The extragradient-Armijo method for pseudomonotone equilibrium problems and strict pseudocontractions

Abstract

In this article, we present a new iteration method for finding a common element of the set of fixed points of p strict pseudocontractions and the set of solutions of equilibrium problems for pseudomonotone bifunctions without Lipschitz-type continuous conditions. The iterative process is based on the extragradient method and Armijo-type linesearch techniques. We obtain weak convergence theorems for the sequences generated by this process in a real Hilbert space.

AMS 2010 Mathematics Subject Classification: 65 K10; 65 K15; 90 C25; 90 C33.

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space and f be a bifunction from C × C to . We consider the following equilibrium problems (shortly EP(f, C)):

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

The set of solutions of Problem EP(f, C) is denoted by Sol(f, C). These problems apprear frequently in many practical problems arising, for instance, physics, engineering, game theory, transportation, economics and network, and become an attractive field for many researchers both theory and applications (see [1–6]). The bifunction f is called

• monotone if

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

• pseudomonotone if

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

• Lipschitz-type continuous with constants c₁> 0 and c₂> 0 if

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

It is clear that every monotone bifunction f is pseudomonotone.

Let C be a nonempty closed convex subset of . A self-mapping S : C → C is called a strict pseudocontraction if there exists a constant 0 ≤ L < 1 such that

$${∥S\left(x\right)-S\left(y\right)∥}^2\le {∥x-y∥}^2+L{∥\left(I-S\right)\left(x\right)-\left(I-S\right)\left(y\right)∥}^2,\forall x,y\in C,$$

where I is the identity mapping on C. The set of fixed points of S is denoted by Fix(S). The following proposition lists some useful properties for strict pseudocontractions.

Proposition 1.1[7]Let C be a nonempty closed convex subset of a real Hilbert space, S : C → C be a L-strict pseudocontraction and for each i = 1, ..., p, S _i : C → C is a L _i -strict pseudocontraction for some 0 ≤ L _i < 1. Then,

(a) S satisfies the Lipschitz condition

$$∥S\left(x\right)-S\left(y\right)∥\le \frac{1+L}{1-L}∥x-y∥,\forall x,y\in C;$$

(b) I - S is demiclosed at 0. That is, if {xⁿ } is a sequence in C such that$x^n\rightharpoonup \bar{x}$and (I - S)(xⁿ ) → 0, then$\left(I-S\right)\left(\bar{x}\right)=0$;

(c) the fixed point set Fix(S) is closed and convex;

(d) if λ _i > 0 and${\sum }_{i=1}^p{\lambda }_i=1$, then${\sum }_{i=1}^p{\lambda }_i{S}_i$is a$\bar{L}$-strict pseudocontraction with$\bar{L}=\mathsf{\text{max}}\left\{L_i:1\le i\le L\right\}$;

(e) if λ _i is given as in (d) and {S _i : i = 1, ..., p} has a common fixed point, then

$$\mathsf{\text{Fix}}\left(\sum _{i=1}^p{\lambda }_i{S}_i\right)={\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right).$$

For finding a common fixed point of p strict pseudocontractions ${\left\{S_i\right\}}_{i=1}^p$, Mastroeni [5] introduced an iterative algorithm in a real Hilbert space. Let sequences {xⁿ } be defined by

$$x^{n+1}={\alpha }_n{x}^n+\left(1-{\alpha }_n\right)\sum _{i=1}^p{\lambda }_{n,i}{S}_i\left(x^n\right),$$

Under appropriate assumptions on the sequence {λ_n,i}, the authors showed that the sequence {xⁿ } converges weakly to the same point $\bar{x}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)$.

For obtaining a common element of set of solutions of Problem EP(f, C) and the set of fixed points of a nonexpansive mapping S in a real Hilbert space , Takahashi and Takahashi [8] first introduced an iterative scheme by the viscosity approximation method. The sequence {xⁿ } is defined by

$$\left\{\begin{array}{c}{x}^0\in \mathcal{H},\hfill \\ f\left(u^n,y\right)+\frac{1}{r_n}〈y-u^n,u^n-x^n〉\ge 0,\forall y\in C,\hfill \\ x^{n+1}={\alpha }_{n}g\left(x^n\right)+\left(1-{\alpha }_n\right)T\left(u^n\right),\forall n\ge 0.\hfill \end{array}\right.$$

The authors showed that under certain conditions over {α _n } and {r _n }, sequences {xⁿ } and {uⁿ } converge strongly to z = Pr_{Fix(T)∩Sol(f,C)}(g(z)), where Pr _C is denoted the projection on C and g : C → C is contractive, i.e., ||g(x) - g(y)|| ≤ δ||x - y|| for all x, y ∈ C.

Recently, for finding a common element of the set of common fixed points of a strict pseudocontraction sequence $\left\{{Ŝ}_i\right\}$ and the set of solutions of Problem EP (f, C), Chen et al. [9] proposed new iterative scheme in a real Hilbert space. Let sequences {xⁿ }, {yⁿ } and {zⁿ } be defined by

$$\left\{\begin{array}{c}{x}^0\in C,\hfill \\ y^n={\alpha }_n{x}^n+\left(1-{\alpha }_n\right){Ŝ}_n\left(x^n\right),\hfill \\ f\left(z^n,y\right)+\frac{1}{r_n}〈y-z^n,z^n-y^n〉\ge 0,\forall y\in C,\hfill \\ C_n=\left\{v\in C:\left|\right|z^n-v\left|\right|\le \left|\right|x^n-v\left|\right|\right\},\hfill \\ x^{n+1}=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C_n}\left(x^0\right).\hfill \end{array}\right.$$

Then, they showed that under certain appropriate conditions imposed on {α _n } and {r _n }, the sequences {xⁿ }, {yⁿ } and {zⁿ } converge strongly to Pr_{Fix(S)∩Sol(f,C)}(x⁰), where S is a mapping of C into itself defined by $S\left(x\right)=\underset{n\to \infty }{\mathsf{\text{lim}}}{Ŝ}_n\left(x\right)$ for all x ∈ C.

There exist some another solution methods for finding a common element of the set of solutions of Problem EP(f, C) and ${\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)$ (see [3, 10–19]). Most of these algorithms are based on solving approximation equilibrium problems for strongly monotone or monotone and Lipschitz-type continuous bifunctions on C. In this article, we introduce a new iteration method for finding a common element of the set of common fixed points of p strict pseudocontractions and the set of solutions of equilibrium problems for pseudomonotone bifunctions. The fundamental difference here is that at each iteration n, we only solve a strongly convex problem and perform a projection on C. The iterative process is based on the extragradient method and Armijo-type linesearch techniques. We obtain weak convergence theorems for sequences generated by this process in a real Hilbert space .

