On boundary value problems for fractional integro-differential equations in Banach spaces

Sabri T. M. Thabet; Machindra B. Dhakne

doi:10.26637/mjm304/012

Abstract

This paper aims to study the existence and uniqueness of solutions of fractional integro-differential equations in Banach spaces by applying a new generalized singular type Gronwall’s inequality, fixed point theorems and Hölder inequality. Example is provided to illustrate the main results.

Keywords:

Fractional integro-differential equations, boundary value problems, existence and uniqueness, generalized singular type Gronwall’s inequality, fixed point theorems

Mathematics Subject Classification:

26A33, 34A08, 34B15

Authors Imprint References Funding Related Articles Downloads

Sabri T. M. Thabet Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad - 431004, Maharashtra, India.
Machindra B. Dhakne Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad - 431004, Maharashtra, India.

Pages: 540-553
Date Published: 01-10-2015
Vol. 3 No. 04 (2015): Malaya Journal of Matematik (MJM)

R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 109(2010), 973-1033. DOI: https://doi.org/10.1007/s10440-008-9356-6

K. Balachandran and J. Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal., 71(2009), 4471-4475. DOI: https://doi.org/10.1016/j.na.2009.03.005

M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractls, 14(2002), 433-440. DOI: https://doi.org/10.1016/S0960-0779(01)00208-9

J. Henderson and A. Ouahab, Fractional functional differential inclusions with finite delay, Nonlinear Anal., 70(2009), 2091-2105. DOI: https://doi.org/10.1016/j.na.2008.02.111

E. Hernandez, D. O'Regan and K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73(2010), 3462-3471. DOI: https://doi.org/10.1016/j.na.2010.07.035

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, (2000). DOI: https://doi.org/10.1142/3779

K. Karthikeyan and J. J. Trujillo, Existenes and uniqueness results for fractional integrodifferential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., 17(2012), 4037-4043. DOI: https://doi.org/10.1016/j.cnsns.2011.11.036

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, (2006).

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, (1993).

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, (1999).

S. K. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Switzerland, (1993).

I. Stakgold and M. Holst, Green's Functions and Boundary Value Problems, 3rd ed. Wiley, Canada, (2011). DOI: https://doi.org/10.1002/9780470906538

J. R. Wang, L. Lv and Y. Zhou, Boundary value problems for fractional differential equations involoving Caputa derivative in Banach spaces, J. Appl. Math. Comput., 38(2012), 209-224. DOI: https://doi.org/10.1007/s12190-011-0474-3

J. R. Wang, W. Wei and Y. L. Yang, Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control, Opusc. Math., 30(2010), 217-234. DOI: https://doi.org/10.7494/OpMath.2010.30.2.217

J. R. Wang, X. Xiang, W. Wei and Q. Chen, The generalized Grownwall inequality and its applications to periodic solutions of integrodifferential impulsive periodic system on Banach space, J. Inequal. Appl., 2008(2008), 22 pages. DOI: https://doi.org/10.1155/2008/430521

J. R. Wang, Y. Yang and W. Wei, Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces, Opusc. Math., 30(2010), 361-381. DOI: https://doi.org/10.7494/OpMath.2010.30.3.361

S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ. Equ., 2006 (2006), 1–12. DOI: https://doi.org/10.1155/ADE/2006/90479