# Algorithms with variant anchors for pseudocontractive mappings

## Abstract

The purpose of this paper is to find the fixed points of pseudocontractive mappings by using the iterative technique. Two algorithms with variant anchors have been introduced. Strong convergence results are given. Especially, we can find the minimum-norm fixed point of pseudocontractive mappings.

MSC:47H05, 47H10, 47H17.

## 1 Introduction

In this paper, we assume that H is a real Hilbert space and $C\subset H$ is a nonempty closed convex subset. Recall that a mapping $T:C\to C$ is said to be Lipschitzian if

$\parallel Tu-T{u}^{†}\parallel \le \kappa \parallel u-{u}^{†}\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u,{u}^{†}\in C,$

where $\kappa >0$ is a constant, which is in general called the Lipschitz constant. If $\kappa =1$, T is called nonexpansive.

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

$〈Tu-T{u}^{†},u-{u}^{†}〉\le {\parallel u-{u}^{†}\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }u,{u}^{†}\in C.$

We use $Fix\left(T\right)$ to denote the set of fixed points of T.

In the literature, there are a large number references associated with the fixed point algorithms for the pseudocontractive mappings. See, for instance, . (The interest of pseudocontractions lies in their connection with monotone operators; namely, T is a pseudocontraction if and only if the complement $I-T$ is a monotone operator.)

Now there exists an example which shows that Mann iteration does not converge for the pseudocontractive mappings . At present, it is still an interesting topic to construct algorithms for finding the fixed points of the pseudocontractive mappings.

On the other hand, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. Recently, in order to find the fixed points of the nonexpansive mappings, Yao and Shahzad  introduced the following algorithms with perturbations and obtained the strong convergence results.

Algorithm 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a nonexpansive mapping. For given ${x}_{0}\in C$, define a sequence $\left\{{x}_{m}\right\}$ in the following manner:

${x}_{m}={proj}_{C}\left[{\alpha }_{m}{u}_{m}+\left(1-{\alpha }_{m}\right)T{x}_{m}\right],\phantom{\rule{1em}{0ex}}m\ge 0,$
(1.1)

where $\left\{{\alpha }_{m}\right\}$ is a sequence in $\left[0,1\right]$ and the sequence $\left\{{u}_{m}\right\}\subset H$ is a small perturbation for the m-step iteration satisfying $\parallel {u}_{m}\parallel \to 0$ as $m\to \mathrm{\infty }$.

Theorem 1.2 Suppose $Fix\left(T\right)\ne \mathrm{\varnothing }$. Then, as ${\alpha }_{m}\to 0$, the sequence $\left\{{x}_{m}\right\}$ generated by the implicit method (1.1) converges to $\stackrel{˜}{x}\in Fix\left(T\right)$, which is the minimum-norm fixed point of T.

Algorithm 1.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a nonexpansive mapping. For given ${x}_{0}\in C$, define a sequence $\left\{{x}_{n}\right\}$ in the following manner:

${x}_{n+1}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}{proj}_{C}\left[{\alpha }_{n}{u}_{n}+\left(1-{\alpha }_{n}\right)T{x}_{n}\right],\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.2)

where $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ are two sequences in $\left(0,1\right)$ and the sequence $\left\{{u}_{n}\right\}\subset H$ is a perturbation for the n-step iteration.

Theorem 1.4 Suppose that $Fix\left(T\right)\ne \mathrm{\varnothing }$. Assume the following conditions are satisfied:

1. (i)

${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$;

2. (ii)

$0<{lim inf}_{n\to \mathrm{\infty }}{\beta }_{n}\le {lim sup}_{n\to \mathrm{\infty }}{\beta }_{n}<1$;

3. (iii)

${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}\parallel {u}_{n}\parallel <\mathrm{\infty }$.

Then the sequence $\left\{{x}_{n}\right\}$ generated by the explicit iterative method (1.2) converges to $\stackrel{˜}{x}\in Fix\left(T\right)$, which is the minimum-norm fixed point of T.

Note that the idea of the iterative algorithms with perturbations has been extended to the other topics, see, for example, .

Motivated by the above ideas and the results in the literature, in the present paper, we present two algorithms with variant anchors for finding the fixed points of the pseudocontractive mappings in Hilbert spaces. Strong convergence results are given. As special cases, we can find the minimum-norm fixed point of the pseudocontractive mappings.

## 2 Preliminaries

Recall that the metric projection ${proj}_{C}:H\to C$ is defined by

${proj}_{C}x:=arg\underset{\mathrm{\forall }y\in C}{min}\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}x\in H.$

It is obvious that ${proj}_{C}$ satisfies

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

and is characterized by

${proj}_{C}x\in C,\phantom{\rule{1em}{0ex}}〈x-{proj}_{C}x,y-{proj}_{C}x〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }y\in C.$

The following two lemmas will be useful for our main results.

