Lyapunov-Type Inequalities for a Conformable Fractional … · 2019. 7. 30. · eorem...

Research ArticleLyapunov-Type Inequalities for a Conformable FractionalBoundary Value Problem of Order 3 lt 120572 le 4

Imed Bachar and Hassan Eltayeb

King Saud University College of Science Mathematics Department PO Box 2455 Riyadh 11451 Saudi Arabia

Correspondence should be addressed to Imed Bachar abacharksuedusa

Received 14 February 2019 Revised 15 March 2019 Accepted 17 March 2019 Published 1 April 2019

Academic Editor Shanhe Wu

Copyright copy 2019 Imed Bachar and Hassan Eltayeb 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

We establish new Lyapunov-type inequalities for the following conformable fractional boundary value problem (BVP) 119879119886120572119906(119905) +119902(119905)119906(119905) = 0 119886 lt 119905 lt 119887 119906(119886) = 1199061015840(119886) = 11990610158401015840(119886) = 11990610158401015840(119887) = 0 where 119879119886120572 is the conformable fractional derivative of order 120572 isin (3 4]and 119902 is a real-valued continuous function Some applications to the corresponding eigenvalue problem are discussed

1 Introduction

In [1] Lyapunov proved that if the boundary value problemBVP

11990610158401015840 (119905) + 119902 (119905) 119906 (119905) = 0 119886 lt 119905 lt 119887119906 (119886) = 119906 (119887) = 0 (1)

where 119902 [119886 119887] 997888rarr R has a nontrivial continuous solutionthen

int119887119886

1003816100381610038161003816119902 (119903)1003816100381610038161003816 119889119903 gt 4119887 minus 119886 (2)

Moreover the constant 4 in (2) is sharp (see [2])We emphasize that the above inequality has been proved

to be very useful in the study of various problems relatedto differential equations see for instance [2ndash5] and thereferences therein

Many researchers have studied generalizations and exten-sions of Lyapunovrsquos inequality

In [6] Wintner improved inequality (2) and obtained thefollowing version

int119887119886119902+ (119903) 119889119903 gt 4

119887 minus 119886 (3)

where 119902+(119903) = max119902(119903) 0

In [2] Hartamn generalized inequality (2) as follows

int119887119886(119887 minus 119903) (119903 minus 119886) 119902+ (119903) 119889119903 gt 119887 minus 119886 (4)

In the frame of fractional differential equations Ferreira(see [7]) proved a Lyapunov-type inequality for the Caputofractional BVP

119862119863120572119886+119906 (119905) + 119902 (119905) 119906 (119905) = 0 119886 lt 119905 lt 119887 1 lt 120572 le 2119906 (119886) = 119906 (119887) = 0 (5)

where 119902 is a real and continuous functionHe showed that if a nontrivial continuous solution to the

above problem exists then

int119887119886

1003816100381610038161003816119902 (119903)1003816100381610038161003816 119889119903 gt Γ (120572) ( 4119887 minus 119886)

120572minus1 (6)

In [8] the same author investigated a Lyapunov-type inequal-ity for the Riemann-Liouville fractional BVP

119863120572119886+119906 (119905) + 119902 (119905) 119906 (119905) = 0 119886 lt 119905 lt 119887 1 lt 120572 le 2119906 (119886) = 119906 (119887) = 0 (7)

He proved that if (7) has a nontrivial continuous solutionthen

int119887119886

1003816100381610038161003816119902 (119903)1003816100381610038161003816 119889119903 gt Γ (120572) 120572120572[(120572 minus 1) (119887 minus 119886)]120572minus1 (8)

HindawiJournal of Function SpacesVolume 2019 Article ID 4605076 5 pageshttpsdoiorg10115520194605076

2 Journal of Function Spaces

For definitions and properties of Caputo fractional deriva-tives and Riemann-Liouville fractional we refer the reader to[9 10]

Observe that inequalities (6) and (8) lead to Lyapunovrsquosclassical inequality (2) when 120572 = 2

Recently Khalil et al [11] introduced a new definition of afractional derivative called conformable fractional derivative(see Definition 1) This derivative is much easier to handle iswell-behaved and obeys the Leibniz rule and chain rule [12]

In short time this new fractional derivative definitionhas attracted many researchers In [13] Chung used theconformable fractional derivative and integral to discussfractional Newtonian mechanics