2 Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space . For each point $x\in \mathcal{H}$, there exists the unique nearest point in C, denoted by Pr _C (x), such that

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

Pr _C is called the metric projection on C. We know that Pr _C is a nonexpansive mapping on C. It is also known that Pr _C is characterized by the following properties

$$\mathsf{\text{P}}{\mathsf{\text{r}}}_C\left(x\right)\in C,〈x-\mathsf{\text{P}}{\mathsf{\text{r}}}_C\left(x\right),\mathsf{\text{P}}{\mathsf{\text{r}}}_C\left(x\right)-y〉\ge 0,$$

(2.1)

for all $x\in \mathcal{H}$, y ∈ C. In the context of the convex optimization, it is also known that if $g:C\to \mathcal{R}$ is convex and subdifferentiable on C, then $\bar{x}$ is a solution to the following convex problem

$$\mathsf{\text{min}}\left\{g\left(x\right):x\in C\right\}$$

if and only if $0\in \partial g\left(\bar{x}\right)+N_C\left(\bar{x}\right)$, where $N_C\left(\bar{x}\right)$ is out normal cone at $\bar{x}$ on C and ∂g(·) denotes the subdifferential of g (see [20]).

Now we are in a position to describe the extragradient-Armijo algorithm for finding a common element of ${\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)$.

Algorithm 2.1 Given a tolerance ε > 0. Choose x⁰ ∈ C, k = 0, γ ∈ (0, 1), $0<\sigma <\frac{\beta }{2}$ and positive sequences{λ_n,i} and {α _n } satisfy the conditions:

$$\left\{\begin{array}{c}\left\{{\alpha }_n\right\}\subset \left[a,b\right]\subset \left(\bar{L},1\right)where\bar{L}:=\mathsf{\text{max}}\left\{L_i:1\le i\le p\right\},\hfill \\ \sum _{i=1}^p{\lambda }_{n,i}=1foralln\ge 1,\underset{n\to \infty }{\mathsf{\text{lim}}}{\lambda }_{n,i}={\lambda }_i\in \left(0,1\right)foralli=1,...,p,\sum _{i=1}^p{\lambda }_i=1.\hfill \end{array}\right.$$

Step 1. Solve the strongly convex problem

$$y^n=argmin\left\{f\left(x^n,y\right)+\frac{\beta }{2}{∥y-x^n∥}^2:y\in C\right\}andsetr\left(x^n\right)=x^n-y^n.$$

If ||r(xⁿ )|| ≠ 0 then go to Step 2. Otherwise, set wⁿ = xⁿ and go to Step 3.

Step 2. (Armijo-type linesearch techniques) Find the smallest positive integer number m _n such that

$$f\left(x^n-{\gamma }^{m_n}r\left(x^n\right),y^n\right)\le -\sigma {∥r\left(x^n\right)∥}^2.$$

(2.2)

Compute

$$w^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C\cap H_n}\left(x^n\right),$$

where$z^n=x^n-{\gamma }^{m_n}r\left(x^n\right),v^n\in {\partial }_{2}f\left(z^k,z^k\right)$ and $H_n=\left\{x\in \mathcal{H}:〈v^n,x-z^n〉\le 0\right\}$, and go to Step 3.

Step 3. Compute

$$x^{n+1}={\alpha }_n{w}^n+\left(1-{\alpha }_n\right)\sum _{i=1}^p{\lambda }_{n,i}{S}_i\left(w^n\right).$$

Increase n by 1 and go back to Step 1.

Remark 2.2 If ||r(xⁿ )|| = 0 then xⁿ is a solution to Problem EP(f, C) but it may be not a common fixed point of${\left\{S_i\right\}}_{i=1}^p$.

Indeed, ||r(xⁿ )|| = 0, i.e., xⁿ is the unique solution to

$$\mathsf{\text{min}}\left\{f\left(x^n,y\right)+\frac{\beta }{2}{∥y-x^n∥}^2:y\in C\right\}.$$

Then

$$0\in {\partial }_{2}f\left(x^n,x^n\right)+N_C\left(x^n\right).$$

Hence

$$〈v^n,x-x^n〉\ge 0,\forall x\in C,v^n\in {\partial }_{2}f\left(x^n,x^n\right).$$

Combining this inequality with f(xⁿ , xⁿ ) = 0 and the convexity of f(xⁿ_, ·), i.e.,

$$f\left(x^n,x\right)-f\left(x^n,x^n\right)\ge 〈v^n,x-x^n〉,\forall x\in C,v^n\in {\partial }_{2}f\left(x^n,x^n\right),$$

we have f(xⁿ , x) ≥ 0 for all x ∈ C. It means that xⁿ is a solution to Problem EP(f, C).

3 Convergence results

In this section, we show the convergence of the sequences {x ⁿ }, {yⁿ } and {wⁿ } defined by Algorithm 2.1 is based on the extragradient method and Armijo-type linesearch techniques which solves the problem of finding a common element of two sets ${\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\mathsf{\text{and Sol}}\left(f,C\right)$. To prove it's convergence, we need the following preparatory result.

Lemma 3.1[21]Let C be a nonempty closed convex subset of a real Hilbert space. Suppose that, for all u ∈ C, the sequence {xⁿ } satisfies

$$∥x^{n+1}-u∥\le ∥x^n-u∥,\forall n\ge 0.$$

Then, the sequence {Pr _C (xⁿ )} converges strongly to$\bar{x}\in C$.

We now state and prove the convergence of the proposed iteration method.

Theorem 3.2 Let C be a nonempty closed convex subset of, S _i : C → C be a L _i -Lipschitz pseudocontractions for all i = 1, ..., p and$f:C\times C\to \mathcal{R}$satisfy the following conditions:

(i) f(x, x) = 0 for all x ∈ C, f is pseudomonotone on C,

(ii) f is continuous on C,

(iii) For each x ∈ C, f(x, ·) is convex and subdifferentiable on C,

(iv) If the sequence{tⁿ } is bounded then {vⁿ } is also bounded, where vⁿ ∈ ∂₂f(tⁿ , tⁿ ),

(v)${\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)\ne \varnothing$.

Then the sequences {xⁿ }, {yⁿ } and {wⁿ } generated by Algorithm 2.1 converge weakly to the point x*, where$x^{*}=\underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{P}}{\mathsf{\text{r}}}_{{\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i,C\right)\cap Sol\left(f,C\right)}\left(x^n\right)$.

Proof. We divide the proof into several steps.

Step 1. If there exists n₀ such that xⁿ = yⁿ for all n ≥ n₀, then the sequences {xⁿ }, {yⁿ } and {wⁿ } generated by Algorithm 2.1 converge weakly to $\bar{x}\in {\cap }_{i=1}^p{S}_i\cap \mathsf{\text{Sol}}\left(f,C\right)$.

Indeed, since xⁿ = yⁿ for all n ≥ n₀, we have wⁿ = xⁿ and

$$x^{n+1}={\alpha }_n{x}^n+\left(1-{\alpha }_n\right)\sum _{i=1}^p{S}_i\left(x^n\right),\forall n\ge n_0.$$

This iteration process is originally introduced by Marino and Xu in a real Hilbert space (see [5]). Under assumptions of Algorithm 2.1 on the sequence {λ_n,i}, the author showed that the sequence {xⁿ } converges weakly to the same point $\bar{x}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)$. Then, the sequence {xⁿ } converges weakly to $\bar{x}\in {\cap }_{i=1}^p{S}_i\cap \mathsf{\text{Sol}}\left(f,C\right)$ in . Consequently, the sequences {yⁿ } and {wⁿ } also converge weakly to $\bar{x}$ as n → ∝. In this case, the sequences {zⁿ } and {vⁿ } might not converge weakly to the point $\bar{x}$.

