Hilfer fractional differential inclusions with Erd\xe9lyi\u2013Kober fractional integral boundary condition

Adel Lachouri,Abdelouaheb Ardjouni,Thabet Abdeljawad,Mohammed S Abdo,Bahaaeldin Abdalla

doi:10.1186/s13662-021-03397-7

Abstract

In this article, we debate the existence of solutions for a nonlinear Hilfer fractional differential inclusion with nonlocal Erdélyi–Kober fractional integral boundary conditions (FIBC). Both cases of convex- and nonconvex-valued right-hand side are considered. Our obtained results are new in the framework of Hilfer fractional derivative and Erdélyi–Kober fractional integral with FIBC via the fixed point theorems (FPTs) for a set-valued analysis. Some pertinent examples demonstrating the effectiveness of the theoretical results are presented.

Highlights

In recent years, fractional differential equation (FDE) theory has received very broad attention in the fields of both pure and applied mathematics, see [27, 35, 39, 45]
Our main concern in this manuscript is to obtain the existence results for the Hilfer inclusion problem (1.2) involving convex, nonconvex set-valued maps via some fixed point theorems (FPTs) of Leray–Schauder type, as well as those of Covitz and Nadler, where some pertinent examples are built for the demonstration of our findings
5 Conclusions We have studied a class of BVPs for Hilfer fractional differential inclusions (FDIs) with nonlocal fractional IBC

Summary

Introduction

Fractional differential equation (FDE) theory has received very broad attention in the fields of both pure and applied mathematics, see [27, 35, 39, 45]. Our main concern in this manuscript is to obtain the existence results for the Hilfer inclusion problem (1.2) involving convex, nonconvex set-valued maps via some FPTs of Leray–Schauder type, as well as those of Covitz and Nadler, where some pertinent examples are built for the demonstration of our findings. If N is completely continuous and has a closed graph, it is u.s.c. Lemma 6 ([40]) Let υ be a separable Banach space, Q : P × υ → Ocp,c(υ) be an L1Carathéodory set-valued map, and Z : L1(P, υ) → C(P, υ) be a linear continuous mapping. Definition 7 A set-valued operator S : υ → Ocl(υ) is said to be κ-Lipschitz if and only if there exists κ > 0 such that. Since Sis a contraction, in the light of Covitz and Nadler theorem, we infer that Shas an FP υ which is a solution of (1.2)

Examples Let us first consider the following FDI:

Conclusions