Strong convergence by the shrinking effect of two half-spaces and applications

Abstract

This paper provides a new hybrid-type shrinking projection method for strong convergence results in a Hilbert space. The paper continues - by utilizing the proposed hybrid algorithm - with a strong convergence towards an approximate common element of the set of solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two finite families of k-strict pseudo-contractions in a Hilbert space. Comparatively, our results improve and extend various results announced in the current literature.

MSC:47H05, 47H09, 49H05.

1 Introduction

Let H be a real Hilbert space equipped with the inner product $〈\cdot ,\cdot 〉$ and the induced norm $\parallel \cdot \parallel$ and let C be a nonempty closed convex subset of H. A map $T:C\to C$ is said to be (i) Lipschitzian if $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel$ for some $L>0$ and for all $x,y\in C$; (ii) nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in C$; (iii) k-strict pseudo-contraction if there exists a constant $k\in [0,1)$ such that

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel (I-T)x-(I-T)y\parallel }^2\text{for all }x,y\in C.$$

Denote by $F(T)$ the set of all fixed points of T.

In 1967, Browder and Petryshyn [1] introduced the class of strict pseudo-contractions as an important generalization of the class of nonexpansive maps. Clearly, T is nonexpansive if and only if T is a 0-strict pseudo-contraction. Moreover, T is said to be pseudo-contraction if $k=1$ and a strong pseudo-contraction if there exists a positive constant $\lambda \in (0,1)$ such that $T-\lambda I$ is a pseudo-contraction. Therefore, the class of k-strict pseudo-contractions falls into the one between the classes of nonexpansive maps and pseudo-contractions. We further remark that the class of strong pseudo-contractions is independent of the class of k-strict pseudo-contractions. If T is a k-strict pseudo-contraction, then T satisfies the Lipschitz condition

$$\parallel Tx-Ty\parallel \le \frac{1+k}{1-k}\parallel x-y\parallel \text{for all }x,y\in C.$$

Recent developments in fixed point theory reflect that an iterative construction of fixed points of nonlinear maps is an active area of research. Note that the class of k-strict pseudo-contractions is prominent among the classes of nonlinear maps in the current literature. Although strict pseudo-contractions have more powerful applications than nonexpansive maps for solving inverse problems, iterative algorithms for strict pseudo-contractions are far less developed than those for nonexpansive maps.

In fixed point theory, various iterative schemas for computing approximate fixed points of nonlinear maps have been proposed and analyzed. Such iterative schemas can be compared w.r.t. their efficiency and convergence properties. In many situations, strong convergence of an iterative algorithm of a nonlinear map is much more desirable than weak convergence. Obviously, a trivial choice for approximation is the classical Picard algorithm (i.e., $x_{n+1}=T^n(x_0)$). On the other hand, if we take $X=\mathbb{R}$, $C:=[0,1]$, $T(x):=1-x$ and $x_0:=0$, then the Picard algorithm alternates between 0 and 1 and does not converge to the fixed point $\frac{1}{2}$. The classical Mann algorithm, which prevails the Picard algorithm, exhibits weak convergence even in the setting of a Hilbert space. Moreover, Chidume and Mutangadura [2] constructed an example for a Lipschitz pseudo-contraction with a unique fixed point for which the Mann algorithm fails to converge. These facts indicate that the iterative schemas should be modified for the desirable strong convergence properties.

The hybrid projection algorithm in mathematical programming was introduced by Haugazeau [3] in 1968. Later, many researchers studied the hybrid projection method and its various modifications for strong convergence results. In 2000, Solodov and Svaiter [4] established a strong convergence result for finding zeros of maximal monotone operators. They proposed and analyzed the following algorithm:

$$\begin{array}{r}{v}_k\in A{y}_k,v_k+{\mu }_k(y_k-x_k)={ϵ}_k;\\ \parallel {ϵ}_k\parallel \le \sigma max\{\parallel v_k\parallel ,{\mu }_k\parallel y_k-x_k\parallel \};\\ H_k=\{z\in H:〈z-y_k,v_k〉\le 0\};\\ W_k=\{z\in H:〈x_k-z,x_0-x_k〉\ge 0\};\\ x_{k+1}=P_{H_k\cap W_k}{x}_0.\end{array}$$

(1.1)

In fact, algorithm (1.1) enforces strong convergence by combining the classical proximal point algorithm with simple projection steps onto intersection of two half-spaces $H_k$ and $W_k$ containing the solution set. Inspired by the seminal work of Solodov and Svaiter [4], Nakajo and Takahashi [5] proposed the following hybrid method for nonexpansive maps in Hilbert spaces:

$$\begin{array}{r}{x}_0\in C\text{chosen arbitrarily},\\ y_k={\alpha }_k{x}_k+(1-{\alpha }_k)T{x}_k,\\ C_k=\{z\in C:\parallel y_k-z\parallel \le \parallel x_k-z\parallel \},\\ Q_k=\{z\in C:〈x_n-z,x-x_n〉\ge 0\},\\ x_{k+1}=P_{C_k\cap Q_k}{x}_0.\end{array}$$

(1.2)

They showed that algorithm (1.2) converges strongly to $P_{F(T)}x$ under some appropriate conditions. Moreover, algorithm (1.2) is also known as the CQ-method for the Mann algorithm because, at each step, the Mann algorithm is used to construct the sets $C_k$ and $Q_k$ which are in turn used to construct the next iterate of $x_{k+1}$ and hence the name.

