Boundary value problem for an infinite system of second order differential equations in $\ell_p$ spaces

TL;DR

This paper establishes the existence of solutions for an infinite system of second order differential equations in l_p spaces by transforming the boundary value problem into integral equations and applying a fixed point theorem.

Contribution

It introduces a method using measure of noncompactness and Darbo's fixed point theorem to solve boundary value problems in infinite-dimensional l_p spaces, extending previous finite-dimensional approaches.

Findings

Proved existence of solutions for the boundary value problem.

Applied the method to an example demonstrating its effectiveness.

Extended the theory to infinite systems in l_p spaces.

Abstract

In this paper the concept of measure of noncompactness is applied to prove the existence of solution for a boundary value problem for an infinite system of second order differential equations in $\ell_{p}$ space. We change the boundary value problem into an equivalent system of infinite integral equations and obtain the result for the system of integral equations, using Darbo type fixed point theorem. The result is applied to an example to illustrate the concept.