Otherwise, we consider the following steps.

Step 2. If ||r(xⁿ )|| ≠ 0, then there exists the smallest nonnegative integer m _n such that

$$f\left(x^n-{\gamma }^{m_n}r\left(x^n\right),y^n\right)\le -\sigma {∥r\left(x^n\right)∥}^2.$$

For ||r(xⁿ )|| ≠ 0 and γ ∈ (0, 1), we suppose for contradiction that for every nonnegative integer m, we have

$$f\left(x^n-{\gamma }^{m}r\left(x^n\right),y^n\right)+\sigma {∥r\left(x^n\right)∥}^2>0.$$

Passing to the limit above inequality as m → ∝, by continuity of f, we obtain

$$f\left(x^n,y^n\right)+\sigma {∥r\left(x^n\right)∥}^2\ge 0.$$

(3.1)

On the other hand, since yⁿ is the unique solution of the strongly convex problem

$$\mathsf{\text{min}}\left\{f\left(x^n,y\right)+\frac{\beta }{2}{∥y-x^n∥}^2:y\in C\right\},$$

we have

$$f\left(x^n,y\right)+\frac{\beta }{2}{∥y-x^n∥}^2\ge f\left(x^n,y^n\right)+\frac{\beta }{2}{∥y^n-x^n∥}^2,\forall y\in C.$$

With y = xⁿ , the last inequality implies

$$f\left(x^n,y^n\right)+\frac{\beta }{2}{∥r\left(x^n\right)∥}^2\le 0.$$

(3.2)

Combining (3.1) with (3.2), we obtain

$$\sigma {∥r\left(x^n\right)∥}^2\ge \frac{\beta }{2}{∥r\left(x^n\right)∥}^2.$$

Hence it must be either ||r(xⁿ)|| = 0 or $\sigma \ge \frac{\beta }{2}$. The first case contradicts to ||r(xⁿ )|| ≠ 0, while the second one contradicts to the fact $\sigma <\frac{\beta }{2}$.

Step 3. We claim that if ||r(xⁿ )|| ≠ 0 then xⁿ∉ H _n .

From $z^n=x^n-{\gamma }^{m_n}r\left(x^n\right)$, it follows that

$$y^n-z^n=\frac{1-{\gamma }^{m_n}}{{\gamma }^{m_n}}\left(z^n-x^n\right).$$

Then using (4.1) and the assumption f(x, x) = 0 for all x ∈ C, we have

$$\begin{array}{ll}\hfill 0& >-\sigma {∥r\left(x^n\right)∥}^2\\ \ge f\left(z^n,y^n\right)\\ =f\left(z^n,y^n\right)-f\left(z^n,z^n\right)\\ \ge 〈v^n,y^n-z^n〉\\ =\frac{1-{\gamma }^{m_n}}{{\gamma }^{m_n}}〈z^n-x^n,v^n〉.\end{array}$$

Hence

$$〈x^n-z^n,v^n〉>0.$$

This implies that xⁿ ∉ H _n .

Step 4. We claim that if ||r(xⁿ )|| ≠ 0 then $w^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C\cap H_n}\left({ȳ}^n\right)$, where ${ȳ}^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{H_n}\left(x^n\right)$.

For $K=\left\{x\in \mathcal{H}:〈w,x-x^0〉\le 0\right\}$ and ||w|| ≠ 0, we know that

$$\mathsf{\text{P}}{\mathsf{\text{r}}}_K\left(y\right)=y-\frac{〈w,y-x^0〉}{{∥w∥}^2}w,$$

Hence,

$$\begin{array}{ll}\hfill {ȳ}^n& =\mathsf{\text{P}}{\mathsf{\text{r}}}_{H_n}\left(x^n\right)\\ =x^n-\frac{〈v^n,x^n-z^n〉}{{∥v^n∥}^2}{v}^n\\ =x^n-\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{{∥v^n∥}^2}{v}^n.\end{array}$$

Otherwise, for every y ∈ C ∩ H _n there exists λ ∈ (0, 1) such that

$$\widehat{x}=\lambda x^n+\left(1-\lambda \right)y\in C\cap \partial H_n,$$

where

$$\partial H_n=\left\{x\in \mathcal{H}:〈v^n,x-z^n〉=0\right\}.$$

From Step 2, it follows that xⁿ ∈ C but xⁿ ∉ H _n . Therefore, we have

$$\begin{array}{ll}\hfill {∥y-{ȳ}^n∥}^2& \ge {\left(1-\lambda \right)}^2{∥y-{ȳ}^n∥}^2\\ ={∥\widehat{x}-\lambda x^n-\left(1-\lambda \right){ȳ}^n∥}^2\\ ={∥\left(\widehat{x}-{ȳ}^n\right)-\lambda \left(x^n-{ȳ}^n\right)∥}^2\\ ={∥\widehat{x}-{ȳ}^n∥}^2+{\lambda }^2{∥x^n-{ȳ}^n∥}^2-2\lambda 〈\widehat{x}-{ȳ}^n,x^n-{ȳ}^n〉\\ ={∥\widehat{x}-{ȳ}^n∥}^2+{\lambda }^2{∥x^n-{ȳ}^n∥}^2\\ \ge {∥\widehat{x}-{ȳ}^n∥}^2,\end{array}$$

(3.3)

because ${ȳ}^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{H_n}\left(x^n\right)$. Also we have

$$\begin{array}{ll}\hfill {∥\widehat{x}-x^n∥}^2& ={∥\widehat{x}-{ȳ}^n+{ȳ}^n-x^n∥}^2\\ ={∥\widehat{x}-{ȳ}^n∥}^2-2〈\widehat{x}-{ȳ}^n,x^n-{ȳ}^n〉+{∥{ȳ}^n-x^n∥}^2\\ ={∥\widehat{x}-{ȳ}^n∥}^2+{∥{ȳ}^n-x^n∥}^2.\end{array}$$

Using $w^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C\cap H_n}\left(x^n\right)$ and the Pythagorean theorem, we can reduce that

$$\begin{array}{ll}\hfill {∥\widehat{x}-{ȳ}^n∥}^2& ={∥\widehat{x}-x^n∥}^2-{∥{ȳ}^n-x^n∥}^2\\ \ge {∥w^n-x^n∥}^2-{∥{ȳ}^n-x^n∥}^2\\ ={∥w^n-{ȳ}^n∥}^2.\end{array}$$

(3.4)

From (3.3) and (3.4), we have

$$∥w^n-{ȳ}^n∥\le ∥y-{ȳ}^n∥,\forall y\in C\cap H_n,$$

which means

$$w^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C\cap H_n}\left({ȳ}^n\right).$$

Step 5. We claim that if ||r(xⁿ )|| ≠ 0 then Sol(f, C) ⊆ C ∩ H _n .

