Multivalued Variational Inequalities with Generalized Fractional Φ-Laplacians

Abstract

In this article, we examine variational inequalities of the form ⟨A(u),v−u⟩+⟨F(u),v−u⟩≥0,∀v∈Ku∈K,, where A is a generalized fractional Φ-Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and F is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term F such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions.