Existence theory for nonlocal boundary value problems involving mixed fractional derivatives

Bashir Ahmad,K Ntouyas,Ahmed Alsaedi

doi:10.15388/na.2019.6.6

Abstract

In this paper, we develop the existence theory for a new kind of nonlocal three-point boundary value problems for differential equations and inclusions involving both left Caputo and right Riemann–Liouville fractional derivatives. The Banach and Krasnoselskii fixed point theorems and the Leray–Schauder nonlinear alternative are used to obtain the desired results for the singlevalued problem. The existence of solutions for the multivalued problem concerning the upper semicontinuous and Lipschitz cases is proved by applying nonlinear alternative for Kakutani maps and Covitz and Nadler fixed point theorem. Examples illustrating the main results are also presented.

Highlights

Fractional differential equations and inclusions involving different kinds of fractional derivatives (Caputo, Riemann–Liouville, Hadamard to name a few) supplemented with a variety of boundary conditions have been investigated by many researchers, and one can find many interesting results on the topic in the related literature
It is imperative to mention that fractional differential equations containing left and right Riemann–Liouville fractional derivatives appear as the Euler–Lagrange equations in the study of variational principles, for details, see [6] and the references cited therein
In [15], the authors studied the existence of solutions for a nonlinear higher-order fractional boundary value problem involving both the left Riemann–Liouville and the right Caputo fractional derivatives: (−1)m CD1α−D0β+ + f t, u(t) = 0, 0 t 1, u(0) = u(i)(0) = 0, i = 1, . . . , m + n − 2, D0β++m−1u(1) = 0, where CD1α− and D0β+ respectively denote the left Caputo fractional derivative of order α ∈ (m − 1, m) and the right Riemann–Liouville fractional derivative of order β ∈ (n − 1, n), m, n 2, are integers

Summary

Introduction

Fractional differential equations and inclusions involving different kinds of fractional derivatives (Caputo, Riemann–Liouville, Hadamard to name a few) supplemented with a variety of boundary conditions have been investigated by many researchers, and one can find many interesting results on the topic in the related literature. There are fewer results on boundary value problems of fractional-order differential equations involving both right and left fractional derivatives. In [15], the authors studied the existence of solutions for a nonlinear higher-order fractional boundary value problem involving both the left Riemann–Liouville and the right Caputo fractional derivatives:. In [19], the authors proved the existence of solutions for the following boundary value problem involving both left Caputo and right Riemann– Liouville fractional derivatives:. We introduce a new class of nonlocal boundary value problems (BVP for short) of mixed fractional differential equations and inclusions involving both left Caputo and right Riemann–Liouville fractional derivatives and obtain some existence and uniqueness results for the problems at hand. Though the tools of the fixed point theory employed in the present analysis are the standard ones, their exposition is proved to be of substantial value in achieving the desired results

Preliminaries

Existence results

Examples

The upper semicontinuous case

The Lipschitz case