A new iterative scheme with nonexpansive mappings for equilibrium problems

Abstract

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.

1 Introduction

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

$$\left|\right|S\left(x\right)-S\left(y\right)\left|\right|\le \delta \left|\right|x-y\left|\right|,\forall x,y\in C.$$

If δ = 1, then S is called nonexpansive on C. Fix(S) is denoted by the set of fixed points of S. Let $f:C\times C\to \mathcal{R}$ 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:

$$\mathsf{\text{Find}}{x}^{*}\in C\mathsf{\text{such}}\mathsf{\text{that}}f\left(x^{*},y\right)\ge 0\mathsf{\text{for}}\mathsf{\text{all}}y\in C.EP\left(f,C\right)$$

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

$$f\left(x,y\right)+f\left(y,x\right)\le -\beta \left|\right|x-y|{|}^2,\forall x,y\in C;$$

monotone on C, if

$$f\left(x,y\right)+f\left(y,x\right)\le 0,\forall x,y\in C;$$

pseudomonotone on C, if

$$f\left(x,y\right)\ge 0\mathsf{\text{implies}}f\left(y,x\right)\le 0,\forall x,y\in C;$$

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

$$f\left(x,y\right)+f\left(y,z\right)\ge f\left(x,z\right)-c_1\left|\right|x-y|{|}^2-c_2||y-z|{|}^2,\forall x,y,z\in C.$$

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.

2 Preliminaries

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

$$\left|\right|x-P{r}_C\left(x\right)\left|\right|\le \left|\right|x-y\left|\right|,\forall y\in C.$$

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

$$〈x-y,P{r}_C\left(x\right)-P{r}_C\left(y\right)〉\ge \left|\right|P{r}_C\left(x\right)-P{r}_C\left(y\right)|{|}^2,\forall x,y\in \mathcal{H},$$

(2.1)

$$〈x-P{r}_C\left(x\right),P{r}_C\left(x\right)-y>〉\ge 0,\forall x\in \mathcal{H},y\in C,$$

(2.2)

$$\left|\right|x-y|{|}^2\ge ||x-P{r}_C\left(x\right)|{|}^2+\left|\right|y-P{r}_C\left(x\right)|{|}^2,\forall x\in \mathcal{H},y\in C.$$

(2.3)

Let us assume that a bifunction $f:C\times C\to \mathcal{R}$ 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:

$$\left\{\begin{array}{c}{x}^0\in \mathcal{H},\hfill \\ \mathsf{\text{Find}}{u}^k\in C\mathsf{\text{such}}\mathsf{\text{that}}f\left(u^k,y\right)+\frac{1}{r_k}〈y-u^k,u^k-x^k〉\ge 0,\forall y\in C,\hfill \\ x^{k+1}={\alpha }_{k}g\left(x^k\right)+\left(1-{\alpha }_k\right)S\left(u^k\right),\forall k\ge 0,\hfill \end{array}\right.$$

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

$$\left\{\begin{array}{c}{y}^k=\mathsf{\text{argmin}}\left\{{\lambda }_{k}f\left(x^k,y\right)+\frac{1}{2}\left|\right|y-x^k|{|}^2:y\in C\right\},\hfill \\ t^k=\mathsf{\text{argmin}}\left\{{\lambda }_{k}f\left(y^k,y\right)+\frac{1}{2}\left|\right|y-x^k|{|}^2:y\in C\right\},\hfill \end{array}\right.$$

(2.4)

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

$$x^{k+1}={\alpha }_{k}g\left(x^k\right)+\left(1-{\alpha }_k\right)S\left(t^k\right),$$

(2.5)

where g : C → C is a δ-contraction with $0<\delta <\frac{1}{2}$.

