# Convergence analysis of an iterative algorithm for monotone operators

## Abstract

In this paper, an iterative algorithm is proposed to study some nonlinear operators which are inverse-strongly monotone, maximal monotone, and strictly pseudocontractive. Strong convergence of the proposed iterative algorithm is obtained in the framework of Hilbert spaces.

MSC:47H05, 47H09.

## 1 Introduction

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation, and this operator is decomposed as the sum of two nonlinear operators. Study of fixed (zero) point approximation algorithms for computing fixed (zero) points constitutes now a topic of intensive research efforts. Many well-known problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex sets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed (zero) point sets of a family of nonlinear mappings; see, for example, [1â€“16].

In this paper, we will investigate the problem of finding a common solution to inclusion problems and fixed point problems based on an iterative algorithm. Strong convergence of the proposed iterative algorithm has been obtained in the framework of Hilbert spaces.

The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, an iterative algorithm is proposed and analyzed. Some subresults of the main results are also discussed in this section.

## 2 Preliminaries

From now on, we always assume that H is a real Hilbert space with the inner product $ã€ˆâ‹\dots ,â‹\dots ã€‰$ and the norm $âˆ¥â‹\dots âˆ¥$, respectively. Let C be a nonempty closed convex subset of H.

Let $S:Câ†’C$ be a mapping. $F\left(S\right)$ stands for the fixed point set of S; that is, $F\left(S\right):=\left\{xâˆˆC:x=Sx\right\}$.

Recall that S is said to be nonexpansive iff

$âˆ¥Sxâˆ’Syâˆ¥â‰¤âˆ¥xâˆ’yâˆ¥,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

S is said to be asymptotically nonexpansive iff there exists a sequence $\left\{{k}_{n}\right\}âŠ‚\left[1,\mathrm{âˆž}\right)$ with ${lim}_{nâ†’\mathrm{âˆž}}{k}_{n}=1$ such that

$âˆ¥{S}^{n}xâˆ’{S}^{n}yâˆ¥â‰¤{k}_{n}âˆ¥xâˆ’yâˆ¥,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

Recall that S is said to be strictly pseudocontractive iff there exits a positive constant Îº such that

${âˆ¥Sxâˆ’Syâˆ¥}^{2}â‰¤{âˆ¥xâˆ’yâˆ¥}^{2}+\mathrm{Îº}{âˆ¥\left(xâˆ’Sx\right)âˆ’\left(yâˆ’Sy\right)âˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

S is said to be asymptotically strictly pseudocontractive iff there exits a positive constant Îº and a sequence $\left\{{k}_{n}\right\}âŠ‚\left[1,\mathrm{âˆž}\right)$ with ${lim}_{nâ†’\mathrm{âˆž}}{k}_{n}=1$ such that

${âˆ¥{S}^{n}xâˆ’{S}^{n}yâˆ¥}^{2}â‰¤{k}_{n}{âˆ¥xâˆ’yâˆ¥}^{2}+\mathrm{Îº}{âˆ¥\left(xâˆ’{S}^{n}x\right)âˆ’\left(yâˆ’{S}^{n}y\right)âˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

Let $A:Câ†’H$ be a mapping. Recall that A is said to be monotone iff

$ã€ˆAxâˆ’Ay,xâˆ’yã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

A is said to be inverse-strongly monotone iff there exists a constant $\mathrm{Î±}>0$ such that

$ã€ˆAxâˆ’Ay,xâˆ’yã€‰â‰¥\mathrm{Î±}{âˆ¥Axâˆ’Ayâˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

For such a case, A is also said to be Î±-inverse-strongly monotone. It is not hard to see that inverse-strongly monotone mappings are Lipschitz continuous.

A multivalued operator $T:Hâ†’{2}^{H}$ with the domain and the range $R\left(T\right)=\left\{Tx:xâˆˆD\left(T\right)\right\}$ is said to be monotone if for ${x}_{1}âˆˆD\left(T\right)$, ${x}_{2}âˆˆD\left(T\right)$, ${y}_{1}âˆˆT{x}_{1}$ and ${y}_{2}âˆˆT{x}_{2}$, we have $ã€ˆ{x}_{1}âˆ’{x}_{2},{y}_{1}âˆ’{y}_{2}ã€‰â‰¥0$. A monotone operator T is said to be maximal if its graph $G\left(T\right)=\left\{\left(x,y\right):yâˆˆTx\right\}$ is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and $T:Hâ†’{2}^{H}$ be a maximal monotone operator. Then we can define, for each $\mathrm{Î»}>0$, a nonexpansive single-valued mapping ${J}_{\mathrm{Î»}}:Hâ†’H$ by ${J}_{\mathrm{Î»}}={\left(I+\mathrm{Î»}T\right)}^{âˆ’1}$. It is called the resolvent of T. We know that ${T}^{âˆ’1}0=F\left({J}_{\mathrm{Î»}}\right)$ for all $\mathrm{Î»}>0$ and ${J}_{\mathrm{Î»}}$ is firmly nonexpansive; see [17â€“23] and the references therein.

Recently, many authors have investigated the solution problems of nonlinear operator equations or inequalities based on iterative methods; see, for instance, [24â€“33] and the references therein. In [19], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator via the following iterative algorithm:

${x}_{0}âˆˆH,\phantom{\rule{2em}{0ex}}{x}_{n+1}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right){J}_{{\mathrm{Î»}}_{n}}{x}_{n},\phantom{\rule{1em}{0ex}}n=0,1,2,â€¦,$
(2.1)

where $\left\{{\mathrm{Î±}}_{n}\right\}$ is a sequence in $\left(0,1\right)$, $\left\{{\mathrm{Î»}}_{n}\right\}$ is a positive sequence, $T:Hâ†’{2}^{H}$ is a maximal monotone and ${J}_{{\mathrm{Î»}}_{n}}={\left(I+{\mathrm{Î»}}_{n}T\right)}^{âˆ’1}$. They showed that the sequence $\left\{{x}_{n}\right\}$ generated in (2.1) converges weakly to some $zâˆˆ{T}^{âˆ’1}\left(0\right)$ provided that the control sequence satisfies some restrictions.

