article · Computation
Proximal Z-condensing operators are formulated in strictly convex Banach spaces by merging simulation functions with measures of noncompactness. Under this setup, a Darbo-type best proximity point theorem is established, leading to several outcomes for nonlinear condensing conditions. The framework addresses systems of nonlinear ordinary differential equations by translating them into non-self operator problems across enlarged product spaces, demonstrating that best proximity points correspond directly to classical solutions. Additionally, a Krasnoselskii-type best proximity point theorem is established for operators combining a simulation-function contraction with a compact mapping, which is applied to nonlinear matrix-valued integral equations. Finally, a multiplicative best proximity point theorem is developed within strictly convex Banach algebras and implemented to solve nonlinear integral equations, establishing a unified operator-theoretic structure for additive and multiplicative equations involving non-self mappings.
Complex mathematical models across engineering and physics frequently rely on nonlinear differential and integral equations that lack standard solutions. By establishing unified conditions for finding optimal solutions where traditional mappings fail, this theoretical work expands analytical methods for resolving intricate coupled systems and matrix-valued equations.
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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.
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DOI: 10.3390/computation14080188
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