An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings

Let HH be a Hilbert space, {Ti}i∈N{Ti}i∈N a family of nonexpansive mappings from HH into itself, Gi:C×C→RGi:C×C→R a finite family of equilibrium functions (i∈{1,2,…,K})(i∈{1,2,…,K}), AA a strongly positive bounded linear operator with coefficient γ̄ and ff an αα-contraction on HH. Let WnWn be the mapping generated by {Ti}{Ti} and {λn}{λn} as in (1.5), let Srk,nk be the resolvent generated by GkGk and rk,nrk,n as in Lemma 2.4. Moreover, let {rk,n}k=1K, {ϵn}{ϵn} and {λn}{λn} satisfy appropriate conditions and F≔(⋂k=1KSEP(Gk))∩(⋂n∈NFix(Tn))≠0̸; we introduce an explicit scheme which defines a suitable sequence as follows: zn+1=ϵnγf(zn)+(I−ϵnA)WnSr1,n1Sr2,n2⋯SrK,nKzn∀n∈N and {zn}{zn} strongly converges to x∗∈Fx∗∈F which satisfies the variational inequality 〈(A−γf)x∗,x−x∗〉≥0〈(A−γf)x∗,x−x∗〉≥0 for all x∈Fx∈F. The results presented in this paper mainly extend and improve some recent results in [Vittorio Colao, et al., An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Anal. 71 (2009) 2708–2715; S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007) 455–469; S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515].