An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of 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, a finite family of equilibrium functions {Gi}i=1,…,K{Gi}i=1,…,K, 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. Assuming there are common equilibrium points of the family {Gi}i=1,…,K{Gi}i=1,…,K which are also fixed points for {Tn}{Tn}, we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality 〈(A−γf)x∗,x−x∗〉≥0〈(A−γf)x∗,x−x∗〉≥0 for all the xx in the intersection of the equilibrium points and the common fixed points of the sequence {Tn}{Tn}.