Fixed Point Theory for Variational Inequalities with Fractional Functional Constraints: A Resolvent-Regularized Approach

This paper develops a fixed point framework for solving variational inequality problems whose feasible sets are described by functional constraints involving fractional-order derivatives in the sense of Caputo. Classical variational inequality theory, built on the Stampacchia and Browder--Minty theorems, does not directly accommodate feasible sets defined through non-local fractional operators, since the associated projection operators lose the elementary geometric structure available in the integer-order case. We introduce a new operator, the \emph{fractional resolvent regularizer} $R_\lambda^{\alpha}$, built from the Mittag-Leffler representation of the solution of a fractional relaxation equation, and we show that it is nonexpansive on $L^2(0,T)$ for every regularization parameter $\lambda>0$ and every order $\alpha\in(0,1)$. Using this operator we construct a composite fixed point map $T_{\lambda,\rho}$ that unifies metric projection, monotone operator evaluation and fractional regularization into a single nonexpansive (and, under a monotonicity/Lipschitz balance condition, contractive) map. We prove existence and uniqueness of solutions to the resulting problem, which we call a \emph{Variational Inequality with Fractional Functional Constraints} (VIFFC), establish well-posedness with respect to data perturbations, and propose the \emph{Fractional Resolvent-regularized Extragradient Algorithm} (FREGA), for which we prove global linear convergence with an explicit, rigorously derived rate. A worked numerical example involving a fractional obstacle-type constraint illustrates the theory, and applications to anomalous diffusion and viscoelastic unilateral contact problems are discussed.