New Iterative Algorithm for Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces

access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce a new iterative scheme with a countable family of nonexpansive mappings for variational inequality problem in a Hilbert space and prove a strong convergence theorem for the iterative scheme.


Introduction
Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let F : H → H be a nonlinear operator.The variational inequality problem on F is to find a point x * ∈ C such that The variational inequality problem is denoted by V I(F, C) [5].
F is called to be κ-Lischitzian and η-strongly monotone if there exist constants κ, η > 0 such that F x − F y ≤ κ x − y , for all x, y ∈ H, and F x − F y, x − y ≥ η x − y 2 , for all x, y ∈ H, respectively.
It is well known that if F is strongly monotone and Lipschitzian on C, then V I(F, C) has a unique solution.An important problem is how to find a solution of V I(F, C).
It is known that the V I(F, C) is equivalent to the fixed point equation [16] u * = P C (u * − μF (u * )), (1.1) where P C is the projection from H onto C; i.e., x − y , ∀x ∈ H, and where μ > 0 is an arbitrarily fixed constant.So, the fixed point methods can be implemented to find a solution of the V I(F, C).
The fixed point formulation (1.1) involves the projection P C , which may not be easy to compute, due to the complexity of the convex set C. Hence, in order to reduce the complexity probably caused by the projection P C , Yamada [16] introduced the hybrid steepest-descent method for solving the V I(F, C) by replacing P C with a nonexpansive mapping T .Recall that a mapping T : H → H is called nonexpansive if for all x, y ∈ H, one holds The set of fixed points of T is denoted by F ix(T ).More precisely, Yamada [16] gave the following iterative scheme: where T : H → H is a nonexpansive mapping, F is η-strongly monotone and κ-Lipschitzian on C = F (T ), {λ n } is a sequence in (0, 1) and μ is a fixed number with 0 < μ < 2η/κ 2 .Yamada [16] proved that if the sequence {λ n } satisfies the conditions: n+1 = 0, then {u n } converges strongly to the unique solution of V I(F, C).Since Yamada's hybrid steepest-descent method for solving variational inequalities [16], there are much research on this aspect; see, e.g., [15,19,20,21].
In 2011, Buong and Duong [2] introduced a new iterative algorithm, based on a combination of thy hybrid steepest-descent method for variational inequalities with the Krasnosel'skii-Mann type algorithm for fixed point problems.Very recently Zhou and Wang [23] simplified the algorithm of Buong and Duong and proved a strong convergence theorem for the variational inequality problem and fixed point problem on finite nonexpansive mappings in Hilbert space.In this paper, motivated by the work of Zhou and Wang [23], we introduce a new iterative algorithm for solving the solution of variational inequality problem and prove the solution is the common fixed point of a countable family of nonexpansive mappings in Hilbert space.

Preliminaries
Let H be a Hilbert space.Let T : H → H be nonexpansive mapping and let F : H → H be a κ-Lipschitzian and η-strong monotone nonlinear operator.

Lemma 2.2 ([7]
).Let {s n }, {c n } be the sequences of nonnegative real numbers and let {a n } ⊂ (0, 1).Suppose {b n } is a real number sequence such that Then the following results hold: Let T : H → H be a nonexpansive mapping.Let α ∈ (0, 1).Define the mapping S = αI + (1 − α)T .Then the mapping S is said to be an averaged mapping with F ix(T ) = F ix(S).Furthermore, on a family of averaged mappings, we have the following conclusion.

Lemma 2.3 ([1, 12]). Let {T i } N
i=1 be the averaged mappings on H and assume that Lemma 2.4 ([13]).Let {a n } be a sequence of nonnegative real numbers satisfying the following condition where {b n } and {c n } are sequences of real numbers such that Then lim n→∞ a n = 0.

Main results
Theorem 3.1.Let H be a real Hilbert space and F : H → H be an L-Lipschitz continuous and η-strongly monotone mapping.
be sequences in (0, 1).Assume that {β n } is strictly decreasing and let β 0 = 1 and μ ∈ (0, 2η/L 2 ).Suppose that the following conditions hold: Then the sequence {x n } generated by the following manner: ) where U i = α i I + (1 − α i )T i , converges strongly to the unique solution x * of the variational inequality: Proof.
So, {x n } is bounded and so are {U i x n } for each i ≥ 1.
Next we prove that We rewrite (3.1) as follows: Then where M = sup n≥1 { F y n }.By Lemma 2.2 we conclude that Note that and {β n } is a strictly decreasing sequence.So, for each i ≥ 1, we have Next we prove that lim sup n→∞ −F (x * ), x n+1 − x * ≤ 0. To prove this, we pick a subsequence Without loss of generality, we may further assume that x n i x for some x ∈ C. By (3.7) and demiclosed principle we get x ∈ F ix(W i ) for each i ≥ 1.Furthermore, by the definition of W i and Lemma 2.3 we conclude that x ∈ ∩ ∞ j=1 F ix(T j ).Since x * is the unique solution of (3.2), we obtain Finally, we prove that x n → x * as n → ∞.
By virtue of (3.3) and Lemma 2.1, we have n=1 be sequences in (0, 1).Assume that μ ∈ (0, 2η/L 2 ).Suppose that the following conditions hold: Then the sequence {x n } generated by the following manner: where U i = α i I + (1 − α i )T i (i = 1, 2, ...s, N), converges strongly to the unique solution x * of the variational inequality: Corollary 3.2.Let H be a real Hilbert space and F : H → H be an L-Lipschitz continuous and η-strongly monotone mapping.Let T be a nonexpansive of H such that C = F ix(T ) = ∅.Let {λ n } ∞ n=1 and {β n } ∞ n=1 be sequences in (0, 1).Assume that μ ∈ (0, 2η/L 2 ).Suppose that the following conditions hold: Then the sequence {x n } generated by the following manner: where U = αI + (1 − α)T and 0 < α < 1, converges strongly to the unique solution x * of the variational inequality:
Corollary 3.1.Let H be a real Hilbert space and F : H → H be an L-Lipschitz continuous and η-strongly monotone mapping.Let {T i } N i=1 be a family of nonexpansive of H such that C