Hybrid shrinking iterative solutions to convex feasibility problems for countable families of relatively nonexpansive mappings and a system of generalized mixed equilibrium problems

Abstract

We propose a new hybrid shrinking iterative scheme for approximating common elements of the set of solutions to convex feasibility problems for countable families of relatively nonexpansive mappings of a set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of other authors, in which the involved mappings consist of just finitely many ones.

MSC:47H09, 47H10, 47J25.

1 Introduction

Throughout this paper we assume that E is a real Banach space with its dual $E^{\ast }$, C is a nonempty, closed, convex subset of E, and $J:E\to 2^{E^{\ast }}$ is the normalized duality mapping defined by

$$Jx=\{f\in E^{\ast }:〈x,f〉={\parallel x\parallel }^2={\parallel f\parallel }^2\},\mathrm{\forall }x\in E.$$

(1.1)

In the sequel, we use $F(T)$ to denote the set of fixed points of a mapping T. A point p in C is said to be an asymptotic fixed point of T if C contains a sequence $\{x_n\}$ which converges weakly to p such that the ${lim}_{n\to \mathrm{\infty }}(x_n-T{x}_n)=0$. The set of asymptotic fixed points of T will be denoted by $\hat{F}(T)$. A mapping $T:C\to C$ is said to be nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

(1.2)

A mapping $T:C\to C$ is said to be relatively nonexpansive if $F(T)=\hat{F}(T)\ne \mathrm{\varnothing }$ and

$$\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,p\in F(T),$$

(1.3)

where $\varphi :E\times E\to {\mathbb{R}}^1$ denotes the Lyapunov functional defined by

$$\varphi (x,y)={\parallel x\parallel }^2-2〈x,Jy〉+{\parallel y\parallel }^2,\mathrm{\forall }x,y\in E.$$

(1.4)

It is obvious from the definition of ϕ that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2,$$

(1.5)

$$\varphi (x,y)=\varphi (x,z)+\varphi (z,y)+2〈x-z,Jz-Jy〉,$$

(1.6)

and

$$\varphi (x,y)=〈x,Jx-Jy〉+〈y-x,Jy〉\le \parallel x\parallel \parallel Jx-Jy\parallel +\parallel y-x\parallel \parallel y\parallel .$$

(1.7)

The asymptotic behavior of a relatively nonexpansive mapping was studied in [1–4]. In 1953, Mann [5] introduced the iteration as follows: a sequence $\{x_n\}$ is defined by

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,$$

(1.8)

where the initial element $x_0\in C$ is arbitrary and $\{{\alpha }_n\}$ is a sequence of real numbers in $[0,1]$. The Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [6]. In an infinite-dimensional Hilbert space, a Mann iteration can yield only weak convergence (see [7, 8]). Attempts to modify the Mann iteration method (1.8) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [9] proposed the following modification of Mann iteration method (1.8) for a nonexpansive mapping T from C into itself in a Hilbert space: from an arbitrary $x_0\in C$,

$$\{\begin{array}{l}{y}_n={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,\\ C_n=\{z\in C:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\\ Q_n=\{z\in C:〈x_n-z,x_0-x_n〉\ge 0\},\\ x_{n+1}=P_{C_n\cap Q_n}{x}_0,\mathrm{\forall }n\in \mathbb{N}\cup \{0\},\end{array}$$

(1.9)

where $P_K$ denotes the metric projection from a Hilbert space H onto a closed convex subset K of H and proved that the sequence $\{x_n\}$ converges strongly to $P_{F(T)}{x}_0$. A projection onto the intersection of two half-spaces is computed by solving a linear system of two equations with two unknowns (see [[10], Section 3]).

Let $\theta :C\times C\to {\mathbb{R}}^1$ be a bifunction, $\psi :C\to {\mathbb{R}}^1$ a real-valued function, and $B:C\to E^{\ast }$ a nonlinear mapping. The so-called generalized mixed equilibrium problem $(GMEP)$ is to find an $u\in C$ such that

$$\theta (u,y)+〈y-u,Bu〉+\psi (y)-\psi (u)\ge 0,\mathrm{\forall }y\in C,$$

(1.10)

whose set of solutions is denoted by $\mathrm{\Omega }(\theta ,B,\psi )$.

The equilibrium problem is a unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Many authors have proposed some useful methods to solve the EP (equilibrium problem), GEP (generalized equilibrium problem), MEP (mixed equilibrium problem), and GMEP.

In 2007, Plubtieng and Ungchittrakool [11] established strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the following hybrid method in mathematical programming:

$$\{\begin{array}{l}{x}_0=x\in C,\\ y_n=J^{-1}[{\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{z}_n],\\ z_n=J^{-1}[{\beta }_n^{(1)}J{x}_n+{\beta }_n^{(2)}JT{x}_n+{\beta }_n^{(3)}JS{x}_n],\\ H_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)\},\\ W_n=\{z\in C:〈x_n-z,Jx-Jy〉\ge 0\},\\ x_{n+1}=P_{H_n\cap W_n}x,\mathrm{\forall }n\in \mathbb{N}\cup \{0\}.\end{array}$$

(1.11)

Their results extended and improved the corresponding ones announced by Nakajo and Takahashi [9], Martinez-Yanes and Xu [12], and Matsushita and Takahashi [4].

