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Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

1
KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
2
KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
3
Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa P.M.B 048, Jigawa State, Nigeria
4
Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok 1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800, Thailand
*
Authors to whom correspondence should be addressed.
Symmetry 2019, 11(6), 829; https://doi.org/10.3390/sym11060829
Received: 10 May 2019 / Revised: 14 June 2019 / Accepted: 18 June 2019 / Published: 22 June 2019
(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)
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Abstract

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D 0 + q u ( t ) = f ( t , u ( t ) ) , t [ 0 , T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = i = 1 m β i R L I 0 + p i u ( η i ) , where n 1 < q < n , n 2 , m , n N , ξ k , β i R , k = 0 , 1 , , n 2 , i = 1 , 2 , , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ( [ 0 , T ] , E ) E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ( [ 0 , T ] , R ) with the supremum-norm. RL I 0 + p i is the Riemann–Liouville fractional integral of order p i > 0 , η i ( 0 , T ) , and i = 1 m β i η i p i + n 1 Γ ( n ) Γ ( n + p i ) T n 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results. View Full-Text
Keywords: Caputo fractional derivative; existence of a solution; fixed point theorem; integral boundary value problems Caputo fractional derivative; existence of a solution; fixed point theorem; integral boundary value problems
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Borisut, P.; Kumam, P.; Ahmed, I.; Sitthithakerngkiet, K. Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems. Symmetry 2019, 11, 829.

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