Indeed, suppose x* ∈ Sol(f, C). Using the definition of x*, f(x*, x) ≥ 0 for all x ∈ C and f is pseudomonotone on C, we get

$$f\left(z^n,x^{*}\right)\le 0.$$

(3.5)

It follows from vⁿ ∈ ∂₂f (zⁿ , zⁿ ) that

$$\begin{array}{ll}\hfill f\left(z^n,x^{*}\right)& =f\left(z^n,x^{*}\right)-f\left(z^n,z^n\right)\\ \ge 〈v^n,x^{*}-z^n〉.\end{array}$$

(3.6)

Combining (3.5) and (3.6), we have

$$〈v^n,x^{*}-z^n〉\le 0.$$

By the definition of H _n , we have x* ∈ H _n . Thus Sol(f, C) ⊆ C ∩ H _n .

Step 6. We claim that if ||r(xⁿ )|| ≠ 0 and the sequence {vⁿ } is uniformly bounded by M > 0 then the sequence {||xⁿ - x*||} is nonincreasing and hence convergent. Moreover, we have

$$\begin{array}{ll}\hfill {∥x^{n+1}-x^{*}∥}^2\le {∥x^n-x^{*}∥}^2& -\left(1-{\alpha }_n\right){∥w^n-{ȳ}^n∥}^2-\left(1-{\alpha }_n\right){\left(\frac{{\gamma }^{m_n}\sigma }{M\left(1-{\gamma }^{m_n}\right)}\right)}^2{∥r\left(x^n\right)∥}^4\\ -\left(1-{\alpha }_n\right)\left({\alpha }_n-\bar{L}\right){∥{\bar{S}}_n\left(w^n\right)-w^n∥}^2,\end{array}$$

(3.7)

where ${ȳ}^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{H_n}\left(x^n\right)$, ${\bar{S}}_n:={\sum }_{i=1}^p{\lambda }_{n,i}{S}_i$ and $x^{*}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)$.

In the case ||r(xⁿ )|| ≠ 0, by Step 4, we have $w^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C\cap H_n}\left({ȳ}^n\right)$, i.e.,

$$〈{ȳ}^n-w^n,z-w^n〉\le 0,\forall z\in C\cap H_n,$$

where ${ȳ}^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{H_n}\left(x^n\right)$. Substituting z = x* ∈ Sol(f, C) ⊆ C ∩ H _n by Step 5, then we have

$$〈{ȳ}^n-w^n,x^{*}-w^n〉\le 0\iff 〈y^n-w^n,x^{*}-{ȳ}^n+{ȳ}^n-w^n〉\le 0,$$

which implies that

$${∥w^n-{ȳ}^n∥}^2\le 〈w^n-{ȳ}^n,x^{*}-{ȳ}^n〉.$$

Hence

$$\begin{array}{ll}\hfill {∥w^n-x^{*}∥}^2& ={∥w^n-{ȳ}^n+{ȳ}^n-x^{*}∥}^2\\ ={∥w^n-{ȳ}^n∥}^2+{∥{ȳ}^n-x^{*}∥}^2+2〈w^n-{ȳ}^n,{ȳ}^n-x^{*}〉\\ \le 〈x^{*}-{ȳ}^n,w^n-{ȳ}^n〉+{∥{ȳ}^n-x^{*}∥}^2+2〈w^n-{ȳ}^n,{ȳ}^n-x^{*}〉\\ ={∥{ȳ}^n-x^{*}∥}^2+〈w^n-{ȳ}^n,{ȳ}^n-x^{*}〉\\ \le {∥{ȳ}^n-x^{*}∥}^2-{∥w^n-{ȳ}^n∥}^2.\end{array}$$

(3.8)

Since $z^n=x^n-{\gamma }^{m_n}r\left(x^n\right)$ and

$${ȳ}^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{H_n}\left(x^n\right)=x^n-\frac{〈v^n,x^n-z^n〉}{{∥v^n∥}^2}{v}^n,$$

we have

$$\begin{array}{l}{∥{ȳ}^n-x^{*}∥}^2\\ ={∥x^n-x^{*}∥}^2+\frac{{〈v^n,x^n-z^n〉}^2}{{∥v^n∥}^4}{∥v^n∥}^2-\frac{2〈v^n,x^n-z^n〉}{{∥v^n∥}^2}〈v^n,x^n-x^{*}〉\\ ={∥x^n-x^{*}∥}^2+{\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{∥v^n∥}\right)}^2-\frac{2{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{{∥v^n∥}^2}〈v^n,x^n-x^{*}〉\\ ={∥x^n-x^{*}∥}^2+{\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{∥v^n∥}\right)}^2\\ -2\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{{∥v^n∥}^2}〈v^n,x^n-x^{*}〉-{\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{∥v^n∥}\right)}^2\right)\\ ={∥x^k-x^{*}∥}^2-{\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{∥v^n∥}\right)}^2\\ -\frac{2{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{{∥v^n∥}^2}\left(〈v^n,x^n-x^{*}〉-{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉\right)\\ ={∥x^n-x^{*}∥}^2-{\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{∥v^n∥}\right)}^2\\ -\frac{2{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{{∥v^n∥}^2}〈v^n,x^n-x^{*}-{\gamma }^{m_n}r\left(x^n\right)〉\\ ={∥x^n-x^{*}∥}^2-{\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{∥v^n∥}\right)}^2-\frac{2{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{{∥v^n∥}^2}〈v^n,z^n-x^{*}〉.\end{array}$$

(3.9)

It follows from vⁿ ∈ ∂₂f(zⁿ , zⁿ ) that

$$f\left(z^n,y\right)-f\left(z^n,z^n\right)\ge 〈v^n,y-z^n〉,\forall y\in C.$$

(3.10)

Replacing y by yⁿ and combining with assumptions f(zⁿ , zⁿ ) = 0 and $z^n=x^n-{\gamma }^{m_n}r\left(x^n\right)$,

we have

$$\begin{array}{ll}\hfill f\left(z^n,y^n\right)& \ge 〈v^n,y^n-z^n〉\\ =-\left(1-{\gamma }^{m_n}\right)〈v^n,r\left(x^n\right)〉.\end{array}$$

Combining this inequality with (4.1) and assumption γ ∈ (0, 1), we obtain

$$〈v^n,r\left(x^n\right)〉\ge \frac{\sigma }{1-{\gamma }^{m_n}}{∥r\left(x^n\right)∥}^2.$$

(3.11)

Substituting y = x* into (3.10) and using f(zⁿ , zⁿ ) = 0, we have

$$f\left(z^n,x^{*}\right)\ge 〈v^n,x^{*}-z^n〉.$$

(3.12)

Since f is pseudomonotone on C and f(x*, x) ≥ 0, ∀x ∈ C, we have

$$f\left(z^n,x^{*}\right)\le 0.$$

Combining this with (3.12), we get

$$0\ge 〈v^n,x^{*}-z^n〉.$$

(3.13)

