On the Some Nonlinear Boundary Value Problems of Type Vibration in Sobolev Spaces with Variable Exponents

Abstract

In this work, we investigate a class of nonlinear boundary value problems within the framework of Sobolev spaces with variable exponents. After establishing precise formulations of these problems, we reformulate them as nonlinear hyperbolic-type equations. For each of the six problems considered, we establish both existence and uniqueness results. Furthermore, we address the associated stationary problems and prove the existence of solutions by applying a suitable variant of Brouwer’s fixed-point theorem.