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:

$$a_{n+1}\le \left(1-{\alpha }_n\right)a_n+{\beta }_n,n\ge 0,$$

where {α _n }, and {ß _n } satisfy the conditions:

(i) α _n ⊂ (0, 1) and $\sum _{n=1}^{\infty }{\alpha }_n=\infty$;

(ii) $\underset{n\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}\frac{{\beta }_n}{{\alpha }_n}\le 0or\sum _{n=1}^{\infty }\left|{\beta }_n\right|<\infty .$

Then

$$\underset{n\to \infty }{\mathsf{\text{lim}}}{a}_n=0.$$

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 $\bar{x}\in C$ and the sequence{(I - S)(x^k)} strongly converges to some $y¯$, it follows that $\left(I--S\right)\left(\bar{x}\right)=ȳ$. 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 $f:C\times C\to \mathcal{R}$ 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

$$\left|\right|t^k-x^{*}|{|}^2\le ||x^k-x^{*}|{|}^2-\left(1-2{\lambda }_k{c}_1\right)\left|\right|x^k-y^k|{|}^2-\left(1-2{\lambda }_k{c}_2\right)||y^k-t^k|{|}^2,\forall k\ge 0.$$

3 Main results

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:

$$\left\{\begin{array}{c}\sum _{k=0}^{\infty }|{\alpha }_{k+1}-{\alpha }_k|<\infty ,\hfill \\ \underset{k\to \infty }{\mathsf{\text{lim}}}{\alpha }_k=0,\hfill \\ \sum _{k=0}^{\infty }{\alpha }_k=\infty ,\hfill \\ \sum _{k=0}^{\infty }\sqrt{|{\lambda }_{k+1}-{\lambda }_k|}<\infty ,\hfill \\ \left\{{\lambda }_k\right\}\subset \left[a,b\right]forsomea,b\in \left(0,\frac{1}{L}\right),whereL=\mathsf{\text{max}}\left\{2c_1,2c_2\right\}.\hfill \end{array}\right.$$

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

$$x^{*}=P{r}_{Fix\left(S\right)\cap Sol\left(f,C\right)}g\left(x^{*}\right).$$

The proof of this theorem is divided into several steps.

Step 1. Claim that

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|x^k-t^k\left|\right|=0.$$

Proof of Step 1. For each x* ∈ Fix(S) ∩ Sol(f, C), it follows from x^k+1= a _k g(x^k) + (1 - a _k )S(t^k), Lemma 2.3 and $\delta \in \left(0,\frac{1}{2}\right)$ that

$$\begin{array}{ll}\hfill \left|\right|x^{k+1}-x^{*}|{|}^2& =\left|\right|{\alpha }_k\left(g\left(x^k\right)-x^{*}\right)+\left(1-{\alpha }_k\right)\left(S\left(t^k\right)-S\left(x^{*}\right)\right)|{|}^2\\ \le {\alpha }_k\left|\right|g\left(x^k\right)-x^{*}|{|}^2+\left(1-{\alpha }_k\right)||S\left(t^k\right)-S\left(x^{*}\right)|{|}^2\\ ={\alpha }_k\left|\right|\left(g\left(x^k\right)-g\left(x^{*}\right)\right)+\left(g\left(x^{*}\right)-x^{*}\right)|{|}^2+\left(1-{\alpha }_k\right)||S\left(t^k\right)-S\left(x^{*}\right)|{|}^2\\ \le 2{\delta }^2{\alpha }_k\left|\right|x^k-x^{*}|{|}^2+2{\alpha }_k||g\left(x^{*}\right)-x^{*}|{|}^2+\left(1-{\alpha }_k\right)\left|\right|t^k-x^{*}|{|}^2\\ \le 2{\delta }^2{\alpha }_k\left|\right|x^k-x^{*}|{|}^2+2{\alpha }_k||g\left(x^{*}\right)-x^{*}|{|}^2+\left(1-{\alpha }_k\right)\left|\right|x^k-x^{*}|{|}^2\\ -\left(1-{\alpha }_k\right)\left(1-2{\lambda }_k{c}_1\right)\left|\right|x^k-y^k|{|}^2-\left(1-{\alpha }_k\right)\left(1-2{\lambda }_k{c}_2\right)||y^k-t^k|{|}^2\\ \le \left|\right|x^k-x^{*}|{|}^2+2{\alpha }_k||g\left(x^{*}\right)-x^{*}|{|}^2-\left(1-{\alpha }_k\right)\left(1-2{\lambda }_k{c}_1\right)\left|\right|x^k-y^k|{|}^2\\ -\left(1-{\alpha }_k\right)\left(1-2{\lambda }_k{c}_2\right)\left|\right|y^k-t^k|{|}^2.\end{array}$$

