Existence of Mild Solutions for Multi-Term Time-Fractional Random Integro-Differential Equations with Random Carathéodory Conditions

Amadou Diop,Wei-Shih Du

doi:10.3390/axioms10040252

Amadou Diop, Wei-Shih Du

Open Access

https://doi.org/10.3390/axioms10040252

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Abstract

In this paper, we investigate the existence of mild solutions to a multi-term fractional integro-differential equation with random effects. Our results are mainly relied upon stochastic analysis, Mönch’s fixed point theorem combined with a random fixed point theorem with stochastic domain, measure of noncompactness and resolvent family theory. Under the condition that the nonlinear term is of Carathéodory type and satisfies some weakly compactness condition, we establish the existence of random mild solutions. A nontrivial example illustrating our main result is also given.

Highlights

For more than three decades, fractional calculus has played an important role in the study of linear and nonlinear fractional integro-differential equations that arise from the modeling of nonlinear phenomena, optimal control of complex systems and other scientific research
Random fractional differential equations exist in many applications and have been studied by many authors and more details from historical points of view and recent developments of such equations are reported to the monographs [7,8], papers [9,10,11], and the references cited therein
Pardo and Lizama [6] studied the existence of mild solutions under Carathéodory type conditions by using measure of noncompactness techniques, Singh and Pandey [14] have established the existence and uniqueness of mild solutions for multi-term time-fractional differential systems with non-instantaneous impulses and finite delay by using Banach fixed point theorem whereas Chang and Ponce [15], with the help of the theory of fractional resolvent families, established the existence of mild solutions to a multi-term fractional differential equation

Summary

Introduction

For more than three decades, fractional calculus has played an important role in the study of linear and nonlinear fractional integro-differential equations that arise from the modeling of nonlinear phenomena, optimal control of complex systems and other scientific research (see, e.g., [1,2]). To the best of our knowledge, the study of the existence of mild solutions of multi-term time-fractional integro-differential equations with random effects by the abstract form (1). The main contributions of this paper are: Firstly, the study of existence of random multi-term time-fractional integro-differential equations of the form (1) via measure of noncompactness is an untreated topic in the literature. The nonlinear term satisfies a weak compactness condition that does not require the compactness of the resolvent family and sufficient conditions for the existence of mild solutions where the solution operators are only equicontinuous, are established by means of Mönch fixed point theorem and a random fixed point theorem with stochastic domain via the noncompactness measure. A nontrivial example illustrating our main result (Theorem 2; see below) is given

Preliminaries

Resolvent Family

Measures of Noncompactness

Some Existence Results

A Nontrivial Example of Application of Theorem 2

Conclusions