In 2008, Takahashi et al. [6] introduced another type of the hybrid method which guarantees strong convergence by the shrinking effect of a sequence of closed convex sets $\{C_{k+1}\}$. More precisely, their algorithm reads as follows:

$$\begin{array}{r}{x}_0\in C\text{chosen arbitrarily where }{C}_0=C,\\ y_k={\alpha }_k{x}_k+(1-{\alpha }_k)T{x}_k,\\ C_{k+1}=\{z\in C_k:\parallel z-y_k\parallel \le \parallel z-x_k\parallel \},\\ x_{k+1}=P_{C_{k+1}}{x}_0,k\ge 0.\end{array}$$

(1.3)

Very recently, Dong et al. [7] proposed a shrinking projection method similar to (1.3) for nonexpansive maps in a Hilbert space setting. They, in fact, established a strong convergence result by the shrinking effect of one of the half-spaces as defined in (1.2), namely $Q_k$.

For $i=1,2,3,\dots ,N$ , let $A_i:C\to H$ be a finite family of nonlinear maps and $f_i:C\times C\to \mathbb{R}$ (the set of reals) be a finite family of bifunctions. A generalized equilibrium problem is to find the set

$$GEP(f_i)=\{x\in C:f_i(x,y)+〈A_{i}x,y-x〉\ge 0\}\text{for all }y\in C.$$

(1.4)

Note that if $A_i\equiv 0$ and $f_i(x,y)=f(x,y)$ for all $i\ge 1$, then the problem (1.4) reduces to the classical equilibrium problem $EP(f)$. That is, to find $x\in C$ such that $f(x,y)\ge 0$. Moreover, if $f_i(x,y)\equiv 0$ and $A_i=A$ for all $i\ge 1$, then the problem (1.4) reduces to the classical variational inequality problem $VI(C,A)$. That is, to find $x\in C$ such that $〈Ax,y-x〉\ge 0$.

Equilibrium problems provide a unified approach to address a variety of problems arising in various disciplines of science. The problem (1.4) is very general in the sense that it includes - as special cases - optimization problem, minimax problem, variational inequality problem, Nash equilibrium problem in noncooperative games and others; see, for instance, [8–10]. Combettes and Hirstoaga [9] introduced an iterative method to find an approximate solution of $EP(f)$. Since then, numerous algorithms have been analyzed to find a common element of the set of solutions of $EP(f)$ or $GEP(f)$ and the set of fixed points of a nonlinear map; see, for example, [10–16] and the references therein.

In 2009, Ceng et al. [17] introduced an implicit-type algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction in a real Hilbert space. Recently, Kim et al. [18] and Kangtunyakarn [19] approximated a common element of the set of solutions of two generalized equilibrium problems and the set of fixed points of a strict pseudo-contraction using the shrinking projection algorithm as defined in (1.3). Quite recently, Cholamjiak and Suantai [20] established a strong convergence result regarding a system of generalized equilibrium problems and a countable family of strict pseudo-contractions in a real Hilbert space.

Inspired and motivated by the work of Nakajo and Takahashi [5], Takahashi et al. [6], Dong et al. [7], Ceng et al. [17] and Cholamjiak and Suantai [20], we propose a hybrid method based on the shrinking effect of the two half-spaces, namely $C_n$ and $Q_n$, of the underlying Hilbert space H. The proposed algorithm approximates a common element of the set of solutions of a finite family of generalized equilibrium problems and the set of common fixed points of two finite families of k-strict pseudo-contractions. Our results refine and improve various results announced in the current literature.

2 Preliminaries

Throughout the paper, we write $x_n\to x$ (resp. $x_n\rightharpoonup x$) to indicate strong convergence (resp. weak convergence). Let H be a real Hilbert space, a map $P_C:H\to C$ defined by

$$\parallel x-P_{C}x\parallel \le \parallel x-y\parallel \text{for all }y\in C$$

is known as a metric projection or a nearest point projection of H onto C. Moreover, $P_C$ is characterized as nonexpansive in Hilbert spaces. We know that for $x\in H$ and $z\in C$, $z=P_{C}x$ is equivalent to $〈x-z,z-u〉\ge 0$ for all $u\in C$.

