An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Abstract

Let H be a Hilbert space and C be a nonempty closed convex subset of H , { T i } i ∈ N be a family of nonexpansive mappings from C into H , G i : C × C → R be a finite family of equilibrium functions ( i ∈ { 1 , 2 , … , K } ) , A be a strongly positive bounded linear operator with a coefficient γ ̄ and B λ -Lipschitzian, relaxed ( μ , ν ) -cocoercive map of C into H . Moreover, let { r k , n } k = 1 K , { α n } satisfy appropriate conditions and F ≔ ( ∩ k = 1 K E P ( G k ) ) ∩ V I ( C , B ) ∩ ( ∩ n ∈ N F i x ( T n ) ) ≠ 0̸ ; we introduce an explicit scheme which defines a suitable sequence as follows: x n + 1 = α n γ f ( x n ) + ( 1 − α n A ) W n P C ( I − s n B ) S r 1 , n 1 S r 2 , n 2 ⋯ S r K , n K x n ∀ n ∈ N and { x 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 a recent result of Colao [V. Colao, 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 Analysis (2009), doi:10.1016/j.na.2009.01.115] and Qin [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis 69 (2008) 3897–3909].