Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations
In this paper, we introduce a new class of boundary value problems consisting of a fractional differential equation of Riemann-Liouville type, D q R L x ( t ) = f ( t , x ( t ) ) , t ∈ [ 0 , T ] , subject to the Hadamard fractional integral conditions x ( 0 ) = 0 , x ( T ) = ∑ i = 1 n α i H I p i x...
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| Published in | Boundary value problems Vol. 2014; no. 1; pp. 1 - 16 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
09.12.2014
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1687-2770 1687-2770 |
| DOI | 10.1186/s13661-014-0253-9 |
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| Abstract | In this paper, we introduce a new class of boundary value problems consisting of a fractional differential equation of Riemann-Liouville type,
D
q
R
L
x
(
t
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=
f
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t
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x
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t
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)
,
t
∈
[
0
,
T
]
, subject to the Hadamard fractional integral conditions
x
(
0
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=
0
,
x
(
T
)
=
∑
i
=
1
n
α
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p
i
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η
i
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. Existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating the results obtained are also presented.
MSC:
34A08, 34B15. |
|---|---|
| AbstractList | In this paper, we introduce a new class of boundary value problems consisting of a fractional differential equation of Riemann-Liouville type,
D
q
R
L
x
(
t
)
=
f
(
t
,
x
(
t
)
)
,
t
∈
[
0
,
T
]
, subject to the Hadamard fractional integral conditions
x
(
0
)
=
0
,
x
(
T
)
=
∑
i
=
1
n
α
i
H
I
p
i
x
(
η
i
)
. Existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating the results obtained are also presented.
MSC:
34A08, 34B15. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).In this paper, we introduce a new class of boundary value problems consisting of a fractional differential equation of Riemann-Liouville type, ..., ..., subject to the Hadamard fractional integral conditions ..., ... Existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating the results obtained are also presented. |
| ArticleNumber | 253 |
| Author | Tariboon, Jessada Sudsutad, Weerawat Ntouyas, Sotiris K |
| Author_xml | – sequence: 1 givenname: Jessada surname: Tariboon fullname: Tariboon, Jessada email: jessadat@kmutnb.ac.th organization: Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok – sequence: 2 givenname: Sotiris K surname: Ntouyas fullname: Ntouyas, Sotiris K organization: Department of Mathematics, University of Ioannina – sequence: 3 givenname: Weerawat surname: Sudsutad fullname: Sudsutad, Weerawat organization: Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok |
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| Copyright | Tariboon et al.; licensee Springer. 2014. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. |
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| Keywords | existence uniqueness fixed point theorems Riemann-Liouville fractional derivative Hadamard fractional integral |
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| References | LiuXJiaMGeWMultiple solutions of a p-Laplacian model involving a fractional derivativeAdv. Differ. Equ201310.1186/1687-1847-2013-126 BoydDWWongJSWOn nonlinear contractionsProc. Am. Math. Soc19692045846423955910.1090/S0002-9939-1969-0239559-9 JaradFAbdeljawadTBaleanuDCaputo-type modification of the Hadamard fractional derivativesAdv. Differ. Equ201210.1186/1687-1847-2012-142 ButzerPLKilbasAATrujilloJJFractional calculus in the Mellin setting and Hadamard-type fractional integralsJ. Math. Anal. Appl2002269127190787110.1016/S0022-247X(02)00001-X GamboYYJaradFBaleanuDAbdeljawadTOn Caputo modification of the Hadamard fractional derivativesAdv. Differ. Equ201410.1186/1687-1847-2014-10 ButzerPLKilbasAATrujilloJJCompositions of Hadamard-type fractional integration operators and the semigroup propertyJ. Math. Anal. Appl2002269387400190712010.1016/S0022-247X(02)00049-5 KilbasAASrivastavaHMTrujilloJJTheory and Applications of Fractional Differential