# Viscosity iteration algorithm for a ϱ-strictly pseudononspreading mapping in a Hilbert space

## Abstract

In this paper, we discuss the strong convergence of the viscosity approximation method in Hilbert spaces relatively to the computation of fixed points of an operator in ϱ-strictly pseudononspreading. Under suitable conditions, some strong convergence theorems are proved. Our work improves previous results for nonspreading mappings.

## 1 Introduction

Throughout this paper, we always assume that H is a real Hilbert space endowed with an inner product and its induced norm denoted by $〈\cdot ,\cdot 〉$ and $|\cdot |$, respectively. Let C be a nonempty, closed and convex subset of H and let $A:C\to H$ be a nonlinear mapping.

Definition 1.1 A is said to be

1. (i)

monotone if

$〈Ax-Ay,x-y〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C;$
1. (ii)

strongly monotone if there exists a constant $\alpha >0$ such that

$〈Ax-Ay,x-y〉\ge \alpha {\parallel x-y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

For such a case, A is said to be α-strongly-monotone;

1. (iii)

inverse-strongly monotone if there exists a constant $\alpha >0$ such that

$〈Ax-Ay,x-y〉\ge \alpha {\parallel Ax-Ay\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

For such a case, A is said to be α-inverse-strongly-monotone;

1. (iv)

k-Lipschitz continuous if there exists a constant $k\ge 0$ such that

$\parallel Ax-Ay\parallel \le k\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

Remark 1.2 Let $F=\mu B-\gamma f$, where B is a θ-Lipschitz and η-strongly monotone operator on H with $\theta >0$ and f is a Lipschitz mapping on H with coefficient $L>0$, $0<\gamma \le \frac{\mu \eta }{L}$. It is a simple matter to see that the operator F is $\left(\mu \eta -\gamma L\right)$-strongly monotone over H, i.e.,

$〈Fx-Fy,x-y〉\ge \left(\mu \eta -\gamma L\right){\parallel x-y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(x,y\right)\in H×H.$

The classical variational inequality, which is denoted by $VI\left(A,C\right)$, is to find $x\in C$ such that

$〈Ax,y-x〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$
(1.1)

The variational inequality has been extensively studied in literature (see  and the references therein).

A mapping $T:C\to C$ is said to be nonexpansive if

$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

A mapping T is said to be firmly nonexpansive if

${\parallel Tx-Ty\parallel }^{2}\le 〈x-Tx,y-Ty〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C;$

see, for instance, . It is known that a mapping $T:C\to C$ is firmly nonexpansive if and only if

${\parallel Tx-Ty\parallel }^{2}+{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2}\le {\parallel x-y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

T is said to be nonspreading in  if

${\parallel Tx-Ty\parallel }^{2}\le {\parallel Tx-y\parallel }^{2}+{\parallel Ty-x\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$
(1.2)

It is shown in  that (1.2) is equivalent to

${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+2〈x-Tx,y-Ty〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

These mappings are generalization of a firmly nonexpansive mapping in a Hilbert space. $T:C\to C$ is said to be firmly nonexpansive if

${\parallel Tx-Ty\parallel }^{2}\le 〈x-y,Tx-Ty〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C.$

Definition 1.3 $T:H\to H$ is called demicontractive on H if there exists a constant $\alpha <1$ such that

${\parallel Tx-q\parallel }^{2}\le {\parallel x-q\parallel }^{2}+\alpha {\parallel x-Tx\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(x,q\right)\in H×{F}_{ix}\left(T\right).$
(1.3)

Definition 1.4 

$T:D\left(T\right)\subseteq H\to H$ is ϱ-strictly pseudononspreading if there exists $\varrho \in \left[0,1\right)$ such that

${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+\varrho {\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}+2〈x-Tx,y-Ty〉,$
(1.4)

for all $x,y\in D\left(T\right)$.

Remark 1.5 It is easy to claim that firmly nonexpansive mapping nonspreading mapping ϱ-strictly pseudononspreading mapping.

Indeed, from the definition of those mappings, $\mathrm{\forall }x,y\in C$, we obtain Clearly, every nonspreading mapping is ϱ-strictly pseudononspreading. The following example shows that the class of ϱ-strictly pseudononspreading mappings is more general than the class of nonspreading mappings. Let us give an example of a ϱ-strictly pseudononspreading mapping satisfying the condition of Definition 1.4.