Lemma 2.1 ()

Let C be a closed convex subset of a Hilbert space H. Let $T:C\to C$ be a Lipschitzian pseudocontractive mapping. Then $Fix\left(T\right)$ is a closed convex subset of C and the mapping $I-T$ is demiclosed at 0, i.e., whenever $\left\{{x}_{n}\right\}\subset C$ is such that ${x}_{n}⇀x$ and $\left(I-T\right){x}_{n}\to 0$, then $\left(I-T\right)x=0$.

Lemma 2.2 ()

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

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

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

1. (i)

${\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_{n}=\mathrm{\infty }$;

2. (ii)

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

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

## 3 Main results

In the sequel, we assume that C is a nonempty closed convex subset of a real Hilbert space H and $T:C\to C$ is a κ-Lipschitzian pseudocontractive mapping with nonempty fixed points set $Fix\left(T\right)$.

The first result is on the convergence of the path for the pseudocontractive mappings. Now, we define our path as follows.

For fixed $\zeta ,t\in \left(0,1\right)$ and ${u}_{t}\in H$, we define a mapping ${G}_{t}:C\to C$ by

${G}_{t}x=\left(1-\zeta \right){proj}_{C}\left[t{u}_{t}+\left(1-t\right)x\right]+\zeta Tx,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in C,$

where ${proj}_{C}:H\to C$ is the metric projection from H on C.

Next, we show that the mapping ${G}_{t}$ is strongly pseudocontractive. Indeed, for $x,y\in C$, we have

$\begin{array}{rcl}〈{G}_{t}x-{G}_{t}y,x-y〉& =& \left(1-\zeta \right)〈{proj}_{C}\left[t{u}_{t}+\left(1-t\right)x\right]-{proj}_{C}\left[t{u}_{t}+\left(1-t\right)y\right],x-y〉\\ +\zeta 〈Tx-Ty,x-y〉\\ \le & \left(1-\zeta \right)\parallel {proj}_{C}\left[t{u}_{t}+\left(1-t\right)x\right]-{proj}_{C}\left[t{u}_{t}+\left(1-t\right)y\right]\parallel \parallel x-y\parallel \\ +\zeta {\parallel x-y\parallel }^{2}\\ \le & \left(1-\zeta \right)\left(1-t\right){\parallel x-y\parallel }^{2}+\zeta {\parallel x-y\parallel }^{2}\\ =& \left[1-\left(1-\zeta \right)t\right]{\parallel x-y\parallel }^{2}.\end{array}$

Since $\zeta ,t\in \left(0,1\right)$, $1-\left(1-\zeta \right)t\in \left(0,1\right)$. Hence, ${G}_{t}$ is a strongly pseudocontractive mapping. By , ${G}_{t}$ has a unique fixed point ${x}_{t}\in C$. That is, ${x}_{t}$ satisfies

${x}_{t}=\left(1-\zeta \right){proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]+\zeta T{x}_{t},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left(0,1\right).$
(3.1)

Remark 3.1 ${u}_{t}\in H$ can be seen as a perturbation.

Next, we prove the convergence of the path (3.1).

Theorem 3.2 If ${lim}_{t\to 0}{u}_{t}=u\in H$, then the path $\left\{{x}_{t}\right\}$ defined by (3.1) converges strongly to ${proj}_{Fix\left(T\right)}\left(u\right)$.

Proof Let $p\in Fix\left(T\right)$. We get from (3.1) that

$\begin{array}{rcl}{\parallel {x}_{t}-p\parallel }^{2}& =& \left(1-\zeta \right)〈{proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]-p,{x}_{t}-p〉+\zeta 〈T{x}_{t}-p,{x}_{t}-p〉\\ \le & \left(1-\zeta \right)\parallel {proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]-p\parallel \parallel {x}_{t}-p\parallel +\zeta {\parallel {x}_{t}-p\parallel }^{2}\\ \le & \left(1-\zeta \right)\parallel t\left({u}_{t}-p\right)+\left(1-t\right)\left({x}_{t}-p\right)\parallel \parallel {x}_{t}-p\parallel +\zeta {\parallel {x}_{t}-p\parallel }^{2}\\ \le & \left(1-\zeta \right)\left[\left(1-t\right)\parallel {x}_{t}-p\parallel +t\parallel {u}_{t}-p\parallel \right]\parallel {x}_{t}-p\parallel +\zeta {\parallel {x}_{t}-p\parallel }^{2}.\end{array}$

