A Unified Two-Step Inertial Viscosity-Type Scheme for Monotone Generalized Mixed Equilibrium and Fixed-Point Problems

We introduce and analyze a two-step inertial viscosity-type algorithm for approximating a common solution of a generalized mixed equilibrium problem and the fixed-point problems associated with a family of quasi-nonexpansive mappings in a real Hilbert space. The generalized mixed equilibrium model contains a monotone operator and a proper convex lower semicontinuous function inside an implicit equilibrium resolvent, while a second inverse strongly monotone operator is treated explicitly through two successive forward–resolvent steps. The method further incorporates two inertial corrections, a family-dependent fixed-point step, and viscosity regularization by a contraction. We first establish boundedness of the generated sequence under summability conditions on the inertial parameters. We then derive a uniform descent inequality that simultaneously controls the two equilibrium residuals and the fixed-point residual. Using a residual-based sequence convergence criterion and an appropriate weak regularity condition for the mapping family, we prove that the iterates converge strongly to the unique point q* of the common solution set Ω satisfying q*=PΩf(q*). The abstract theorem is specialized to composite convex minimization, mixed variational inequalities, countable convex feasibility problems, and finite families under cyclic control. A five-dimensional example verifies the assumptions explicitly and illustrates convergence to the viscosity-selected solution.