Proximal Z-Condensing Operators via Simulation Functions and Applications

In this paper, we introduce and study proximal Z-condensing operators in strictly convex Banach spaces by combining simulation functions with measures of noncompactness. A Darbo-type best proximity point theorem is established, and several consequences corresponding to nonlinear condensing conditions are obtained. As an application, a system of nonlinear ordinary differential equations is embedded into a non-self operator problem on an enlarged product space; in this formulation, best proximity points are shown to be equivalent to classical solutions of the system. We also prove a Krasnoselskii-type best proximity point theorem for the sum of a simulation-function contraction and a compact operator and apply it to a nonlinear matrix-valued integral equation. Finally, a multiplicative best proximity point theorem is obtained in strictly convex Banach algebras and is used to study a nonlinear integral equation. The results provide a unified operator-theoretic framework for additive and multiplicative equations involving non-self mappings.