Equations2006AmsterdamElsevier AhmadBNietoJJBoundary value problems for a class of sequential integrodifferential equations of fractional orderJ. Funct. Spaces Appl201310.1155/2013/149659 BaleanuDDiethelmKScalasETrujilloJJFractional Calculus Models and Numerical Methods2012BostonWorld Scientific AhmadBNietoJJRiemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditionsBound. Value Probl201110.1186/1687-2770-2011-36 ZhangLAhmadBWangGAgarwalRPNonlinear fractional integro-differential equations on unbounded domains in a Banach spaceJ. Comput. Appl. Math20132495156303780510.1016/j.cam.2013.02.010 O’ReganDStanekSFractional boundary value problems with singularities in space variablesNonlinear Dyn201371641652303012710.1007/s11071-012-0443-x ButzerPLKilbasAATrujilloJJMellin transform analysis and integration by parts for Hadamard-type fractional integralsJ. Math. Anal. Appl2002270115191174810.1016/S0022-247X(02)00066-5 AgarwalRPZhouYHeYExistence of fractional neutral functional differential equationsComput. Math. Appl20105910951100257947410.1016/j.camwa.2009.05.010 KilbasAAHadamard-type fractional calculusJ. Korean Math. Soc200138119112041858760 PodlubnyIFractional Differential Equations1999San DiegoAcademic Press KrasnoselskiiMATwo remarks on the method of successive approximationsUsp. Mat. Nauk19551012312768119 BaleanuDMustafaOGAgarwalRPOn Lp-solutions for a class of sequential fractional differential equationsAppl. Math. Comput201121820742081283148310.1016/j.amc.2011.07.024 AhmadBNtouyasSKAlsaediANew existence results for nonlinear fractional differential equations with three-point integral boundary conditionsAdv. Differ. Equ201110.1155/2011/107384 HadamardJEssai sur l’étude des fonctions données par leur développement de TaylorJ. Math. Pures Appl18928101186 GranasADugundjiJFixed Point Theory2003New YorkSpringer10.1007/978-0-387-21593-8 AhmadBNtouyasSKAlsaediAA study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditionsMath. Probl. Eng2013 KilbasAATrujilloJJHadamard-type integrals as G-transformsIntegral Transforms Spec. Funct200314413427200599910.1080/1065246031000074443 SudsutadWTariboonJExistence results of fractional integro-differential equations with m-point multi-term fractional order integral boundary conditionsBound. Value Probl201210.1186/1687-2770-2012-94 SudsutadWTariboonJBoundary value problems for fractional differential equations with three-point fractional integral boundary conditionsAdv. Differ. Equ201210.1186/1687-1847-2012-93 B Ahmad (253_CR9) 2013 L Zhang (253_CR11) 2013; 249 W Sudsutad (253_CR14) 2012 AA Kilbas (253_CR19) 2001; 38 B Ahmad (253_CR7) 2011 DW Boyd (253_CR23) 1969; 20 J Hadamard (253_CR15) 1892; 8 B Ahmad (253_CR10) 2013 X Liu (253_CR12) 2013 YY Gambo (253_CR22) 2014 D Baleanu (253_CR3) 2012 I Podlubny (253_CR1) 1999 MA Krasnoselskii (253_CR24) 1955; 10 PL Butzer (253_CR16) 2002; 269 PL Butzer (253_CR18) 2002; 270 D Baleanu (253_CR5) 2011; 218 PL Butzer (253_CR17) 2002; 269 D O’Regan (253_CR8) 2013; 71 W Sudsutad (253_CR13) 2012 B Ahmad (253_CR6) 2011 RP Agarwal (253_CR4) 2010; 59 F Jarad (253_CR21) 2012 AA Kilbas (253_CR20) 2003; 14 A Granas (253_CR25) 2003 AA Kilbas (253_CR2) 2006 |
| References_xml | – reference: KilbasAASrivastavaHMTrujilloJJTheory and Applications of Fractional Differential Equations2006AmsterdamElsevier – reference: BaleanuDDiethelmKScalasETrujilloJJFractional Calculus Models and Numerical Methods2012BostonWorld Scientific – reference: KrasnoselskiiMATwo remarks on the method of successive approximationsUsp. Mat. Nauk19551012312768119 – reference: BoydDWWongJSWOn nonlinear contractionsProc. Am. Math. Soc19692045846423955910.1090/S0002-9939-1969-0239559-9 – reference: O’ReganDStanekSFractional boundary value problems with singularities in space variablesNonlinear Dyn201371641652303012710.1007/s11071-012-0443-x – reference: HadamardJEssai sur l’étude des fonctions données par leur développement de TaylorJ. Math. Pures Appl18928101186 – reference: BaleanuDMustafaOGAgarwalRPOn Lp-solutions for a class of sequential fractional differential equationsAppl. Math. Comput201121820742081283148310.1016/j.amc.2011.07.024 – reference: ZhangLAhmadBWangGAgarwalRPNonlinear fractional integro-differential equations on unbounded domains in a Banach spaceJ. Comput. Appl. Math20132495156303780510.1016/j.cam.2013.02.010 – reference: ButzerPLKilbasAATrujilloJJCompositions of