A fixed point approach for finding approximate solutions to second order non-instantaneous impulsive abstract differential equations

TL;DR

This paper develops a fixed point method using Faedo-Galerkin approximations to find and analyze the convergence of solutions for second order non-instantaneous impulsive abstract differential equations.

Contribution

It introduces a fixed point approach with F-G approximations for these equations and proves existence and convergence results using advanced functional analysis tools.

Findings

Established convergence of approximate solutions to the exact solution.

Demonstrated the effectiveness of the fixed point approach with an example.

Extended the application of cosine function theory and fixed point theorems to impulsive differential equations.

Abstract

This paper is concerned with the approximation of solutions to a class of second order non linear abstract differential equations. The finite-dimensional approximate solutions of the given system are built with the aid of the projection operator. We investigate the connection between the approximate solution and exact solution, and the question of convergence. Moreover, we define the Faedo-Galerkin(F-G) approximations and prove the existence and convergence results. The results are obtained by using the theory of cosine functions, Banach fixed point theorem and fractional power of closed linear operators. At last, an example of abstract formulation is provided.