Recall that the classical variational inequality is to find an $xâˆˆC$ such that

$ã€ˆAx,yâˆ’xã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆC.$
(2.2)

In this paper, we use $VI\left(C,A\right)$ to denote the solution set of (2.2). It is known that $xâˆˆC$ is a solution to (2.1) iff x is a fixed point of the mapping ${P}_{C}\left(Iâˆ’\mathrm{Î»}A\right)$, where $\mathrm{Î»}>0$ is a constant, I stands for the identity mapping, and ${P}_{C}$ stands for the metric projection from H onto C. If A is Î±-inverse-strongly monotone and $\mathrm{Î»}âˆˆ\left(0,2\mathrm{Î±}\right]$, then the mapping ${P}_{C}\left(Iâˆ’rA\right)$ is nonexpansive; see [28] for more details. It follows that $VI\left(C,A\right)$ is closed and convex.

In [28], Takahashi an Toyoda investigated the problem of finding a common solution of variational inequality problem (2.1) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:

${x}_{0}âˆˆC,\phantom{\rule{2em}{0ex}}{x}_{n+1}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{P}_{C}\left({x}_{n}âˆ’{\mathrm{Î»}}_{n}A{x}_{n}\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâ‰¥0,$
(2.3)

where $\left\{{\mathrm{Î±}}_{n}\right\}$ is a sequence in $\left(0,1\right)$, $\left\{{\mathrm{Î»}}_{n}\right\}$ is a positive sequence, $S:Câ†’C$ is a nonexpansive mapping and $A:Câ†’H$ is an inverse-strongly monotone mapping. They proved that the sequence $\left\{{x}_{n}\right\}$ generated in (2.3) converges weakly to some $zâˆˆVI\left(C,A\right)âˆ©F\left(S\right)$ provided that the control sequence satisfies some restrictions.

In [29], Tada and Takahashi investigated the problem of finding a common solution of an equilibrium problem and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:

$\left\{\begin{array}{c}{u}_{n}âˆˆC\phantom{\rule{1em}{0ex}}\text{such that}\phantom{\rule{1em}{0ex}}F\left({u}_{n},u\right)+\frac{1}{{r}_{n}}ã€ˆuâˆ’{u}_{n},{u}_{n}âˆ’{x}_{n}ã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}uâˆˆC,\hfill \\ {x}_{n+1}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{u}_{n}\hfill \end{array}$
(2.4)

for each $nâ‰¥1$, where $\left\{{\mathrm{Î±}}_{n}\right\}$ is a sequence in $\left(0,1\right)$, $\left\{{r}_{n}\right\}$ is a positive sequence, $S:Câ†’C$ is a nonexpansive mapping and $F:CÃ—Câ†’R$ is a bifunction. They showed that the sequence $\left\{{x}_{n}\right\}$ generated in (2.4) converges weakly to some $zâˆˆEP\left(F\right)âˆ©F\left(S\right)$, where $EP\left(F\right)$ stands for the solution set of the equilibrium problem, provided that the control sequence satisfies some restrictions.

In [30], Manaka and Takahashi introduced the following iteration:

${x}_{1}âˆˆC,\phantom{\rule{2em}{0ex}}{x}_{n+1}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{J}_{{\mathrm{Î»}}_{n}}\left(Iâˆ’{\mathrm{Î»}}_{n}A\right){x}_{n},\phantom{\rule{1em}{0ex}}nâ‰¥1,$
(2.5)

where $\left\{{\mathrm{Î±}}_{n}\right\}$ is a sequence in $\left(0,1\right)$, $\left\{{\mathrm{Î»}}_{n}\right\}$ is a positive sequence, $S:Câ†’C$ is a nonexpansive mapping, $A:Câ†’H$ is an inversely-strongly monotone mapping, $B:D\left(B\right)âŠ‚Câ†’{2}^{H}$ is a maximal monotone operator, ${J}_{{\mathrm{Î»}}_{n}}={\left(I+{\mathrm{Î»}}_{n}B\right)}^{âˆ’1}$ is the resolvent of B. They showed that the sequence $\left\{{x}_{n}\right\}$ generated in (2.5) converges weakly to some $zâˆˆ{\left(A+B\right)}^{âˆ’1}\left(0\right)âˆ©F\left(S\right)$ provided that the control sequence satisfies some restrictions.

In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problems involving two monotone operators and fixed point problems involving asymptotically strictly pseudocontractive mappings based on a one-step iterative method. Weak convergence theorems are established in the framework of Hilbert spaces.

In order to obtain our main results in this paper, we need the following lemmas.

Recall that a space is said to satisfy Opialâ€™s property [34] if, for any sequence $\left\{{x}_{n}\right\}âŠ‚H$ with ${x}_{n}â‡€x$, where â‡€ denotes the weak convergence, the inequality

$\underset{nâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{n}âˆ’xâˆ¥<\underset{nâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{n}âˆ’yâˆ¥$

holds for every $yâˆˆH$ with . Indeed, the above inequality is equivalent to the following:

$\underset{nâ†’\mathrm{âˆž}}{limâ€‰sup}âˆ¥{x}_{n}âˆ’xâˆ¥<\underset{nâ†’\mathrm{âˆž}}{limâ€‰sup}âˆ¥{x}_{n}âˆ’yâˆ¥.$

Lemma 2.1 [20]

Let C be a nonempty, closed, and convex subset of H, $A:Câ†’H$ be a mapping, and $B:Hâ‡‰H$ be a maximal monotone operator. Then $F\left({J}_{r}\left(Iâˆ’\mathrm{Î»}A\right)\right)={\left(A+B\right)}^{âˆ’1}\left(0\right)$.