In [14] the authors proved a generalized Lyapunov-typeinequality for a conformable BVP of order 1 lt 120572 le 2 Theyhave established that if the BVP

119879119886120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 119886 lt 119905 lt 119887 1 lt 120572 le 2119906 (119886) = 119906 (119887) = 0 (9)

where 119879119886120572 is the conformable derivative of order 120572 isin (1 2]has a nontrivial continuous solution then

int119887119886

1003816100381610038161003816119902 (119903)1003816100381610038161003816 119889119903 gt 120572120572[(120572 minus 1) (119887 minus 119886)]120572minus1 (10)

For other generalizations and extensions of the classicalLyapunovrsquos inequality we refer the reader to [2 5 15ndash23] andthe references therein

In this paper we establish new Hartman-type and Lya-punov-type inequalities for the following conformable frac-tional BVP

119879119886120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 119886 lt 119905 lt 119887 3 lt 120572 le 4119906 (119886) = 1199061015840 (119886) = 11990610158401015840 (119886) = 11990610158401015840 (119887) = 0 (11)

where 119879119886120572 is the conformable derivative starting at 119886 of order3 lt 120572 le 4 and 119902 is a real-valued continuous function on [119886 119887]Some applications to the corresponding eigenvalue problemare discussed The obtained results are new in the context ofconformable fractional derivatives

The outline of the paper is as follows In Section 2 werecall and collect basic properties on conformable derivativesThis allows us to construct Greenrsquos function of the corre-sponding linear problem Some properties of this Greenrsquosfunction are established In Section 3 we state and prove ourmain results Some applications are discussed

2 Preliminaries on Conformable Derivatives

In this section we recall some basic definitions and lemmaswhich will be very useful to state our results

Definition 1 (see [11 12]) For a given function ℎ [119886infin) 997888rarrR the conformable fractional derivative of ℎ of order 120572 isin(0 1] is defined by

119879119886120572ℎ (119905) = lim120576997888rarr0

ℎ (119905 + 120576 (119905 minus 119886)1minus120572) minus ℎ (119905)120576

for all 119905 gt 119886(12)

If 119886 = 0 we write 119879120572 If 119879119886120572ℎ(119905) exists on (119886 119887) thendefine 119879119886120572ℎ(119886) = lim119905997888rarr119886+119879119886120572ℎ(119905)The geometric and physicalinterpretation of the conformable fractional derivatives wasgiven in Zhao [24]

Remark 2 (i) Let 0 lt 120572 le 1 and ℎ be differentiable functionat 119905 gt 119886 and then

119879119886120572ℎ (119905) = (119905 minus 119886)1minus120572 ℎ1015840 (119905) (13)

(ii) For ℎ(119905) = 2radic119905 minus 119886 we have 11987912ℎ(119905) = 1 for all 119905 gt 119886Therefore 11987912ℎ(119886) = 1 but ℎ is not differentiable at 119886

Some important properties for the conformable frac-tional derivative given in [11 12] are as follows

Theorem 3 Let 120572 isin (0 1] and 119891 119892 be 120572-differentiable at apoint 119905 and then

(i) 119879119886120572(120582119891 + 120583119892) = 120582119879119886120572(119891) + 120583119879119886120572(119892) for all 120582 120583 isin R(ii) 119879119886120572((119904 minus 119886)120583)(119905) = 120583(119905 minus 119886)120583minus120572 for all 119905 gt 119886 and 120583 isin R(iii) 119879119886120572(119891119892) = 119891119879119886120572(119892) + 119892119879119886120572(119891)(iv) 119879119886120572(119891119892) = (119891119879119886120572(119892) minus 119892119879119886120572(119891))1198922(v) Assume further that the function 119892 is defined in the

range of 119891 and then for all 119905 with 119905 = 119886 and 119892(119905) = 0 onehas the following Chain Rule

119879119886120572 (119891 ∘ 119892) (119905) = 119879119886120572119891 (119892 (119905)) 119879119886120572119892 (119905) (119892 (119905))120572minus1 (14)

The following conformable fractional derivatives of cer-tain functions [11] are worth noting

(i) 119879119886120572((1120572)(119905 minus 119886)120572) = 1(ii) 119879119886120572(sin(1120572)(119905 minus 119886)120572) = cos(1120572)(119905 minus 119886)120572(iii) 119879119886120572(cos(1120572)(119905 minus 119886)120572) = minus sin(1120572)(119905 minus 119886)120572(iv) 119879119886120572(119890(1120572)(119905minus119886)120572) = 119890(1120572)(119905minus119886)120572