Recently, Su and Qin [13] modified iteration (1.9), the so-called monotone CQ method for nonexpansive mapping, as follows: from an arbitrary $x_0\in C$,

$$\{\begin{array}{l}{y}_n={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,\\ C_0=\{z\in C:\parallel y_0-z\parallel \le \parallel x_0-z\parallel \},Q_0=C,\\ C_n=\{z\in C_{n-1}\cap Q_{n-1}:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\\ Q_n=\{z\in C_{n-1}\cap Q_{n-1}:〈x_n-z,x_0-x_n〉\ge 0\},\\ x_{n+1}=P_{C_n\cap Q_n}{x}_0,\mathrm{\forall }n\in \mathbb{N}\cup \{0\},\end{array}$$

(1.12)

and proved that the sequence $\{x_n\}$ converges strongly to $P_{F(T)}{x}_0$.

Inspired and motivated by the studies mentioned above, in this paper, we use a modified hybrid iteration scheme for approximating common elements of the set of solutions to convex feasibility problem for a countable families of relatively nonexpansive mappings, of set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of the authors, in which the involved mappings consist of just finitely many ones.

2 Preliminaries

We say that E is strictly convex if the following implication holds for $x,y\in E$:

$$\parallel x\parallel =\parallel y\parallel =1,x\ne y\Rightarrow \parallel \frac{x+y}{2}\parallel <1.$$

(2.1)

It is also said to be uniformly convex if for any $ϵ>0$, there exists $\delta >0$ such that

$$\parallel x\parallel =\parallel y\parallel =1,\parallel x-y\parallel \ge ϵ\Rightarrow \parallel \frac{x+y}{2}\parallel \le 1-\delta .$$

(2.2)

It is well known that if E is a uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

(2.3)

exists for each $x,y\in S(E):=\{x\in E:\parallel x\parallel =1\}$. E is said to be uniformly smooth if the limit (2.3) is attained uniformly for $x,y\in S(E)$.

Following Alber [14], the generalized projection $P_C:E\to C$ is defined by

$$P_C=arg\underset{y\in C}{inf}\varphi (y,x),\mathrm{\forall }x\in E.$$

(2.4)

Lemma 2.1 [14]

Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty, closed, convex subset of E. Then the following conclusions hold:

• (1) $\varphi (x,P_{C}y)+\varphi (P_{C}y,y)\le \varphi (x,y)$ for all $x\in C$ and $y\in E$.

• (2) If $x\in E$ and $z\in C$, then $z=P_{C}x\iff 〈z-y,Jx-Jz〉\ge 0$, $\mathrm{\forall }y\in C$.

• (3) For $x,y\in E$, $\varphi (x,y)=0$ if and only if $x=y$.

Lemma 2.2 [15]

Let E be a uniformly convex and smooth Banach space and let $r>0$. Then there exists a continuous, strictly increasing, and convex function $h:[0,2r]\to [0,\mathrm{\infty })$ such that $h(0)=0$ and

$$h(\parallel x\parallel -\parallel y\parallel )\le \varphi (x,y)$$

(2.5)

for all $x,y\in B_r:=\{z\in E:\parallel z\parallel \le r\}$.

Lemma 2.3 [16]

Let E be a uniformly convex and smooth Banach space and let $\{x_n\}$ and $\{y_n\}$ be two sequences of E. If $\varphi (x_n,y_n)\to 0$, where ϕ is the function defined by (1.4), and either $\{x_n\}$ or $\{y_n\}$ is bounded, then $\parallel x_n-y_n\parallel \to 0$.

Remark 2.4 The following basic properties for a Banach space E can be found in Cioranescu [17].

• (i) If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E.

• (ii) If E is reflexive and strictly convex, then $J^{-1}$ is norm-weak-continuous.

• (iii) If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping $J:E\to 2^{E^{\ast }}$ is single valued, one-to-one, and onto.

• (iv) A Banach space E is uniformly smooth if and only if $E^{\ast }$ is uniformly convex.

• (v) Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence $\{x_n\}\subset E$, if $x_n\rightharpoonup x\in E$ and $\parallel x_n\parallel \to \parallel x\parallel$, then $x_n\to x$ as $n\to \mathrm{\infty }$.

Lemma 2.5 [18]

Let E be a real uniformly convex Banach space and let $B_r(0)$ be the closed ball of E with center at the origin and radius $r>0$. Then there exists a continuous strictly increasing convex function $g:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $g(0)=0$ such that

$${\parallel \lambda x+\mu y+\gamma z\parallel }^2\le \lambda {\parallel x\parallel }^2+\mu {\parallel y\parallel }^2+\gamma {\parallel z\parallel }^2-\lambda \mu g(\parallel x-y\parallel )$$

(2.6)

for all $x,y,z\in B_r(0)$ and $\lambda ,\mu ,\gamma \in [0,1]$ with $\lambda +\mu +\gamma =1$.