Lemma 2.2 Let H be a real Hilbert space. For any $aâˆˆ\left(0,1\right)$ and $x,yâˆˆH$, the following holds:

${âˆ¥ax+\left(1âˆ’a\right)yâˆ¥}^{2}=a{âˆ¥xâˆ¥}^{2}+\left(1âˆ’a\right){âˆ¥yâˆ¥}^{2}âˆ’a\left(1âˆ’a\right){âˆ¥xâˆ’yâˆ¥}^{2}.$

Lemma 2.3 [35]

Let $\left\{{a}_{n}\right\}$, $\left\{{b}_{n}\right\}$, and $\left\{{c}_{n}\right\}$ be three nonnegative sequences satisfying the following condition:

${a}_{n+1}â‰¤\left(1+{b}_{n}\right){a}_{n}+{c}_{n},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}nâ‰¥{n}_{0},$

where ${n}_{0}$ is some nonnegative integer, ${âˆ‘}_{n=1}^{\mathrm{âˆž}}{b}_{n}<\mathrm{âˆž}$ and ${âˆ‘}_{n=1}^{\mathrm{âˆž}}{c}_{n}<\mathrm{âˆž}$. Then the limit ${lim}_{nâ†’\mathrm{âˆž}}{a}_{n}$ exists.

Lemma 2.4 [36]

Let C be a nonempty closed convex subset of H and S be an asymptotically Îº-strictly pseudocontractive mapping. Then we have

1. (a)

S is uniformly Lipschitz continuous;

2. (b)

$Iâˆ’S$ is demiclosed at zero, that is, if $\left\{{x}_{n}\right\}$ is a sequence in C with ${x}_{n}â‡€x$ and ${x}_{n}âˆ’S{x}_{n}â†’0$, then $xâˆˆF\left(S\right)$.

The following lemma can be obtained from [37] immediately.

Lemma 2.5 Let H be a real Hilbert space. The following holds:

where $Nâ‰¥2$ denotes some positive integer, ${a}_{1},{a}_{2},â€¦,{a}_{N}$ are real numbers with ${âˆ‘}_{i=1}^{N}{a}_{i}=1$ in $\left(0,1\right)$ and ${x}_{1},{x}_{2},â€¦,{x}_{N}âˆˆH$.

## 3 Main results

Theorem 3.1 Let C be a nonempty closed convex subset of H. Let $Nâ‰¥2$ be some positive integer and $S:Câ†’C$ be an asymptotically strictly pseudocontractive mapping with the constant Îº and the sequence $\left\{{k}_{n}\right\}$. Let ${A}_{m}:Câ†’H$ be an inverse-strongly monotone mapping with the constant ${\mathrm{Î±}}_{m}$ and ${B}_{m}$ be a maximal monotone operator on H such that the domain of ${B}_{m}$ is included in C for each $mâˆˆ\left\{2,3,â€¦,N\right\}$. Assume . Let $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$ and $\left\{{\mathrm{Î²}}_{n}\right\}$ are real number sequences in $\left(0,1\right)$. Let $\left\{{r}_{n,2}\right\},â€¦$â€‰, and $\left\{{r}_{n,N}\right\}$ be positive real number sequences. Let $\left\{{x}_{n}\right\}$ be a sequence in C generated in the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {y}_{n}={\mathrm{Î²}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){S}^{n}{x}_{n},\hfill \\ {x}_{n+1}={\mathrm{Î±}}_{n,1}{y}_{n}+{âˆ‘}_{m=2}^{N}{\mathrm{Î±}}_{n,m}{J}_{{r}_{n,m}}\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâ‰¥1,\hfill \end{array}$
(3.1)

where ${J}_{{r}_{n,m}}={\left(I+{r}_{n,m}{B}_{m}\right)}^{âˆ’1}$ is the resolvent of ${B}_{m}$. Assume that the sequences $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$, $\left\{{\mathrm{Î²}}_{n}\right\}$, $\left\{{r}_{n,2}\right\},â€¦,\left\{{r}_{n,N}\right\}$, and $\left\{{k}_{n}\right\}$ satisfy the following restrictions:

1. (a)

${âˆ‘}_{m=1}^{N}{\mathrm{Î±}}_{n,m}=1$ and $0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

2. (b)

$0â‰¤\mathrm{Îº}â‰¤{\mathrm{Î²}}_{n}â‰¤b<1$;

3. (c)

$0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

4. (d)

${âˆ‘}_{n=1}^{\mathrm{âˆž}}\left({k}_{n}âˆ’1\right)<\mathrm{âˆž}$,

where a, b, c, and d are positive real numbers. Then the sequence $\left\{{x}_{n}\right\}$ generated in (3.1) converges weakly to some point in â„±.