Then, we have

$$\begin{array}{ll}\hfill \left(1-{\alpha }_k\right)\left(1-2b{c}_1\right)\left|\right|x^k-y^k|{|}^2& \le \left(1-{\alpha }_k\right)\left(1-2{\lambda }_k{c}_1\right)\left|\right|x^k-y^k|{|}^2\\ \le \left|\right|x^k-x^{*}|{|}^2-||x^{k+1}-x^{*}|{|}^2+2{\alpha }_k\left|\right|g\left(x^{*}\right)-x^{*}|{|}^2\\ \to 0\mathsf{\text{as}}k\to \infty ,\end{array}$$

and

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|x^k-y^k\left|\right|=0.$$

(3.1)

By the similar way, also

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|y^k-t^k\left|\right|=0.$$

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

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|x^k-t^k\left|\right|=0.$$

(3.2)

Step 2. Claim that

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|x^{k+1}-x^k\left|\right|=0.$$

Proof of Step 2. It is easy to see that $t^k=\mathsf{\text{argmin }}\left\{\frac{1}{2}\left|\right|t-x^k|{|}^2+{\lambda }_{k}f\left(y^k,t\right):t\in C\right\}$ if and only if

$$0\in {\partial }_2\left({\lambda }_{k}f\left(y^k,y\right)+\frac{1}{2}\left|\right|y-x^k|{|}^2\right)\left(t^k\right)+N_C\left(t^k\right),$$

where N _C (x) is the (outward) normal cone of C at x ∈ C. This means that $0={\lambda }_{k}w+t^k-x^k+\bar{w}$, where w ∈ ∂₂f(y^k, t^k) and $\bar{w}\in N_C\left(t^k\right)$. By the definition of the normal cone N _C we have, from this relation that

$$〈t^k-x^k,t-t^k〉\ge {\lambda }_k〈w,t^k-t〉\forall t\in C.$$

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

$$〈t^k-x^k,t^{k+1}-t^k〉\ge {\lambda }_k〈w,t^k-t^{k+1}〉.$$

(3.3)

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

$$f\left(y^k,t\right)-f\left(y^k,t^k\right)\ge 〈w,t-t^k〉\forall t\in C,w\in {\partial }_{2}f\left(y^k,t^k\right).$$

Using this and (3.3), we have

$$\begin{array}{ll}\hfill 〈t^k-x^k,t^{k+1}-t^k〉& \ge {\lambda }_k〈w,t^k-t^{k+1}〉\\ \ge {\lambda }_k\left(f\left(y^k,t^k\right)-f\left(y^k,t^{k+1}\right)\right).\end{array}$$

(3.4)

By the similar way, we also have

$$〈t^{k+1}-x^{k+1},t^k-t^{k+1}〉\ge {\lambda }_{k+1}\left(f\left(y^{k+1},t^{k+1}\right)-f\left(y^{k+1},t^k\right)\right).$$

(3.5)

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

$$\begin{array}{ll}\hfill \frac{1}{2}\left|\right|x^{k+1}& -x^k|{|}^2-\frac{1}{2}||t^{k+1}-t^k|{|}^2\\ \ge 〈t^{k+1}-t^k,t^k-x^k-t^{k+1}+x^{k+1}〉\\ \ge {\lambda }_k\left(f\left(y^k,t^k\right)-f\left(y^k,t^{k+1}\right)\right)\\ +{\lambda }_{k+1}\left(f\left(y^{k+1},t^{k+1}\right)-f\left(y^{k+1},t^k\right)\right)\\ \ge {\lambda }_k\left(-f\left(t^k,t^{k+1}\right)-c_1\left|\right|y^k-t^k|{|}^2-c_2||t^k-t^{k+1}|{|}^2\right)\\ +{\lambda }_{k+1}\left(-f\left(t^{k+1},t^k\right)-c_1\left|\right|y^{k+1}-t^{k+1}|{|}^2-c_2||t^k-t^{k+1}|{|}^2\right)\\ \ge \left({\lambda }_{k+1}-{\lambda }_k\right)f\left(t^k,t^{k+1}\right)\\ \ge -|{\lambda }_{k+1}-{\lambda }_k|\left|f\left(t^k,t^{k+1}\right)\right|.\end{array}$$