Let $A:C\to H$ be a δ-inverse-strongly monotone map, if there exists $\delta >0$ such that

$$〈x-y,Ax-Ay〉\ge \delta {\parallel Ax-Ay\parallel }^2\text{for all }x,y\in C.$$

If A is a δ-inverse-strongly monotone map of C onto H, then A is $(\frac{1}{\delta })$-Lipschitz continuous. Moreover, for $x,y\in C$ and $r>0$, we have

$$\begin{array}{rcl}{\parallel (I-rA)x-(I-rA)y\parallel }^2& =& {\parallel x-y-r(Ax-Ay)\parallel }^2\\ =& {\parallel x-y\parallel }^2-2r〈x-y,Ax-Ay〉+r^2{\parallel Ax-Ay\parallel }^2\\ \le & {\parallel x-y\parallel }^2+r(r-2\delta ){\parallel Ax-Ay\parallel }^2.\end{array}$$

(2.1)

If $r\le 2\delta$, then $I-rA$ is a nonexpansive map from C onto H.

The following crucial results for a k-strict pseudo-contraction can be found in [[21], Proposition 2.1].

Lemma 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H. If $T:C\to H$ is a k-strict pseudo-contraction, then the fixed point set $F(T)$ is closed and convex so that the projection $P_{F(T)}$ is well defined.

Lemma 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H and $T:C\to C$ be a k-strict pseudo-contraction. Then $(I-T)$ is demiclosed, that is, if $\{x_n\}$ is a sequence in C with $x_n\rightharpoonup x$ and $x_n-T{x}_n\to 0$, then $x\in F(T)$.

The following lemma is well known in the context of a real Hilbert space.

Lemma 2.3 Let H be a real Hilbert space, then the following identity holds:

$${\parallel \alpha x+(1-\alpha )y\parallel }^2=\alpha {\parallel x\parallel }^2+(1-\alpha ){\parallel y\parallel }^2-\alpha (1-\alpha ){\parallel x-y\parallel }^2.$$

For solving the equilibrium problem, let us assume that the bifunction f satisfies the following conditions (cf. [8]and [9]):

• (A1) $f(x,x)=0$ for all $x\in C$;

• (A2) f is monotone, i.e., $f(x,y)+f(y,x)\le 0$ for all $x,y\in C$;

• (A3) for all $x,y,z\in C$, ${lim\hspace{0.17em}sup}_{t\downarrow 0}f(tz+(1-t)x,y)\le f(x,y)$;

• (A4) $f(x,\cdot )$ is convex and lower semicontinuous for all $x\in C$.

A central result in the theory of equilibrium problems - for an approximate solution of $EP(f)$ - is the following result due to [8].

Lemma 2.4 Let C be a closed convex subset of a real Hilbert space H, and let $f:C\times C\to \mathbb{R}$ be a bifunction satisfying (A1)-(A4). For $r>0$ and $x\in H$, there exists $z\in C$ such that

$$f(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0\mathit{\text{for all}}y\in C.$$

Moreover, define a map $V_r:H\to C$ by

$$V_r(x)=\{z\in C:f(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0\mathit{\text{for all}}y\in C\}$$

for all $x\in H$. Then the following hold:

• (1) $EP(f)$ is closed and convex;

• (2) $V_r$ is single-valued;

• (3) $V_r$ is a firmly nonexpansive-type map, i.e.,

  $${\parallel V_{r}x-V_{r}y\parallel }^2\le 〈V_{r}x-V_{r}y,x-y〉\mathit{\text{for all}}x,y\in H;$$

• (4) $F(V_r)=EP(f)$.

3 Main result

Let H be a real Hilbert space and C be its nonempty closed convex subset. Let $T_i(modN),S_i(modN):C\to C$ be two finite families of k-strict pseudo-contractions such that $k=max\{k_i:1\le i\le N\}$. Let $f_i(modN):C\times C\to \mathbb{R}$ be a finite family of bifunctions and $A_i(modN):C\to H$ be a finite family of δ-inverse-strongly monotone maps such that $\delta =max\{{\delta }_i:1\le i\le N\}$.

Throughout this section, we assume that the modN function takes values in the indexing set $I=\{1,2,3,\dots ,N\}$ and the set of common fixed points of two finite families of k-strict pseudo-contractions ${\{T_i\}}_{i=1}^N$ and ${\{S_i\}}_{i=1}^N$ is nonempty, that is, $F=({\bigcap }_{i=1}^{N}F(T_i))\cap ({\bigcap }_{i=1}^{N}F(S_i))\ne \mathrm{\varnothing }$.

Algorithm

Our hybrid algorithm reads as follows:

(3.1)

It is worth mentioning that at each iteration, algorithm (3.1) computes a projection onto intersection of two half-spaces. Since these half-spaces are closed and convex, so one can follow Dykstra’s algorithm [22] for the computation of such a projection.