Definition 4 (see [11 12]) Let 119899 lt 120572 le 119899+1 and ℎ [119886infin) 997888rarrR be a function such that ℎ(119899)(119905) exists The conformablefractional derivative of ℎ of order 120572 is defined by

119879119886120572ℎ (119905) = (119879119886120574ℎ(119899)) (119905) for which 120574 = 120572 minus 119899 (15)

Definition 5 (see [11 12]) Let 119899 lt 120572 le 119899 + 1 The fractionalintegral of a function ℎ [119886infin) 997888rarr R of order 120572 is definedby

(119868119886120572ℎ) (119905) = 1119899 int119905

119886(119905 minus 119904)119899 (119904 minus 119886)120572minus119899minus1 ℎ (119904) 119889119904 (16)

Lemma 6 (see [11 12]) Let 120572 isin (119899 119899 + 1](i) If ℎ is continuous on [119886infin) then

119879119886120572 (119868119886120572ℎ) (119905) = ℎ (119905) for all 119905 ge 119886 (17)

(ii) 119879119886120572ℎ(119905) = 0 if and only if ℎ(119905) = sum119899119896=0 119888119896(119905 minus 119886)119896where 119888119896 isin R for 119896 = 0 1 119899

(iii) If 119879119886120572ℎ is continuous on [119886infin) then for 119905 gt 119886

Journal of Function Spaces 3

119868119886120572 (119879119886120572ℎ) (119905) = ℎ (119905) + 1198880 + 1198881 (119905 minus 119886) + sdot sdot sdot + 119888119899 (119905 minus 119886)119899 (18)

where 119888119896 isin R for 119896 = 0 1 119899Lemma 7 Let 120572 isin (3 4] and ℎ isin 119862([119886 119887])Then the BVP

119879119886120572119906 (119905) = minusℎ (119905) 119886 lt 119905 lt 119887119906 (119886) = 1199061015840 (119886) = 11990610158401015840 (119886) = 11990610158401015840 (119887) = 0 (19)

admits a solution 119906 isin 119862 ([119886 119887]R) if and only if119906 (119905) = int119887

119886119866 (119905 119904) ℎ (119904) 119889119904 (20)

where 119866(119905 119904) is Greenrsquos function defined as

119866 (119905 119904) = 16 (119904 minus 119886)120572minus4

sdot

(119905 minus 119886)3 (119887 minus 119904)119887 minus 119886 minus (119905 minus 119904)3 119886 le 119904 le 119905 le 119887

(119905 minus 119886)3 (119887 minus 119904)119887 minus 119886 119886 le 119905 le 119904 le 119887

(21)

Proof Using Lemma 6 and Definition 5 we deduce that 119906 isin119862([119886 119887]R) is a solution of problem (19) if and only if

119906 (119905) = 1198880 + 1198881 (119905 minus 119886) + 1198882 (119905 minus 119886)2 + 1198883 (119905 minus 119886)3

minus 16 int119905119886(119905 minus 119904)3 (119904 minus 119886)120572minus4 ℎ (119904) 119889119904 (22)

where (1198880 1198881 1198882 1198883) isin R4This together with the boundary conditions implies 1198880 =1198881 = 1198882 = 0 and

1198883 = 16 (119887 minus 119886) int

119887

119886(119887 minus 119904) (119904 minus 119886)120572minus4 ℎ (119904) 119889119904 (23)

Hence

119906 (119905) = 16 (119887 minus 119886) int

119887

119886(119905 minus 119886)3 (119887 minus 119904) (119904 minus 119886)120572minus4 ℎ (119904) 119889119904

minus 16 int119905119886(119905 minus 119904)3 (119904 minus 119886)120572minus4 ℎ (119904) 119889119904

= int119887119886119866 (119905 119904) ℎ (119904) 119889119904

(24)

where 119866(119905 119904) is given in (21)

Lemma 8 Let 120572 isin (3 4]The following property is satisfied byGreenrsquos function (21) For any (119905 119904) in [119886 119887] times [119886 119887]

0 le 119866 (119905 119904) le 119866 (119887 119904)= 16 (119904 minus 119886)120572minus3 (119887 minus 119904) (2119887 minus 119886 minus 119904) (25)