Example 1.6 Let $X={l}^{2}$ with the norm $\parallel \cdot \parallel$ defined by

$\parallel x\parallel =\sqrt{\sum _{i=1}^{\mathrm{\infty }}{x}_{i}^{2}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x=\left({x}_{1},{x}_{2},\dots ,{x}_{n},\dots \right)\in X,$

$C=\left\{x=\left({x}_{1},{x}_{2},\dots ,{x}_{n},\dots \right)|{x}_{i}\in {R}^{1},i=1,2,\dots \right\}$, and let C be an orthogonal subspace of X (i.e., $\mathrm{\forall }x,y\in C$, we have $〈x,y〉=0$). Then it is obvious that C is a nonempty closed convex subset of X. Now, for any $x=\left({x}_{1},{x}_{2},\dots ,{x}_{n},\dots \right)\in C$, define a mapping $T:C\to C$ as follows:

$Tx=\left\{\begin{array}{cc}\left({x}_{1},{x}_{2},\dots ,{x}_{n},\dots \right),\hfill & {\prod }_{i=1}^{\mathrm{\infty }}{x}_{i}<0,\hfill \\ \left(-2{x}_{1},-2{x}_{2},\dots ,-2{x}_{n},\dots \right),\hfill & {\prod }_{i=1}^{\mathrm{\infty }}{x}_{i}\ge 0.\hfill \end{array}$
(1.5)

To see that T is $\frac{1}{3}$-strictly pseudononspreading, we break the process of proof into three cases. $\mathrm{\forall }x,y\in C$,

Case 1: ${\prod }_{i=1}^{\mathrm{\infty }}{x}_{i}<0$ and ${\prod }_{i=1}^{\mathrm{\infty }}{y}_{i}<0$, observe that

${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+\frac{1}{3}{\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}+2〈x-Tx,y-Ty〉,\phantom{\rule{1em}{0ex}}\frac{1}{3}\in \left[0,1\right),$

since ${\parallel Tx-Ty\parallel }^{2}={\parallel x-y\parallel }^{2}$ and $\frac{1}{3}{\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}=2〈x-Tx,y-Ty〉=0$.

Case 2: ${\prod }_{i=1}^{\mathrm{\infty }}{x}_{i}\le 0$ and ${\prod }_{i=1}^{\mathrm{\infty }}{y}_{i}\ge 0$, we obtain ${\parallel Tx-Ty\parallel }^{2}={\parallel x+2y\parallel }^{2}={\parallel x\parallel }^{2}+4〈x,y〉+4{\parallel y\parallel }^{2}$, $2〈x-Tx,y-Ty〉=0$ and $\frac{1}{3}{\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}=3{\parallel y\parallel }^{2}$.

Hence, Case 3: ${\prod }_{i=1}^{\mathrm{\infty }}{x}_{i}\ge 0$ and ${\prod }_{i=1}^{\mathrm{\infty }}{y}_{i}\ge 0$, we have ${\parallel Tx-Ty\parallel }^{2}=4{\parallel x-y\parallel }^{2}$, ${\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}=9{\parallel x-y\parallel }^{2}$ and $2〈x-Tx,y-Ty〉=18〈x,y〉=0$. Thus

$\begin{array}{rcl}{\parallel Tx-Ty\parallel }^{2}& =& 4{\parallel x-y\parallel }^{2}\\ =& {\parallel x-y\parallel }^{2}+\frac{1}{3}{\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}+\frac{1}{3}{\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}+2〈x-Tx,y-Ty〉.\end{array}$

From (1), (2) and (3), we obtain that T is $\frac{1}{3}$-strictly pseudononspreading, i.e.,

${\parallel Tx-Ty\parallel }^{2}={\parallel x-y\parallel }^{2}+\frac{1}{3}{\parallel x-Tx-\left(y-Ty\right)\parallel }^{2}+2〈x-Tx,y-Ty〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in R.$

We can easily know that ${F}_{ix}\left(T\right)=\left\{\left({x}_{1},{x}_{2},\dots ,{x}_{n},\dots \right),{\prod }_{i=1}^{\mathrm{\infty }}{x}_{i}<0\right\}\cup \left\{0\right\}$, where ${F}_{ix}\left(T\right)$ is defined by the set of fixed points of T.

T is not nonspreading, since for $x=\left\{0,0,\dots ,0,\dots \right\}$, $y=\left\{1,0,\dots ,0,\dots \right\}$, we have ${\parallel Tx-Ty\parallel }^{2}=4$, ${\parallel x-y\parallel }^{2}=1$ and $2〈x-Tx,y-Ty〉=0$, we obtain

${\parallel Tx-Ty\parallel }^{2}=4>1={\parallel x-y\parallel }^{2}+2〈x-Tx,y-Ty〉.$

Since our class of maps contains the class of nonspreading mappings, it also contains the class of firmly nonexpansive mappings.

Remark 1.7 

Let T be an α-demicontractive mapping on H with ${F}_{ix}\left(T\right)\ne \mathrm{\varnothing }$ and ${T}_{\omega }=\left(1-\omega \right)I+\omega T$ for $\omega \in \left(0,\mathrm{\infty }\right)$:

(A1) T α-demicontractive is equivalent to

$〈x-{T}_{\omega }x,x-q〉\ge \frac{\omega }{2}{\parallel x-Tx\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(x,q\right)\in H×{F}_{ix}\left(T\right).$

(A2) ${F}_{ix}\left(T\right)={F}_{ix}\left({T}_{\omega }\right)$ if $\omega \ne 0$.

