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

Abstract

Let H be a Hilbert space, { T i } i ∈ N a family of nonexpansive mappings from H into itself, G i : C × C → R a finite family of equilibrium functions ( i ∈ { 1 , 2 , … , K } ) , A a strongly positive bounded linear operator with coefficient γ ̄ and f an α -contraction on H . Let W n be the mapping generated by { T i } and { λ n } as in (1.5), let S r k , n k be the resolvent generated by G k and r k , n as in Lemma 2.4. Moreover, let { r k , n } k = 1 K , { ϵ n } and { λ n } satisfy appropriate conditions and F ≔ ( ⋂ k = 1 K S E P ( G k ) ) ∩ ( ⋂ n ∈ N F i x ( T n ) ) ≠ 0̸ ; we introduce an explicit scheme which defines a suitable sequence as follows: z n + 1 = ϵ n γ f ( z n ) + ( I − ϵ n A ) W n S r 1 , n 1 S r 2 , n 2 ⋯ S r K , n K z n ∀ n ∈ N and { z n } strongly converges to x ∗ ∈ F which satisfies the variational inequality 〈 ( A − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 for all x ∈ 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].