Proof Fix 119904 in [119886 119887] By differentiating 119866(119905 119904) with respect to119905 we obtain120597119905119866 (119905 119904) = 1

2 (119904 minus 119886)120572minus4

sdot



(26)

Hence 119886 le 119905 le 119904 le 119887 and we have

120597119905119866 (119905 119904) = 12 (119904 minus 119886)120572minus4 (119905 minus 119886)2 (119887 minus 119904)

119887 minus 119886 ge 0 (27)

and for 119886 le 119904 le 119905 le 119887 we have120597119905119866 (119905 119904) = 1

2 (119904 minus 119886)120572minus4((119905 minus 119886)2 (119887 minus 119904)119887 minus 119886 minus (119905 minus 119904)2)

= 12 (119904 minus 119886)120572minus4 (119905 minus 119886)2 (119887 minus 119904)

119887 minus 119886 [1

minus ( 119887 minus 119904119887 minus 119886)((119887 minus 119886) (119905 minus 119904)

(119887 minus 119904) (119905 minus 119886))2]

(28)

Using the fact that ((119887minus119904)(119887minus119886)) and (119887minus119886)(119905minus119904)(119887minus119904)(119905minus119886)are in [0 1] we deduce that

120597119905119866 (119905 119904) ge 0 for 119886 le 119904 le 119905 le 119887 (29)

So the function 119905 997888rarr 119866(119905 119904) is nondecreasing on [119886 119887]Thisimplies that

0 = 119866 (119886 119904) le 119866 (119905 119904) le 119866 (119887 119904) (119905 119904) isin [119886 119887] times [119886 119887] (30)

The proof is completed

3 Main Results

Theorem9 (Hartman-type inequality) Assume that the BVP(11) has a nontrivial continuous solution then

int119887119886(119904 minus 119886)120572minus3 (119887 minus 119904) (2119887 minus 119886 minus 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904 ge 6 (31)

Proof Let 120572 isin (3 4] 119886 119887 isin R with 119886 lt 119887 and 119902 isin119862 ([119886 119887]R)Consider the Banach space 119862([119886 119887]R) equipped with

the uniform norm 119906infin = sup119905isin[119886119887]|119906(119905)|Assume that problem (11) has a nontrivial solution 119906 isin119862([119886 119887]R)By (20) we have

119906 (119905) = int119887119886119866 (119905 119904) 119902 (119904) 119906 (119904) 119889119904 119905 isin [119886 119887] (32)


Note that since 119906 is nontrivial then 119902 cannot be the zerofunction on [119886 119887] This with Lemma 8 implies for all 119905 isin[119886 119887]

|119906 (119905)| le int119887119886119866 (119905 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 |119906 (119904)| 119889119904

le 119906infin (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904)

(33)

Therefore

119906infin le 119906infin (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904) (34)

Now since 119906infin = 0 then we deduce that

1 le (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904) (35)

where

119866 (119887 119904) = 16 (119904 minus 119886)120572minus3 (119887 minus 119904) (2119887 minus 119886 minus 119904)

for 119904 isin [119886 119887] (36)


Corollary 10 Assume that the BVP (11) has a nontrivialcontinuous solution then

int119887119886(119904 minus 119886)120572minus3 (119887 minus 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904 ge 3

119887 minus 119886 (37)

Proof The property follows fromTheorem 9 and the fact

(2119887 minus 119886 minus 119904) le 2 (119887 minus 119886) for 119904 isin [119886 119887] (38)

Theorem 11 (Lyapunov-type inequality) Assume that theBVP (11) has a nontrivial continuous solution then

int119887119886

1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904 ge 3 (120572 minus 2)120572minus2(120572 minus 3)120572minus3 (119887 minus 119886)120572minus1 (39)

Proof From Corollary 10 we have

int119887119886120593 (119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904 ge 3

119887 minus 119886 (40)

where 120593(119904) = (119904 minus 119886)120572minus3(119887 minus 119904) 119904 isin [119886 119887]By simple computation one can check that

max119904isin[119886119887]

120593 (119904) = 120593(119886 + (120572 minus 3) 119887120572 minus 2 )

= (120572 minus 3)120572minus3 (119887 minus 119886120572 minus 2)

120572minus2 (41)