Using (3.9), (3.11) and (3.13), we have

$$\begin{array}{ll}\hfill {∥{ȳ}^n-x^{*}∥}^2& \le {∥x^n-x^{*}∥}^2-{\left(\frac{{\gamma }^{m_n}〈v^n,r\left(x^n\right)〉}{∥v^n∥}\right)}^2\\ \le {∥x^n-x^{*}∥}^2-{\left(\frac{{\gamma }^{m_n}\sigma }{∥v^n∥\left(1-{\gamma }^{m_n}\right)}\right)}^2{∥r\left(x^n\right)∥}^4.\end{array}$$

(3.14)

Combining (3.8) with (3.14), we obtain

$${∥w^n-x^{*}∥}^2\le {∥x^n-x^{*}∥}^2-{∥w^n-{ȳ}^n∥}^2-{\left(\frac{{\gamma }^{m_n}\sigma }{∥v^n∥\left(1-{\gamma }^{m_n}\right)}\right)}^2{∥r\left(x^n\right)∥}^4.$$

(3.15)

Using ${\bar{S}}_n:={\sum }_{i=1}^p{\lambda }_{n,i}{S}_i,\left(3.15\right),x^{n+1}={\alpha }_n{w}^n+\left(1-{\alpha }_n\right){\sum }_{i=1}^p{\lambda }_{n,i}{S}_i\left(w^n\right)$ and the equality

$${∥\lambda x+\left(1-\lambda \right)y∥}^2=\lambda {∥x∥}^2+\left(1-\lambda \right){∥y∥}^2-\lambda \left(1-\lambda \right){∥x-y∥}^2,\forall \lambda \in \left[0,1\right],x,y\in {\mathcal{R}}^n,$$

(3.16)

we have

$$\begin{array}{l}{∥x^{n+1}-x^{*}∥}^2\\ ={∥{\alpha }_n{w}^n+\left(1-{\alpha }_n\right){\bar{S}}_n\left(w^n\right)-x^{*}∥}^2\\ ={∥{\alpha }_n\left(w^n-x^{*}\right)+\left(1-{\alpha }_n\right)\left({\bar{S}}_n\left(w^n\right)-x^{*}\right)∥}^2\\ ={\alpha }_n{∥w^n-x^{*}∥}^2+\left(1-{\alpha }_n\right){∥{\bar{S}}_n\left(w^n\right)-{\bar{S}}_n\left(x^{*}\right)∥}^2-{\alpha }_n\left(1-{\alpha }_n\right){∥{\bar{S}}_n\left(w^n\right)-w^n∥}^2\\ \le {\alpha }_n{∥w^n-x^{*}∥}^2+\left(1-{\alpha }_n\right)\left({∥w^n-x^{*}∥}^2+\bar{L}{∥\left(I-{\bar{S}}_n\right)\left(w^n\right)-\left(I-{\bar{S}}_n\right)\left(x^{*}\right)∥}^2\right)\\ -{\alpha }_n\left(1-{\alpha }_n\right){∥{\bar{S}}_n\left(w^n\right)-w^n∥}^2\\ ={∥w^n-x^{*}∥}^2+\left(1-{\alpha }_n\right)\left(\bar{L}-{\alpha }_n\right){∥{\bar{S}}_n\left(w^n\right)-w^n∥}^2\\ \le {∥x^n-x^{*}∥}^2-\left(1-{\alpha }_n\right){∥w^n-{ȳ}^n∥}^2-\left(1-{\alpha }_n\right){\left(\frac{{\gamma }^{m_n}\sigma }{∥v^n∥\left(1-{\gamma }^{m_n}\right)}\right)}^2{∥r\left(x^n\right)∥}^4\\ -\left(1-{\alpha }_n\right)\left({\alpha }_n-\bar{L}\right){∥{\bar{S}}_n\left(w^n\right)-w^n∥}^2.\end{array}$$

(3.17)

In the case ||r(xⁿ )|| = 0, by Algorithm 2.1 and (3.16), we have wⁿ = xⁿ and

$$\begin{array}{ll}\hfill {∥x^{n+1}-x^{*}∥}^2& ={∥{\alpha }_n{x}^n+\left(1-{\alpha }_n\right){\bar{S}}_n\left(x^n\right)-x^{*}∥}^2\\ ={∥{\alpha }_n\left(x^n-x^{*}\right)+\left(1-{\alpha }_n\right)\left({\bar{S}}_n\left(x^n\right)-{\bar{S}}_n\left(x^{*}\right)\right)∥}^2\\ ={\alpha }_n{∥x^n-x^{*}∥}^2+\left(1-{\alpha }_n\right){∥{\bar{S}}_n\left(x^n\right)-{\bar{S}}_n\left(x^{*}\right)∥}^2\\ -{\alpha }_n\left(1-{\alpha }_n\right){∥\left(I-{\bar{S}}_n\right)\left(x^n\right)-\left(I-{\bar{S}}_n\right)\left(x^{*}\right)∥}^2\\ \le {\alpha }_n{∥x^n-x^{*}∥}^2+\left(1-{\alpha }_n\right)\left({∥x^n-x^{*}∥}^2+\bar{L}{∥\left(I-{\bar{S}}_n\right)\left(x^n\right)-\left(I-{\bar{S}}_n\right)\left(x^{*}\right)∥}^2\right)\\ -{\alpha }_n\left(1-{\alpha }_n\right){∥\left(I-{\bar{S}}_n\right)\left(x^n\right)-\left(I-{\bar{S}}_n\right)\left(x^{*}\right)∥}^2\\ =∥x^n-x^{*}∥-\left(1-{\alpha }_n\right)\left({\alpha }_n-\bar{L}\right)∥{\bar{S}}_n\left(x^n\right)-x^n∥\\ \le ∥x^n-x^{*}∥.\end{array}$$

Combining this and (3.17), we get

$$∥x^{n+1}-x^{*}∥\le ∥x^n-x^{*}∥,\forall n\ge 0.$$

So the sequence {||xⁿ - x*||} is nonincreasing and hence convergent. Since (3.17) and the sequence {vⁿ } is uniformly bounded by M > 0, i.e.,

$$∥v^n∥\le M,\forall n\ge 0,$$

we obtain (3.7).

Step 7. We claim that there exists $c=\underset{n\to \infty }{\mathsf{\text{lim}}}∥x^n-x^{*}∥=\underset{n\to \infty }{\mathsf{\text{lim}}}∥w^n-x^{*}∥$, where $x^{*}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)$. Consequently, the sequences {xⁿ }, {yⁿ }, {zⁿ }, {vⁿ } and {wⁿ } are bounded.

By Step 6, there exists

$$c=\underset{n\to \infty }{\mathsf{\text{lim}}}∥x^n-x^{*}∥.$$

(3.18)

From $w^n=x^n\mathsf{\text{if}}∥r\left(x^n\right)∥=0,w^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C\cap H_n}\left(x^n\right)\mathsf{\text{if}}∥r\left(x^n\right)∥\ne 0$ and Step 6, it follows that

$$∥w^n-x^{*}∥\le ∥x^n-x^{*}∥,\forall n\ge 0.$$

Hence

$$\underset{n\to \infty }{\mathsf{\text{lim}}}∥w^n-x^{*}∥\le \underset{n\to \infty }{\mathsf{\text{lim}}}∥x^n-x^{*}∥=c.$$

(3.19)

Using $x^{n+1}={\alpha }_n{w}^n+\left(1-{\alpha }_n\right){\bar{S}}_n\left(w^n\right)$, we have

$$\begin{array}{ll}\hfill {∥x^{n+1}-x^{*}∥}^2& ={∥{\alpha }_n{w}^n+\left(1-{\alpha }_n\right){\bar{S}}_n\left(w^n\right)-x^{*}∥}^2\\ ={∥{\alpha }_n\left(w^n-x^{*}\right)+\left(1-{\alpha }_n\right)\left({\bar{S}}_n\left(w^n\right)-x^{*}\right)∥}^2\\ ={\alpha }_n{∥w^n-x^{*}∥}^2+\left(1-{\alpha }_n\right){∥{\bar{S}}_n\left(w^n\right)-{\bar{S}}_n\left(x^{*}\right)∥}^2\\ -{\alpha }_n\left(1-{\alpha }_n\right){∥\left(I-{\bar{S}}_n\right)\left(w^n\right)-\left(I-{\bar{S}}_n\right)\left(x^{*}\right)∥}^2\\ \le {\alpha }_n{∥w^n-x^{*}∥}^2+\left(1-{\alpha }_n\right)\left({∥w^n-x^{*}∥}^2+\bar{L}{∥{\bar{S}}_n\left(w^n\right)-w^n∥}^2\right)\\ -{\alpha }_n\left(1-{\alpha }_n\right){∥\left(I-{\bar{S}}_n\right)\left(w^n\right)-\left(I-{\bar{S}}_n\right)\left(x^{*}\right)∥}^2\\ ={∥w^n-x^{*}∥}^2-\left(1-{\alpha }_n\right)\left({\alpha }_n-\bar{L}\right){∥{\bar{S}}_n\left(w^n\right)-w^n∥}^2\\ \le ∥w^n-x^{*}∥.\end{array}$$

(3.20)

Hence

$$c\le \underset{n\to \infty }{\mathsf{\text{lim}}}∥w^n-x^{*}∥.$$

(3.21)

From (3.21) and (3.19), it follows that

$$c=\underset{n\to \infty }{\mathsf{\text{lim}}}∥w^n-x^{*}∥.$$

Since yⁿ is the unique solution to

$$\mathsf{\text{min}}\left\{f\left(x^n,y\right)+\frac{\beta }{2}{∥y-x^n∥}^2:y\in C\right\},$$

we have

$$f\left(x^n,y\right)+\frac{\beta }{2}{∥y-x^n∥}^2\ge f\left(x^n,y^n\right)+\frac{\beta }{2}{∥y^n-x^n∥}^2,\forall y\in C.$$

With y = xⁿ ∈ C and f(xⁿ , xⁿ ) = 0, we have

$$0\ge f\left(x^n,y^n\right)+\frac{\beta }{2}{∥y^n-x^n∥}^2.$$

(3.22)

Since f (xⁿ , ·) is convex and subdifferentiable on C, i.e.,

$$f\left(x^n,y\right)-f\left(x^n,x^n\right)\ge 〈u^n,y-x^n〉\forall y\in C,$$

where uⁿ ∈ ∂₂f (xⁿ , xⁿ ). Using y = yⁿ , we have

$$f\left(x^n,y^n\right)\ge 〈u^n,y^n-x^n〉.$$

Combining this and (3.22), we obtain

$$〈u^n,y^n-x^n〉+\frac{\beta }{2}\parallel x^n-y^n{\parallel }^2\le 0.$$

Hence

$$∥x^n-y^n+\frac{1}{\beta }{u}^n∥\le \frac{1}{\beta }\parallel u^n\parallel .$$

(3.23)

From the assumption (iv) and (3.18), it implies that the sequence {uⁿ } is bounded. Then, it follows from (3.23) that {yⁿ } is bounded and hence $z^n=x^n-{\gamma }^{m_n}\left(x^n-y^n\right)$ is also bounded. Also the sequences {vⁿ } and {wⁿ } are bounded.

Step 8. We claim that there exists a subsequence of the sequence {xⁿ } which converges weakly to $\bar{x}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\mathsf{\text{Sol}}\left(f,C\right)$ and hence the whole sequence {xⁿ } converges weakly to $\bar{x}$.

Suppose that $\left\{x^{n_k}\right\}$ is a subsequence of {xⁿ } such that

$$\parallel r\left(x^{n_k}\right)\parallel \ne 0.$$

By Step 7 and the assumption (iv), the sequence {vⁿ } is bounded by M > 0. We show that

$$\begin{array}{ll}\hfill \parallel x^{n_{k+1}}-x^{*}{\parallel }^2\le \parallel x^{n_k}-x^{*}{\parallel }^2& -\left(1-b\right)\parallel w^{n_k+p}-{ȳ}^{n_k+p}{\parallel }^2\\ -b\left(a-\bar{L}\right)\parallel {\bar{S}}_{n_k+p}\left(w^{n_k+p}\right)-w^{n_k+p}{\parallel }^2\\ -\left(1-b\right){\left(\frac{{\gamma }^{m_{n_k+p}}\sigma }{M\left(1-{\gamma }^{m_{n_k+p}}\right)}\right)}^2\parallel r\left(x^{n_k+p}\right){\parallel }^4,\end{array}$$

(3.24)

where $p=n_{k+1}-n_k-1,{ȳ}^{n_k+p}=\mathsf{\text{P}}{\mathsf{\text{r}}}_{H_{n_k+p}}\left(x^{n_k+p}\right)$, $x^{*}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)$ and ${\bar{S}}_n:={\sum }_{i=1}^p{\lambda }_{n,i}{S}_i$. Indeed, if n_k+1= n _k + 1 then it is clear from Step 6. Otherwise, we suppose that there exists a positive integer p such that n _k + p + 1 = n_k+1. Note that $\parallel r\left(x^{n_k+i}\right)\parallel =0$ for all i = 0, 1, ..., p - 1. Using $r\left(x^{n_k+p}\right)\ne 0$, (3.17) and Step 6, we have

$$\begin{array}{ll}\hfill \parallel x^{n_{k+1}}-x^{*}{\parallel }^2& =\parallel x^{n_k+p+1}-x^{*}{\parallel }^2\\ \le \parallel x^{n_k+p}-x^{*}{\parallel }^2-\left(1-{\alpha }_{n_k+p}\right)\parallel w^{n_k+p}-{ȳ}^{n_k+p}{\parallel }^2\\ -\left(1-{\alpha }_{n_k+p}\right){\left(\frac{{\gamma }^{m_{n_k+p}}\sigma }{\parallel v^{n_k+p}\parallel \left(1-{\gamma }^{m_{n_k+p}}\right)}\right)}^2\parallel r\left(x^{n_k+p}\right){\parallel }^4\\ -\left(1-{\alpha }_{n_k+p}\right)\left({\alpha }_{n_k+p}-\bar{L}\right)\parallel {\bar{S}}_{n_k+p}\left(w^{n_k+p}\right)-w^{n_k+p}{\parallel }^2\\ \le \cdots \\ \le \parallel x^{n_k}-x^{*}{\parallel }^2-\left(1-b\right)\parallel w^{n_k+p}-{ȳ}^{n_k+p}{\parallel }^2\\ -b\left(a-\bar{L}\right)\parallel {\bar{S}}_{n_k+p}\left(w^{n_k+p}\right)-w^{n_k+p}{\parallel }^2\\ -\left(1-b\right){\left(\frac{{\gamma }^{m_{n_k+p}}\sigma }{M\left(1-{\gamma }^{m_{n_k+p}}\right)}\right)}^2\parallel r\left(x^{n_k+p}\right){\parallel }^4.\end{array}$$

This implies (3.24). Then, since {||xⁿ - x*||} is convergent, it is easy to see that

$$\underset{k\to \infty }{\mathsf{\text{lim}}}{\gamma }^{m_{n_k+p}}\parallel r\left(x^{n_k+p}\right)\parallel =0.$$

The cases remaining to consider are the following.

Case 1.$\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{sup}}}{\gamma }^{m_{n_k+p}}>0$. This case must follow that $\underset{k\to \infty }{\mathsf{\text{lim}}\mathsf{\text{inf}}}\parallel r\left(x^{n_k+p}\right)\parallel =0$. Since $\left\{x^{n_k+p}\right\}$ is bounded, there exists an accumulation point $\bar{x}$ of $\left\{x^{n_k+p}\right\}$. In other words, a subsequence $\left\{x^{n_{k_j}}\right\}$ converges weakly to some $\bar{x}$, as j → ∝ such that $r\left(\bar{x}\right)=0$. Then by Remark 2.2, we have $\bar{x}\in \mathsf{\text{Sol}}\left(f,C\right)$.

Case 2. $\underset{k\to \infty }{\mathsf{\text{lim}}}{\gamma }^{m_{n_k+p}}=0$. Since $\left\{\left|\right|x^{n_k+p}-x^{*}\left|\right|\right\}$ is convergent, there is the subsequence $\left\{x^{n_{k_j}}\right\}$of $\left\{x^{n_k+p}\right\}$ which converges weakly to $\bar{x}$, as j → ∝. Since $m_{n_k+p}$ is the smallest nonnegative integer, $m_{n_k+p}-1$ does not satisfy (4.1). Hence, we have

$$f\left(x^{n_{k_j}}-{\gamma }^{m_{n_{k_j}-1}}r\left(x^{n_{k_j}}\right),y^{n_{k_j}}\right)>-\sigma \left|\right|r\left(x^{n_{k_j}}\right)|{|}^2.$$

Passing onto the limit, as j → ∝ and using the continuity of f, we have $y^{n_{k_j}}\to ȳ$ and

$$f\left(\bar{x},ȳ\right)\ge -\sigma \left|\right|r\left(\bar{x}\right)|{|}^2,$$

(3.25)

where $r\left(\bar{x}\right)=\bar{x}-ȳ$. It follows from (3.2) that

$$f\left(x^{n_{k_j}-1}-{\gamma }^{m_{n_{k_j}}}r\left(x^{n_{k_j}}\right),y^{n_{k_j}-1}\right)+\frac{\beta }{2}\left|\right|r\left(x^{n_{k_j}-1}\right)|{|}^2\le 0.$$

Since f is continuous and passing onto the limit, as j → ∝, we obtain

$$f\left(\bar{x},ȳ\right)+\frac{\beta }{2}\left|\right|r\left(\bar{x}\right)|{|}^2\le 0.$$

Combining this with (3.25), we have

$$\sigma \parallel r\left(\bar{x}\right){\parallel }^2\ge -f\left(\bar{x},ȳ\right)\ge \frac{\beta }{2}\parallel r\left(\bar{x}\right){\parallel }^2.$$

which implies $r\left(\bar{x}\right)=0$, and hence $\bar{x}=ȳ\in \mathsf{\text{Sol}}\left(f,C\right)$. Thus every cluster point of the sequence $\left\{x^{n_k+p}\right\}$ is a solution to Problem EP(f, C).

Now we show that every cluster point of $\left\{x^{n_k+p}\right\}$ is a fixed point of p strict pseudocontractions ${\left\{S_i\right\}}_{i=1}^p$. Suppose that there exists a subsequence $\left\{x^{n_{k_j}}\right\}$of $\left\{x^{n_k+p}\right\}$ which converges weakly to $\bar{x}$, as j → ∝. By the above proof, we have $\bar{x}\in \mathsf{\text{Sol}}\left(f,C\right)$. Then $\left\{y^{n_{k_j}}\right\}$ and $\left\{w^{n_{k_j}}\right\}$ converge weakly also to $\bar{x}$, as j → ∝. For each i = 1, ..., p, we suppose that ${\lambda }_{n_{k_j},i}$ converges ${\lambda }_i$ as i → ∝ such that

$$\sum _{i=1}^p{\lambda }_i=1.$$

Then, we have

$${\bar{S}}_{n_{k_j}}\left(x\right)\to S\left(x\right):=\sum _{i=1}^p{\lambda }_i{S}_i\left(x\right)\left(\mathsf{\text{as}}j\to \infty \right),\forall x\in C.$$

For each $x^{*}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)$, it follows from (3.20) that

$$\left(1-{\alpha }_n\right)\left({\alpha }_n-\bar{L}\right)\parallel {\bar{S}}_n\left(w^n\right)-w^n{\parallel }^2\le \parallel w^n-x^{*}\parallel -\parallel x^{n+1}-x^{*}{\parallel }^2.$$

Combining this and Step 6, we get

$$\begin{array}{ll}\hfill \parallel {\bar{S}}_n\left(w^n\right)-w^n{\parallel }^2& \le \frac{1}{\left(1-{\alpha }_n\right)\left({\alpha }_n-\bar{L}\right)}\left(\parallel w^n-x^{*}{\parallel }^2-\parallel x^{n+1}-x^{*}{\parallel }^2\right)\\ \le \frac{1}{\left(1-b\right)\left(a-\bar{L}\right)}\left(\parallel w^n-x^{*}{\parallel }^2-\parallel x^{n+1}-x^{*}{\parallel }^2\right)\\ \to 0\mathsf{\text{as }}n\to \infty .\end{array}$$