The main result of this section is as follows.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let $T_i(modN),S_i(modN):C\to C$ be two finite families of k-strict pseudo-contractions. Let $f_i(modN):C\times C\to \mathbb{R}$ be a finite family of bifunctions satisfying (A1)-(A4) and let $A_i(modN):C\to H$ be a finite family of δ-inverse-strongly monotone maps. Let $\{r_{n,i}\}\subset (0,\mathrm{\infty })$ and $\{{\alpha }_{n,i}\}$, $\{{\beta }_{n,i}\}$ be two control sequences such that

• (C1) $0\le k<a\le {\alpha }_{n,i},{\beta }_{n,i}\le b<1$;

• (C2) $0<r<r_{n,i}<s<2\delta$ for all $i\ge 1$.

Assume that $\mathcal{F}:=[{\bigcap }_{i=1}^{N}F(T_i)]\cap [{\bigcap }_{i=1}^{N}F(S_i)]\cap [{\bigcap }_{i=1}^{N}GEP(f_i,A_i)]\ne \mathrm{\varnothing }$, then the sequence $\{x_n\}$ generated by (3.1) converges strongly to $x=P_{\mathcal{F}}{x}_1$, where $P_{\mathcal{F}}$ is the metric projection of H onto ℱ.

Proof It follows from Lemma 2.4 that $u_{n,i}$ can be written as $u_{n,i}=V_{r_{n,i}}(x_n-r_{n,i}{A}_i{x}_n)$ for all $n\ge 1$. Moreover, it follows from the nonexpansiveness of $I-r_{n,i}{A}_i$ that

$$\parallel u_{n,i}-p\parallel \le \parallel x_n-p\parallel ,$$

where $p={\bigcap }_{i=1}^N{T}_{i}p={\bigcap }_{i=1}^N{S}_{i}p={\bigcap }_{i=1}^N{V}_{r_{n,i}}(p-r_{n,i}{A}_{i}p)$.

We proceed to show that algorithm (3.1) is well defined. It is obvious from the definitions of respective sets that $C_{n+1}$ is closed and $Q_{n+1}$ is closed and convex. Now, we show that $C_{n+1}$ is convex. Since $C_1=C$ is convex, we assume that $C_k$ is convex for some $k\ge 2$. For any $z\in C_k$, the inequality $\parallel y_{n,i}-z\parallel \le \parallel x_n-z\parallel$ is equivalent to

$${\parallel y_{n,i}\parallel }^2-{\parallel x_n\parallel }^2-2〈z,y_{n,i}-x_n,〉\ge 0.$$

Hence, $C_{k+1}$ is convex, and consequently $C_{n+1}$ is convex for each $n\ge 1$.

Next, we show by induction that $\mathcal{F}\subset C_{n+1}\cap Q_{n+1}$ for all $n\ge 1$. Obviously, $\mathcal{F}\subset C_1\cap Q_1=C$. Let $p\in \mathcal{F}$. From (2.1), (3.1) and Lemma 2.3, we have the following estimate:

$$\begin{array}{rcl}{\parallel z_{n,i}-p\parallel }^2& =& {\parallel {\alpha }_{n,i}(u_{n,i}-p)+(1-{\alpha }_{n,i})(T_i{u}_{n,i}-p)\parallel }^2\\ =& {\alpha }_{n,i}{\parallel u_{n,i}-p\parallel }^2+(1-{\alpha }_{n,i}){\parallel T_i{u}_{n,i}-p\parallel }^2\\ -{\alpha }_{n,i}(1-{\alpha }_{n,i}){\parallel u_{n,i}-T_i{u}_{n,i}\parallel }^2\\ \le & {\alpha }_{n,i}{\parallel u_{n,i}-p\parallel }^2+(1-{\alpha }_{n,i})({\parallel u_{n,i}-p\parallel }^2+k{\parallel u_{n,i}-T_i{u}_{n,i}\parallel }^2)\\ -{\alpha }_{n,i}(1-{\alpha }_{n,i}){\parallel u_{n,i}-T_i{u}_{n,i}\parallel }^2\\ =& {\parallel u_{n,i}-p\parallel }^2-(1-{\alpha }_{n,i})({\alpha }_{n,i}-k){\parallel u_{n,i}-T_i{u}_{n,i}\parallel }^2.\end{array}$$

(3.2)

Since ${\alpha }_{n,i}-k>0$ (by (C1)), therefore the above estimate (3.2) yields

$${\parallel z_{n,i}-p\parallel }^2\le {\parallel u_{n,i}-p\parallel }^2\le {\parallel x_n-p\parallel }^2.$$

(3.3)

