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.