Proof First, we show $Iâˆ’{r}_{n,m}{A}_{m}$ is nonexpansive. In view of the restriction (c), we find that

$\begin{array}{c}{âˆ¥\left(Iâˆ’{r}_{n,m}{A}_{m}\right)xâˆ’\left(Iâˆ’{r}_{n,m}{A}_{m}\right)yâˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={âˆ¥xâˆ’yâˆ¥}^{2}âˆ’2{r}_{n,m}ã€ˆxâˆ’y,{A}_{m}xâˆ’{A}_{m}yã€‰+{r}_{n,m}^{2}{âˆ¥{A}_{m}xâˆ’{A}_{m}yâˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}â‰¤{âˆ¥xâˆ’yâˆ¥}^{2}âˆ’{r}_{n,m}\left(2{\mathrm{Î±}}_{m}âˆ’{r}_{n,m}\right){âˆ¥{A}_{m}xâˆ’{A}_{m}yâˆ¥}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}â‰¤{âˆ¥xâˆ’yâˆ¥}^{2}.\hfill \end{array}$

This proves that $Iâˆ’{r}_{n,m}{A}_{m}$ is nonexpansive. Let $pâˆˆ\mathcal{F}$. In view of Lemma 2.1, we find that

$p=Sp={J}_{{r}_{n,m}}\left(pâˆ’{r}_{n,m}{A}_{m}p\right).$

Putting ${u}_{n,m}={J}_{{r}_{n,m}}\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right)$, we find that

$\begin{array}{rl}âˆ¥{u}_{n,m}âˆ’pâˆ¥& â‰¤âˆ¥\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right)âˆ’\left(pâˆ’{r}_{n,m}{A}_{m}p\right)âˆ¥\\ â‰¤âˆ¥{x}_{n}âˆ’pâˆ¥.\end{array}$
(3.2)

In view of Lemma 2.2, we find from the restriction (b) that

$\begin{array}{rcl}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}& =& {âˆ¥{\mathrm{Î²}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){S}^{n}{x}_{n}âˆ’pâˆ¥}^{2}\\ =& {\mathrm{Î²}}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){âˆ¥{S}^{n}{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{\mathrm{Î²}}_{n}\left(1âˆ’{\mathrm{Î²}}_{n}\right){âˆ¥{x}_{n}âˆ’{S}^{n}{x}_{n}âˆ¥}^{2}\\ â‰¤& {\mathrm{Î²}}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\left(\mathrm{Îº}âˆ’{\mathrm{Î²}}_{n}\right){âˆ¥{x}_{n}âˆ’{S}^{n}{x}_{n}âˆ¥}^{2}\\ â‰¤& {k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}.\end{array}$
(3.3)

From (3.2) and (3.3), we have

$\begin{array}{rl}{âˆ¥{x}_{n+1}âˆ’pâˆ¥}^{2}& ={âˆ¥{\mathrm{Î±}}_{n,1}\left({y}_{n}âˆ’p\right)+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}\left({u}_{n,m}âˆ’p\right)âˆ¥}^{2}\\ â‰¤{\mathrm{Î±}}_{n,1}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}\\ â‰¤{\mathrm{Î±}}_{n,1}{k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}\\ â‰¤{k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}.\end{array}$
(3.4)

We draw the conclusion that ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’pâˆ¥$ exists with the aid of Lemma 2.3. This implies that the sequence $\left\{{x}_{n}\right\}$ is bounded. In view of Lemma 2.5, we find that

$\begin{array}{rcl}{âˆ¥{x}_{n+1}âˆ’pâˆ¥}^{2}& =& {âˆ¥{\mathrm{Î±}}_{n,1}\left({y}_{n}âˆ’p\right)+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}\left({u}_{n,m}âˆ’p\right)âˆ¥}^{2}\\ â‰¤& {\mathrm{Î±}}_{n,1}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}\\ âˆ’{\mathrm{Î±}}_{n,1}{\mathrm{Î±}}_{n,r}{âˆ¥{y}_{n}âˆ’{u}_{n,r}âˆ¥}^{2}\\ â‰¤& {\mathrm{Î±}}_{n,1}{k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}\\ âˆ’{\mathrm{Î±}}_{n,1}{\mathrm{Î±}}_{n,r}{âˆ¥{y}_{n}âˆ’{u}_{n,r}âˆ¥}^{2}\\ â‰¤& {k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{\mathrm{Î±}}_{n,1}{\mathrm{Î±}}_{n,r}{âˆ¥{y}_{n}âˆ’{u}_{n,r}âˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}râˆˆ\left\{2,3,â€¦,N\right\},\end{array}$
(3.5)

which yields

${\mathrm{Î±}}_{n,1}{\mathrm{Î±}}_{n,r}{âˆ¥{y}_{n}âˆ’{u}_{n,r}âˆ¥}^{2}â‰¤{k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{x}_{n+1}âˆ’pâˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}râˆˆ\left\{2,3,â€¦,N\right\}.$

In view of the restriction (a), we find that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{y}_{n}âˆ’{u}_{n,m}âˆ¥=0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}râˆˆ\left\{2,3,â€¦,N\right\}.$
(3.6)

On the other hand, we have

$\begin{array}{rl}{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}& â‰¤{âˆ¥\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right)âˆ’\left(pâˆ’{r}_{n,m}{A}_{m}p\right)âˆ¥}^{2}\\ ={âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’2{r}_{n,m}ã€ˆ{x}_{n}âˆ’p,{A}_{m}{x}_{n}âˆ’{A}_{m}pã€‰+{r}_{n,m}^{2}{âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥}^{2}\\ â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{r}_{n,m}\left(2{\mathrm{Î±}}_{m}âˆ’{r}_{n,m}\right){âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥}^{2}.\end{array}$
(3.7)

It follows that

$\begin{array}{rl}{âˆ¥{x}_{n+1}âˆ’pâˆ¥}^{2}& â‰¤{\mathrm{Î±}}_{n,1}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}\\ â‰¤{\mathrm{Î±}}_{n,1}{k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}\\ â‰¤{k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{r}_{n,m}\left(2{\mathrm{Î±}}_{m}âˆ’{r}_{n,m}\right){âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥}^{2}.\end{array}$

This in turn implies that

$\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{r}_{n,m}\left(2{\mathrm{Î±}}_{m}âˆ’{r}_{n,m}\right){âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥}^{2}â‰¤{k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{x}_{n+1}âˆ’pâˆ¥}^{2}.$

It follows from the restrictions (b) and (d) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥=0.$
(3.8)

