Application of Measure of Non-compactness for the Existence of Solutions of an Infinite System of Differential Equations in the Sequence Spaces of Convergent and Bounded Series

Abstract

AbstractThe aim of this paper is to study the infinite system of second-order differential equations along with the given boundary conditions for its solvability in the sequence spaces of convergent and bounded series. The result is achieved with the analytical tool, namely the measure of non-compactness along with the idea of Meir–Keeler condensing operator and provides the realization of the sufficient conditions for the existence results in the Banach sequence spaces cs and bs. The solvability conditions are further tested upon and illustrated by using an example in the cs space. For the bs space due to the benefit of analogy to cs space, solvability conditions have been discussed giving a remark. The Equation that concerns here is $$\frac{d^2}{dt^2}u_n(\xi )-u_n(\xi )=f_n(\xi ,u_1,u_2,\cdots );~~~n\in \mathbb {N},~\xi \in [0,\tau ]$$along with the boundary conditions \(u_n(0)=0=u_n(\tau )\). Here the family of functions \(f_n(\xi ,u_1,u_2,\cdots )\) belongs to \(C([0,\tau ]\times \mathbb {R},\mathbb {R})\).KeywordsInfinite system of differential equationsMeir–keeler condensing operatorMeasure of non-compactnesscs and bs spaces