Remark 1.8 Observe that if T is ϱ-strictly pseudononspreading and ${F}_{ix}\left(T\right)\ne \mathrm{\varnothing }$, then $\mathrm{\forall }x\in D\left(T\right)$ and $\mathrm{\forall }p\in {F}_{ix}\left(T\right)$, we obtain

${\parallel Tx-p\parallel }^{2}\le {\parallel x-p\parallel }^{2}+\varrho {\parallel x-Tx\parallel }^{2}.$

Thus, every ϱ-strictly pseudononspreading mapping with a nonempty fixed point set ${F}_{ix}\left(T\right)$ is demicontractive (see [20, 21]).

Remark 1.9 According to Remark 1.7(A1) and the fact that the ϱ-strictly pseudononspreading mapping of T is demicontractive, let $I-{T}_{\omega }=\omega \left(I-T\right)$. Then we obtain

$〈x-{T}_{\omega }x,x-q〉\ge \frac{\omega \left(1-\varrho \right)}{2}{\parallel x-Tx\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\left(x,q\right)\in H×{F}_{ix}\left(T\right).$
(1.6)

In 2011, Osilike and Isiogugu  introduced the following propositions and proved a strong convergence theorem somewhat related to a Halpern-type iteration algorithm for a ϱ-strictly pseudononspreading mapping in Hilbert spaces.

Proposition 1.10 

Let C be a nonempty closed convex subset of a real Hilbert space H and let $T:C\to C$ be a ϱ-strictly pseudononspreading mapping. If ${F}_{ix}\left(T\right)\ne \mathrm{\varnothing }$, then it is closed and convex.

Proposition 1.11 

Let C be a nonempty closed convex subset of a real Hilbert space H and let $T:C\to C$ be a ϱ-strictly pseudononspreading mapping. Then $\left(I-T\right)$ is demiclosed at 0.

Theorem 1.12 

Let C be a nonempty closed convex subset of a real Hilbert space H and let $T:C\to C$ be a ϱ-strictly pseudononspreading mapping with a nonempty fixed point set ${F}_{ix}\left(T\right)$. Let $\alpha \in \left[\varrho ,1\right)$ and let ${\left\{{\alpha }_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ be a real sequence in $\left[0,1\right)$ such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$. Let $u\in C$, $\left\{{x}_{n}\right\}$ and $\left\{{z}_{n}\right\}$ be sequences in C generated from an arbitrary ${x}_{1}\in C$ by

$\left\{\begin{array}{c}{x}_{n+1}={\alpha }_{n}u+\left(I-{\alpha }_{n}\right){z}_{n},\phantom{\rule{1em}{0ex}}n>0,\hfill \\ {z}_{n}=\frac{1}{n}{\sum }_{k=1}^{n-1}{T}_{\alpha }^{k}{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1.\hfill \end{array}$
(1.7)

Then $\left\{{x}_{n}\right\}$ and $\left\{{z}_{n}\right\}$ converge strongly to ${P}_{{F}_{ix}\left(T\right)}u$, where ${P}_{{F}_{ix}\left(T\right)}:H\to {F}_{ix}\left(T\right)$ is a metric projection of H onto ${F}_{ix}\left(T\right)$.

In 2010, Tian  introduced the following theorem for finding an element of a set of solutions to the fixed point of a nonexpansive mapping in a Hilbert space.

Theorem 1.13 

Let f be a contraction on a real Hilbert space H and T be a nonexpansive mapping on H. Starting with an arbitrary initial ${x}_{0}\in H$, define a sequence $\left\{{x}_{n}\right\}$ generated by

${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-\mu {\alpha }_{n}B\right)T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.8)

where B is a θ-Lipschitz and η-strongly monotone operator on H with $\theta >0$, $\eta >0$ and $0<\mu <2\eta /{\theta }^{2}$. Assume also that a sequence $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left(0,1\right)$ satisfying the following conditions:

(c1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$,

(c2) ${\sum }_{n=0}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_{n}|<\mathrm{\infty }$ or ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n+1}/{\alpha }_{n}=1$.

Then the sequence $\left\{{x}_{n}\right\}$ generated by (1.8) converges strongly to the unique solution ${x}^{\ast }\in {F}_{ix}\left(T\right)$ of the variational inequality

$〈\left(\gamma f-\mu B\right){x}^{\ast },x-{x}^{\ast }〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in {F}_{ix}\left(T\right).$
(1.9)

In this paper, we combine Theorem 1.12 and Theorem 1.13 and introduce the following general iterative algorithm for a ϱ-strictly pseudononspreading mapping T.

Algorithm 1.14 Let ${x}_{0}\in H$ be arbitrary

$\left\{\begin{array}{c}{x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-\mu {\alpha }_{n}B\right){z}_{n},\phantom{\rule{1em}{0ex}}n>0,\hfill \\ {z}_{n}=\frac{1}{n}{\sum }_{k=1}^{n}{T}_{\alpha }^{k}{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$

where $B:H\to H$ is η-strongly monotone and boundedly Lipschitzian, f is an L-Lipschitz mapping on H with coefficient $L>0$ and ${T}_{\alpha }^{k}=\left(1-\alpha \right)I+\alpha {T}^{k}$, $\alpha \in \left({\varrho }_{k},\frac{1}{2}\right)$.