Notice that

$\begin{array}{rcl}{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}& â‰¤& ã€ˆ\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right)âˆ’\left(pâˆ’{r}_{n,m}{A}_{m}p\right),{u}_{n,m}âˆ’pã€‰\\ =& \frac{1}{2}\left({âˆ¥\left({x}_{n}âˆ’{r}_{n}{A}_{m}{x}_{n}\right)âˆ’\left(pâˆ’{r}_{n}{A}_{m}p\right)âˆ¥}^{2}+{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}\\ âˆ’{âˆ¥\left({x}_{n}âˆ’{r}_{n}{A}_{m}{x}_{n}\right)âˆ’\left(pâˆ’{r}_{n}{A}_{m}p\right)âˆ’\left({u}_{n,m}âˆ’p\right)âˆ¥}^{2}\right)\\ â‰¤& \frac{1}{2}\left({âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ’{r}_{n}\left({A}_{m}{x}_{n}âˆ’{A}_{m}p\right)âˆ¥}^{2}\right)\\ â‰¤& \frac{1}{2}\left({âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥}^{2}âˆ’{r}_{n}^{2}{âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥}^{2}\\ +2{r}_{n}âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥\right)\\ â‰¤& \frac{1}{2}\left({âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥}^{2}\\ +2{r}_{n}âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥\right).\end{array}$

It follows that

${âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥}^{2}+2{r}_{n,m}âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥.$
(3.9)

This implies that

$\begin{array}{rcl}{âˆ¥{x}_{n+1}âˆ’pâˆ¥}^{2}& =& {âˆ¥{\mathrm{Î±}}_{n,1}\left({y}_{n}âˆ’p\right)+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}\left({u}_{n,m}âˆ’p\right)âˆ¥}^{2}\\ â‰¤& {\mathrm{Î±}}_{n,1}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{u}_{n,m}âˆ’pâˆ¥}^{2}\\ â‰¤& {k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥}^{2}\\ +2\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{r}_{n,m}âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥,\end{array}$

which finds that

$\begin{array}{rcl}\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥}^{2}& â‰¤& {k}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{x}_{n+1}âˆ’pâˆ¥}^{2}\\ +2\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{r}_{n,m}âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥âˆ¥{A}_{m}{x}_{n}âˆ’{A}_{m}pâˆ¥.\end{array}$

In view of the restriction (a), we find from (3.8) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥=0.$
(3.10)

Notice that

$âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥â‰¤âˆ¥{x}_{n}âˆ’{u}_{n,m}âˆ¥+âˆ¥{u}_{n,m}âˆ’{y}_{n}âˆ¥.$

From (3.6) and (3.10), we obtain that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥=0.$
(3.11)

On the other hand, we have

$\begin{array}{rcl}âˆ¥{S}^{n}{x}_{n}âˆ’{x}_{n}âˆ¥& â‰¤& âˆ¥{S}^{n}{x}_{n}âˆ’\left({\mathrm{Î²}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){S}^{n}{x}_{n}\right)âˆ¥+âˆ¥\left({\mathrm{Î²}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){S}^{n}{x}_{n}\right)âˆ’{x}_{n}âˆ¥\\ =& {\mathrm{Î²}}_{n}âˆ¥{S}^{n}{x}_{n}âˆ’{x}_{n}âˆ¥+âˆ¥{y}_{n}âˆ’{x}_{n}âˆ¥,\end{array}$

which yields

$\left(1âˆ’{\mathrm{Î²}}_{n}\right)âˆ¥{S}^{n}{x}_{n}âˆ’{x}_{n}âˆ¥â‰¤âˆ¥{y}_{n}âˆ’{x}_{n}âˆ¥.$

This implies from the restriction (c) and (3.11) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{S}^{n}{x}_{n}âˆ’{x}_{n}âˆ¥=0.$
(3.12)

Notice that

$âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥â‰¤{\mathrm{Î±}}_{n,1}âˆ¥{y}_{n}âˆ’{x}_{n}âˆ¥+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}âˆ¥{u}_{n,m}âˆ’{x}_{n}âˆ¥.$

This implies from (3.10) and (3.11) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n+1}âˆ’{x}_{n}âˆ¥=0.$
(3.13)

On the other hand, we have

$\begin{array}{rcl}âˆ¥{x}_{n}âˆ’S{x}_{n}âˆ¥& â‰¤& âˆ¥{x}_{n}âˆ’{x}_{n+1}âˆ¥+âˆ¥{x}_{n+1}âˆ’{S}^{n+1}{x}_{n+1}âˆ¥\\ +âˆ¥{S}^{n+1}{x}_{n+1}âˆ’{S}^{n+1}{x}_{n}âˆ¥+âˆ¥{S}^{n+1}{x}_{n}âˆ’S{x}_{n}âˆ¥.\end{array}$

Since S is uniformly continuous, we obtain from (3.12) and (3.13) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥S{x}_{n}âˆ’{x}_{n}âˆ¥=0.$
(3.14)

Since $\left\{{x}_{n}\right\}$ is bounded, there exists a subsequence $\left\{{x}_{{n}_{i}}\right\}$ of $\left\{{x}_{n}\right\}$ such that ${x}_{{n}_{i}}â‡€\mathrm{Ï‰}âˆˆC$. We find that $\mathrm{Ï‰}âˆˆF\left(S\right)$ with the aid of Lemma 2.4.