It follows that

$\parallel {x}_{t}-p\parallel \le \parallel {u}_{t}-p\parallel .$

Since ${lim}_{t\to 0}{u}_{t}=u\in H$, there exists a constant $M>0$ such that ${sup}_{t\in \left(0,1\right)}\parallel {u}_{t}-u\parallel \le M$. So,

$\parallel {x}_{t}-p\parallel \le \parallel {u}_{t}-p\parallel \le \parallel {u}_{t}-u\parallel +\parallel u-p\parallel \le M+\parallel u-p\parallel .$

Thus, $\left\{{x}_{t}\right\}$ is bounded.

By (3.1), we have

$\begin{array}{rcl}\parallel {x}_{t}-T{x}_{t}\parallel & =& \parallel \left(1-\zeta \right){proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]+\zeta T{x}_{t}-T{x}_{t}\parallel \\ \le & \left(1-\zeta \right)\parallel {proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]-T{x}_{t}\parallel \\ \le & \left(1-\zeta \right)\left[\parallel {x}_{t}-T{x}_{t}\parallel +t\parallel {u}_{t}-{x}_{t}\parallel \right].\end{array}$

Therefore,

(3.2)

Let $\left\{{t}_{n}\right\}\subset \left(0,1\right)$ be a sequence satisfying ${t}_{n}\to {0}^{+}$ as $n\to \mathrm{\infty }$. Put ${x}_{n}:={x}_{{t}_{n}}$. By (3.2), we get

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-T{x}_{n}\parallel =0.$
(3.3)

By (3.1), we obtain

$\begin{array}{rcl}{\parallel {x}_{t}-p\parallel }^{2}& =& \left(1-\zeta \right)〈{proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]-p,{x}_{t}-p〉+\zeta 〈T{x}_{t}-p,{x}_{t}-p〉\\ \le & \left(1-\zeta \right)\parallel {proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]-p\parallel \parallel {x}_{t}-p\parallel +\zeta {\parallel {x}_{t}-p\parallel }^{2}\\ \le & \frac{1-\zeta }{2}\left({\parallel {proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]-p\parallel }^{2}+{\parallel {x}_{t}-p\parallel }^{2}\right)+\zeta {\parallel {x}_{t}-p\parallel }^{2}.\end{array}$

Hence,

$\begin{array}{rcl}{\parallel {x}_{t}-p\parallel }^{2}& \le & {\parallel {proj}_{C}\left[t{u}_{t}+\left(1-t\right){x}_{t}\right]-p\parallel }^{2}\\ \le & {\parallel {x}_{t}-p+t\left({u}_{t}-{x}_{t}\right)\parallel }^{2}\\ =& {\parallel {x}_{t}-p\parallel }^{2}+2t〈{u}_{t}-{x}_{t},{x}_{t}-p〉+{t}^{2}{\parallel {u}_{t}-{x}_{t}\parallel }^{2}\\ =& {\parallel {x}_{t}-p\parallel }^{2}-2t〈{x}_{t}-p,{x}_{t}-p〉+2t〈{u}_{t}-p,{x}_{t}-p〉+{t}^{2}{\parallel {u}_{t}-{x}_{t}\parallel }^{2}\\ =& \left(1-2t\right){\parallel {x}_{t}-p\parallel }^{2}+2t〈{u}_{t}-p,{x}_{t}-p〉+{t}^{2}{\parallel {u}_{t}-{x}_{t}\parallel }^{2}.\end{array}$

It follows that

$\begin{array}{rcl}{\parallel {x}_{t}-p\parallel }^{2}& \le & 〈{u}_{t}-p,{x}_{t}-p〉+\frac{t}{2}{\parallel {u}_{t}-{x}_{t}\parallel }^{2}\\ \le & 〈{u}_{t}-p,{x}_{t}-p〉+t{M}_{1}.\end{array}$
(3.4)

Here ${M}_{1}>0$ is a constant such that ${sup}_{t\in \left(0,1\right)}\frac{{\parallel {u}_{t}-{x}_{t}\parallel }^{2}}{2}\le {M}_{1}$. In particular, we obtain

${\parallel {x}_{n}-p\parallel }^{2}\le 〈{u}_{n}-p,{x}_{n}-p〉+{t}_{n}{M}_{1},\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in Fix\left(T\right).$
(3.5)

Since $\left\{{x}_{n}\right\}$ is bounded, there exists a subsequence $\left\{{x}_{{n}_{i}}\right\}$ of $\left\{{x}_{n}\right\}$ satisfying ${x}_{{n}_{i}}\to {x}^{\ast }\in C$ weakly. By (3.3), we get

$\underset{i\to \mathrm{\infty }}{lim}\parallel {x}_{{n}_{i}}-T{x}_{{n}_{i}}\parallel =0.$
(3.6)

Applying Lemma 2.1 to (3.6) to deduce ${x}^{\ast }\in Fix\left(T\right)$.