Moreover,

$$\begin{array}{rcl}{\parallel y_{n,i}-p\parallel }^2& =& {\parallel {\beta }_{n,i}(z_{n,i}-p)+(1-{\beta }_{n,i})(S_i{z}_{n,i}-p)\parallel }^2\\ =& {\beta }_{n,i}{\parallel z_{n,i}-p\parallel }^2+(1-{\beta }_{n,i}){\parallel S_i{z}_{n,i}-p\parallel }^2\\ -{\beta }_{n,i}(1-{\beta }_{n,i}){\parallel z_{n,i}-S_i{z}_{n,i}\parallel }^2\\ \le & {\beta }_{n,i}{\parallel z_{n,i}-p\parallel }^2+(1-{\beta }_{n,i})({\parallel z_{n,i}-p\parallel }^2+k{\parallel z_{n,i}-S_i{z}_{n,i}\parallel }^2)\\ -{\beta }_{n,i}(1-{\beta }_{n,i}){\parallel z_{n,i}-S_i{z}_{n,i}\parallel }^2\\ \le & {\parallel z_{n,i}-p\parallel }^2-(1-{\beta }_{n,i})({\beta }_{n,i}-k){\parallel z_{n,i}-S_i{z}_{n,i}\parallel }^2.\end{array}$$

(3.4)

Reasoning as above and utilizing (3.3), the estimate (3.4) implies that

$$\parallel y_{n,i}-p\parallel \le \parallel x_n-p\parallel .$$

(3.5)

This implies that $p\in C_{n+1}$ for all $n\ge 1$. It suffices to show that $p\in Q_{n+1}$ for all $n\ge 1$. We prove this by induction. Note that $\mathcal{F}\subset Q_1=C$ is obvious. Assume that $\mathcal{F}\subset Q_k$ also $\mathcal{F}\subset C_k\cap Q_k$ for some $k\ge 2$. This implies $x_k$ is a projection of $x_1$ onto $C_k\cap Q_k$, and consequently we have

$$〈x_k-z,x_1-x_k〉\ge 0\text{for all }z\in C_k\cap Q_k.$$

Since $\mathcal{F}\subset C_k\cap Q_k$, we have

$$〈x_k-p,x_1-x_k〉\ge 0\text{for all }p\in \mathcal{F}.$$

Hence, $p\in Q_{k+1}$, and consequently $\mathcal{F}\subset C_{n+1}\cap Q_{n+1}$ for all $n\ge 1$. Since ℱ is now closed and convex, so it follows from Lemma 2.1 that $P_{\mathcal{F}}$ is well defined. Note that $x_{n+1}=P_{C_{n+1}\cap Q_{n+1}}{x}_1$, therefore $\parallel x_{n+1}-x_1\parallel \le \parallel p-x_1\parallel$ for all $p\in \mathcal{F}\subset C_{n+1}\cap Q_{n+1}$. In particular, we have $\parallel x_{n+1}-x_1\parallel \le \parallel P_{\mathcal{F}}{x}_1-x_1\parallel$. This implies that $\{x_n\}$ is bounded. On the other hand, $x_n=P_{C_n\cap Q_n}{x}_1$ and $x_{n+1}\in C_{n+1}\cap Q_{n+1}\subset Q_{n+1}$, we have

$$\parallel x_n-x_1\parallel \le \parallel x_{n+1}-x_1\parallel .$$

That is, the sequence $\{\parallel x_n-x_1\parallel \}$ is nondecreasing. This implies ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_1\parallel$ exists. Note that

$$\begin{array}{rcl}{\parallel x_{n+1}-x_n\parallel }^2& =& {\parallel x_{n+1}-x_1+x_1-x_n\parallel }^2\\ =& {\parallel x_{n+1}-x_1\parallel }^2+{\parallel x_n-x_1\parallel }^2-2〈x_n-x_1,x_{n+1}-x_1〉\\ =& {\parallel x_{n+1}-x_1\parallel }^2+{\parallel x_n-x_1\parallel }^2-2〈x_n-x_1,x_{n+1}-x_n+x_n-x_1〉\\ =& {\parallel x_{n+1}-x_1\parallel }^2-{\parallel x_n-x_1\parallel }^2-2〈x_n-x_1,x_{n+1}-x_n〉\\ \le & {\parallel x_{n+1}-x_1\parallel }^2-{\parallel x_n-x_1\parallel }^2.\end{array}$$

Taking lim sup on both sides of the above estimate, we have ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\parallel x_{n+1}-x_n\parallel }^2=0$. That is,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0.$$

(3.6)

Since $x_{n+1}\in C_{n+1}$, we have $\parallel y_{n,i}-x_{n+1}\parallel \le \parallel x_n-x_{n+1}\parallel$. This implies

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_{n,i}-x_{n+1}\parallel =0.$$

(3.7)

Moreover, it follows from (3.6), (3.7) and the following inequality:

$$\parallel y_{n,i}-x_n\parallel \le \parallel y_{n,i}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel$$

that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_{n,i}-x_n\parallel =0.$$

(3.8)