Next, we show $\mathrm{Ï‰}âˆˆ{\left({A}_{m}+{B}_{m}\right)}^{âˆ’1}0$ for every $mâˆˆ\left\{1,2,â€¦,N\right\}$. In view of (3.10), we can choose a subsequence $\left\{{u}_{{n}_{i},m}\right\}$ of $\left\{{u}_{n,m}\right\}$ such that ${u}_{{n}_{i},m}â‡€\mathrm{Ï‰}$. Notice that

${u}_{n,m}={J}_{{r}_{n,m}}\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right).$

This implies that

${x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}âˆˆ\left(I+{r}_{n,m}{B}_{m}\right){u}_{n,m}.$

That is,

$\frac{{x}_{n}âˆ’{u}_{n,m}}{{r}_{n,m}}âˆ’{A}_{m}{x}_{n}âˆˆ{B}_{m}{u}_{n,m}.$

Since ${B}_{m}$ is monotone, we get for any $\left({u}_{m},{v}_{m}\right)âˆˆG\left({B}_{m}\right)$ that

$ã€ˆ{u}_{n,m}âˆ’{u}_{m},\frac{{x}_{n}âˆ’{u}_{n,m}}{{r}_{n,m}}âˆ’{A}_{m}{x}_{n}âˆ’{v}_{m}ã€‰â‰¥0.$
(3.15)

Replacing n by ${n}_{i}$ and letting $iâ†’\mathrm{âˆž}$, we obtain from (3.10) that

$ã€ˆ\mathrm{Ï‰}âˆ’{u}_{m},âˆ’{A}_{m}\mathrm{Ï‰}âˆ’{v}_{m}ã€‰â‰¤0.$

This means $âˆ’{A}_{m}{\mathrm{Ï‰}}_{m}âˆˆ{B}_{m}\mathrm{Ï‰}$, that is, $0âˆˆ\left({A}_{m}+{B}_{m}\right)\left(\mathrm{Ï‰}\right)$. Hence we get $\mathrm{Ï‰}âˆˆ{\left({A}_{m}+{B}_{m}\right)}^{âˆ’1}\left(0\right)$ for every $mâˆˆ\left\{1,2,â€¦,N\right\}$. This completes the proof that $\mathrm{Ï‰}âˆˆ\mathcal{F}$.

Suppose there is another subsequence $\left\{{x}_{{n}_{j}}\right\}$ of $\left\{{x}_{n}\right\}$ such that ${x}_{{n}_{j}}â‡€{\mathrm{Ï‰}}^{â€²}$. Then we can show that ${\mathrm{Ï‰}}^{â€²}âˆˆ\mathcal{F}$ in the same way. Assume . Since ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’pâˆ¥$ exits for any $pâˆˆ\mathcal{F}$. Put ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’\mathrm{Ï‰}âˆ¥=d$. Since the space satisfies Opialâ€™s condition, we see that

$\begin{array}{rl}d& =\underset{iâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{i}}âˆ’\mathrm{Ï‰}âˆ¥\\ <\underset{iâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{i}}âˆ’{\mathrm{Ï‰}}^{â€²}âˆ¥\\ =\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{\mathrm{Ï‰}}^{â€²}âˆ¥\\ =\underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{j}}âˆ’{\mathrm{Ï‰}}^{â€²}âˆ¥\\ <\underset{jâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{{n}_{j}}âˆ’\mathrm{Ï‰}âˆ¥=d.\end{array}$

This is a contradiction. This shows that $\mathrm{Ï‰}={\mathrm{Ï‰}}^{â€²}$. This proves that the sequence $\left\{{x}_{n}\right\}$ converges weakly to $\mathrm{Ï‰}âˆˆ\mathcal{F}$. This completes the proof.â€ƒâ–¡

If $N=2$, then we have the following.

Corollary 3.2 Let C be a nonempty closed convex subset of H. Let $S:Câ†’C$ be an asymptotically strictly pseudocontractive mapping with the constant Îº and the sequence $\left\{{k}_{n}\right\}$. Let $A:Câ†’H$ be an inverse-strongly monotone mapping with the constant Î±, and B be a maximal monotone operator on H such that the domain of B is included in C. Assume . Let $\left\{{\mathrm{Î±}}_{n,1}\right\}$, $\left\{{\mathrm{Î±}}_{n,2}\right\}$, and $\left\{{\mathrm{Î²}}_{n}\right\}$ be real number sequences in $\left(0,1\right)$. Let $\left\{{r}_{n}\right\}$ be a positive real number sequence. Let $\left\{{x}_{n}\right\}$ be a sequence in C generated in the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {y}_{n}={\mathrm{Î²}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){S}^{n}{x}_{n},\hfill \\ {x}_{n+1}={\mathrm{Î±}}_{n,1}{y}_{n}+{\mathrm{Î±}}_{n,2}{J}_{{r}_{n}}\left({x}_{n}âˆ’{r}_{n}{A}_{2}{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâ‰¥1,\hfill \end{array}$

where ${J}_{{r}_{n}}={\left(I+{r}_{n}B\right)}^{âˆ’1}$ is the resolvent of B. Assume that the sequences $\left\{{\mathrm{Î±}}_{n,1}\right\}$, $\left\{{\mathrm{Î±}}_{n,2}\right\}$, $\left\{{\mathrm{Î²}}_{n}\right\}$, $\left\{{r}_{n}\right\}$, and $\left\{{k}_{n}\right\}$ satisfy the following restrictions:

1. (a)

${âˆ‘}_{m=1}^{2}{\mathrm{Î±}}_{n,m}=1$ and $0, $\mathrm{âˆ€}mâˆˆ\left\{1,2\right\}$;

2. (b)

$0â‰¤\mathrm{Îº}â‰¤{\mathrm{Î²}}_{n}â‰¤b<1$;

3. (c)

$0;

4. (d)

${âˆ‘}_{n=1}^{\mathrm{âˆž}}\left({k}_{n}âˆ’1\right)<\mathrm{âˆž}$,

where a, b, c, and d are positive real numbers. Then the sequence $\left\{{x}_{n}\right\}$ converges weakly to some point in â„±.

If S is asymptotically nonexpansive, then we find from Theorem 3.1 the following by letting ${\mathrm{Î²}}_{n}=0$.