By (3.5), we derive

${\parallel {x}_{{n}_{i}}-{x}^{\ast }\parallel }^{2}\le 〈{u}_{{n}_{i}}-{x}^{\ast },{x}_{{n}_{i}}-{x}^{\ast }〉+{t}_{{n}_{i}}{M}_{1}.$
(3.7)

Since ${u}_{{n}_{i}}-{x}^{\ast }\to u-{x}^{\ast }$ and ${t}_{{n}_{i}}\to 0$, we deduce that ${x}_{{n}_{i}}\to {x}^{\ast }$ by (3.7). By (3.5), we have

${\parallel {x}^{\ast }-p\parallel }^{2}\le 〈u-p,{x}^{\ast }-p〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in Fix\left(T\right).$
(3.8)

Assume that there exists another subsequence $\left\{{x}_{{n}_{j}}\right\}$ of $\left\{{x}_{n}\right\}$ satisfying ${x}_{{n}_{j}}\to {x}^{†}$ weakly. Similarly, we can prove that ${x}_{{n}_{j}}\to {x}^{†}\in Fix\left(T\right)$, which satisfies

${\parallel {x}^{†}-p\parallel }^{2}\le 〈u-p,{x}^{†}-p〉,\phantom{\rule{1em}{0ex}}\mathrm{\forall }p\in Fix\left(T\right).$
(3.9)

In (3.8), we pick up $p={x}^{†}$ to get

${\parallel {x}^{\ast }-{x}^{†}\parallel }^{2}\le 〈u-{x}^{†},{x}^{\ast }-{x}^{†}〉.$
(3.10)

In (3.9), we pick up $p={x}^{\ast }$ to get

${\parallel {x}^{†}-{x}^{\ast }\parallel }^{2}\le 〈u-{x}^{\ast },{x}^{†}-{x}^{\ast }〉.$
(3.11)

Adding (3.10) and (3.11), we deduce

${\parallel {x}^{†}-{x}^{\ast }\parallel }^{2}\le 0.$

Thus, ${x}^{\ast }={x}^{†}$. This indicates that the weak limit set of $\left\{{x}_{n}\right\}$ is singleton and the path $\left\{{x}_{t}\right\}$ converges strongly to ${x}^{\ast }={proj}_{Fix\left(T\right)}\left(u\right)$ by (3.8). This completes the proof. □

Corollary 3.3 The path $\left\{{x}_{t}\right\}$ defined by

${x}_{t}=\left(1-\zeta \right){proj}_{C}\left[\left(1-t\right){x}_{t}\right]+\zeta T{x}_{t},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in \left(0,1\right),$

converges strongly to ${proj}_{Fix\left(T\right)}\left(0\right)$, which is the minimum-norm fixed point of T.

Now, we introduce another algorithm, which is an explicit manner.

Algorithm 3.4 Let $\left\{{\varsigma }_{n}\right\}$ and $\left\{{\zeta }_{n}\right\}$ be two real number sequences in $\left(0,1\right)$. Let $\left\{{u}_{n}\right\}\subset H$ be a sequence. For ${x}_{0}\in C$ arbitrarily, let the sequence $\left\{{x}_{n}\right\}$ be generated by

${x}_{n+1}=\left(1-{\zeta }_{n}\right){proj}_{C}\left[{\varsigma }_{n}{u}_{n}+\left(1-{\varsigma }_{n}\right){x}_{n}\right]+{\zeta }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.$
(3.12)

Theorem 3.5 Assume the following conditions are satisfied:

1. (C1)

${lim}_{n\to \mathrm{\infty }}{\varsigma }_{n}={lim}_{n\to \mathrm{\infty }}\frac{{\varsigma }_{n}}{{\zeta }_{n}}={lim}_{n\to \mathrm{\infty }}\frac{{\zeta }_{n}^{2}}{{\varsigma }_{n}}=0$;

2. (C2)

${lim}_{n\to \mathrm{\infty }}{u}_{n}=u\in H$.

Then we have

1. (1)

the sequence $\left\{{x}_{n}\right\}$ is bounded;

2. (2)

the sequence $\left\{{x}_{n}\right\}$ is asymptotically regular, that is, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n+1}-{x}_{n}\parallel =0$.