Hence

$$\begin{array}{ll}\hfill \left|\right|t^{k+1}-t^k\left|\right|& \le \sqrt{\left|\right|x^{k+1}-x^k|{|}^2+2|{\lambda }_{k+1}-{\lambda }_k\left|\right|f\left(t^k,t^{k+1}\right)|}\\ \le \left|\right|x^{k+1}-x^k\left|\right|+\sqrt{2|{\lambda }_{k+1}-{\lambda }_k|\left|f\left(t^k,t^{k+1}\right)\right|}\end{array}$$

(3.6)

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+1that x^k+1= a _k g(x^k) + a _k S(t^k), we have

$$\begin{array}{ll}\hfill \left|\right|x^{k+1}-x^k\left|\right|& =\left|\right|{\alpha }_{k}g\left(x^k\right)+{\alpha }_{k}S\left(t^k\right)-{\alpha }_{k-1}g\left(x^{k-1}\right)-{\alpha }_{k-1}S\left(t^{k-1}\right)\left|\right|\\ =\left|\right|\left({\alpha }_k-{\alpha }_{k-1}\right)\left(g\left(x^{k-1}\right)-S\left(t^{k-1}\right)\right)+\left(1-{\alpha }_k\right)\left(S\left(t^k\right)-S\left(t^{k-1}\right)\right)\\ +{\alpha }_k\left(g\left(x^k\right)-g\left(x^{k-1}\right)\right)\left|\right|\\ \le |{\alpha }_k-{\alpha }_{k-1}|\left|\right|g\left(x^{k-1}\right)-S\left(t^{k-1}\right)\left|\right|+\left(1-{\alpha }_k\right)\left|\right|t^k-t^{k-1}\left|\right|+{\alpha }_k\delta \left|\right|x^k-x^{k-1}\left|\right|\\ \le |{\alpha }_k-{\alpha }_{k-1}|\left|\right|g\left(x^{k-1}\right)-S\left(t^{k-1}\right)\left|\right|+\left(1-{\alpha }_k\right)\left(\right||x^k-x^{k-1}||\\ +\sqrt{2|{\lambda }_k-{\lambda }_{k-1}|\left|f\left(t^{k-1},t^k\right)\right|})+{\alpha }_k\delta ||x^k-x^{k-1}||\\ =\left(1-\left(1-\delta \right){\alpha }_k\right)\left|\right|x^k-x^{k-1}\left|\right|+|{\alpha }_k-{\alpha }_{k-1}|\left|\right|g\left(x^{k-1}\right)-S\left(t^{k-1}\right)\left|\right|\\ +\left(1-{\alpha }_k\right)\sqrt{2|{\lambda }_k-{\lambda }_{k-1}|\left|f\left(t^{k-1},t^k\right)\right|}\\ \le \left(1-\left(1-\delta \right){\alpha }_k\right)\left|\right|x^k-x^{k-1}\left|\right|+M|{\alpha }_k-{\alpha }_{k-1}|+K\left(1-{\alpha }_k\right)\sqrt{2|{\lambda }_k-{\lambda }_{k-1}|},\end{array}$$

where δ is contractive constant of the mapping g, M = sup{||g(x^{k - 1}) - S(t^{k - 1})||: k = 0, 1, ...} and $K=\mathsf{\text{sup}}\left\{\sqrt{\left|f\left(t^{k-1},t^k\right)\right|}:k=0,1,\cdots \right\}$, since $\sum _{k=0}^{\infty }|{\alpha }_k-{\alpha }_{k-1}|<\infty$ and $\sum _{k=0}^{\infty }\sqrt{|{\lambda }_k-{\lambda }_{k-1}|}<\infty$, in view of Lemma 2.1, we have $\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|x^{k+1}-x^k\left|\right|=0$.

Step 3. Claim that

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|t^k-S\left(t^k\right)\left|\right|=0.$$

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

$$\begin{array}{ll}\hfill x^{k+1}-x^k& ={\alpha }_{k}g\left(x^k\right)+\left(1-{\alpha }_k\right)S\left(t^k\right)-x^k\\ ={\alpha }_k\left(g\left(x^k\right)-x^k\right)+\left(1-{\alpha }_k\right)\left(t^k-x^k\right)+\left(1-{\alpha }_k\right)\left(S\left(t^k\right)-t^k\right)\end{array}$$

and hence

$$\left(1-{\alpha }_k\right)\left|\right|S\left(t^k\right)-t^k\left|\right|\le \left|\right|x^{k+1}-x^k\left|\right|+{\alpha }_k\left|\right|g\left(x^k\right)-x^k\left|\right|+\left(1-{\alpha }_k\right)\left|\right|t^k-x^k\left|\right|.$$