Under suitable conditions, some strong convergence theorems are proved in the following chapter.

## 2 Preliminaries

Throughout this paper, we write ${x}_{n}⇀x$ to indicate that the sequence $\left\{{x}_{n}\right\}$ converges weakly to x. ${x}_{n}\to x$ implies that $\left\{{x}_{n}\right\}$ converges strongly to x. The following lemmas are useful for main results.

Definition 2.1 A mapping T is said to be demiclosed if for any sequence $\left\{{x}_{n}\right\}$ which weakly converges to y, and if the sequence $\left\{T{x}_{n}\right\}$ strongly converges to z, then $T\left(y\right)=z$.

Lemma 2.2 

Assume $\left\{{\alpha }_{n}\right\}$ is a sequence of nonnegative real numbers such that

${\alpha }_{n+1}\le \left(1-{\gamma }_{n}\right){\alpha }_{n}+{\delta }_{n},\phantom{\rule{1em}{0ex}}n\ge 0,$

where $\left\{{\gamma }_{n}\right\}$ is a sequence in $\left(0,1\right)$ and $\left\{{\delta }_{n}\right\}$ is a sequence in such that

1. (i)

${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_{n}=\mathrm{\infty }$,

2. (ii)

${lim sup}_{n\to \mathrm{\infty }}\frac{{\delta }_{n}}{{\gamma }_{n}}=0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n}|<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$.

Lemma 2.3 

Let $\left\{{\mathcal{T}}_{n}\right\}$ be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence ${\left\{{\mathcal{T}}_{{n}_{j}}\right\}}_{j\ge 0}$ of $\left\{{\mathcal{T}}_{n}\right\}$ which satisfies ${\mathcal{T}}_{{n}_{j}}<{\mathcal{T}}_{{n}_{j}+1}$ for all $j\ge 0$. Also, consider the sequence of integers ${\left\{\delta \left(n\right)\right\}}_{n\ge {n}_{0}}$ defined by

$\delta \left(n\right)=max\left\{k\le n|{\mathcal{T}}_{k}<{\mathcal{T}}_{k+1}\right\}.$
(2.1)

Then ${\left\{\delta \left(n\right)\right\}}_{n\ge {n}_{0}}$ is a nondecreasing sequence verifying ${lim}_{n\to \mathrm{\infty }}\delta \left(n\right)=\mathrm{\infty }$, $\mathrm{\forall }n\ge {n}_{0}$. It holds that ${T}_{\delta \left(n\right)}<{\mathcal{T}}_{\delta \left(n\right)+1}$, and we have

${\mathcal{T}}_{n}<{\mathcal{T}}_{\delta \left(n\right)+1}.$

Lemma 2.4 Let K be a closed convex subset of a real Hilbert space H given $x\in H$ and $y\in K$. Then $y={P}_{K}x$ if and only if the following inequality holds:

$〈x-y,y-z〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }z\in K.$

## 3 Main results

Let C be a nonempty closed convex subset of a real Hilbert space H and let ${T}^{k}:C\to C$ be a ${\varrho }_{k}$-strictly pseudononspreading mapping with a common nonempty fixed point set ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$. Let f be an L-Lipschitz mapping on H with coefficient $L>0$. Assume the set ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$ is nonempty. Since ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$ is closed and convex, the nearest point projection from C onto ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$ is well defined. Recall $B:H\to H$ is η-strongly monotone and θ-Lipschitzian on H with $\theta >0$, $\eta >0$. Let $0<\mu <2\eta /{\theta }^{2}$, $0<\gamma <\mu \left(\eta -\frac{\mu {\theta }^{2}}{2}\right)/L=\tau /L$, consider the following sequence $\left\{{x}_{n}\right\}$ defined by

$\left\{\begin{array}{c}{x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-\mu {\alpha }_{n}B\right){z}_{n},\phantom{\rule{1em}{0ex}}n>0,\hfill \\ {z}_{n}=\frac{1}{n}{\sum }_{k=1}^{n}{T}_{\alpha }^{k}{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$
(3.1)

where ${T}_{\alpha }^{k}=\left(1-\alpha \right)I+\alpha {T}^{k}$, $\alpha \in \left({\varrho }_{k},\frac{1}{2}\right)$, $k=\left\{1,2,\dots ,n\right\}$, and $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left(0,1\right)$ satisfying the following conditions:

(c1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$,

(c2) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$ or ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n+1}/{\alpha }_{n}=1$.