Further, if ${\sum }_{n=0}^{\mathrm{\infty }}{\varsigma }_{n}=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}\frac{\parallel {x}_{n+1}-{x}_{n}\parallel }{{\zeta }_{n}}=0$, then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${proj}_{Fix\left(T\right)}\left(u\right)$.

Proof By the condition (C1), we can find a sufficiently large positive integer m such that

$1-\frac{1}{1/2-{\zeta }_{m}}\left(\kappa +1\right)\left(\kappa +2\right)\left({\varsigma }_{m}+2{\zeta }_{m}+\frac{{\zeta }_{m}^{2}}{{\varsigma }_{m}}\right)>0.$
(3.13)

Let $p\in Fix\left(T\right)$. For fixed m, we pick up a constant ${M}_{2}>0$ such that

$max\left\{\parallel {x}_{0}-p\parallel ,\parallel {x}_{1}-p\parallel ,\dots ,\parallel {x}_{m-1}-p\parallel ,4\parallel {x}_{m}-p\parallel +4\parallel {u}_{m}-p\parallel \right\}\le {M}_{2}.$
(3.14)

Next, we show that $\parallel {x}_{m+1}-p\parallel \le {M}_{2}$. Set ${y}_{n}={proj}_{C}\left[{\varsigma }_{n}{u}_{n}+\left(1-{\varsigma }_{n}\right){x}_{n}\right]$ for all $n\ge 0$. Thus, we have ${x}_{n+1}=\left(1-{\zeta }_{n}\right){y}_{n}+{\zeta }_{n}T{x}_{n}$ for all $n\ge 0$.

Since $I-T$ is monotone, we have

$〈\left(I-T\right){x}_{m+1},{x}_{m+1}-p〉=〈\left(I-T\right){x}_{m+1}-\left(I-T\right)p,{x}_{m+1}-p〉\ge 0.$

By (3.12), we obtain

$\begin{array}{rcl}{\parallel {x}_{m+1}-p\parallel }^{2}& =& \left(1-{\zeta }_{m}\right)〈{y}_{m}-p,{x}_{m+1}-p〉+{\zeta }_{m}〈T{x}_{m}-p,{x}_{m+1}-p〉\\ =& \left(1-{\zeta }_{m}\right)〈{y}_{m}-{\varsigma }_{m}{u}_{m}-\left(1-{\varsigma }_{m}\right){x}_{m},{x}_{m+1}-p〉\\ +\left(1-{\zeta }_{m}\right)〈{\varsigma }_{m}{u}_{m}+\left(1-{\varsigma }_{m}\right){x}_{m}-p,{x}_{m+1}-p〉\\ +{\zeta }_{m}〈T{x}_{m}-p,{x}_{m+1}-p〉\\ =& \left(1-{\zeta }_{m}\right)〈{y}_{m}-{\varsigma }_{m}{u}_{m}-\left(1-{\varsigma }_{m}\right){x}_{m},{x}_{m+1}-p〉\\ +\left(1-{\zeta }_{m}\right)〈{x}_{m}-p,{x}_{m+1}-p〉+\left(1-{\zeta }_{m}\right){\varsigma }_{m}〈{u}_{m}-{x}_{m},{x}_{m+1}-p〉\\ +{\zeta }_{m}〈T{x}_{m}-p,{x}_{m+1}-p〉\\ =& \left(1-{\zeta }_{m}\right)〈{y}_{m}-{\varsigma }_{m}{u}_{m}-\left(1-{\varsigma }_{m}\right){x}_{m},{x}_{m+1}-p〉\\ +〈{x}_{m}-p,{x}_{m+1}-p〉-\left(1-{\zeta }_{m}\right){\varsigma }_{m}〈{x}_{m+1}-p,{x}_{m+1}-p〉\\ -\left(1-{\zeta }_{m}\right){\varsigma }_{m}〈{x}_{m}-{x}_{m+1},{x}_{m+1}-p〉-\left(1-{\zeta }_{m}\right){\varsigma }_{m}〈p-{u}_{m},{x}_{m+1}-p〉\\ +{\zeta }_{m}〈T{x}_{m}-T{x}_{m+1},{x}_{m+1}-p〉+{\zeta }_{m}〈{x}_{m+1}-{x}_{m},{x}_{m+1}-p〉\\ -{\zeta }_{m}〈{x}_{m+1}-T{x}_{m+1},{x}_{m+1}-p〉.\end{array}$

