Dirichlet boundary conditionsrdquoAdvances in Difference Equations vol 2014 article 199 2014
[6] B Ahmad S KNtouyas andAAlsaedi ldquoExistence of solutionsfor fractional q-integro-difference inclusions with fractional q-integral boundary conditionsrdquo Advances in Difference Equa-tions vol 2014 article 257 2014
[7] R P Agarwal B Ahmad A Alsaedi and H Al-Hutami ldquoOnnonlinear fractional q-difference equations involving two frac-tional orders with three-point nonlocal boundary conditionsrdquoDynamics of Continuous Discrete amp Impulsive Systems Series AMathematical Analysis vol 21 no 1 pp 135ndash151 2014
[8] M El-Shahed andM A Al-Yami ldquoPositive solutions of bound-ary value problems for nth order q-differential equationsrdquoInternational Journal of Mathematical Archive vol 2 pp 521ndash532 2011
[9] J Ma and J Yang ldquoExistence of solutions for multi-pointboundary value problem of fractional q-difference equationrdquoElectronic Journal ofQualitativeTheory ofDifferential Equationsvol 92 pp 1ndash10 2011
[10] P M Rajkovic S D Marinkovic and M S Stankovic ldquoFrac-tional integrals and derivatives in q-calculusrdquo Applicable Anal-ysis and Discrete Mathematics vol 1 no 1 pp 311ndash323 2007
[11] Y Zhao H Chen and B Qin ldquoMultiple solutions for acoupled systemof nonlinear fractional differential equations viavariational methodsrdquo Applied Mathematics and Computationvol 257 pp 417ndash427 2015
[12] W-X Zhou X Liu and J-G Zhang ldquoSome new existenceand uniqueness results of solutions to semilinear impulsivefractional integro-differential equationsrdquoAdvances inDifferenceEquations vol 2015 article 38 2015
[13] Q Yuan andW Yang ldquoPositive solutions of nonlinear boundaryvalue problems for delayed fractional q-difference systemsrdquoAdvances in Difference Equations vol 2014 no 1 article 51 16pages 2014
[14] W Yang ldquoPositive solutions for nonlinear semipositone frac-tional q-difference system with coupled integral boundaryconditionsrdquo Applied Mathematics and Computation vol 244pp 702ndash725 2014
[15] J R L Webb and K Q Lan ldquoEigenvalue criteria for existenceof multiple positive solutions of nonlinear boundary valueproblems of local and nonlocal typerdquo Topological Methods inNonlinear Analysis vol 27 no 1 pp 91ndash116 2006
[16] J R L Webb ldquoNonlocal conjugate type boundary value prob-lems of higher orderrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 1933ndash1940 2009
[17] J R L Webb and G Infante ldquoNon-local boundary value prob-lems of arbitrary orderrdquo Journal of the London MathematicalSociety vol 79 no 1 pp 238ndash258 2009
10 Abstract and Applied Analysis
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
[18] J R Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems involving integral conditionsrdquo Non-linear Differential Equations and Applications NoDEA vol 15no 1-2 pp 45ndash67 2008
[19] J R L Webb and G Infante ldquoPositive solutions of nonlocalboundary value problems a unified approachrdquo Journal of theLondon Mathematical Society vol 74 no 3 pp 673ndash693 2006
[20] V Kac and P Cheung Quantum Calculus Springer Science ampBusiness Media 2002
[21] R A C Ferreira ldquoNontrivial solutions for fractional q-difference boundary value problemsrdquo Electronic Journal ofQualitative Theory of Differential Equations no 70 pp 1ndash102010
[22] M S Stankovic P M Rajkovic and S D Marinkovic ldquoOn q-fractional derivatives of Riemann-Liouville and Caputo typerdquohttparxivorgabs09090387
[23] M H Annaby and Z S Mansour q-Fractional Calculus andEquations vol 2056 of Lecture Notes in Mathematics SpringerBerlin Germany 2012
[24] M El-Shahed and W M Shammakh ldquoMultiple positive solu-tions for nonlinear fractional eigenvalue problemwith non localconditionsrdquo Fractional Calculus and Applied Analysis vol 3 pp1ndash13 2012