Using this, $\underset{k\to \infty }{\mathsf{\text{lim}}}{\alpha }_k=0$, Step 1 and Step 2, we have

$$\underset{k\to \infty }{\mathsf{\text{lim}}}\left|\right|t^k-S\left(t^k\right)\left|\right|=0.$$

Step 4. Claim that

$$\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈x^{*}-g\left(x^{*}\right),S\left(t^k\right)-x^{*}〉\ge 0.$$

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

$$\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈x^{*}-g\left(x^{*}\right),t^k-x^{*}〉=\underset{i\to \infty }{\mathsf{\text{lim}}}〈x^{*}-g\left(x^{*}\right),t^{k_i}-x^{*}〉.$$

Since the sequence $\left\{t^{k_i}\right\}$ is bounded, there exists a subsequence $\left\{t^{k_{i_j}}\right\}$ of $\left\{t^{k_i}\right\}$ which converges weakly to $\bar{t}$. Without loss of generality we suppose that the sequence $\left\{t^{k_i}\right\}$ converges weakly to $\bar{t}$ such that

$$\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈x^{*}-g\left(x^{*}\right),t^k-x^{*}〉=\underset{i\to \infty }{\mathsf{\text{lim}}}〈x^{*}-g\left(x^{*}\right),t^{k_i}-x^{*}〉.$$

(3.7)

Since Lemma 2.2 and Step 3, we have

$$S\left(\bar{t}\right)=\bar{t}\iff \bar{t}\in Fix\left(S\right).$$

(3.8)

Now we show that $\bar{t}\in Sol\left(f,C\right)$. By Step 1, we also have

$$x^{k_i}\rightharpoonup \bar{t},y^{k_i}\rightharpoonup \bar{t}.$$

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

$$\mathsf{\text{min}}\left\{\frac{1}{2}\left|\right|y-x^k|{|}^2+f\left(x^k,y\right):y\in C\right\},$$

we have

$$0\in {\partial }_2\left({\lambda }_{k}f\left(x^k,y\right)+\frac{1}{2}\left|\right|y-x^k|{|}^2\right)\left(y^k\right)+N_C\left(y^k\right).$$

This follows that

$$0={\lambda }_{k}w+y^k-x^k+w^k,$$

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

$$〈y^k-x^k,y-y^k〉\ge {\lambda }_k〈w,y^k-y〉,\forall y\in C.$$

(3.9)

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

$$f\left(x^k,y\right)-f\left(x^k,y^k\right)\ge 〈w,y-y^k〉,\forall y\in C.$$

Combining this with (3.9), we have

$${\lambda }_k\left(f\left(x^k,y\right)-f\left(x^k,y^k\right)\right)\ge 〈y^k-x^k,y^k-y〉,\forall y\in C.$$

Hence

$${\lambda }_{k_j}\left(f\left(x^{k_j},y\right)-f\left(x^{k_j},y^{k_j}\right)\right)\ge 〈y^{k_j}-x^{k_j},y^{k_j}-y〉,\forall y\in C.$$

Then, using $\left\{{\lambda }_k\right\}\subset \left[a,b\right]\subset \left(0,\frac{1}{L}\right)$ and the continuity of f , we have

$$f\left(\bar{t},y\right)\ge 0,\forall y\in C.$$

Combining this and (3.8), we obtain

$$t^{k_i}\rightharpoonup \bar{t}\in Fix\left(S\right)\cap Sol\left(f,C\right).$$

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

$$\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈x^{*}-g\left(x^{*}\right),t^k-x^{*}〉=〈x^{*}-g\left(x^{*}\right),\bar{t}-x^{*}〉\ge 0.$$

Using this and Step 3, we get

$$\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈x^{*}-g\left(x^{*}\right),S\left(t^k\right)-x^{*}〉=〈x^{*}-g\left(x^{*}\right),\bar{t}-x^{*}〉\ge 0.$$