This fact with (40) gives the required result

Corollary 12 If 120582 is an eigenvalue to the fractional BVP

119879120572119906 (119905) + 120582119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (42)

then

|120582| ge 6120572 (120572 minus 1) (120572 minus 2)120572 + 2 (43)

Proof By using Theorem 9 we obtain

|120582| int10119904120572minus3 (1 minus 119904) (2 minus 119904) 119889119904 ge 6 (44)

Now by simple computation we have

int10119904120572minus3 (1 minus 119904) (2 minus 119904) 119889119904 = 120572 + 2

120572 (120572 minus 1) (120572 minus 2) (45)

This gives inequality (43)

Corollary 13 Let 3 lt 120572 le 4 and 119902 isin 119862 ([0 1]R) such thatint10

1003816100381610038161003816119902 (119903)1003816100381610038161003816 119889119903 lt 3 (120572 minus 2)120572minus2(120572 minus 3)120572minus3 (46)

Then the fractional BVP

119879120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (47)

has no nontrivial solution

Proof The assertion follows fromTheorem 11

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Theauthors would like to extend their sincere appreciation tothe Deanship of Scientific Research at King Saud Universityfor funding this Research group NO (RG-1435-043) Theauthors would like to thank the anonymous referees for theircareful reading of the paper

References

[1] A M Liapounoff ldquoProbleme general de la stabilite du mou-vemenrdquo Annales de la Faculte des sciences de lrsquoUniversite deToulouse vol 9 no 2 pp 203ndash474 1907

[2] P Hartman Ordinary Differential Equations John Wiley ampSons New York NY USA 1964


[3] R C Brown and D B Hinton ldquoLyapunov inequalities and theirapplicationsrdquo in Survey on Classical Inequalities Mathematicsand Its Applications TM Rassias Ed vol 517 pp 1ndash25 KluwerAcademic Publishers Dordrecht 2000

[4] S-S Cheng ldquoLyapunov inequalities for differential and differ-ence equationsrdquo Fasciculi Mathematici no 23 pp 25ndash41 1991

[5] P Hartman and A Wintner ldquoOn an oscillation criterion ofLiapounoffrdquo American Journal of Mathematics vol 73 pp 885ndash890 1951

[6] A Wintner ldquoOn the non-existence of conjugate pointsrdquo Amer-ican Journal of Mathematics vol 73 pp 368ndash380 1951

[7] R A C Ferreira ldquoA Lyapunov-type inequality for a fractionalboundary value problemrdquo Fractional Calculus and AppliedAnalysis vol 16 no 4 pp 978ndash984 2013

[8] R A Ferreira ldquoOn a Lyapunov-type inequality and the zerosof a certain Mittag-LEFfler functionrdquo Journal of MathematicalAnalysis and Applications vol 412 no 2 pp 1058ndash1063 2014

[9] I Podlubny ldquoFractional differential equationsrdquo inMathematicsin Sciences and Engineering vol 198 of Mathematics in Scienceand Engineering Academic Press San Diego Calif USA 1999

[10] S Samko A Kilbas and O Marichev Fractional Integrals andDerivative Theory and Applications Gordon amp Breach SciencePublishers Yverdon 1993

[11] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

[12] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[13] W S Chung ldquoFractional Newton mechanics with conformablefractional derivativerdquo Journal of Computational and AppliedMathematics vol 290 pp 150ndash158 2015

[14] T Abdeljawad J Alzabut and F Jarad ldquoA generalizedLyapunov-type inequality in the frame of conformable deriva-tivesrdquo Advances in Difference Equations vol 2017 article 321 p10 2017

[15] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[16] T Abdeljawad R P Agarwal J Alzabut F Jarad and AOzbekler ldquoLyapunov-type inequalities for mixed non-linearforced differential equations within conformable derivativesrdquoJournal of Inequalities and Applications vol 2018 article 1432018

[17] M F Aktas D Cakmak and A Ahmetoglu ldquoLyapunov-typeinequalities for fourth-order boundary value problemsrdquo Revistade la Real Academia de Ciencias Exactas Fısicas y NaturalesSerie A Matematicas pp 1ndash11 2018

[18] S Dhar andQKong ldquoLyapunov-type inequalities for a-th orderfractional differential equations with 2lt 120572 le3 and fractionalboundary conditionsrdquo Electronic Journal of Differential Equa-tions vol 2017 no 203 p 15 2017

