Random Coupled Hilfer and Hadamard Fractional Differential Systems in Generalized Banach Spaces

Saïd Abbas,Yong Zhou,Mouffak Benchohra,Nassir Arifi

doi:10.3390/math7030285

Abstract

This article deals with some existence and uniqueness result of random solutions for some coupled systems of Hilfer and Hilfer–Hadamard fractional differential equations with random effects. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces.

Highlights

Fractional calculus is an extension of the ordinary differentiation and integration to arbitrary non-integer order
We denote by C; the Banach space of all continuous functions from I into Rm with the supremum norm k · k∞
We show that N satisfies all conditions of Theorem 1

Summary

Introduction

Fractional calculus is an extension of the ordinary differentiation and integration to arbitrary non-integer order In recent years, this theory has become an important object of investigations due to its demonstrated applications in different areas of physics and engineering (see, for example, [1,2] and the references therein). Rm ; i = 1, 2 are given functions, Rm ; m ∈ N∗ , H I11−γi is the left-sided mixed Hadamard integral of α ,β i order 1 − γi , and H D1 i is the Hilfer–Hadamard fractional derivative of order αi and type β i ; i = 1, 2

Preliminaries

Coupled Hilfer Fractional Differential Systems

Coupled Hilfer–Hadamard Fractional Differential Systems

An Example