Note that

$\begin{array}{rl}\parallel {y}_{m}-{\varsigma }_{m}{u}_{m}-\left(1-{\varsigma }_{m}\right){x}_{m}\parallel & \le \parallel {y}_{m}-{x}_{m}\parallel +{\varsigma }_{m}\parallel {x}_{m}-{u}_{m}\parallel \\ =\parallel {proj}_{C}\left[{\varsigma }_{m}{u}_{m}+\left(1-{\varsigma }_{m}\right){x}_{m}\right]-{x}_{m}\parallel +{\varsigma }_{m}\parallel {x}_{m}-{u}_{m}\parallel \\ \le 2{\varsigma }_{m}\parallel {x}_{m}-{u}_{m}\parallel .\end{array}$

Then we have

$\begin{array}{rcl}{\parallel {x}_{m+1}-p\parallel }^{2}& \le & \left(1-{\zeta }_{m}\right)\parallel {y}_{m}-{\varsigma }_{m}{u}_{m}-\left(1-{\varsigma }_{m}\right){x}_{m}\parallel \parallel {x}_{m+1}-p\parallel \\ +\parallel {x}_{m}-p\parallel \parallel {x}_{m+1}-p\parallel -\left(1-{\zeta }_{m}\right){\varsigma }_{m}{\parallel {x}_{m+1}-p\parallel }^{2}\\ +\left(1-{\zeta }_{m}\right){\varsigma }_{m}\left(\parallel {x}_{m+1}-{x}_{m}\parallel +\parallel {u}_{m}-p\parallel \right)\parallel {x}_{m+1}-p\parallel \\ +{\zeta }_{m}\left(\parallel T{x}_{m}-T{x}_{m+1}\parallel +\parallel {x}_{m+1}-{x}_{m}\parallel \right)\parallel {x}_{m+1}-p\parallel \\ \le & 2\left(1-{\zeta }_{m}\right){\varsigma }_{m}\parallel {x}_{m}-{u}_{m}\parallel \parallel {x}_{m+1}-p\parallel +\parallel {x}_{m}-p\parallel \parallel {x}_{m+1}-p\parallel \\ +\left(1-{\zeta }_{m}\right){\varsigma }_{m}\left(\parallel {x}_{m+1}-{x}_{m}\parallel +\parallel {u}_{m}-p\parallel \right)\parallel {x}_{m+1}-p\parallel \\ -\left(1-{\zeta }_{m}\right){\varsigma }_{m}{\parallel {x}_{m+1}-p\parallel }^{2}+{\zeta }_{m}\left(\kappa +1\right)\parallel {x}_{m+1}-{x}_{m}\parallel \parallel {x}_{m+1}-p\parallel \\ \le & \parallel {x}_{m}-p\parallel \parallel {x}_{m+1}-p\parallel +2\left(1-{\zeta }_{m}\right){\varsigma }_{m}\left(\parallel {x}_{m}-p\parallel +\parallel {u}_{m}-p\parallel \right)\parallel {x}_{m+1}-p\parallel \\ -\left(1-{\zeta }_{m}\right){\varsigma }_{m}{\parallel {x}_{m+1}-p\parallel }^{2}+\left({\varsigma }_{m}+{\zeta }_{m}\right)\left(\kappa +1\right)\parallel {x}_{m+1}-{x}_{m}\parallel \parallel {x}_{m+1}-p\parallel .\end{array}$

Hence,

$\begin{array}{rcl}\left[1+\left(1-{\zeta }_{m}\right){\varsigma }_{m}\right]\parallel {x}_{m+1}-p\parallel & \le & \parallel {x}_{m}-p\parallel +2{\varsigma }_{m}\left(\parallel {x}_{m}-p\parallel +\parallel {u}_{m}-p\parallel \right)\\ +\left(\kappa +1\right)\left({\varsigma }_{m}+{\zeta }_{m}\right)\parallel {x}_{m+1}-{x}_{m}\parallel .\end{array}$
(3.15)

By (3.12), we have

$\begin{array}{rcl}\parallel {x}_{m+1}-{x}_{m}\parallel & \le & \left(1-{\zeta }_{m}\right)\parallel {proj}_{C}\left[{\varsigma }_{m}{u}_{m}+\left(1-{\varsigma }_{m}\right){x}_{m}\right]-{x}_{m}\parallel +{\zeta }_{m}\parallel T{x}_{m}-{x}_{m}\parallel \\ \le & \left(1-{\zeta }_{m}\right){\varsigma }_{m}\left(\parallel {x}_{m}-p\parallel +\parallel {u}_{m}-p\parallel \right)+{\zeta }_{m}\left(\parallel T{x}_{m}-p\parallel +\parallel p-{x}_{m}\parallel \right)\\ \le & {\varsigma }_{m}\left(\parallel {x}_{m}-p\parallel +\parallel {u}_{m}-p\parallel \right)+{\zeta }_{m}\left(\kappa +1\right)\parallel {x}_{m}-p\parallel \\ \le & \left(\kappa +1\right)\left({\varsigma }_{m}+{\zeta }_{m}\right)\parallel {x}_{m}-p\parallel +{\varsigma }_{m}\parallel {u}_{m}-p\parallel \\ \le & \left(\kappa +2\right)\left({\varsigma }_{m}+{\zeta }_{m}\right){M}_{2}.\end{array}$
(3.16)