Lemma 2.6 [19]

The unique solutions to the positive integer equation

$$n=i_n+\frac{(m_n-1)m_n}{2},m_n\ge i_n,n=1,2,3,\dots$$

(2.7)

are

$$i_n=n-\frac{(m_n-1)m_n}{2},m_n=-[\frac{1}{2}-\sqrt{2n+\frac{1}{4}}],n=1,2,3,\dots ,$$

(2.8)

where $[x]$ denotes the maximal integer that is not larger than x.

3 Main results

Theorem 3.1 Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty, closed, convex subset of E. Let $\{T_i\}:C\to C$ and $\{S_i\}:C\to C$ be two sequences of relatively nonexpansive mappings with $F:={\bigcap }_{i=1}^{\mathrm{\infty }}(F(T_i)\cap F(S_i))\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by

$$\{\begin{array}{l}{x}_0=x\in C,H_{-1}=W_{-1}=C,\\ y_n=J^{-1}[{\lambda }_{n}J{x}_n+(1-{\lambda }_n)J{z}_n],\\ z_n=J^{-1}[{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n],\\ H_n=\{z\in H_{n-1}\cap W_{n-1}:\varphi (z,y_n)\le \varphi (z,x_n)\},\\ W_n=\{z\in H_{n-1}\cap W_{n-1}:〈x_n-z,Jx-Jy〉\ge 0\},\\ x_{n+1}=P_{H_n\cap W_n}x,\mathrm{\forall }n\in \mathbb{N}\cup \{0\},\end{array}$$

(3.1)

where $\{{\lambda }_n\}$, $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ are sequences in $[0,1]$ satisfying

• (1) $0\le {\lambda }_n<1$, $\mathrm{\forall }n\in \mathbb{N}\cup \{0\}$; ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\lambda }_n<1$;

• (2) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$; ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n{\gamma }_n>0$;

and $i_n$ is the solution to the positive integer equation $n=i_n+\frac{(m_n-1)m_n}{2}$ ($m_n\ge i_n$, $n=1,2,\dots$), that is, for each $n\ge 1$, there exists a unique $i_n$ such that

$$\begin{array}{c}{i}_1=1,i_2=1,i_3=2,i_4=1,i_5=2,i_6=3,\hfill \\ i_7=1,i_8=2,i_9=3,i_{10}=4,i_{11}=1,\dots .\hfill \end{array}$$

Then $\{x_n\}$ converges strongly to $P_{F}x$, where $P_{F}x$ is the generalized projection from C onto F.

Proof We divide the proof into several steps.

• (I) $H_n$ and $W_n$ ($\mathrm{\forall }n\in \mathbb{N}\cup \{0\}$) both are closed and convex subsets in C.

This follows from the fact that $\varphi (z,y_n)\le \varphi (z,x_n)$ is equivalent to

$$2〈z,J{x}_n-J{y}_n〉\le {\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2.$$

(3.2)

(II) F is a subset of ${\bigcap }_{n=0}^{\mathrm{\infty }}(H_n\cap W_n)$.

In fact, we note by [[4], Proposition 2.4] that for each $i\ge 1$, $F(S_i)$ and $F(T_i)$ are closed convex sets and so is F. It is clear that $F\subset C=H_{-1}\cap W_{-1}$. Suppose that $F\subset C_{n-1}\cap Q_{n-1}$ for some $n\in \mathbb{N}$. For any $u\in F$, by the convexity of ${\parallel \cdot \parallel }^2$, we have

$$\begin{array}{rcl}\varphi (u,z_n)& =& \varphi (u,J^{-1}[{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n])\\ =& {\parallel u\parallel }^2-2〈u,{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n〉\\ +{\parallel {\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n\parallel }^2\\ \le & {\parallel u\parallel }^2-2{\alpha }_n〈u,J{x}_n〉-2{\beta }_n〈u,J{T}_{i_n}{x}_n〉-2{\gamma }_n〈u,J{S}_{i_n}{x}_n〉\\ +{\alpha }_n{\parallel x_n\parallel }^2+{\beta }_n{\parallel T_{i_n}{x}_n\parallel }^2+{\gamma }_n{\parallel S_{i_n}{x}_n\parallel }^2\\ =& {\alpha }_n\varphi (u,x_n)+{\beta }_n\varphi (u,T_{i_n}{x}_n)+{\gamma }_n\varphi (u,S_{i_n}{x}_n)\\ \le & {\alpha }_n\varphi (u,x_n)+{\beta }_n\varphi (u,x_n)+{\gamma }_n\varphi (u,x_n)\\ =& \varphi (u,x_n),\end{array}$$