Note that

$$\begin{array}{rcl}{\parallel u_{n,i}-p\parallel }^2& =& {\parallel V_{r_{n,i}}(I-r_{n,i}{A}_i)x_n-V_{r_{n,i}}(I-r_{n,i}{A}_i)p\parallel }^2\\ =& {\parallel (I-r_{n,i}{A}_i)x_n-(I-r_{n,i}{A}_i)p\parallel }^2\\ =& {\parallel x_n-p-r_{n,i}(A_i{x}_n-A_{i}p)\parallel }^2\\ =& {\parallel x_n-p\parallel }^2-2r_{n,i}〈A_i{x}_n-A_{i}p,x_n-p〉+r_{n,i}^2{\parallel A_i{x}_n-A_{i}p\parallel }^2\\ \le & {\parallel x_n-p\parallel }^2+r_{n,i}(r_{n,i}-2\delta ){\parallel A_i{x}_n-A_{i}p\parallel }^2.\end{array}$$

(3.9)

Since ${\parallel y_{n,i}-p\parallel }^2\le {\parallel z_{n,i}-p\parallel }^2\le {\parallel u_{n,i}-p\parallel }^2$, therefore utilizing (3.9), we get

$${\parallel y_{n,i}-p\parallel }^2\le {\parallel x_n-p\parallel }^2-r_{n,i}(2\delta -r_{n,i}){\parallel A_i{x}_n-A_{i}p\parallel }^2.$$

Re-arranging the terms in the above estimate and utilizing (C2), we have

$$r(2\delta -s){\parallel A_i{x}_n-A_{i}p\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel y_{n,i}-p\parallel }^2\le (\parallel x_n-p\parallel +\parallel y_{n,i}-p\parallel )\parallel x_n-y_{n,i}\parallel .$$

Hence, (3.8) implies that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel A_i{x}_n-A_{i}p\parallel =0\text{for all }i\ge 1.$$

(3.10)

Since $k<a\le {\beta }_{n,i}\le b<1$, then the following variant of the estimate (3.4) implies that

$$\begin{array}{rcl}(1-b)(a-k){\parallel z_{n,i}-S_i{z}_{n,i}\parallel }^2& \le & {\parallel x_n-p\parallel }^2-{\parallel y_{n,i}-p\parallel }^2\\ \le & (\parallel x_n-p\parallel +\parallel y_{n,i}-p\parallel )\parallel x_n-y_{n,i}\parallel .\end{array}$$

Again, utilizing (3.8), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel z_{n,i}-S_i{z}_{n,i}\parallel =0\text{for all }i\ge 1.$$

(3.11)

Observe that $\parallel y_{n,i}-z_{n,i}\parallel =(1-{\beta }_{n,i})\parallel S_i{z}_{n,i}-z_{n,i}\parallel$. It follows from the fact that $k<a\le {\beta }_{n,i}\le b<1$ and (3.11) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_{n,i}-z_{n,i}\parallel =0\text{for all }i\ge 1.$$

(3.12)

Moreover, $\parallel x_n-z_{n,i}\parallel \le \parallel x_n-y_{n,i}\parallel +\parallel y_{n,i}-z_{n,i}\parallel$. Letting $n\to \mathrm{\infty }$ on both sides and utilizing (3.8) and (3.12), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-z_{n,i}\parallel =0\text{for all }i\ge 1.$$

(3.13)

Reasoning as above, that is, $k<a\le {\alpha }_{n,i}\le b<1$, we consider the following variant of the estimate (3.2):

$$\begin{array}{rcl}(1-b)(a-k){\parallel u_{n,i}-T_i{u}_{n,i}\parallel }^2& \le & {\parallel x_n-p\parallel }^2-{\parallel z_{n,i}-p\parallel }^2\\ \le & (\parallel x_n-p\parallel +\parallel z_{n,i}-p\parallel )\parallel x_n-z_{n,i}\parallel .\end{array}$$

Hence, we conclude from the above estimate and (3.13) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_{n,i}-T_i{u}_{n,i}\parallel =0\text{for all }i\ge 1.$$

(3.14)

On the other hand, $\parallel z_{n,i}-u_{n,i}\parallel =(1-{\alpha }_{n,i})\parallel T_i{u}_{n,i}-u_{n,i}\parallel$. Making use of the fact that $k<a\le {\alpha }_{n,i}\le b<1$ and (3.14), we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel z_{n,i}-u_{n,i}\parallel =0\text{for all }i\ge 1.$$

(3.15)

Moreover, we conclude from the estimates (3.12) and (3.15) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_{n,i}-u_{n,i}\parallel =0\text{for all }i\ge 1.$$

(3.16)

As a direct consequence of the estimates (3.8) and (3.16), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-u_{n,i}\parallel =0\text{for all }i\ge 1.$$

(3.17)

Next, we show that $\omega (x_n)\subset \mathcal{F}$, where $\omega (x_n)$ is the set of all weak ω-limit of $\{x_n\}$. Since $\{x_n\}$ is bounded, therefore $\omega (x_n)\ne \mathrm{\varnothing }$. Let $q\in \omega (x_n)$, there exists a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ such that $x_{n_j}\rightharpoonup q$. It follows from the estimate (3.17) that $u_{n_j,i}\rightharpoonup q$. We first show that $q\in GEP(f_1,A_1)$, where $f_1=f_{n_j}$ for some $j\ge 1$. From $u_{n_j,i}=V_{r_{n_j,i}}(I-r_{n_j,i}{A}_i)x_n$, for all $n\ge 1$, we have

$$f_1(u_{n_j,i},y)+〈A_1{x}_{n_j},y-u_{n_j,i}〉+\frac{1}{r_{n_j,i}}〈y-u_{n_j,i},u_{n_j,i}-x_{n_j}〉\ge 0\text{for all }y\in C.$$