Corollary 3.3 Let C be a nonempty closed convex subset of H. Let $Nâ‰¥2$ be some positive integer and $S:Câ†’C$ be an asymptotically nonexpansive mapping with the sequence $\left\{{k}_{n}\right\}$. Let ${A}_{m}:Câ†’H$ be an inverse-strongly monotone mapping with the constant ${\mathrm{Î±}}_{m}$ and let ${B}_{m}$ be a maximal monotone operator on H such that the domain of ${B}_{m}$ is included in C for each $mâˆˆ\left\{2,3,â€¦,N\right\}$. Assume . Let $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$, and $\left\{{\mathrm{Î²}}_{n}\right\}$ be real number sequences in $\left(0,1\right)$. Let $\left\{{r}_{n,2}\right\},â€¦$â€‰, and $\left\{{r}_{n,N}\right\}$ be positive real number sequences. Let $\left\{{x}_{n}\right\}$ be a sequence in C generated in the following iterative process:

${x}_{1}âˆˆC,\phantom{\rule{2em}{0ex}}{x}_{n+1}={\mathrm{Î±}}_{n,1}{S}^{n}{x}_{n}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{J}_{{r}_{n,m}}\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâ‰¥1,$

where ${J}_{{r}_{n,m}}={\left(I+{r}_{n,m}{B}_{m}\right)}^{âˆ’1}$ is the resolvent of ${B}_{m}$. Assume that the sequences $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$, $\left\{{\mathrm{Î²}}_{n}\right\}$, $\left\{{r}_{n,2}\right\},â€¦,\left\{{r}_{n,N}\right\}$, and $\left\{{k}_{n}\right\}$ satisfy the following restrictions:

1. (a)

${âˆ‘}_{m=1}^{N}{\mathrm{Î±}}_{n,m}=1$ and $0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

2. (b)

$0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

3. (c)

${âˆ‘}_{n=1}^{\mathrm{âˆž}}\left({k}_{n}âˆ’1\right)<\mathrm{âˆž}$,

where a, b and c are positive real numbers. Then the sequence $\left\{{x}_{n}\right\}$ converges weakly to some point in â„±.

If S is the identity mapping, then we draw from Theorem 3.1 the following.

Corollary 3.4 Let C be a nonempty closed convex subset of H. Let $Nâ‰¥2$ be some positive integer. Let ${A}_{m}:Câ†’H$ be an inverse-strongly monotone mapping with the constant ${\mathrm{Î±}}_{m}$ and let ${B}_{m}$ be a maximal monotone operator on H such that the domain of ${B}_{m}$ is included in C for each $mâˆˆ\left\{2,3,â€¦,N\right\}$. Assume . Let $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦$â€‰, and $\left\{{\mathrm{Î±}}_{n,N}\right\}$ be real number sequences in $\left(0,1\right)$. Let $\left\{{r}_{n,2}\right\},â€¦$â€‰, and $\left\{{r}_{n,N}\right\}$ be positive real number sequences. Let $\left\{{x}_{n}\right\}$ be a sequence in C generated in the following iterative process:

${x}_{1}âˆˆC,\phantom{\rule{2em}{0ex}}{x}_{n+1}={\mathrm{Î±}}_{n,1}{x}_{n}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{J}_{{r}_{n,m}}\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâ‰¥1,$

where ${J}_{{r}_{n,m}}={\left(I+{r}_{n,m}{B}_{m}\right)}^{âˆ’1}$ is the resolvent of ${B}_{m}$. Assume that the sequences $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$, $\left\{{r}_{n,2}\right\},â€¦$â€‰, and $\left\{{r}_{n,N}\right\}$ satisfy the following restrictions:

1. (a)

${âˆ‘}_{m=1}^{N}{\mathrm{Î±}}_{n,m}=1$ and $0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

2. (b)

$0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$,

where a, b, and c are positive real numbers. Then the sequence $\left\{{x}_{n}\right\}$ converges weakly to some point in â„±.

Let $f:Hâ†’\left(âˆ’\mathrm{âˆž},\mathrm{âˆž}\right]$ be a proper lower semicontinuous convex function. Define the subdifferential

$\mathrm{âˆ‚}f\left(x\right)=\left\{zâˆˆH:f\left(x\right)+ã€ˆyâˆ’x,zã€‰â‰¤f\left(y\right),\mathrm{âˆ€}yâˆˆH\right\}$

for all $xâˆˆH$. Then âˆ‚f is a maximal monotone operator of H into itself; see [38] for more details. Let C be a nonempty closed convex subset of H and ${i}_{C}$ be the indicator function of C, that is,

${i}_{C}x=\left\{\begin{array}{cc}0,\hfill & xâˆˆC,\hfill \\ \mathrm{âˆž},\hfill & xâˆ‰C.\hfill \end{array}$

Furthermore, we define the normal cone ${N}_{C}\left(v\right)$ of C at v as follows:

${N}_{C}v=\left\{zâˆˆH:ã€ˆz,yâˆ’vã€‰â‰¤0,\mathrm{âˆ€}yâˆˆH\right\}$

for any $vâˆˆC$. Then ${i}_{C}:Hâ†’\left(âˆ’\mathrm{âˆž},\mathrm{âˆž}\right]$ is a proper lower semicontinuous convex function on H and $\mathrm{âˆ‚}{i}_{C}$ is a maximal monotone operator. Let ${J}_{r}x={\left(I+r\mathrm{âˆ‚}{i}_{C}\right)}^{âˆ’1}x$ for any $r>0$ and $xâˆˆH$. From $\mathrm{âˆ‚}{i}_{C}x={N}_{C}x$ and $xâˆˆC$, we get

$\begin{array}{rcl}v={J}_{r}x\phantom{\rule{1em}{0ex}}& â‡”& \phantom{\rule{1em}{0ex}}xâˆˆv+r{N}_{C}v\\ â‡”& \phantom{\rule{1em}{0ex}}ã€ˆxâˆ’v,yâˆ’vã€‰â‰¤0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆC,\\ â‡”& \phantom{\rule{1em}{0ex}}v={P}_{C}x,\end{array}$

where ${P}_{C}$ is the metric projection from H into C. Similarly, we can get that $xâˆˆ{\left(A+\mathrm{âˆ‚}{i}_{C}\right)}^{âˆ’1}\left(0\right)â‡”xâˆˆVI\left(A,C\right)$.

