Existence and Compactness Results for a System of Fractional Differential Equations

Cheikh Guendouz,Abdelghani Ouahab,Jamal Eddine Lazreg,Juan J Nieto

doi:10.1155/2020/5735140

Abstract

The existence and uniqueness, boundedness, and continuous dependence of solutions for fractional differential equations with Caputo fractional derivative is proven by Perov’s fixed point theorem in vector Banach spaces. We study the existence and compactness of solution sets and the u.s.c. of operator solutions.

Highlights

IntroductionIn the past twenty years, the fractional differential equation has aroused great consideration in its application in mathematics and in other applications in physics, engineering, finance, fluid mechanics, viscoelastic mechanics, electroanalytical chemistry, and biological and other sciences [1,2,3,4,5,6,7]
We study the existence and compactness of solution sets and the u.s.c. of operator solutions
The plan of this paper is as follows: in Section 2, we introduce all the background material used in this paper such as some properties of generalized Banach spaces, fixed point theory, and fractional calculus theory

Summary

Introduction

In the past twenty years, the fractional differential equation has aroused great consideration in its application in mathematics and in other applications in physics, engineering, finance, fluid mechanics, viscoelastic mechanics, electroanalytical chemistry, and biological and other sciences [1,2,3,4,5,6,7]. Perov in 1964 [22] and Perov and Kibenko [23] extended the classical Banach contraction principle for contractive maps on space endowed with a vector-valued metric. Later, they attempted to generalize the Perov fixed point theorem in several directions which has a number of applications in various fields of nonlinear analysis, semilinear differential equations, and system of ordinary differential equations. Journal of Function Spaces equations by using the vector version fixed point theorem; their results are given in [26,27,28,29,30]. By the Leray-Schauder fixed point in vector Banach space, we prove the existence and compactness of solution sets of the above problems

Preliminaries

ΓðaÞ ðt ðt

ΓðβÞ ðt ðt ðt ðt

Existence and Compactness of Solution Sets

Applications

Conclusion