Duality and Weak Compactness in Generalized Orlicz-Bochner Spaces with Applications to Operator Equations

Abstract

In this paper, we investigate the duality structure and weak compactness properties of generalized Orlicz–Bochner spaces LΦ(X,µ;E), where (X,µ) is a finite measure space, E is a Banach space, and Φ is a convex modular function satisfying a generalized ∆2-condition. Unlike classical Lebesgue and Bochner spaces, these spaces accommodate variable nonlinearity and non-standard growth, thus providing a more flexible functional analytic framework for studying nonlinear phenomena. We first characterize the dual of LΦ(X,µ;E) under modular convergence and develop criteria for reflexivity and weak compactness based on modular and geometric conditions on Φ and E. Furthermore, we establish sufficient conditions for the compactness and continuity of integral operators acting on these spaces. As an application, we analyze the solvability of nonlinear operator equations and integrodifferential equations with kernel-type operators, where standard Lp or Sobolev methods fail. Our results extend classical duality and compactness theory and open new avenues for solving evolution equations in variable exponent and Orlicz-type frameworks.