[19] D OrsquoRegan and B Samet ldquoLyapunov-type inequalities for aclass of fractional differential equationsrdquo Journal of Inequalitiesand Applications vol 2015 no 247 p 10 2015

[20] X H Tang and X He ldquoLower bounds for generalized eigenval-ues of the quasilinear systemsrdquo Journal ofMathematical Analysisand Applications vol 385 no 1 pp 72ndash85 2012

[21] A Tiryaki ldquoRecent developments of Lyapunov-type inequali-tiesrdquo Advances in Dynamical Systems and Applications (ADSA)vol 5 no 2 pp 231ndash248 2010

[22] X Wang and R Xu ldquoLyapunov-type inequalities for con-formable BVPrdquo Journal of AppliedMathematics and Physics vol06 no 07 pp 1549ndash1557 2018

[23] X Yang Y-I Kim and K Lo ldquoLyapunov-type inequalityfor a class of odd-order differential equationsrdquo Journal ofComputational and Applied Mathematics vol 234 no 10 pp2962ndash2968 2010

[24] DZhao andMLuo ldquoGeneral conformable fractional derivativeand its physical interpretationrdquo Calcolo vol 54 no 3 pp 903ndash917 2017

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For definitions and properties of Caputo fractional deriva-tives and Riemann-Liouville fractional we refer the reader to[9 10]

Observe that inequalities (6) and (8) lead to Lyapunovrsquosclassical inequality (2) when 120572 = 2

Recently Khalil et al [11] introduced a new definition of afractional derivative called conformable fractional derivative(see Definition 1) This derivative is much easier to handle iswell-behaved and obeys the Leibniz rule and chain rule [12]

In short time this new fractional derivative definitionhas attracted many researchers In [13] Chung used theconformable fractional derivative and integral to discussfractional Newtonian mechanics

In [14] the authors proved a generalized Lyapunov-typeinequality for a conformable BVP of order 1 lt 120572 le 2 Theyhave established that if the BVP

119879119886120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 119886 lt 119905 lt 119887 1 lt 120572 le 2119906 (119886) = 119906 (119887) = 0 (9)

where 119879119886120572 is the conformable derivative of order 120572 isin (1 2]has a nontrivial continuous solution then

int119887119886

1003816100381610038161003816119902 (119903)1003816100381610038161003816 119889119903 gt 120572120572[(120572 minus 1) (119887 minus 119886)]120572minus1 (10)

For other generalizations and extensions of the classicalLyapunovrsquos inequality we refer the reader to [2 5 15ndash23] andthe references therein

In this paper we establish new Hartman-type and Lya-punov-type inequalities for the following conformable frac-tional BVP

119879119886120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 119886 lt 119905 lt 119887 3 lt 120572 le 4119906 (119886) = 1199061015840 (119886) = 11990610158401015840 (119886) = 11990610158401015840 (119887) = 0 (11)

where 119879119886120572 is the conformable derivative starting at 119886 of order3 lt 120572 le 4 and 119902 is a real-valued continuous function on [119886 119887]Some applications to the corresponding eigenvalue problemare discussed The obtained results are new in the context ofconformable fractional derivatives

The outline of the paper is as follows In Section 2 werecall and collect basic properties on conformable derivativesThis allows us to construct Greenrsquos function of the corre-sponding linear problem Some properties of this Greenrsquosfunction are established In Section 3 we state and prove ourmain results Some applications are discussed

2 Preliminaries on Conformable Derivatives

In this section we recall some basic definitions and lemmaswhich will be very useful to state our results

Definition 1 (see [11 12]) For a given function ℎ [119886infin) 997888rarrR the conformable fractional derivative of ℎ of order 120572 isin(0 1] is defined by

119879119886120572ℎ (119905) = lim120576997888rarr0

ℎ (119905 + 120576 (119905 minus 119886)1minus120572) minus ℎ (119905)120576

for all 119905 gt 119886(12)

If 119886 = 0 we write 119879120572 If 119879119886120572ℎ(119905) exists on (119886 119887) thendefine 119879119886120572ℎ(119886) = lim119905997888rarr119886+119879119886120572ℎ(119905)The geometric and physicalinterpretation of the conformable fractional derivatives wasgiven in Zhao [24]

Remark 2 (i) Let 0 lt 120572 le 1 and ℎ be differentiable functionat 119905 gt 119886 and then