From condition (C1), we deduce ${\varsigma }_{m}\to 0$ and ${\zeta }_{m}\to 0$ as $m\to \mathrm{\infty }$. Therefore, we get

$\underset{m\to \mathrm{\infty }}{lim}\parallel {x}_{m+1}-{x}_{m}\parallel =0.$

That is, the sequence $\left\{{x}_{m}\right\}$ is asymptotically regular.

By (3.15) and (3.16), we have

$\begin{array}{r}\left[1+\left(1-{\zeta }_{m}\right){\varsigma }_{m}\right]\parallel {x}_{m+1}-p\parallel \\ \phantom{\rule{1em}{0ex}}\le \parallel {x}_{m}-p\parallel +{\varsigma }_{m}\left(2\parallel {x}_{m}-p\parallel +2\parallel {u}_{m}-p\parallel \right)+\left(\kappa +1\right)\left(\kappa +2\right){\left({\varsigma }_{m}+{\zeta }_{m}\right)}^{2}{M}_{2}\\ \phantom{\rule{1em}{0ex}}\le \left(1+\frac{1}{2}{\varsigma }_{m}\right){M}_{2}+\left(\kappa +1\right)\left(\kappa +2\right){\left({\varsigma }_{m}+{\zeta }_{m}\right)}^{2}{M}_{2}.\end{array}$

This together with (3.13) and (3.14) imply that

$\begin{array}{rcl}\parallel {x}_{m+1}-p\parallel & \le & \left[1-\frac{\left(1/2-{\zeta }_{m}\right){\varsigma }_{m}-\left(\kappa +1\right)\left(\kappa +2\right){\left({\varsigma }_{m}+{\zeta }_{m}\right)}^{2}}{1+\left(1-{\zeta }_{m}\right){\varsigma }_{m}}\right]{M}_{2}\\ =& \left\{1-\frac{\left(1/2-{\zeta }_{m}\right){\varsigma }_{m}\left[1-\frac{1}{1/2-{\zeta }_{m}}\left(\kappa +1\right)\left(\kappa +2\right)\left({\varsigma }_{m}+2{\zeta }_{m}+\left({\zeta }_{m}^{2}/{\varsigma }_{m}\right)\right)\right]}{1+\left(1-{\zeta }_{m}\right){\varsigma }_{m}}\right\}{M}_{2}\\ \le & {M}_{2}.\end{array}$

By induction, we get

$\parallel {x}_{n}-p\parallel \le {M}_{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0.$

So $\left\{{x}_{n}\right\}$ is bounded.

By (3.12), we have

$\begin{array}{rcl}\parallel {x}_{n}-T{x}_{n}\parallel & \le & \parallel {x}_{n}-{x}_{n+1}\parallel +\parallel {x}_{n+1}-T{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{x}_{n+1}\parallel +\left(1-{\zeta }_{n}\right)\parallel {proj}_{C}\left[{\varsigma }_{n}{u}_{n}+\left(1-{\varsigma }_{n}\right){x}_{n}\right]-T{x}_{n}\parallel \\ \le & \parallel {x}_{n}-{x}_{n+1}\parallel +\left(1-{\zeta }_{n}\right)\parallel {x}_{n}-T{x}_{n}\parallel +{\varsigma }_{n}\parallel {x}_{n}-{u}_{n}\parallel .\end{array}$

It follows that

$\parallel {x}_{n}-T{x}_{n}\parallel \le \frac{1}{{\zeta }_{n}}\parallel {x}_{n}-{x}_{n+1}\parallel +\frac{{\varsigma }_{n}}{{\zeta }_{n}}\parallel {x}_{n}-{u}_{n}\parallel .$

By the condition ${lim}_{n\to \mathrm{\infty }}\frac{{\varsigma }_{n}}{{\zeta }_{n}}=0$ and the assumption ${lim}_{n\to \mathrm{\infty }}\frac{\parallel {x}_{n+1}-{x}_{n}\parallel }{{\zeta }_{n}}=0$, we deduce

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{n}-T{x}_{n}\parallel =0.$
(3.17)

