Analysis of two-variable infinite system of nonlinear integral equations using measures of non-compactness: existence and approximation in $BC(\mathbb {R}_{+}\times \mathbb {R}_{+}, \ell _{\infty })$

Abstract

This paper investigates the existence and approximation of solutions for a generalized infinite system of two-variable nonlinear integral equations in the Banach space $BC(\mathbb {R}_{+}\times \mathbb {R}_{+}, \ell _{\infty })$. This space comprises bounded and continuous functions defined on $\mathbb {R}_{+}\times \mathbb {R}_{+}$ with values in the sequence space $\ell _{\infty }$, equipped with the supremum norm. To establish solvability, we construct a novel measure of non-compactness (MNC) for the space $BC(\mathbb {R}_{+}\times \mathbb {R}_{+}, \ell _{\infty })$ based on the MNC in the space $\ell _{\infty }$. Employing Darbo’s fixed point theorem (DFPT), we rigorously demonstrate the existence of solutions under appropriate conditions. This methodological approach provides a robust framework for handling infinite-dimensional systems within the given functional space. To validate the theoretical results, we present a practical example and develop an iterative algorithm for approximating the solutions. The numerical results illustrate the effectiveness and accuracy of the proposed method, highlighting its potential for broader applications. This work not only generalizes existing theories but also lays a foundation for solving complex infinite systems in functional analysis and applied mathematics.