Remark 3.1 

Let H be a real Hilbert space. Let B be a θ-Lipschitzian and η-strongly monotone operator on H with $\theta >0$, $\eta >0$. Let $0<\mu <2\eta /{\theta }^{2}$, let $S=\left(I-t\mu B\right)$ and $\mu \left(\eta -\frac{\mu {\theta }^{2}}{2}\right)=\tau$, then for $t\in \left(0,min\left\{1,\frac{1}{\tau }\right\}\right)$, S is a contraction with a constant $1-t\tau$.

Before stating our main result, we introduce some lemmas for algorithm (3.1) as follows.

Lemma 3.2 The sequence $\left\{{x}_{n}\right\}$ is generated by (3.1) with ${T}^{k}$ being a ϱ-strictly pseudononspreading mapping on H and $\left\{{\alpha }_{n}\right\}\subset \left(0,1\right)$. Then $\left\{{x}_{n}\right\}$ is bounded.

Proof Let ${T}_{\alpha }^{k}x=\left(1-\alpha \right)x+\alpha {T}^{k}x$ and $0<{\varrho }_{k}<\alpha <\frac{1}{2}$. Then $\mathrm{\forall }x,y\in C$, we have

$\begin{array}{rcl}{\parallel {T}_{\alpha }^{k}x-{T}_{\alpha }^{k}y\parallel }^{2}& =& \alpha {\parallel x-y\parallel }^{2}+\left(1-\alpha \right){\parallel {T}^{k}x-{T}^{k}y\parallel }^{2}-\alpha \left(1-\alpha \right){\parallel x-{T}^{k}x-\left(y-{T}^{k}y\right)\parallel }^{2}\\ \le & \alpha {\parallel x-y\parallel }^{2}+\left(1-\alpha \right)\left[{\parallel x-y\parallel }^{2}+{\varrho }_{k}{\parallel x-{T}^{k}x-\left(y-{T}^{k}y\right)\parallel }^{2}\\ +2〈x-{T}^{k}x,y-{T}^{k}y〉\right]-\alpha \left(1-\alpha \right){\parallel x-{T}^{k}x-\left(y-{T}^{k}y\right)\parallel }^{2}\\ =& {\parallel x-y\parallel }^{2}+2\left(1-\alpha \right)〈x-{T}^{k}x,y-{T}^{k}y〉\\ -\left(1-\alpha \right)\left(\alpha -{\varrho }_{k}\right){\parallel x-{T}^{k}x-\left(y-{T}^{k}y\right)\parallel }^{2}\\ \le & {\parallel x-y\parallel }^{2}+2\left(1-\alpha \right)〈x-{T}^{k}x,y-{T}^{k}y〉\\ =& {\parallel x-y\parallel }^{2}+\frac{2\left(1-\alpha \right)}{{\alpha }^{2}}〈x-{T}_{\alpha }^{k}x,y-{T}_{\alpha }^{k}y〉.\end{array}$
(3.2)

From $p\in {\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$ and (3.2), we also have

$\parallel {T}_{\alpha }^{k}{x}_{n}-p\parallel \le \parallel {x}_{n}-p\parallel .$
(3.3)

According to (3.3), (3.1) and Remark 3.1, we obtain

$\parallel {z}_{n}-p\parallel =\parallel \frac{1}{n}\sum _{k=1}^{n}{T}_{\alpha }^{k}{x}_{n}-p\parallel \le \frac{1}{n}\sum _{k=1}^{n}\parallel {T}_{\alpha }^{k}{x}_{n}-p\parallel \le \frac{1}{n}\sum _{k=1}^{n}\parallel {x}_{n}-p\parallel =\parallel {x}_{n}-p\parallel .$
(3.4)

Thus,

$\begin{array}{rcl}\parallel {x}_{n+1}-p\parallel & =& \parallel {\alpha }_{n}\gamma \left(f\left({x}_{n}\right)-f\left(p\right)\right)+{\alpha }_{n}\left(\gamma f\left(p\right)-\mu Bp\right)+\left(I-\mu {\alpha }_{n}B\right)\left({z}_{n}-p\right)\parallel \\ \le & {\alpha }_{n}\gamma \parallel f\left({x}_{n}\right)-f\left(p\right)\parallel +{\alpha }_{n}\parallel \gamma f\left(p\right)-\mu Bp\parallel +\left(1-{\alpha }_{n}\tau \right)\parallel {z}_{n}-p\parallel \\ \le & {\alpha }_{n}\gamma \parallel f\left({x}_{n}\right)-f\left(p\right)\parallel +{\alpha }_{n}\parallel \gamma f\left(p\right)-\mu Bp\parallel +\left(1-{\alpha }_{n}\tau \right)\parallel {x}_{n}-p\parallel ,\end{array}$
(3.5)

which combined with $\parallel f\left({x}_{n}\right)-f\left(p\right)\parallel \le L\parallel {x}_{n}-p\parallel$ amounts to

$\parallel {x}_{n+1}-p\parallel \le \left(1-{\alpha }_{n}\left(\tau -\gamma L\right)\right)\parallel {x}_{n}-p\parallel +{\alpha }_{n}\parallel \gamma f\left(p\right)-\mu Bp\parallel .$
(3.6)

Putting ${M}_{1}=max\left\{\parallel {x}_{0}-p\parallel ,\parallel \gamma f\left(p\right)-\mu Bp\parallel \right\}$, we clearly obtain $\parallel {x}_{n}-p\parallel \le {M}_{1}$. Hence ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ and ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ are bounded. From (3.3), we have that ${\left\{{T}_{\alpha }^{k}{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ is also bounded. □

Now we are in a position to claim the main result.

