Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

Xue Wang,Bo Zhu

doi:10.1155/2021/9449270

Abstract

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

Highlights

The fractional evolution equation has been applied to many fields, and scholars have obtained abundant research achievements [1,2,3,4,5,6,7,8,9,10,11,12,13]
Impulsive fractional integrodifferential equations can describe some phenomena which often occur in physics, geology, and economics, for instance, earthquake, the closing of the switch in the circuit, and so on
Based on the fact that nonlocal initial conditions are more effective than classical initial conditions in applied physics, the study of differential equations with nonlocal conditions has attracted more and more researchers’ attention [8,9,10,11,12,13]

Summary

Introduction

The fractional evolution equation has been applied to many fields, and scholars have obtained abundant research achievements [1,2,3,4,5,6,7,8,9,10,11,12,13]. ( cDβt uðtÞ = AðtÞuðtÞ + f ðt, uðtÞ, GuðtÞ, SuðtÞÞ, t ∈ 1⁄20, T0, uð0Þ + gðuÞ = u0, ð2Þ where 0 < β ≤ 1, AðtÞ is a closed linear operator with domain DðAÞ defined on a Banach space E; the existence and uniqueness of mild solutions have been established by k-set contraction and β-resolvent family. Gou and Li [16] studied the fractional impulsive integrodifferential equations in Banach space E; local and global existences of mild solutions have been proved by measure of noncompactness and Sadovskii’s fixed point theorem:. Journal of Function Spaces where 0 < β < 1, A : DðAÞ ⊂ E ⟶ E is a closed linear operator and −A generates a uniformly bounded C0-semigroup T ðt Þ Inspired by these contributions, we consider the following impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions: cDβt xðtÞ.

Preliminaries

Existence and Uniqueness of Mild Solution

Conclusion