Existence and Uniqueness Results for Solutions to Fractional $p(·, ·)$-Laplacian Problems with a Variable-Order Derivative

Abstract

This paper investigates a class of fractional problems involving the variable-order $p(·, ·)$-Laplacian with homogeneous Dirichlet boundary conditions. Under suitable assumptions on the nonlinear term, we establish novel existence and uniqueness results for weak solutions. We achieve this by combining variational techniques with a result from the theory of monotone operators. Additionally, we reveal several interesting properties of the solution.