A new iterative scheme with nonexpansive mappings for equilibrium problems

In this paper, we suggest a new iteration scheme for finding a common of the solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed points of a nonexpansive mapping. The scheme is based on both hybrid method and extragradient-type method. We obtain a strong convergence theorem for the sequences generated by these processes in a real Hilbert space. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature.

AMS 2010 Mathematics subject classification: 65 K10, 65 K15, 90 C25, 90 C33.

Let be a real Hilbert space with inner product 〈·,·〉 and norm || · ||. Let C be a nonempty closed convex subset of a real Hilbert space . A mapping S : C → C is a contraction with a constant δ ∈ (0, 1), if

If δ = 1, then S is called nonexpansive on C. Fix(S) is denoted by the set of fixed points of S. Let be a bifunction such that f(x, x) = 0 for all x ∈ C. We consider the equilibrium problem in the sense of Blum and Oettli (see [1]) which is presented as follows:

The set of solutions of EP(f, C) is denoted by Sol(f, C). The bifunction f is called strongly monotone on C with ß > 0, if

monotone on C, if

pseudomonotone on C, if

Lipschitz-type continuous on C with constants c₁ > 0 and c₂ > 0 (see [2]), if

It is well-known that Problem EP(f, C) includes, as particular cases, the optimization problem, the variational inequality problem, the Nash equilibrium problem in noncooperative games, the fixed point problem, the nonlinear complementarity problem and the vector minimization problem (see [2-6]).

In recent years, the problem to find a common point of the solution set of problem (EP) and the set of fixed points of a nonexpansive mapping becomes an attractive field for many researchers (see [7-15]). An important special case of equilibrium problems is the variational inequalities (shortly (VIP)), where F : C → and f(x, y) = 〈F(x), y - x〉. Various methods have been developed for finding a common point of the solution set of problem (VIP) and the set of fixed points of a nonexpansive mapping when F is monotone (see [16-18]).

Motivated by fixed point techniques of Takahashi and Takahashi in [19] and an improvement set of extragradient-type iteration methods in [20], we introduce a new iteration algorithm for finding a common of the solution set of equilibrium problems with a monotone and Lipschitz-type continuous bifunction and the set of fixed points of a nonexpansive mapping. We show that all of the iterative sequences generated by this algorithm convergence strongly to the common element in a real Hilbert space.

Let C be a nonempty closed convex subset of a Hilbert space . We write xⁿ ⇀ x to indicate that the sequence {xⁿ} converges weakly to x as n → ∞, xⁿ → x implies that {xⁿ} converges strongly to x. For any x ∈ , there exists a nearest point in C, denoted by Pr_C(x), such that

Pr_C is called the metric projection of to C. It is well known that Pr_C satisfies the following properties:

Let us assume that a bifunction and a nonexpansive mapping S : C → C satisfy the following conditions:

A₁. f is Lipschitz-type continuous on C;

A₂. f is monotone on C;

A₃ . for each x ∈ C, f (x, ·) is subdifferentiable and convex on C;

A₄. Fix(S) ∩ Sol(f, C) ≠ ∅.

Recently, Takahashi and Takahashi in [19] first introduced an iterative scheme by the viscosity approximation method. The sequence {x^k} is defined by:

where C is a nonempty closed convex subset of and g is a contractive mapping of into itself. The authors showed that under certain conditions over {α_k} and {r_k}, sequences {x_k} and {u_k} converge strongly to z = Pr_{Sol(f,C)∩Fix(S)} (g(x⁰)). Recently, iterative methods for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonexpansive mapping have further developed by many authors. These methods require to solve approximation auxilary equilibrium problems.

In this paper, we introduce a new iteration method for finding a common point of the set of fixed points of a nonexpansive mapping S and the set of solutions of problem EP(f, C). At each our iteration, the main steps are to solve two strongly convex problems

and compute the next iteration point by Mann-type fixed points

where g : C → C is a δ-contraction with .

To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.

Lemma 2.1 (see [21]) Let {a_n} be a sequence of nonnegative real numbers such that:

where {α_n}, and {ß_n} satisfy the conditions:

(i) α_n ⊂ (0, 1) and ;

(ii)

Then

Lemma 2.2 ([22]) Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space . If Fix(S) ≠ Ø, then I - S is demiclosed; that is, whenever {x^k} is a sequence in C weakly converging to some and the sequence {(I - S)(x^k)} strongly converges to some , it follows that . Here I is the identity operator of .

Lemma 2.3 (see [20], Lemma 3.1) Let C be a nonempty closed convex subset of a real Hilbert space . Let be a pseudomonotone, Lipschitz-type continuous bifunction with constants c₁ > 0 and c₂ > 0. For each × ∈ C, let f(x, ·) be convex and subdifferentiable on C. Suppose that the sequences {x^k}, {y^k}, {t^k} generated by Scheme (2.4) and x* ∈ Sol(f, C). Then

Now, we prove the main convergence theorem.