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 _k g(x^k) + (1 - a _k )S(t^k) and Lemma 2.3, we have

$$\begin{array}{ll}\hfill \left|\right|x^{k+1}-x^{*}|{|}^2& =\left|\right|{\alpha }_k\left(g\left(x^k\right)-x^{*}\right)+\left(1-{\alpha }_k\right)\left(S\left(t^k\right)-x^{*}\right)|{|}^2\\ ={\alpha }_k^2\left|\right|g\left(x^k\right)-x^{*}|{|}^2+{\left(1-{\alpha }_k\right)}^2||S\left(t^k\right)-x^{*}|{|}^2\\ +2{\alpha }_k\left(1-{\alpha }_k\right)〈g\left(x^k\right)-x^{*},S\left(t^k\right)-x^{*}〉\\ \le {\alpha }_k^2\left|\right|g\left(x^k\right)-x^{*}|{|}^2+{\left(1-{\alpha }_k\right)}^2||x^k-x^{*}|{|}^2\\ +2{\alpha }_k\left(1-{\alpha }_k\right)〈g\left(x^k\right)-x^{*},S\left(t^k\right)-x^{*}〉\\ ={\alpha }_k^2\left|\right|g\left(x^k\right)-x^{*}|{|}^2+{\left(1-{\alpha }_k\right)}^2||x^k-x^{*}|{|}^2\\ +2{\alpha }_k\left(1-{\alpha }_k\right)〈g\left(x^k\right)-g\left(x^{*}\right),S\left(t^k\right)-x^{*}〉\\ +2{\alpha }_k\left(1-{\alpha }_k\right)〈g\left(x^{*}\right)-x^{*},S\left(t^k\right)-x^{*}〉\\ \le {\alpha }_k^2\left|\right|g\left(x^k\right)-x^{*}|{|}^2+{\left(1-{\alpha }_k\right)}^2||x^k-x^{*}|{|}^2\\ +2\delta {\alpha }_k\left(1-{\alpha }_k\right)\left|\right|x^k-x^{*}\left|\right|\left|\right|\left(t^k\right)-x^{*}\left|\right|\\ +2{\alpha }_k\left(1-{\alpha }_k\right)〈g\left(x^{*}\right)-x^{*},S\left(t^k\right)-x^{*}〉\\ \le {\alpha }_k^2\left|\right|g\left(x^k\right)-x^{*}|{|}^2+\left({\left(1-{\alpha }_k\right)}^2+2\delta {\alpha }_k\left(1-{\alpha }_k\right)\right)||x^k-x^{*}|{|}^2\\ +2{\alpha }_k\left(1-{\alpha }_k\right)〈g\left(x^{*}\right)-x^{*},S\left(t^k\right)-x^{*}〉\\ \le \left(1-{\alpha }_k+2\delta {\alpha }_k\right)\left|\right|x^k-x^{*}|{|}^2+{\alpha }_k^2||g\left(x^k\right)-x^{*}|{|}^2\\ +2{\alpha }_k\left(1-{\alpha }_k\right)\mathsf{\text{max}}\left\{0,〈g\left(x^{*}\right)-x^{*},S\left(t^k\right)-x^{*}〉\right\}\\ =\left(1-A_k\right)\left|\right|x^k-x^{*}|{|}^2+B_k,\end{array}$$

where A _k and B _k are defined by

$$\left\{\begin{array}{c}{A}_k={\alpha }_k\left(1-2\delta \right),\hfill \\ B_k={\alpha }_k^2\left|\right|g\left(x^k\right)-x^{*}|{|}^2+2{\alpha }_k\left(1-{\alpha }_k\right)\mathsf{\text{max}}\left\{0,〈g\left(x^{*}\right)-x^{*},S\left(t^k\right)-x^{*}〉\right\}.\hfill \end{array}\right.$$

Since $\underset{k\to \infty }{\mathsf{\text{lim}}}{\alpha }_k=0,\sum _{k=1}^{\infty }{\alpha }_k=\infty$, Step 4, we have $\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈x^{*}-g\left(x^{*}\right),S\left(t^k\right)-x^{*}〉\ge 0$ and hence

$$B_k=o\left(A_k\right),\underset{k\to \infty }{\mathsf{\text{lim}}}{A}_k=0,\sum _{k=1}^{\infty }{A}_k=\infty .$$

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*).

