A hybrid method for countable equilibrium, variational inequality and maximal monotone inclusion problems with fixed point constraints

Abstract

Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^{*}$. We introduce a hybrid projection method for approximating a common element of four classes of constraints: the set of fixed points of a countable family of generalized nonexpansive-type maps, the solution sets of countably many equilibrium problems, the solution sets of countably many variational inequality problems, and the zero sets of countably many maximal monotone operators. The algorithm combines equilibrium resolvents, variational inequality resolvents, generalized resolvents of maximal monotone operators and a shrinking projection step. Under precise monotonicity, continuity and closedness assumptions, we prove that the generated sequence converges strongly to the generalized projection of the initial point onto the common solution set. We also establish residual convergence, derive convex minimization consequences, present a finite-truncation result, and give an illustrative Hilbert-space specialization showing why the countable setting cannot, in general, be reduced to a finite-family theorem.