119879119886120572ℎ (119905) = (119905 minus 119886)1minus120572 ℎ1015840 (119905) (13)

(ii) For ℎ(119905) = 2radic119905 minus 119886 we have 11987912ℎ(119905) = 1 for all 119905 gt 119886Therefore 11987912ℎ(119886) = 1 but ℎ is not differentiable at 119886

Some important properties for the conformable frac-tional derivative given in [11 12] are as follows

Theorem 3 Let 120572 isin (0 1] and 119891 119892 be 120572-differentiable at apoint 119905 and then

(i) 119879119886120572(120582119891 + 120583119892) = 120582119879119886120572(119891) + 120583119879119886120572(119892) for all 120582 120583 isin R(ii) 119879119886120572((119904 minus 119886)120583)(119905) = 120583(119905 minus 119886)120583minus120572 for all 119905 gt 119886 and 120583 isin R(iii) 119879119886120572(119891119892) = 119891119879119886120572(119892) + 119892119879119886120572(119891)(iv) 119879119886120572(119891119892) = (119891119879119886120572(119892) minus 119892119879119886120572(119891))1198922(v) Assume further that the function 119892 is defined in the

range of 119891 and then for all 119905 with 119905 = 119886 and 119892(119905) = 0 onehas the following Chain Rule

119879119886120572 (119891 ∘ 119892) (119905) = 119879119886120572119891 (119892 (119905)) 119879119886120572119892 (119905) (119892 (119905))120572minus1 (14)

The following conformable fractional derivatives of cer-tain functions [11] are worth noting

(i) 119879119886120572((1120572)(119905 minus 119886)120572) = 1(ii) 119879119886120572(sin(1120572)(119905 minus 119886)120572) = cos(1120572)(119905 minus 119886)120572(iii) 119879119886120572(cos(1120572)(119905 minus 119886)120572) = minus sin(1120572)(119905 minus 119886)120572(iv) 119879119886120572(119890(1120572)(119905minus119886)120572) = 119890(1120572)(119905minus119886)120572

Definition 4 (see [11 12]) Let 119899 lt 120572 le 119899+1 and ℎ [119886infin) 997888rarrR be a function such that ℎ(119899)(119905) exists The conformablefractional derivative of ℎ of order 120572 is defined by

119879119886120572ℎ (119905) = (119879119886120574ℎ(119899)) (119905) for which 120574 = 120572 minus 119899 (15)

Definition 5 (see [11 12]) Let 119899 lt 120572 le 119899 + 1 The fractionalintegral of a function ℎ [119886infin) 997888rarr R of order 120572 is definedby

(119868119886120572ℎ) (119905) = 1119899 int119905

119886(119905 minus 119904)119899 (119904 minus 119886)120572minus119899minus1 ℎ (119904) 119889119904 (16)

Lemma 6 (see [11 12]) Let 120572 isin (119899 119899 + 1](i) If ℎ is continuous on [119886infin) then

119879119886120572 (119868119886120572ℎ) (119905) = ℎ (119905) for all 119905 ge 119886 (17)

(ii) 119879119886120572ℎ(119905) = 0 if and only if ℎ(119905) = sum119899119896=0 119888119896(119905 minus 119886)119896where 119888119896 isin R for 119896 = 0 1 119899

(iii) If 119879119886120572ℎ is continuous on [119886infin) then for 119905 gt 119886




119879119886120572119906 (119905) = minusℎ (119905) 119886 lt 119905 lt 119887119906 (119886) = 1199061015840 (119886) = 11990610158401015840 (119886) = 11990610158401015840 (119887) = 0 (19)


119886119866 (119905 119904) ℎ (119904) 119889119904 (20)


119866 (119905 119904) = 16 (119904 minus 119886)120572minus4

sdot



(21)





1198883 = 16 (119887 minus 119886) int

119887


Hence

119906 (119905) = 16 (119887 minus 119886) int

119887



= int119887119886119866 (119905 119904) ℎ (119904) 119889119904

(24)

where 119866(119905 119904) is given in (21)




2 (119904 minus 119886)120572minus4

sdot



(26)



119887 minus 119886 ge 0 (27)




119887 minus 119886 [1


(119887 minus 119904) (119905 minus 119886))2]

(28)


120597119905119866 (119905 119904) ge 0 for 119886 le 119904 le 119905 le 119887 (29)