Corollary 3.5 Let C be a nonempty closed convex subset of H. Let $Nâ‰¥2$ be some positive integer and $S:Câ†’C$ be an asymptotically strictly pseudocontractive mapping with the constant Îº and the sequence $\left\{{k}_{n}\right\}$. Let ${A}_{m}:Câ†’H$ be an inverse-strongly monotone mapping with the constant ${\mathrm{Î±}}_{m}$ for each $mâˆˆ\left\{2,3,â€¦,N\right\}$. Assume . Let $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$, and $\left\{{\mathrm{Î²}}_{n}\right\}$ be real number sequences in $\left(0,1\right)$. Let $\left\{{r}_{n,2}\right\},â€¦$â€‰, and $\left\{{r}_{n,N}\right\}$ be positive real number sequences. Let $\left\{{x}_{n}\right\}$ be a sequence in C generated in the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {y}_{n}={\mathrm{Î²}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î²}}_{n}\right){S}^{n}{x}_{n},\hfill \\ {x}_{n+1}={\mathrm{Î±}}_{n,1}{y}_{n}+{âˆ‘}_{m=2}^{N}{\mathrm{Î±}}_{n,m}{P}_{C}\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâ‰¥1.\hfill \end{array}$

Assume that the sequences $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$, $\left\{{\mathrm{Î²}}_{n}\right\}$, $\left\{{r}_{n,2}\right\},â€¦,\left\{{r}_{n,N}\right\}$, and $\left\{{k}_{n}\right\}$ satisfy the following restrictions:

1. (a)

${âˆ‘}_{m=1}^{N}{\mathrm{Î±}}_{n,m}=1$ and $0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

2. (b)

$0â‰¤\mathrm{Îº}â‰¤{\mathrm{Î²}}_{n}â‰¤b<1$;

3. (c)

$0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

4. (d)

${âˆ‘}_{n=1}^{\mathrm{âˆž}}\left({k}_{n}âˆ’1\right)<\mathrm{âˆž}$,

where a, b, c, and d are positive real numbers. Then the sequence $\left\{{x}_{n}\right\}$ converges weakly to some point in â„±.

Proof Putting ${B}_{m}=\mathrm{âˆ‚}{i}_{C}$ for every $mâˆˆ\left\{2,3,â€¦,N\right\}$, we see ${J}_{{r}_{n,m}}={P}_{C}$. We can immediately draw from Theorem 3.1 the desired conclusion.â€ƒâ–¡

If S is the identity mapping, then we find from Corollary 3.5 the following.

Corollary 3.6 Let C be a nonempty closed convex subset of H. Let $Nâ‰¥2$ be some positive integer. Let ${A}_{m}:Câ†’H$ be an inverse-strongly monotone mapping with the constant ${\mathrm{Î±}}_{m}$ for each $mâˆˆ\left\{2,3,â€¦,N\right\}$. Assume . Let $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦$â€‰, and $\left\{{\mathrm{Î±}}_{n,N}\right\}$ be real number sequences in $\left(0,1\right)$. Let $\left\{{r}_{n,2}\right\},â€¦$â€‰, and $\left\{{r}_{n,N}\right\}$ be positive real number sequences. Let $\left\{{x}_{n}\right\}$ be a sequence in C generated in the following iterative process:

${x}_{1}âˆˆC,\phantom{\rule{2em}{0ex}}{x}_{n+1}={\mathrm{Î±}}_{n,1}{x}_{n}+\underset{m=2}{\overset{N}{âˆ‘}}{\mathrm{Î±}}_{n,m}{P}_{C}\left({x}_{n}âˆ’{r}_{n,m}{A}_{m}{x}_{n}\right),\phantom{\rule{1em}{0ex}}nâ‰¥1.$

Assume that the sequences $\left\{{\mathrm{Î±}}_{n,1}\right\},\left\{{\mathrm{Î±}}_{n,2}\right\},â€¦,\left\{{\mathrm{Î±}}_{n,N}\right\}$, $\left\{{\mathrm{Î²}}_{n}\right\}$, $\left\{{r}_{n,2}\right\},â€¦,\left\{{r}_{n,N}\right\}$, and $\left\{{k}_{n}\right\}$ satisfy the following restrictions:

1. (a)

${âˆ‘}_{m=1}^{N}{\mathrm{Î±}}_{n,m}=1$ and $0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

2. (b)

$0, $\mathrm{âˆ€}mâˆˆ\left\{2,â€¦,N\right\}$;

3. (c)

${âˆ‘}_{n=1}^{\mathrm{âˆž}}\left({k}_{n}âˆ’1\right)<\mathrm{âˆž}$,

where a, b, and c are positive real numbers. Then the sequence $\left\{{x}_{n}\right\}$ converges weakly to some point in â„±.

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## Acknowledgements

The authors are grateful to the editor and the reviewersâ€™ suggestions which improved the contents of the article.

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Correspondence to Shin Min Kang.

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All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Cho, S.Y., Li, W. & Kang, S.M. Convergence analysis of an iterative algorithm for monotone operators. J Inequal Appl 2013, 199 (2013). https://doi.org/10.1186/1029-242X-2013-199