(3.3)

and then

$$\begin{array}{rcl}\varphi (u,y_n)& =& \varphi (u,J^{-1}[{\lambda }_{n}J{x}_n+(1-{\lambda }_n)J{z}_n])\\ =& {\parallel u\parallel }^2-2〈u,{\lambda }_{n}J{x}_n+(1-{\lambda }_n)J{z}_n〉+{\parallel {\lambda }_{n}J{x}_n+(1-{\lambda }_n)J{z}_n\parallel }^2\\ \le & {\parallel u\parallel }^2-2{\lambda }_n〈u,J{x}_n〉-2(1-{\lambda }_n)〈u,J{z}_n〉+{\lambda }_n{\parallel x_n\parallel }^2+(1-{\lambda }_n){\parallel z_n\parallel }^2\\ =& {\lambda }_n({\parallel u\parallel }^2-2〈u,J{x}_n〉+{\parallel x_n\parallel }^2)+(1-{\lambda }_n)({\parallel u\parallel }^2-2〈u,J{z}_n〉+{\parallel z_n\parallel }^2)\\ =& {\lambda }_n\varphi (u,x_n)+(1-{\lambda }_n)\varphi (u,z_n)\\ \le & {\lambda }_n\varphi (u,x_n)+(1-{\lambda }_n)\varphi (u,x_n)\\ =& \varphi (u,x_n).\end{array}$$

(3.4)

This implies that $F\subset H_n$. It follows from $x_n=P_{H_{n-1}\cap W_{n-1}}x$ and Lemma 2.1(2) that

$$〈x_n-z,Jx-J{x}_n〉\ge 0,\mathrm{\forall }z\in H_{n-1}\cap W_{n-1}.$$

(3.5)

Particularly,

$$〈x_n-z,Jx-J{x}_n〉\ge 0,\mathrm{\forall }u\in F,$$

(3.6)

and hence $F\subset W_n$, which yields $F\subset H_n\cap W_n$. By induction, $F\subset {\bigcap }_{n=0}^{\mathrm{\infty }}(H_n\cap W_n)$.

• (III) ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T_{i_n}{x}_n\parallel ={lim}_{n\to \mathrm{\infty }}\parallel x_n-S_{i_n}{x}_n\parallel =0$.

In view of $x_{n+1}=P_{H_n\cap W_n}x\in H_n$ and the definition of $H_n$, we also have

$$\varphi (x_{n+1},y_n)\le \varphi (x_{n+1},x_n),\mathrm{\forall }n\in \mathbb{N}.$$

(3.7)

This implies that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},y_n)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},x_n)=0.$$

(3.8)

It follows from Lemma 2.2 that

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

(3.9)

Since J is uniformly norm-to-norm continuous on bounded sets, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n+1}-J{y}_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n+1}-J{x}_n\parallel =0$$

(3.10)

and

$$\parallel J{x}_{n+1}-J{y}_n\parallel \ge (1-{\lambda }_n)\parallel J{x}_{n+1}-J{z}_n\parallel -{\lambda }_n\parallel J{x}_{n+1}-J{x}_n\parallel ,\mathrm{\forall }n\in \mathbb{N}\cup \{0\}.$$

(3.11)

This implies that

$$\begin{array}{rcl}\parallel J{x}_{n+1}-J{z}_n\parallel & \le & \frac{1}{1-{\lambda }_n}(\parallel J{x}_{n+1}-J{y}_n\parallel +{\lambda }_n\parallel J{x}_{n+1}-J{x}_n\parallel )\\ \le & \frac{1}{1-{\lambda }_n}(\parallel J{x}_{n+1}-J{y}_n\parallel +\parallel J{x}_{n+1}-J{x}_n\parallel ).\end{array}$$

(3.12)

From (3.10) and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\lambda }_n<1$, we have ${lim}_{n\to \mathrm{\infty }}\parallel J{x}_{n+1}-J{z}_n\parallel =0$. Since $J^{-1}$ is also uniformly norm-to-norm continuous on bounded sets, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-z_n\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel J^{-1}(J{x}_{n+1})-J^{-1}(J{z}_n)\parallel =0.$$

(3.13)

From $\parallel x_n-z_n\parallel \le \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-z_n\parallel$ we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n-z_n\parallel =0$. Since $\{x_n\}$ is bounded, $\varphi (p,T_{i_n}{x}_n)\le \varphi (p,x_n)$ and $\varphi (p,S_{i_n}{x}_n)\le \varphi (p,x_n)$ for any $p\in F$. We also find that $\{J{x}_n\}$, $\{J{T}_{i_n}{x}_n\}$ and $\{J{S}_{i_n}{x}_n\}$ are bounded, and then there exists an $r>0$ such that $\{J{x}_n\},\{J{T}_{i_n}{x}_n\},\{J{S}_{i_n}{x}_n\}\subset B_r(0)$. Therefore Lemma 2.5 is applicable and we observe that

