Strong convergence for countable generalized Bregman nonexpansive-type mappings with equilibrium and variational inequality constraints

Abstract

We develop a Legendre--Bregman outer-approximation method for a common-solution problem in a uniformly smooth and uniformly convex Banach space. The constraint system consists of countable families of fixed-point-type mappings, equilibrium bifunctions, and monotone variational-inequality operators. Strong convergence to the Bregman projection of the initial point onto the common solution set is obtained without a family-level NST compatibility condition. A localized theorem covers negative-entropy generators, and Hilbert-space and Alber-functional cases are also derived.