   □

4 Applications

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:

$$\mathsf{\text{Find}}{x}^{*}\in C\mathsf{\text{such}}\mathsf{\text{that}}〈F\left(x^{*}\right),x-x^{*}〉\ge 0\mathsf{\text{for all }}x\in C.VI\left(F,C\right)$$

Let $f:C\times C\to \mathcal{R}$ 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

$$〈F\left(x\right)-F\left(y\right),x-y〉\ge \beta \left|\right|x-y|{|}^2,\forall x,y\in C;$$

monotone on C if

$$〈F\left(x\right)-F\left(y\right),x-y〉\ge 0,\forall x,y\in C;$$

pseudomonotone on C if

$$〈F\left(y\right),x-y〉\ge 0\Rightarrow 〈F\left(x\right),x-y〉\ge 0,\forall x,y\in C;$$

Lipschitz continuous on C with constants L > 0 if

$$\left|\right|F\left(x\right)-F\left(y\right)\left|\right|\le L\left|\right|x-y\left|\right|,\forall x,y\in C.$$

Since

$$\begin{array}{ll}\hfill y^k& =\mathsf{\text{argmin}}\left\{{\lambda }_{k}f\left(x^k,y\right)+\frac{1}{2}\left|\right|y-x^k|{|}^2:y\in C\right\}\\ =\mathsf{\text{argmin}}\left\{{\lambda }_k〈F\left(x^k\right),y-x^k〉+\frac{1}{2}\left|\right|y-x^k|{|}^2:y\in C\right\}\\ =P{r}_C\left(x^k-{\lambda }_{k}F\left(x^k\right)\right),\end{array}$$

(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 $\delta \in \left(0,\frac{1}{2}\right),$ S: C→ C be nonexpansive and positive sequences {a _k } and {λ _k } satisfy the following restrictions

$$\left\{\begin{array}{c}\sum _{k=0}^{\infty }|{\alpha }_{k+1}-{\alpha }_k|<\infty ,\hfill \\ \underset{k\to \infty }{\mathsf{\text{lim}}}{\alpha }_k=0,\hfill \\ \sum _{k=0}^{\infty }{\alpha }_k=\infty ,\hfill \\ \sum _{k=0}^{\infty }\sqrt{|{\lambda }_{k+1}-{\lambda }_k|}<\infty ,\hfill \\ \left\{{\lambda }_k\right\}\subset \left[a,b\right]forsomea,b\in \left(0,\frac{1}{L}\right).\hfill \end{array}\right.$$

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

$$\left\{\begin{array}{c}{y}^k=P{r}_C\left(x^k-{\lambda }_{k}F\left(x^k\right)\right),\hfill \\ t^k=P{r}_C\left(x^k-{\lambda }_{k}F\left(y^k\right)\right),\hfill \\ x^{k+1}={\alpha }_{k}g\left(x^k\right)+\left(1-{\alpha }_k\right)S\left(t^k\right),\hfill \end{array}\right.$$

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:

$$\left\{\begin{array}{c}\sum _{k=0}^{\infty }|{\alpha }_{k+1}-{\alpha }_k|<\infty ,\hfill \\ \underset{k\to \infty }{\mathsf{\text{lim}}}{\alpha }_k=0,\hfill \\ \sum _{k=0}^{\infty }{\alpha }_k=\infty ,\hfill \\ \sum _{k=0}^{\infty }\sqrt{|{\lambda }_{k+1}-{\lambda }_k|}<\infty ,\hfill \\ \left\{{\lambda }_k\right\}\subset \left[a,b\right]forsomea,b\in \left(0,\frac{1}{L}\right),whereL=\mathsf{\text{max}}\left\{2c_1,2c_2\right\}.\hfill \end{array}\right.$$

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

$$\left\{\begin{array}{c}{y}^k=argmin\left\{{\lambda }_{k}f\left(x^k,y\right)+\frac{1}{2}\left|\right|y-x^k|{|}^2:y\in C\right\},\hfill \\ t^k=argmin\left\{{\lambda }_{k}f\left(y^k,y\right)+\frac{1}{2}\left|\right|y-x^k|{|}^2:y\in C\right\}\hfill \\ x^{k+1}={\alpha }_{k}g\left(x^k\right)+\left(1-{\alpha }_k\right)t^k,\hfill \end{array}\right.$$

where g : C → C is a δ-contraction with $0<\delta <\frac{1}{2}$, converge strongly to the same point x*=Pr_Sol(f,C)g(x*).