$$\begin{array}{rcl}\varphi (p,z_n)& =& {\parallel p\parallel }^2-2〈p,{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n〉\\ +{\parallel {\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n\parallel }^2\\ \le & {\parallel p\parallel }^2-2{\alpha }_n〈p,J{x}_n〉-2{\beta }_n〈p,J{T}_{i_n}{x}_n〉-2{\gamma }_n〈p,J{S}_{i_n}{x}_n〉\\ +{\alpha }_n{\parallel x_n\parallel }^2+{\beta }_n{\parallel T_{i_n}{x}_n\parallel }^2+{\gamma }_n{\parallel S_{i_n}{x}_n\parallel }^2-{\beta }_n{\gamma }_{n}g(\parallel J{T}_{i_n}{x}_n-J{S}_{i_n}{x}_n\parallel )\\ =& {\alpha }_n\varphi (p,x_n)+{\beta }_n\varphi (p,T_{i_n}{x}_n)+{\gamma }_n\varphi (p,S_{i_n}{x}_n)-{\beta }_n{\gamma }_{n}g(\parallel J{T}_{i_n}{x}_n-J{S}_{i_n}{x}_n\parallel )\\ \le & \varphi (p,x_n)-{\beta }_n{\gamma }_{n}g(\parallel J{T}_{i_n}{x}_n-J{S}_{i_n}{x}_n\parallel ).\end{array}$$

(3.14)

That is,

$${\beta }_n{\gamma }_{n}g(\parallel J{T}_{i_n}{x}_n-J{S}_{i_n}{x}_n\parallel )\le \varphi (p,x_n)-\varphi (p,z_n),$$

(3.15)

where $g:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a continuous strictly convex function with $g(0)=0$.

Let $\{\parallel T_{i_{n_k}}{x}_{n_k}-S_{i_{n_k}}{x}_{n_k}\parallel \}$ be any subsequence of $\{\parallel T_{i_n}{x}_n-S_{i_n}{x}_n\parallel \}$. Since $\{x_{n_k}\}$ is bounded, there exists a subsequence $\{x_{n_j}\}$ of $\{x_{n_k}\}$ such that for any $p\in F$,

$$\underset{j\to \mathrm{\infty }}{lim}\varphi (p,x_{n_j})=\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (p,x_{n_k}):=a.$$

(3.16)

From (1.6) we have

$$\begin{array}{rcl}\varphi (p,x_{n_j})& =& \varphi (p,z_{n_j})+\varphi (z_{n_j},x_{n_j})+2〈p-z_{n_j},J{z}_{n_j}-J{x}_{n_j}〉\\ \le & \varphi (p,z_{n_j})+\varphi (z_{n_j},x_{n_j})+M\parallel J{z}_{n_j}-J{x}_{n_j}\parallel \end{array}$$

(3.17)

for some appropriate constant $M>0$. Since

$$\underset{j\to \mathrm{\infty }}{lim}\varphi (z_{n_j},x_{n_j})=0=\underset{j\to \mathrm{\infty }}{lim}\parallel J{z}_{n_j}-J{x}_{n_j}\parallel ,$$

(3.18)

it follows that

$$a=\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (p,x_{n_j})\le \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (p,z_{n_j}).$$

(3.19)

From (3.3), we have

$$\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (p,z_{n_j})\le \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (p,x_{n_j})=a$$

(3.20)

and hence ${lim}_{j\to \mathrm{\infty }}\varphi (p,x_{n_j})=a={lim}_{j\to \mathrm{\infty }}\varphi (p,z_{n_j})$. By (3.15), we observe that, as $j\to \mathrm{\infty }$,

$${\beta }_{n_j}{\gamma }_{n_j}g(\parallel J{T}_{i_{n_j}}{x}_{n_j}-J{S}_{i_{n_j}}{x}_{n_j}\parallel )\le \varphi (p,x_{n_j})-\varphi (p,z_{n_j})\to 0.$$

(3.21)

Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n{\gamma }_n>0$, it follows that ${lim}_{j\to \mathrm{\infty }}g(\parallel J{T}_{i_{n_j}}{x}_{n_j}-J{S}_{i_{n_j}}{x}_{n_j}\parallel )=0$. By the properties of the mapping g, we have ${lim}_{j\to \mathrm{\infty }}\parallel J{T}_{i_{n_j}}{x}_{n_j}-J{S}_{i_{n_j}}{x}_{n_j}\parallel =0$. Since $J^{-1}$ is also uniformly norm-to-norm continuous on bounded sets, we obtain

$$\underset{j\to \mathrm{\infty }}{lim}\parallel T_{i_{n_j}}{x}_{n_j}-S_{i_{n_j}}{x}_{n_j}\parallel =\underset{j\to \mathrm{\infty }}{lim}\parallel J^{-1}(J{T}_{i_{n_j}}{x}_{n_j})-J^{-1}(J{S}_{i_{n_j}}{x}_{n_j})\parallel =0,$$

