Holder's Inequality ρ–Mean Continuity for Existence and Uniqueness Solution of Fractional Multi-Integrodifferential Delay System

Sameer Qasim Hasan

doi:10.1155/2020/1819752

Abstract

We herein present the detailed results for the existence and uniqueness of mild solution for multifractional order impulsive integrodifferential control equations with a nonlocal condition involving several types of semigroups of bounded linear operators, which were established on probability density functions related with the fractional differential equation. Additionally, we present the necessary and sufficient conditions to investigate Schauder’s fixed point theorem with Holder’s inequality ρ–mean continuity and infinite delay parameter to guarantee the uniqueness of a fixed point.

Highlights

Existence and uniqueness investigated in several studies for impulsive fractional order integrodifferential problems are presented in [3, 9,10,11,12,13,14]
We first present the basic theory of existence and uniqueness of mild solution for multifractional order impulsive integrodifferential control equations with a nonlocal condition and infinite delay parameter (1) by defining several types of semigroups of linear-bounded operators established on probability density functions defined on (0, ∞) and consider the necessary and sufficient estimators conditions, which play an important role in investigating Schauder’s fixed point theorem with Holder’s inequality ρ–mean continuity to guarantee the existence and uniqueness of a fixed point
In order to guarantee the existence and uniqueness of mild solution of problem (1), we introduce the following assumptions

Summary

Introduction

Existence and uniqueness investigated in several studies for impulsive fractional order integrodifferential problems are presented in [3, 9,10,11,12,13,14]. We first present the basic theory of existence and uniqueness of mild solution for multifractional order impulsive integrodifferential control equations with a nonlocal condition and infinite delay parameter (1) by defining several types of semigroups of linear-bounded operators established on probability density functions defined on (0, ∞) and consider the necessary and sufficient estimators conditions, which play an important role in investigating Schauder’s fixed point theorem with Holder’s inequality ρ–mean continuity to guarantee the existence and uniqueness of a fixed point.

Results

Conclusion