Theorem 3.1 Suppose that Assumptions A₁-A₄ are satisfied, x⁰ ∈ C and two positive sequences {λ_k}, {a_k} satisfy the following restrictions:

Then the sequences {x^k}, {y^k} and {t^k} generated by (2.4) and (2.5) converge strongly to the same point x*, where

The proof of this theorem is divided into several steps.

Step 1. Claim that

Proof of Step 1. For each x* ∈ Fix(S) ∩ Sol(f, C), it follows from x^k+1 = a_kg(x^k) + (1 - a_k)S(t^k), Lemma 2.3 and that

Then, we have

and

By the similar way, also

Combining this, (3.1) and the inequality ||x^k - t^k|| = ||x^k - y^k || + || y^k - t^k ||, we have

Step 2. Claim that

Proof of Step 2. It is easy to see that if and only if

where N_C(x) is the (outward) normal cone of C at x ∈ C. This means that , where w ∈ ∂₂f(y^k, t^k) and . By the definition of the normal cone N_C we have, from this relation that

Substituting t = t^k+1 into this inequality, we get

Since f(x, ·) is convex on C for all x ∈ C, we have

Using this and (3.3), we have

By the similar way, we also have

Using (3.4), (3.5) and f is Lipschitz-type continuous and monotone, we get

Hence

Since (3.6), a_k+1 - a_k → 0 as k →∞, g is contractive on C, Lemma 2.3, Step 2 and the definition of x^k+1 that x^k+1 = a_kg(x^k) + a_kS(t^k), we have

where δ is contractive constant of the mapping g, M = sup{||g(x^{k - 1}) - S(t^{k - 1})||: k = 0, 1, ...} and , since and , in view of Lemma 2.1, we have .

Step 3. Claim that

Proof of Step 3. From x^k+1 = a_kg(x^k) + (1 - a_k)S(t^k), we have

and hence

Using this, , Step 1 and Step 2, we have

Step 4. Claim that

Proof of Step 4. By Step 1, {t^k} is bounded, there exists a subsequence of {t^k} such that

Since the sequence is bounded, there exists a subsequence of which converges weakly to . Without loss of generality we suppose that the sequence converges weakly to such that

Since Lemma 2.2 and Step 3, we have

Now we show that . By Step 1, we also have

Since y^k is the unique solution of the strongly convex problem

we have

This follows that

where w ∈ ∂₂f (x^k, y^k) and w^k ∈ N_C(y^k). By the definition of the normal cone N_C, we have

On the other hand, since f(x^k, ·) is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem, there exists w ∈ ∂₂f(x^k, y^k) such that

Combining this with (3.9), we have

Hence

Then, using and the continuity of f , we have

Combining this and (3.8), we obtain

By (3.7) and the definition of x*, we have

Using this and Step 3, we get

Step 5. Claim that the sequences {x^k}, {y^k} and {t^k} converge strongly to x*.

Proof of Step 5. Using x^k+1 = α_kg(x^k) + (1 - a_k)S(t^k) and Lemma 2.3, we have

where A_k and B_k are defined by

Since , Step 4, we have and hence

By Lemma 2.1, we obtain that the sequence {x^k} converges strongly to x*. It follows from Step 1 that the sequences {y^k} and {t^k} also converge strongly to the same solution x*= Pr_{Fix(S)∩Sol(f,C)}g(x*).

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Let C be a nonempty closed convex subset of a real Hilbert space and F be a function from C into . In this section, we consider the variational inequality problem which is presented as follows:

Let be defined by f(x, y) = 〈F(x), y - x〉. Then Problem EP(f, C) can be written in VI(F, C). The set of solutions of VI(F, C) is denoted by Sol(F, C). Recall that the function F is called strongly monotone on C with ß > 0 if

monotone on C if

pseudomonotone on C if

Lipschitz continuous on C with constants L > 0 if

Since

(2.4), (2.5) and Theorem 3.1, we obtain that the following convergence theorem for finding a common element of the set of fixed points of a nonexpansive mapping S and the solution set of problem VI(F, C).

Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space , F be a function from C to such that F is monotone and L-Lipschitz continuous on C, g : C → C is contractive with constant S: C → C be nonexpansive and positive sequences {a_k} and {λ_k} satisfy the following restrictions

Then sequences {x^k}, {y^k} and {t^k} generated by

converge strongly to the same point x* ∈ Pr_{Fix(S)∩Sol(F,C)}g(x*).

Thus, this scheme and its convergence become results proposed by Nadezhkina and Takahashi in [23]. As direct consequences of Theorem 3.1, we obtain the following corollary.

Corollary 4.2 Suppose that Assumptions A₁-A₃ are satisfied, Sol(f, C) ≠ Ø, x⁰ ∈ C and two positive sequences {λ_k}, {a_k} satisfy the following restrictions:

Then, the sequences {x^k}, {y^k} and {t^k} generated by

where g : C → C is a δ-contraction with , converge strongly to the same point x*=Pr_Sol(f,C)g(x*).

The authors declare that they have no competing interests.

The main idea of this paper is proposed by P.N. Anh. The revision is made by DDT. PNA and DDT prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.

We are very grateful to the anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.

The work was supported by National Foundation for Science and Technology Development of Vietnam (NAFOSTED).

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