(3.22)

and then ${lim}_{n\to \mathrm{\infty }}\parallel T_{i_n}{x}_n-S_{i_n}{x}_n\parallel =0$. Next, we note by the convexity of ${\parallel \cdot \parallel }^2$ and (1.7) that, as $n\to \mathrm{\infty }$,

$$\begin{array}{rcl}\varphi (T_{i_n}{x}_n,z_n)& =& {\parallel T_{i_n}{x}_n\parallel }^2-2〈T_{i_n}{x}_n,{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n〉\\ +{\parallel {\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{i_n}{x}_n\parallel }^2\\ \le & {\parallel T_{i_n}{x}_n\parallel }^2-2{\alpha }_n〈T_{i_n}{x}_n,J{x}_n〉-2{\beta }_n〈T_{i_n}{x}_n,J{T}_{i_n}{x}_n〉-2{\gamma }_n〈T_{i_n}{x}_n,J{S}_{i_n}{x}_n〉\\ +{\alpha }_n{\parallel x_n\parallel }^2+{\beta }_n{\parallel T_{i_n}{x}_n\parallel }^2+{\gamma }_n{\parallel S_{i_n}{x}_n\parallel }^2\\ =& {\alpha }_n\varphi (T_{i_n}{x}_n,x_n)+{\beta }_n\varphi (T_{i_n}{x}_n,S_{i_n}{x}_n)\to 0,\end{array}$$

(3.23)

since ${\alpha }_n\to 0$. By Lemma 2.3, we have ${lim}_{n\to \mathrm{\infty }}\parallel T_{i_n}{x}_n-z_n\parallel =0$ and hence

$$\parallel T_{i_n}{x}_n-x_n\parallel \le \parallel T_{i_n}{x}_n-z_n\parallel +\parallel z_n-x_n\parallel \to 0$$

(3.24)

as $n\to \mathrm{\infty }$. Moreover, we observe that

$$\parallel S_{i_n}{x}_n-x_n\parallel \le \parallel S_{i_n}{x}_n-T_{i_n}{x}_n\parallel +\parallel T_{i_n}{x}_n-x_n\parallel \to 0$$

(3.25)

as $n\to \mathrm{\infty }$.

• (IV) $x_n\to P_{F}x$ as $n\to \mathrm{\infty }$.

It follows from the definition of $W_n$ and Lemma 2.1(2) that $x_n=P_{W_n}x$. Since $x_{n+1}=P_{H_n\cap W_n}x\in W_n$, we have

$$\varphi (x_n,x)\le \varphi (x_{n+1},x),\mathrm{\forall }n\ge 1.$$

(3.26)

Therefore, $\{\varphi (x_n,x)\}$ is nondecreasing. Using $x_n=P_{W_n}x$ and Lemma 2.1(1), we have

$$\varphi (x_n,x)=\varphi (P_{W_n}x,x)\le \varphi (p,x)-\varphi (p,x_n)\le \varphi (p,x)$$

(3.27)

for all $p\in F$ and for all $n\in \mathbb{N}$, that is, $\{\varphi (x_n,x)\}$ is bounded. Then

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,x)\text{ exists}.$$

(3.28)

In particular, by (1.5), the sequence $\{{(\parallel x_n\parallel -\parallel x\parallel )}^2\}$ is bounded. This implies that $\{x_n\}$ is bounded. Note again that $x_n=P_{W_n}x$ and for any positive integer k, $x_{n+k}\in W_{n+k-1}\subset W_n$. By Lemma 2.1(1),

$$\begin{array}{rcl}\varphi (x_{n+k},x_n)& =& \varphi (x_{n+k},P_{W_n}x)\\ \le & \varphi (x_{n+k},x)-\varphi (P_{W_n}x,x)\\ =& \varphi (x_{n+k},x)-\varphi (x_n,x).\end{array}$$

(3.29)

By Lemma 2.2, we have, for $m,n\in \mathbb{N}$ with $m>n$,

$$h(\parallel x_m-x_n\parallel )\le \varphi (x_m,x_n)\le \varphi (x_m,x)-\varphi (x_n,x),$$

(3.30)

where $h:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a continuous, strictly increasing, and convex function with $h(0)=0$. Then the properties of the function g show that $\{x_n\}$ is a Cauchy sequence in C, so there exists $x^{\ast }\in C$ such that

$$x_n\to x^{\ast }(n\to \mathrm{\infty }).$$

(3.31)

Now, set ${\mathbb{N}}_i=\{k\in \mathbb{N}:k=i+\frac{(m-1)m}{2},m\ge i,m\in \mathbb{N}\}$ for each $i\in \mathbb{N}$. Note that $T_{i_k}=T_i$ and $S_{i_k}=S_i$ whenever $k\in {\mathbb{N}}_i$. By Lemma 2.6 and the definition of ${\mathbb{N}}_i$, we have ${\mathbb{N}}_1=\{1,2,4,7,11,16,\dots \}$ and $i_1=i_2=i_4=i_7=i_{11}=i_{16}=\cdots =1$. Then it follows from (3.15) and (3.24) that

$$\underset{{\mathbb{N}}_i\ni k\to \mathrm{\infty }}{lim}\parallel T_i{x}_k-x_k\parallel =\underset{{\mathbb{N}}_i\ni k\to \mathrm{\infty }}{lim}\parallel S_i{x}_k-x_k\parallel =0,\mathrm{\forall }i\in \mathbb{N}.$$

(3.32)

It then immediately follows from (3.31) and (3.32) that $x^{\ast }\in F(T_i)\cap F(S_i)$ for each $i\in \mathbb{N}$ and hence $x^{\ast }\in F$.