0 = 119866 (119886 119904) le 119866 (119905 119904) le 119866 (119887 119904) (119905 119904) isin [119886 119887] times [119886 119887] (30)


3 Main Results





119906 (119905) = int119887119886119866 (119905 119904) 119902 (119904) 119906 (119904) 119889119904 119905 isin [119886 119887] (32)



|119906 (119905)| le int119887119886119866 (119905 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 |119906 (119904)| 119889119904

le 119906infin (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904)

(33)

Therefore



1 le (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904) (35)

where


for 119904 isin [119886 119887] (36)




119887 minus 119886 (37)




int119887119886



int119887119886120593 (119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904 ge 3

119887 minus 119886 (40)


max119904isin[119886119887]

120593 (119904) = 120593(119886 + (120572 minus 3) 119887120572 minus 2 )


120572minus2 (41)



119879120572119906 (119905) + 120582119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (42)

then

|120582| ge 6120572 (120572 minus 1) (120572 minus 2)120572 + 2 (43)





120572 (120572 minus 1) (120572 minus 2) (45)





119879120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (47)



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018











119879119886120572119906 (119905) = minusℎ (119905) 119886 lt 119905 lt 119887119906 (119886) = 1199061015840 (119886) = 11990610158401015840 (119886) = 11990610158401015840 (119887) = 0 (19)


119886119866 (119905 119904) ℎ (119904) 119889119904 (20)


119866 (119905 119904) = 16 (119904 minus 119886)120572minus4

sdot



(21)





1198883 = 16 (119887 minus 119886) int

119887


Hence

119906 (119905) = 16 (119887 minus 119886) int

119887



= int119887119886119866 (119905 119904) ℎ (119904) 119889119904

(24)

where 119866(119905 119904) is given in (21)




2 (119904 minus 119886)120572minus4

sdot



(26)



119887 minus 119886 ge 0 (27)




119887 minus 119886 [1


(119887 minus 119904) (119905 minus 119886))2]

(28)


120597119905119866 (119905 119904) ge 0 for 119886 le 119904 le 119905 le 119887 (29)


0 = 119866 (119886 119904) le 119866 (119905 119904) le 119866 (119887 119904) (119905 119904) isin [119886 119887] times [119886 119887] (30)


3 Main Results





119906 (119905) = int119887119886119866 (119905 119904) 119902 (119904) 119906 (119904) 119889119904 119905 isin [119886 119887] (32)



|119906 (119905)| le int119887119886119866 (119905 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 |119906 (119904)| 119889119904

le 119906infin (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904)

(33)

Therefore



1 le (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904) (35)

where


for 119904 isin [119886 119887] (36)




119887 minus 119886 (37)




int119887119886



int119887119886120593 (119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904 ge 3

119887 minus 119886 (40)


max119904isin[119886119887]

120593 (119904) = 120593(119886 + (120572 minus 3) 119887120572 minus 2 )


120572minus2 (41)



119879120572119906 (119905) + 120582119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (42)

then

|120582| ge 6120572 (120572 minus 1) (120572 minus 2)120572 + 2 (43)





120572 (120572 minus 1) (120572 minus 2) (45)





119879120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (47)



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018










|119906 (119905)| le int119887119886119866 (119905 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 |119906 (119904)| 119889119904

le 119906infin (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904)

(33)

Therefore



1 le (int119887119886119866 (119887 119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904) (35)

where


for 119904 isin [119886 119887] (36)




119887 minus 119886 (37)




int119887119886



int119887119886120593 (119904) 1003816100381610038161003816119902 (119904)1003816100381610038161003816 119889119904 ge 3

119887 minus 119886 (40)


max119904isin[119886119887]

120593 (119904) = 120593(119886 + (120572 minus 3) 119887120572 minus 2 )


120572minus2 (41)



119879120572119906 (119905) + 120582119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (42)

then

|120582| ge 6120572 (120572 minus 1) (120572 minus 2)120572 + 2 (43)





120572 (120572 minus 1) (120572 minus 2) (45)





119879120572119906 (119905) + 119902 (119905) 119906 (119905) = 0 0 lt 119905 lt 1119906 (0) = 1199061015840 (0) = 11990610158401015840 (0) = 11990610158401015840 (1) = 0 (47)



Data Availability




Acknowledgments


References

































Journal of












Journal of







Volume 2018






Volume 2018






































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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