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

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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

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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

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Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa P.M.B 048, Jigawa State, Nigeria

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Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok 1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800, Thailand

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Authors to whom correspondence should be addressed.

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)

# Abstract

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP):^{c}

^{c}

_{RL}

*Keywords:*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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**MDPI and ACS Style**

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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