From (A2), we have

$$〈A_1{x}_{n_j},y-u_{n_j,i}〉+\frac{1}{r_{n_j,i}}〈y-u_{n_j,i},u_{n_j,i}-x_{n_j}〉\ge f_1(y,u_{n_j,i})\text{for all }y\in C.$$

(3.18)

Let $y_t=ty+(1-t)q$ for $0<t<1$ and $y\in C$. Since $q\in C$, this implies that $y_t\in C$. It follows from the estimate (3.18) that

$$\begin{array}{rcl}〈y_t-u_{n_j,i},A_1{y}_t〉& \ge & 〈y_t-u_{n_j,i},A_1{y}_t〉-〈A_1{x}_{n_j},y_t-u_{n_j,i}〉\\ -〈y_t-u_{n_j,i},\frac{u_{n_j,i}-x_{n_j}}{r_{n_j,i}}〉+f_1(y_t,u_{n_j,i})\\ =& 〈y_t-u_{n_j,i},A_1{y}_t-A_1{u}_{n_j,i}〉+〈y_t-u_{n_j,i},A_1{u}_{n_j,i}-A_1{x}_{n_j}〉\\ -〈y_t-u_{n_j,i},\frac{u_{n_j,i}-x_{n_j}}{r_{n_j,i}}〉+f_1(y_t,u_{n_j,i}).\end{array}$$

(3.19)

Since ${lim}_{n\to \mathrm{\infty }}\parallel x_{n_j}-u_{n_j,i}\parallel =0$, therefore ${lim}_{n\to \mathrm{\infty }}\parallel A_1{x}_{n_j}-A_1{u}_{n_j,i}\parallel =0$. Moreover, it follows from the monotonicity of A that $〈y_t-u_{n_j,i},A_1{y}_t-A_1{u}_{n_j,i}〉\ge 0$. Hence, (A4) implies that

$$〈y_t-q,A_1{y}_t〉\ge f_1(y_t,q).$$

(3.20)

Using (3.20), (A1) and (A4), the following estimate:

$$\begin{array}{rcl}0& =& f_1(y_t,y_t)\le t{f}_1(y_t,y)+(1-t)f_1(y_t,q)\\ \le & t{f}_1(y_t,y)+(1-t)〈y_t-q,A_1{y}_t〉\\ =& t{f}_1(y_t,y)+(1-t)t〈y-q,A_1{y}_t〉,\end{array}$$

implies that

$$f_1(y_t,y)+(1-t)〈y-q,A_1{y}_t〉\ge 0.$$

(3.21)

Letting $t\to 0$, we have $f_1(q,y)+〈y-q,A_{1}q〉\ge 0$ for all $y\in C$. Thus, $q\in GEP(f_1,A_1)$. In a similar fashion, we have some $k\ge 1$ such that $f_2=f_{n_k}$ and $q\in GEP(f_2,A_2)$. Therefore, $q\in {\bigcap }_{i=1}^{N}GEP(f_i,A_i)$. Since $u_{n_j,i}\rightharpoonup q$, so it follows from (3.14) and Lemma 2.2 that $q\in {\bigcap }_{i=1}^{N}F(T_i)$. Reasoning as above, we can show that $q\in {\bigcap }_{i=1}^{N}F(S_i)$. Hence, $q\in \mathcal{F}$. Let $x=P_{\mathcal{F}}{x}_1$, which implies that $x=P_{\mathcal{F}}{x}_1\in C_{n+1}$. Since $x_{n+1}=P_{C_{n+1}\cap Q_{n+1}}{x}_1\in C_{n+1}$, we have

$$\parallel x_{n+1}-x_1\parallel \le \parallel x-x_1\parallel .$$

On the other hand, we have

$$\begin{array}{rcl}\parallel x-x_1\parallel & \le & \parallel q-x_1\parallel \\ \le & \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-x_1\parallel \\ \le & \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_{n_j}-x_1\parallel \\ \le & \parallel x-x_1\parallel .\end{array}$$

That is,

$$\parallel q-x_1\parallel =\underset{j\to \mathrm{\infty }}{lim}\parallel x_{n_j}-x_1\parallel =\parallel x-x_1\parallel .$$

Therefore, we conclude that ${lim}_{j\to \mathrm{\infty }}{x}_{n_j}=q=P_{\mathcal{F}}{x}_1$. From the arbitrariness of $\{x_{n_j}\}$, we get that ${lim}_{n\to \mathrm{\infty }}{x}_n=P_{\mathcal{F}}{x}_1$. This completes the proof. □

In particular, if $T_i$ and $S_i$ - in algorithm (3.1) - are two finite families of nonexpansive maps, then the following result holds.

