Existence Theorem for Noninstantaneous Impulsive Evolution Equations

Gul I Hina Aslam,Amjad Ali,Maimona Rafiq,Ali Jaballah

doi:10.1155/2021/5539306

Abstract

In this note, the variational form of the classical Lax–Milgram theorem is used for the divulgence of variational structure of the first-order noninstantaneous impulsive linear evolution equation. The existence and uniqueness of the weak solution of the problem is obtained. In future, this constructive theory can be used for the corresponding semilinear problems.

Highlights

Evolution equations interpret the differential law of development with respect to time
We intend to acquire the variational structure associated to the following first-order noninstantaneous impulsive linear evolution equations:
Where A: D(A) ⊆ V ⟶ V is the infinitesimal generator of a strongly continuous semigroup of linear operators {Z(t)}t≥0 on a Banach space V endowed with a norm, the impulsive jump operator Bi ∈ B(V), i ∈ N, and the sequences 􏼈ti􏼉i∈N0 and 􏼈si􏼉i∈N0 satisfy the relation ti < si < ti+1, i ∈ N0 and set t0 s0 0

Summary

Introduction

Evolution equations interpret the differential law of development with respect to time. In [10], Hernandaz and O’Regan established the theory of noninstantaneous impulsive equations and showed the existence of corresponding mild solutions. In [11], JinRong Wang acquired ample conditions to guarantee the asymptotic stability of linear and semilinear noninstantaneous impulsive evolution equations. In [12], Tang and Nieto used the variational method to show existence of the solution to impulsive evolution equations. This approach, to the best of our knowledge, has not been used to show the existence (uniqueness) of the solution to noninstantaneous impulsive evolution equations. We intend to acquire the variational structure associated to the following first-order noninstantaneous impulsive linear evolution equations:.

Preliminaries

Main Results

Example of the Main Result