Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings

Let HH be a Hilbert space. Consider on HH a sequence of nonexpansive mappings {Tn}{Tn} with common fixed points, an equilibrium function GG, a contraction ff with coefficient 0<α<10<α<1 and a strongly positive linear bounded operator AA with coefficient γ¯>0. Let 0<γ<γ¯α. We define a suitable Mann type algorithm which strongly converges to the unique solution of the minimization problem minx∈C12〈Ax,x〉−h(x), where hh is a potential function for γfγf and CC is the intersection of the equilibrium points and the common fixed points of the sequence {Tn}{Tn}.