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let $T_i(modN),S_i(modN):C\to C$ be two finite families of nonexpansive maps. Let $f_i(modN):C\times C\to \mathbb{R}$ be a finite family of bifunctions satisfying (A1)-(A4) and let $A_i(modN):C\to H$ be a finite family of δ-inverse-strongly monotone maps. Let $\{r_{n,i}\}\subset (0,\mathrm{\infty })$ and $\{{\alpha }_{n,i}\}$, $\{{\beta }_{n,i}\}$ be two control sequences such that

• (C1) $0\le k<a\le {\alpha }_{n,i},{\beta }_{n,i}\le b<1$;

• (C2) $0<r<r_{n,i}<s<2\delta$ for all $i\ge 1$.

Assume that $\mathcal{F}:=[{\bigcap }_{i=1}^{N}F(T_i)]\cap [{\bigcap }_{i=1}^{N}F(S_i)]\cap [{\bigcap }_{i=1}^{N}GEP(f_i,A_i)]\ne \mathrm{\varnothing }$, then the sequence $\{x_n\}$ generated by (3.1) converges strongly to $x=P_{\mathcal{F}}{x}_1$, where $P_{\mathcal{F}}$ is the metric projection of H onto ℱ.

In order to address variational inequality problems coupled with the fixed point problems, we prove the following result with a slight modification of algorithm (3.1).

Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let $T_i(modN),S_i(modN):C\to C$ be two finite families of k-strict pseudo-contractions. Let $f_i(modN):C\times C\to \mathbb{R}$ be a finite family of bifunctions satisfying (A1)-(A4) and let $A_i(modN):C\to H$ be a finite family of δ-inverse-strongly monotone maps. Let $\{r_{n,i}\}\subset (0,\mathrm{\infty })$ and $\{{\alpha }_{n,i}\}$, $\{{\beta }_{n,i}\}$ be two control sequences such that

• (C1) $0\le k<a\le {\alpha }_{n,i},{\beta }_{n,i}\le b<1$;

• (C2) $0<r<r_{n,i}<s<2\delta$ for all $i\ge 1$.

Assume that $\mathcal{F}:=[{\bigcap }_{i=1}^{N}F(T_i)]\cap [{\bigcap }_{i=1}^{N}F(S_i)]\cap [{\bigcap }_{i=1}^{N}VI(C,A_i)]\ne \mathrm{\varnothing }$, then the sequence $\{x_n\}$ generated by

(3.22)

converges strongly to $x=P_{\mathcal{F}}{x}_1$, where $P_{\mathcal{F}}$ is the metric projection of H onto ℱ.

Proof Set $f_i(x,y)\equiv 0$ for each $i\ge 1$, then

$$〈A_i{x}_n,y-u_{n,i}〉+\frac{1}{r_{n,i}}〈y-u_{n,i},u_{n,i}-x_n〉\ge 0$$

is equivalent to

$$〈x_n-r_{n,i}{A}_i{x}_n-u_{n,i},u_{n,i}-y〉\ge 0.$$

This implies that $h_{n,i}=u_{n,i}:=P_C(x_n-r_{n,i}{A}_i{x}_n)$. The desired result then follows from Theorem 3.1 immediately. □

As an application of Theorem 3.1 - by substituting $A_i\equiv 0$ for all $i\ge 1$ in algorithm (3.1) - we have the following result for a finite family of equilibrium problems.

Theorem 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let $T_i(modN),S_i(modN):C\to C$ be two finite families of k-strict pseudo-contractions. Let $f_i(modN):C\times C\to \mathbb{R}$ be a finite family of bifunctions satisfying (A1)-(A4). Let $\{{\alpha }_{n,i}\}$, $\{{\beta }_{n,i}\}$ be two control sequences such that

(C1) $0\le k<a\le {\alpha }_{n,i},{\beta }_{n,i}\le b<1$.

Assume that $\mathcal{F}:=[{\bigcap }_{i=1}^{N}F(T_i)]\cap [{\bigcap }_{i=1}^{N}F(S_i)]\cap [{\bigcap }_{i=1}^{N}EP(f_i)]\ne \mathrm{\varnothing }$, then the sequence $\{x_n\}$ generated by

(3.23)

converges strongly to $x=P_{\mathcal{F}}{x}_1$, where $P_{\mathcal{F}}$ is the metric projection of H onto ℱ.

Remark 3.5 Additionally - in Theorem 3.4 - if we set ${\beta }_{n,i}\equiv 0$, then Theorem 3.4 sets analogue [[17], Theorem 3.3] in the following aspects:

• (i) from a single k-strict pseudo-contraction to a finite family of maps;

• (ii) from an equilibrium problem to a finite family of generalized equilibrium problems.