Theorem 3.3 Assume C is a nonempty closed convex subset of a real Hilbert space H and let ${T}^{k}:C\to C$ be a ${\varrho }_{k}$-strictly pseudononspreading mapping with a common nonempty fixed point set ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$. Let f be an L-Lipschitz mapping on H with coefficient $L>0$ and $B:H\to H$ be η-strongly monotone and θ-Lipschitzian on H with $\theta >0$, $\eta >0$. Let $0<\mu <2\eta /{\theta }^{2}$, $0<\gamma <\mu \left(\eta -\frac{\mu {\theta }^{2}}{2}\right)/L=\tau /L$. Consider the sequences ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ and ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ to be sequences in C generated from an arbitrary ${x}_{1}\in C$ by (3.1), where ${T}_{\alpha }^{k}=\left(1-\alpha \right)I+\alpha {T}^{k}$, $\alpha \in \left({\varrho }_{k},\frac{1}{2}\right)$, $k=\left\{1,2,\dots ,n\right\}$, ${\left\{{\alpha }_{n}\right\}}_{n=1}^{\mathrm{\infty }}\in \left[0,1\right)$ and ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$. Then ${\left\{{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ and ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ converge strongly to the unique element ${x}^{\ast }$ in ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$ verifying

${P}_{{\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)}\left(I-\mu B+\gamma f\right){x}^{\ast }={x}^{\ast },$
(3.7)

which equivalently solves the following variational inequality problem:

${x}^{\ast }\in \bigcap _{k}^{n}{F}_{ix}\left({T}^{k}\right),\phantom{\rule{1em}{0ex}}〈\left(\gamma f-\mu B\right){x}^{\ast },v-{x}^{\ast }〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }v\in \bigcap _{k}^{n}{F}_{ix}\left({T}^{k}\right).$
(3.8)

Proof According to Lemma 3.2, it is simple to know that ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ and ${\left\{{T}_{\alpha }^{k}{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ are bounded. Thus, for $\mathrm{\forall }y\in C$ and $\mathrm{\forall }k=0,1,2,\dots ,n-1$ and according to (3.2) and (3.1), we have

$\begin{array}{rcl}{\parallel {T}_{\alpha }^{k+1}{x}_{n}-{T}_{\alpha }y\parallel }^{2}& =& {\parallel {T}_{\alpha }\left({T}_{\alpha }^{k}{x}_{n}\right)-{T}_{\alpha }y\parallel }^{2}\\ \le & {\parallel {T}_{\alpha }^{k}{x}_{n}-y\parallel }^{2}+\frac{2}{\left(1-\alpha \right)}〈{T}_{\alpha }^{k}{x}_{n}-{T}_{\alpha }^{k+1}{x}_{n},y-{T}_{\alpha }y〉\\ =& {\parallel {T}_{\alpha }^{k}{x}_{n}-{T}_{\alpha }y\parallel }^{2}+{\parallel {T}_{\alpha }y-y\parallel }^{2}+2〈{T}_{\alpha }^{k}{x}_{n}-{T}_{\alpha }y,{T}_{\alpha }y-y〉\\ +\frac{2}{\left(1-\alpha \right)}〈{T}_{\alpha }^{k}{x}_{n}-{T}_{\alpha }^{k+1}{x}_{n},y-{T}_{\alpha }y〉.\end{array}$
(3.9)

Summing (3.9) from $k=0$ to n and dividing by n, we obtain

$\begin{array}{rcl}\frac{1}{n}\parallel {T}_{\alpha }^{k+1}{x}_{n}-{T}_{\alpha }y\parallel & \le & \frac{1}{n}{\parallel {x}_{n}-{T}_{\alpha }y\parallel }^{2}+{\parallel {T}_{\alpha }y-y\parallel }^{2}+2〈{z}_{n}-{T}_{\alpha }y,{T}_{\alpha }y-y〉\\ +\frac{2}{n\left(1-\alpha \right)}〈{x}_{n}-{T}_{\alpha }^{n}{x}_{n},{T}_{\alpha }y-y〉.\end{array}$
(3.10)

Since ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ is bounded, then there exists a subsequence ${\left\{{z}_{{n}_{j}}\right\}}_{j=1}^{\mathrm{\infty }}$ of ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ which converges weakly to $\omega \in C$. Replacing n by ${n}_{j}$ in (3.10), we obtain (3.11)

Since ${\left\{{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ and ${\left\{{T}_{\alpha }^{n}{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ are bounded, letting $j\to \mathrm{\infty }$ in (3.11) yields

$0\le {\parallel {T}_{\alpha }y-y\parallel }^{2}+2〈\omega -{T}_{\alpha }y,{T}_{\alpha }y-y〉.$
(3.12)

Let $y=\omega$ in (3.12). We obtain that $\omega \in {F}_{ix}\left({T}_{\alpha }\right)={F}_{ix}\left(T\right)$.

