Fixed point theory for variational inequalities with fractional functional constraints: A resolvent-regularized approach

Abstract

Classical variational inequality theory does not directly accommodate feasible sets defined by fractional (Caputo) functional constraints, since the associated projection operators lose the geometric structure available in the integer-order case. We introduce the fractional resolvent regularizer R λ α , built from the Mittag-Leffler representation of a fractional relaxation equation, and show it is nonexpansive on L 2 ( 0 , T ) for every regularization parameter and fractional order. Using this operator we formulate the Variational Inequality with Fractional Functional Constraints (VIFFC), establish existence, uniqueness and well-posedness via a composite nonexpansive fixed point map, and propose an extragradient-type algorithm with proven global linear convergence. A numerical example and applications to anomalous diffusion and viscoelastic contact are discussed.