Generalizations of the Nonlinear Henry Inequality and the Leray–Schauder Type Fixed Point Theorem with Applications to Fractional Differential Inclusions

Zouaoui Hamlat,Abdelghani Ouahab,John R Graef

doi:10.3390/fractalfract5040173

Abstract

The authors give some singular versions of the Gronwall–Bihari–Henry inequalities. They also establish a multivalued version of the Leray–Schauder alternative in strictly star-shaped sets. Based on these new fractional inequalities and fixed point theorem, they study an initial value problem for fractional differential inclusions with delay.

Highlights

Generalizations of the Nonlinear Henry Inequality and the. It is well-known that inequalities, such as the Gronwall–Bellman–Bihari–Henry inequality, play an important role in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations, integral equations, and differential inclusions
The question of the existence of solutions and other mathematical aspects of fractional differential equations and inclusions have been extensively studied and have attracted much attention; many important contributions have been obtained so far
We wish to establish some nonlinear integral inequalities that can be used in the analysis of fractional differential equations and inclusions

Summary

Introduction

It is well-known that inequalities, such as the Gronwall–Bellman–Bihari–Henry inequality, play an important role in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations, integral equations, and differential inclusions (see, for example, [1,2,3,4,5,6,7]). This integral inequality was generalized by many authors. The question of the existence of solutions and other mathematical aspects of fractional differential equations and inclusions have been extensively studied and have attracted much attention; many important contributions have been obtained so far (see the monographs [16,17,18] as well as papers listed in the references below).

A Nonlinear Integral Inequality

Multivalued Analysis

Fixed Point Theory

Fractional Differential Inclusions with Delay

Conclusions