Observe that since ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ and ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ are bounded, and ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$, then

$\begin{array}{rcl}\parallel {x}_{n+1}-{z}_{n}\parallel & =& {\alpha }_{n}\parallel \gamma f\left({x}_{n}\right)-\mu B{z}_{n}\parallel \\ \le & {\alpha }_{n}\gamma \parallel f\left({x}_{n}\right)-f\left(p\right)\parallel +{\alpha }_{n}\parallel \gamma f\left(p\right)-\mu Bp\parallel +{\alpha }_{n}\parallel \mu B\left({z}_{n}-p\right)\parallel \\ \le & {\alpha }_{n}\gamma L\parallel {x}_{n}-p\parallel +{\alpha }_{n}\parallel \gamma f\left(p\right)-\mu Bp\parallel +{\alpha }_{n}\tau \parallel {z}_{n}-p\parallel ,\end{array}$

then

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n+1}-{z}_{n}\parallel =0.$
(3.13)

We next show that

$\underset{n\to \mathrm{\infty }}{lim sup}〈\left(\gamma f-\mu B\right)z,{x}_{n+1}-z〉\le 0.$
(3.14)

Indeed, take ${\left\{{x}_{{n}_{j}+1}\right\}}_{n=1}^{\mathrm{\infty }}$ of ${\left\{{x}_{n+1}\right\}}_{n=1}^{\mathrm{\infty }}$ such that

$\underset{n\to \mathrm{\infty }}{lim sup}〈\left(\gamma f-\mu B\right){x}^{\ast },{x}_{n+1}-{x}^{\ast }〉=\underset{j\to \mathrm{\infty }}{lim}〈\left(\gamma f-\mu B\right){x}^{\ast },{x}_{{n}_{j}+1}-{x}^{\ast }〉,$

where ${x}^{\ast }$ is obtained in (3.7). We may assume that ${x}_{{n}_{j}+1}⇀z$ as $j\to \mathrm{\infty }$. From (3.13), we have ${z}_{{n}_{j}}⇀z$ as $j\to \mathrm{\infty }$, then to arbitrary bounded linear functional g on H, we have

Thus, we obtain ${z}_{{n}_{j}}\to z$ as $j\to 0$, and $z\in {F}_{ix}\left(T\right)$. Hence, we have

$\underset{j\to \mathrm{\infty }}{lim}〈\left(\gamma f-\mu B\right){x}^{\ast },{x}_{{n}_{j}+1}-{x}^{\ast }〉=〈\left(\gamma f-\mu B\right){x}^{\ast },z-{x}^{\ast }〉\le 0.$
(3.15)

Moreover, from (3.1), (3.13) and (3.14), we have (3.16)

As required, finally we show that ${x}_{n}\to {x}^{\ast }$ and ${z}_{n}\to {x}^{\ast }$.

According to (3.1), (3.4) and (3.16), we obtain

$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& =& {\parallel {\alpha }_{n}\left(\gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\right)+\left(I-\mu {\alpha }_{n}B\right){z}_{n}-\left(I-\mu {\alpha }_{n}B\right){x}^{\ast }\parallel }^{2}\\ =& {\alpha }_{n}^{2}{\parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel }^{2}+{\parallel \left(I-\mu {\alpha }_{n}B\right){z}_{n}-\left(I-\mu {\alpha }_{n}B\right){x}^{\ast }\parallel }^{2}\\ +2{\alpha }_{n}〈\left(I-\mu {\alpha }_{n}B\right){z}_{n}-\left(I-\mu {\alpha }_{n}B\right){x}^{\ast },\gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }〉\\ \le & {\alpha }_{n}^{2}{\parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel }^{2}+{\left(1-{\alpha }_{n}\tau \right)}^{2}{\parallel {z}_{n}-{x}^{\ast }\parallel }^{2}\\ +2{\alpha }_{n}\left[〈{z}_{n}-{x}^{\ast },\gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }〉-\mu {\alpha }_{n}〈B{z}_{n}-B{x}^{\ast },\gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }〉\right]\\ \le & \left[{\left(1-{\alpha }_{n}\tau \right)}^{2}+2{\alpha }_{n}\gamma L\right]{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}\left[2〈{z}_{n}-{x}^{\ast },\gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }〉\\ +{\alpha }_{n}{\parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel }^{2}+2\mu {\alpha }_{n}\parallel B{z}_{n}-B{x}^{\ast }\parallel \parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel \right]\\ \le & \left[1-2{\alpha }_{n}\left(\tau -\gamma L\right)\right]{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\alpha }_{n}\left[2〈{x}_{n}-{x}^{\ast },\gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }〉\\ +{\alpha }_{n}{\parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel }^{2}+2\mu {\alpha }_{n}\parallel B{z}_{n}-B{x}^{\ast }\parallel \parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel \\ +{\alpha }_{n}{\tau }^{2}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\right]\\ =& \left(1-\overline{{\alpha }_{n}}\right){\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\overline{{\alpha }_{n}}\overline{{\beta }_{n}},\end{array}$