Then, using (a) of Proposition 1.1, we obtain

$$\begin{array}{ll}\hfill \parallel x^{n_{k_j}}-{\bar{S}}_{n_{k_j}}\left(x^{n_{k_j}}\right)\parallel & \le \parallel x^{n_{k_j}}-w^{n_{k_j}}\parallel +\parallel w^{n_{k_j}}-{\bar{S}}_{n_{k_j}}\left(w^{n_{k_j}}\right)\parallel +\parallel {\bar{S}}_{n_{k_j}}\left(w^{n_{k_j}}\right)-{\bar{S}}_{n_{k_j}}\left(x^{n_{k_j}}\right)\parallel \\ \le \parallel x^{n_{k_j}}-w^{n_{k_j}}\parallel +\parallel w^{n_{k_j}}-{\bar{S}}_{n_{k_j}}\left(w^{n_{k_j}}\right)\parallel +\frac{1+\bar{L}}{1-\bar{L}}\parallel w^{n_{k_j}}-x^{n_{k_j}}\parallel \\ =\frac{2}{1-\bar{L}}\parallel x^{n_{k_j}}-w^{n_{k_j}}\parallel +\parallel w^{n_{k_j}}-{\bar{S}}_{n_{k_j}}\left(w^{n_{k_j}}\right)\parallel \\ \to 0\mathsf{\text{as }}j\to \infty .\end{array}$$

So $\bar{x}\in \mathsf{\text{Fix}}\left(S\right)$. Then, it follows from (e) of Proposition 1.1 that $\bar{x}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)$. Thus $\bar{x}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)$ letting $x^{*}=\bar{x}$ and using Step 7, we have

$$c=\underset{n\to \infty }{\mathsf{\text{lim}}}\parallel x^n-\bar{x}\parallel =\underset{j\to \infty }{\mathsf{\text{lim}}}\parallel x^{n_{k_j}}-\bar{x}\parallel =0.$$

We conclude that the whole sequence {xⁿ } converges weakly to $\bar{x}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)$. Consequently, the sequences {yⁿ } and {wⁿ } also converge weakly to $\bar{x}$.

Step 9. We claim that the sequences {xⁿ }, {yⁿ } and {wⁿ } converge weakly to $\bar{x}$, where $\bar{x}=\underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{P}}{\mathsf{\text{r}}}_{{\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)}\left(x^n\right)$.

By Step 8, we suppose that $t^n:=\mathsf{\text{P}}{\mathsf{\text{r}}}_{{\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)}\left(x^n\right)$ and $x^n\to \bar{x}$ as n → ∝. Using the definition of Pr _C (·), we have

$$〈t^n-x^n,t^n-x〉\le 0,\forall x\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right).$$

(3.26)

It follows from Step 7 that

$$\parallel x^{n+1}-x^{*}\parallel \le \parallel x^n-x^{*}\parallel ,\forall n\ge 0,x^{*}\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right).$$

By Lemma 3.1, we have

$$t^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{{\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)}\left(x^n\right)\to x_1\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)\mathsf{\text{as }}n\to \infty .$$

(3.27)

Pass the limit in (3.26) and combining this with (3.27), we have

$$〈x_1-\bar{x},x_1-x〉\le 0,\forall x\in {\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right).$$

This means that $\bar{x}=x_1$ and

$$\bar{x}=\underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{P}}{\mathsf{\text{r}}}_{{\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)}\left(x^n\right).$$

It follows from Step 8 that the sequences {xⁿ }, {yⁿ } and {wⁿ } converge weakly to $\bar{x}$, where

$$\bar{x}=\underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{P}}{\mathsf{\text{r}}}_{{\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)}\left(x^n\right).$$

The proof is completed.

4 Application to variational inequalities

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

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

Let $f:C\times C\to \mathcal{R}$ be defined by f(x, y) = 〈F(x), y - x〉. Then problem P(f, C) can be written in V I(F, C). The set of solutions of V I(F, C) is denoted by Sol(F, C). Recall that the function F is called

• 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 (shortly, L-Lipschitz continuous) if

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

Since

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

Algorithm 2.1, the convergence algorithm for finding a common element of the set of common fixed points of p strict pseudocontractions and the set of solutions of equilibrium problems for pseudomonotone bifunctions is presented as follows:

Algorithm 4.1 Give a tolerance ε > 0. Choose x⁰ ∈ C, k = 0, γ ∈ (0, 1), $0<\sigma <\frac{\beta }{2}$and positive sequences {λ_n,i} and {α _n } satisfy the conditions:

$$\left\{\begin{array}{c}\left\{{\alpha }_n\right\}\subset \left[a,b\right]\subset \left(\bar{L},1\right)where\bar{L}:=\mathsf{\text{max}}\left\{L_i:1\le i\le p\right\},\hfill \\ \sum _{i=1}^p{\lambda }_{n,i}=1foralln\ge 1,\underset{n\to \infty }{\mathsf{\text{lim}}}{\lambda }_{n,i}={\lambda }_i\in \left(0,1\right)foralli=1,\dots ,p,\sum _{i=1}^p{\lambda }_i=1.\hfill \end{array}\right.$$

Step 1. Compute

$$y^k=\mathsf{\text{P}}{\mathsf{\text{r}}}_C\left(x^k-\frac{1}{\beta }F\left(x^k\right)\right)andsetr\left(x^n\right)=x^n-y^n.$$

If ||r(xⁿ )||≠ 0 then go to Step 2. Otherwise, set wⁿ = xⁿ and go to Step 3.

Step 2. (Armijo-type linesearch techniques) Find the smallest positive integer number m _n such that

$$\left(1-{\gamma }^{m_n}\right)〈F\left(x^n-{\gamma }^{m_n}r\left(x^n\right)\right),r\left(x^n\right)〉\ge \sigma \left|\right|r\left(x^n\right)|{|}^2.$$

(4.1)

Compute

$$w^n=\mathsf{\text{P}}{\mathsf{\text{r}}}_{C\cap H_n}\left(x^n\right),$$

where$z^n=x^n-{\gamma }^{m_n}r\left(x^n\right)$and$H_n=\left\{x\in \mathcal{H}:〈F\left(z^n\right),x-z^n〉\le 0\right\}$, and go to Step 3.

Step 3. Compute

$$x^{n+1}={\alpha }_n{w}^n+\left(1-{\alpha }_n\right)\sum _{i=1}^p{\lambda }_{n,i}{S}_i\left(w^n\right).$$

Increase n by 1 and go back to Step 1.

Using Theorem 3.2, we also have the convergence of Algorithm 4.1 as the follows:

Theorem 4.2 Let C be a nonempty closed convex subset of. Let $F:C\to \mathcal{H}$be continuous and pseudomonotone, and S _i : C → C be a L _i -Lipschitz pseudocontractions for all i = 1, ..., p such that${\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i\right)\cap \mathsf{\text{Sol}}\left(f,C\right)\ne \varnothing$. Then the sequences {xⁿ }, {yⁿ } and {wⁿ } generated by Algorithm 4.1 converge weakly to the point x*, where$x^{*}=\underset{n\to \infty }{\mathsf{\text{lim}}}\mathsf{\text{P}}{\mathsf{\text{r}}}_{{\cap }_{i=1}^p\mathsf{\text{Fix}}\left(S_i,C\right)\cap Sol\left(F,C\right)}\left(x^n\right)$.