Weak Condensing Frameworks and Best Proximity Methods for Nonlinear Operator Equations

Abstract In this paper, we study best proximity point results and the existence of solutions for a class of nonlinear functional and integral equations in strictly convex Banach spaces and Banach algebras. By employing measures of weak noncompactness, weak sequential continuity, and Lipschitz-type conditions, we extend classical fixed point theorems to the setting of non-weakly compact operators. We establish general conditions under which proximal condensing operators admit best proximity points, even in non-reflexive Banach spaces. These results are further applied to nonlinear functional equations in L¹0, 1, where the operators involved are Nemytskii-type or integral operators that are not necessarily weakly compact. We also consider product-type operators in Banach algebras satisfying property (P), demonstrating the applicability of our main theorems to nonlinear integral equations with an explicit example. The results provide a unified framework for analyzing solvability of nonlinear equations, extending existing approaches based on the Schauder–Tychonoff and O’Regan fixed point theorems, and illustrate the role of weak noncompactness measures in obtaining best proximity solutions.