where $\overline{{\alpha }_{n}}=2{\alpha }_{n}\left(\tau -\gamma L\right)$,

$\begin{array}{rcl}\overline{{\beta }_{n}}& =& \frac{1}{2\left(\tau -\gamma L\right)}\left[2〈{x}_{n}-{x}^{\ast },\gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }〉+{\alpha }_{n}{\parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel }^{2}\\ +2\mu {\alpha }_{n}\parallel B{z}_{n}-B{x}^{\ast }\parallel \parallel \gamma f\left({x}_{n}\right)-\mu B{x}^{\ast }\parallel +{\alpha }_{n}{\tau }^{2}{\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}\right].\end{array}$

It is easily seen that ${lim}_{n\to \mathrm{\infty }}\overline{{\alpha }_{n}}$, $\sum \overline{{\alpha }_{n}}=\mathrm{\infty }$ and ${lim sup}_{n\to \mathrm{\infty }}\overline{{\beta }_{n}}\le 0$. By Lemma 2.2, we conclude that ${x}_{n}\to {x}^{\ast }$ as $n\to \mathrm{\infty }$, and ${z}_{n}$ also converges strongly to the unique element ${x}^{\ast }$ in ${F}_{ix}\left(T\right)$. In addition, the variational inequality (3.15) can be written as

$〈\left(I-\mu B+\gamma f\right){x}^{\ast }-{x}^{\ast },z-{x}^{\ast }〉\ge 0,\phantom{\rule{1em}{0ex}}z\in \bigcap _{k}^{n}{F}_{ix}\left({T}^{k}\right).$

So, by Lemma 2.4, it is equivalent to the fixed point equation

${P}_{{\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)}\left(I-\mu B+\gamma f\right){x}^{\ast }={x}^{\ast }.$

□

Remark 3.4 For a nonspreading mapping T, we have $\varrho =0$ in Theorem 3.3 to obtain the following corollary.

Corollary 3.5 Assume C is a nonempty closed convex subset of a real Hilbert space H and let ${T}^{k}:C\to C$ be a nonspreading mapping with a common nonempty fixed point set ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$. Let f be an L-Lipschitz mapping on H with coefficient $L>0$ and $B:H\to H$ be η-strongly monotone and θ-Lipschitzian on H with $\theta >0$, $\eta >0$. Let $0<\mu <2\eta /{\theta }^{2}$, $0<\gamma <\mu \left(\eta -\frac{\mu {\theta }^{2}}{2}\right)/L=\tau /L$, consider the sequences ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ and ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ to be sequences in C generated from an arbitrary ${x}_{1}\in C$ by

$\left\{\begin{array}{c}{x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-\mu {\alpha }_{n}B\right){z}_{n},\phantom{\rule{1em}{0ex}}n>0,\hfill \\ {z}_{n}=\frac{1}{n}{\sum }_{k=1}^{n}{T}_{\alpha }^{k}{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$

where ${T}_{\alpha }^{k}=\left(1-\alpha \right)I+\alpha {T}^{k}$, $\alpha \in \left(0,\frac{1}{2}\right)$, ${\left\{{\alpha }_{n}\right\}}_{n=1}^{\mathrm{\infty }}\in \left[0,1\right)$ and ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$. Then ${\left\{{x}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ and ${\left\{{z}_{n}\right\}}_{n=1}^{\mathrm{\infty }}$ converge strongly to the unique element ${x}^{\ast }$ in ${\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)$ verifying

${P}_{{\bigcap }_{k}^{n}{F}_{ix}\left({T}^{k}\right)}\left(I-\mu B+\gamma f\right){x}^{\ast }={x}^{\ast },$

which equivalently solves the following variational inequality problem:

${x}^{\ast }\in \bigcap _{k}^{n}{F}_{ix}\left({T}^{k}\right),\phantom{\rule{1em}{0ex}}〈\left(\gamma f-\mu B\right){x}^{\ast },v-{x}^{\ast }〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }v\in \bigcap _{k}^{n}{F}_{ix}\left({T}^{k}\right).$

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## Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039) and China Postdoctoral Science Foundation (Grant No. 20100470783).

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Correspondence to Zhi-Fang Li.

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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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Deng, BC., Chen, T. & Li, ZF. Viscosity iteration algorithm for a ϱ-strictly pseudononspreading mapping in a Hilbert space. J Inequal Appl 2013, 80 (2013). https://doi.org/10.1186/1029-242X-2013-80

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