Put $u=P_{F}x$. Since $u\in F\subset H_n\cap W_n$ and $x_{n+1}=P_{H_n\cap W_n}x$, we have $\varphi (x_{n+1},x)\le \varphi (u,x)$, $\mathrm{\forall }n\in \mathbb{N}$. Then

$$\varphi (x^{\ast },x)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},x)\le \varphi (u,x),$$

(3.33)

which implies that $x^{\ast }=u$ since $u=P_{F}x$, and hence $x_n\to x^{\ast }=P_{F}x$ as $n\to \mathrm{\infty }$. This completes the proof. □

Remark 3.2 Note that the algorithm (3.1) is based on the projection onto an intersection of two closed and convex sets. We first give an example [20] of how to compute such a projection onto the intersection of two half-spaces.

Let H be a Hilbert space and suppose that $(x,y,z)\in H^3$ satisfies

$$\{w\in H:〈w-y,x-y〉\le 0\}\cap \{w\in H:〈w-z,y-z〉\le 0\}\ne \mathrm{\varnothing }.$$

(3.34)

Set

$$\pi =〈x-y,y-z〉,\mu ={\parallel x-y\parallel }^2,\nu ={\parallel y-z\parallel }^2,\rho =\mu \nu -{\pi }^2,$$

(3.35)

and

$$Q(x,y,z)=\{\begin{array}{ll}z,& \text{if }\rho =0\text{ and }\pi \ge 0;\\ x+(1+\pi /\nu )(z-y),& \text{if }\rho >0\text{ and }\pi \nu \ge \rho ;\\ y+(\nu /\rho )(\pi (x-y)+\mu (z-y)),& \text{if }\rho >0\text{ and }\pi \nu <\rho .\end{array}$$

(3.36)

In [21], Haugazeau introduced the operator Q as an explicit description of the projector onto the intersection of the two half-spaces defined in (3.34). He proved in [21] that the sequence $\{y_n\}$ defined by $y_0=x$ and

$$(\mathrm{\forall }n\in \mathbb{N})y_{n+1}=Q(x,Q(x,y_n,P_B{y}_n),P_{A}Q(x,y_n,P_B{y}_n))$$

(3.37)

converges strongly to $P_{C}x$.

Since the algorithm (3.1) involves the projection onto the intersection of two convex sets not necessarily half-spaces, we next give an example [22] to explain and illustrate how the projection is calculated in the general convex case.

Dykstra’s algorithm Let ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2,\dots ,{\mathrm{\Omega }}_p$ be closed and convex subsets of ${\mathbb{R}}^n$. For any $i=1,2,\dots ,p$ and $x^0\in {\mathbb{R}}^n$, the sequences $\{x_i^k\}$ are defined by the following recursive formulas:

$$\{\begin{array}{l}{x}_0^k=x_p^{k-1},\\ x_i^k=P_{{\mathrm{\Omega }}_i}(x_{i-1}^k-y_i^{k-1}),i=1,2,\dots ,p,\\ y_i^k=x_i^k-(x_{i-1}^k-y_i^{k-1}),i=1,2,\dots ,p,\end{array}$$

(3.38)

for $k=1,2,\dots$ with initial values $x_p^0=x^0$ and $y_i^0=0$ for $i=1,2,\dots ,p$. If $\mathrm{\Omega }:={\bigcap }_{i=1}^p{\mathrm{\Omega }}_i\ne \mathrm{\varnothing }$, then $\{x_i^k\}$ converges to $x^{\ast }=P_{\mathrm{\Omega }}(x^0)$, where $P_{\mathrm{\Omega }}(x):=arg{inf}_{y\in \mathrm{\Omega }}{\parallel y-x\parallel }^2$, $\mathrm{\forall }x\in {\mathbb{R}}^n$.

Note Another iterative method termed HAAR (Haugazeau-like Averaged Alternating Reflections) for finding the projection onto intersection of finitely many closed convex sets in a Hilbert space can be found in [[20], Remark 3.4(iii)].

4 Applications

The so-called convex feasibility problem for a family of mappings ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ is to find a point in the nonempty intersection ${\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$.

Note Although the problem mentioned above is indeed a convex feasibility problem, it is mainly referred to the finite case.

Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty, closed, convex subset of E. Let ${\{B_i\}}_{i=1}^{\mathrm{\infty }}:C\to E^{\ast }$ be a sequence of ${\beta }_i$-inverse strongly monotone mappings, ${\{\psi \}}_{i=1}^{\mathrm{\infty }}:C\to {\mathbb{R}}^1$ a sequence of lower semi-continuous and convex functions, and ${\{{\theta }_i\}}_{i=1}^{\mathrm{\infty }}:C\times C\to {\mathbb{R}}^1$ a sequence of bifunctions satisfying the conditions:

(A₁) $\theta (x,x)=0$;

(A₂) θ is monotone, i.e., $\theta (x,y)+\theta (y,x)\le 0$;

(A₃) $lim{sup}_{t\downarrow 0}\theta (x+t(z-x),y)\le \theta (x,y)$;

(A₄) the mapping $y\mapsto \theta (x,y)$ is convex and lower semicontinuous.

A system of generalized mixed equilibrium problems $(GMEP)$ for ${\{{\theta }_i\}}_{i=1}^{\mathrm{\infty }}$, ${\{B_i\}}_{i=1}^{\mathrm{\infty }}$ and ${\{{\psi }_i\}}_{i=1}^{\mathrm{\infty }}$ is to find an $x^{\ast }\in C$ such that

$${\theta }_i(x^{\ast },y)+〈y-x^{\ast },B_i{x}^{\ast }〉+{\psi }_i(y)-{\psi }_i\left(x^{\ast }\right)\ge 0,\mathrm{\forall }y\in C,i\in \mathbb{N},$$

(4.1)

whose set of common solutions is denoted by $\mathrm{\Omega }:={\bigcap }_{i=1}^{\mathrm{\infty }}{\mathrm{\Omega }}_i$, where ${\mathrm{\Omega }}_i$ denotes the set of solutions to generalized mixed equilibrium problem for ${\theta }_i$, $B_i$, and ${\psi }_i$.

Define a countable family of mappings ${\{S_{r,i}\}}_{i=1}^{\mathrm{\infty }}:E\to C$ with $r>0$ as follows:

$$S_{r,i}(x)=\{z\in C:{\tau }_i(z,y)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\mathrm{\forall }y\in C\},\mathrm{\forall }i\in \mathbb{N},$$

(4.2)

where ${\tau }_i(x,y)={\theta }_i(x,y)+〈y-x,B_{i}x〉+{\psi }_i(y)-{\psi }_i(x)$, $\mathrm{\forall }x,y\in C$, $i\in \mathbb{N}$. It has been shown by Zhang [23] that

• (1) ${\{S_{r,i}\}}_{i=1}^{\mathrm{\infty }}$ is a sequence of single-valued mappings;

• (2) ${\{S_{r,i}\}}_{i=1}^{\mathrm{\infty }}$ is a sequence of closed relatively nonexpansive mappings;

• (3) ${\bigcap }_{i=1}^{\mathrm{\infty }}F(S_{r,i})=\mathrm{\Omega }$.

Theorem 4.1 Let E be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty, closed, convex subset of E. Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a sequence of relatively nonexpansive mappings and ${\{S_{r,i}\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a sequence of mappings defined by (4.2) with $F:={\bigcap }_{i=1}^{\mathrm{\infty }}(F(T_i)\cap F(S_{r,i}))\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by

$$\{\begin{array}{l}{x}_0=x\in C,H_{-1}=W_{-1}=C,\\ y_n=J^{-1}[{\lambda }_{n}J{x}_n+(1-{\lambda }_n)J{z}_n],\\ z_n=J^{-1}[{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}_{i_n}{x}_n+{\gamma }_{n}J{S}_{r,i_n}{x}_n],\\ H_n=\{z\in H_{n-1}\cap W_{n-1}:\varphi (z,y_n)\le \varphi (z,x_n)\},\\ W_n=\{z\in H_{n-1}\cap W_{n-1}:〈x_n-z,Jx-Jy〉\ge 0\},\\ x_{n+1}=P_{H_n\cap W_n}x,\mathrm{\forall }n\in \mathbb{N}\cup \{0\},\end{array}$$

(4.3)

where $\{{\lambda }_n\}$, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\gamma }_n\}$ are sequences in $[0,1]$ satisfying

• (1) $0\le {\lambda }_n<1$, $\mathrm{\forall }n\in \mathbb{N}\cup \{0\}$; ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\lambda }_n<1$;

• (2) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$; ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n{\gamma }_n>0$;

and $i_n$ satisfies the equation $n=i_n+\frac{(m_n-1)m_n}{2}$ ($m_n\ge i_n$, $n=1,2,\dots$). Then $\{x_n\}$ converges strongly to $P_{F}x$, which is some common solution to the convex feasibility problem for ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ and a system of generalized mixed equilibrium problems for ${\{S_{r,i}\}}_{i=1}^{\mathrm{\infty }}$.