Hadamard-type fractional integration operators and the semigroup propertyJ. Math. Anal. Appl2002269387400190712010.1016/S0022-247X(02)00049-5 – reference: AgarwalRPZhouYHeYExistence of fractional neutral functional differential equationsComput. Math. Appl20105910951100257947410.1016/j.camwa.2009.05.010 – reference: SudsutadWTariboonJBoundary value problems for fractional differential equations with three-point fractional integral boundary conditionsAdv. Differ. Equ201210.1186/1687-1847-2012-93 – reference: AhmadBNtouyasSKAlsaediANew existence results for nonlinear fractional differential equations with three-point integral boundary conditionsAdv. Differ. Equ201110.1155/2011/107384 – reference: AhmadBNtouyasSKAlsaediAA study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditionsMath. Probl. Eng2013 – reference: KilbasAATrujilloJJHadamard-type integrals as G-transformsIntegral Transforms Spec. Funct200314413427200599910.1080/1065246031000074443 – reference: SudsutadWTariboonJExistence results of fractional integro-differential equations with m-point multi-term fractional order integral boundary conditionsBound. Value Probl201210.1186/1687-2770-2012-94 – reference: LiuXJiaMGeWMultiple solutions of a p-Laplacian model involving a fractional derivativeAdv. Differ. Equ201310.1186/1687-1847-2013-126 – reference: JaradFAbdeljawadTBaleanuDCaputo-type modification of the Hadamard fractional derivativesAdv. Differ. Equ201210.1186/1687-1847-2012-142 – reference: ButzerPLKilbasAATrujilloJJFractional calculus in the Mellin setting and Hadamard-type fractional integralsJ. Math. Anal. Appl2002269127190787110.1016/S0022-247X(02)00001-X – reference: ButzerPLKilbasAATrujilloJJMellin transform analysis and integration by parts for Hadamard-type fractional integralsJ. Math. Anal. Appl2002270115191174810.1016/S0022-247X(02)00066-5 – reference: GamboYYJaradFBaleanuDAbdeljawadTOn Caputo modification of the Hadamard fractional derivativesAdv. Differ. Equ201410.1186/1687-1847-2014-10 – reference: KilbasAAHadamard-type fractional calculusJ. Korean Math. Soc200138119112041858760 – reference: PodlubnyIFractional Differential Equations1999San DiegoAcademic Press – reference: AhmadBNietoJJRiemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditionsBound. Value Probl201110.1186/1687-2770-2011-36 – reference: GranasADugundjiJFixed Point Theory2003New YorkSpringer10.1007/978-0-387-21593-8 – reference: AhmadBNietoJJBoundary value problems for a class of sequential integrodifferential equations of fractional orderJ. Funct. Spaces Appl201310.1155/2013/149659 – volume-title: Fixed Point Theory year: 2003 ident: 253_CR25 doi: 10.1007/978-0-387-21593-8 – volume-title: Math. Probl. Eng year: 2013 ident: 253_CR9 – volume: 269 start-page: 387 year: 2002 ident: 253_CR16 publication-title: J. Math. Anal. Appl doi: 10.1016/S0022-247X(02)00049-5 – volume: 59 start-page: 1095 year: 2010 ident: 253_CR4 publication-title: Comput. Math. Appl doi: 10.1016/j.camwa.2009.05.010 – volume-title: Adv. Differ. Equ year: 2012 ident: 253_CR13 – volume-title: Adv. Differ. Equ year: 2014 ident: 253_CR22 – volume-title: Theory and Applications of Fractional Differential Equations year: 2006 ident: 253_CR2 – volume: 8 start-page: 101 year: 1892 ident: 253_CR15 publication-title: J. Math. Pures Appl – volume: 269 start-page: 1 year: 2002 ident: 253_CR17 publication-title: J. Math. Anal. Appl doi: 10.1016/S0022-247X(02)00001-X – volume: 218 start-page: 2074 year: 2011 ident: 253_CR5 publication-title: Appl. Math. 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| Snippet | In this paper, we introduce a new class of boundary value problems consisting of a fractional differential equation of Riemann-Liouville type,
D
q
R
L
x
(
t
)... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).In this paper, we introduce a new class of boundary value problems consisting of a... |
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| SubjectTerms | Analysis Approximations and Expansions Boundary value problems Difference and Functional Equations Differential equations Existence theorems Integrals Mathematical analysis Mathematics Mathematics and Statistics Nonlinearity Ordinary Differential Equations Partial Differential Equations Texts Theorems Uniqueness |
| Title | Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations |
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