Let the net $\left\{{z}_{t}\right\}$ be defined by ${z}_{t}=\left(1-\zeta \right){proj}_{C}\left[t{u}_{t}+\left(1-t\right){z}_{t}\right]+\zeta T{z}_{t}$. By Theorem 3.2, we know that ${z}_{t}$ converges strongly to ${proj}_{Fix\left(T\right)}\left(u\right)$. Next, we prove

$\underset{n\to \mathrm{\infty }}{lim sup}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{proj}_{Fix\left(T\right)}\left(u\right)-{y}_{n}〉\le 0.$

By the definition of $\left\{{z}_{t}\right\}$, we have

${z}_{t}-{x}_{n}=\left(1-\zeta \right)\left({proj}_{C}\left[t{u}_{t}+\left(1-t\right){z}_{t}\right]-{x}_{n}\right)+\zeta \left(T{z}_{t}-T{x}_{n}\right)+\zeta \left(T{x}_{n}-{x}_{n}\right).$

It follows that

$\begin{array}{rcl}{\parallel {z}_{t}-{x}_{n}\parallel }^{2}& =& \left(1-\zeta \right)〈{proj}_{C}\left[t{u}_{t}+\left(1-t\right){z}_{t}\right]-{x}_{n},{z}_{t}-{x}_{n}〉+\zeta 〈T{z}_{t}-T{x}_{n},{z}_{t}-{x}_{n}〉\\ +\zeta 〈T{x}_{n}-{x}_{n},{z}_{t}-{x}_{n}〉\\ =& \left(1-\zeta \right)〈{proj}_{C}\left[t{u}_{t}+\left(1-t\right){z}_{t}\right]-t{u}_{t}-\left(1-t\right){z}_{t},{z}_{t}-{x}_{n}〉\\ +\left(1-\zeta \right)〈t{u}_{t}+\left(1-t\right){z}_{t}-{x}_{n},{z}_{t}-{x}_{n}〉+\zeta 〈T{z}_{t}-T{x}_{n},{z}_{t}-{x}_{n}〉\\ +\zeta 〈T{x}_{n}-{x}_{n},{z}_{t}-{x}_{n}〉.\end{array}$

Since ${x}_{n}\in C$, by the characteristic inequality of metric projection, we have

$〈{proj}_{C}\left[t{u}_{t}+\left(1-t\right){z}_{t}\right]-t{u}_{t}-\left(1-t\right){z}_{t},{z}_{t}-{x}_{n}〉\le 0.$

Then

$\begin{array}{rcl}{\parallel {z}_{t}-{x}_{n}\parallel }^{2}& \le & \left(1-\zeta \right)〈t{u}_{t}+\left(1-t\right){z}_{t}-{x}_{n},{z}_{t}-{x}_{n}〉+\zeta {\parallel {z}_{t}-{x}_{n}\parallel }^{2}\\ +\zeta \parallel T{x}_{n}-{x}_{n}\parallel \parallel {z}_{t}-{x}_{n}\parallel \\ =& \left(1-\zeta \right){\parallel {z}_{t}-{x}_{n}\parallel }^{2}-\left(1-\zeta \right)t〈{z}_{t}-{u}_{t},{z}_{t}-{x}_{n}〉+\zeta {\parallel {z}_{t}-{x}_{n}\parallel }^{2}\\ +\zeta \parallel T{x}_{n}-{x}_{n}\parallel \parallel {z}_{t}-{x}_{n}\parallel ,\end{array}$

which implies that

$〈{z}_{t}-{u}_{t},{z}_{t}-{x}_{n}〉\le \frac{\zeta }{\left(1-\zeta \right)t}\parallel T{x}_{n}-{x}_{n}\parallel \parallel {z}_{t}-{x}_{n}\parallel .$

By (3.17), we deduce

$\underset{t\to 0}{lim sup}\phantom{\rule{0.2em}{0ex}}\underset{n\to \mathrm{\infty }}{lim sup}〈{z}_{t}-{u}_{t},{z}_{t}-{x}_{n}〉\le 0.$
(3.18)

Note the fact that the two limits ${lim sup}_{t\to 0}$ and ${lim sup}_{n\to \mathrm{\infty }}$ are interchangeable. This together with ${z}_{t}\to {proj}_{Fix\left(T\right)}\left(u\right)$, ${u}_{t}\to u$ and (3.18) implies that

$\underset{n\to \mathrm{\infty }}{lim sup}〈{proj}_{Fix\left(T\right)}\left(u\right)-u,{proj}_{Fix\left(T\right)}\left(u\right)-{x}_{n}〉\le 0.$

Note that $\parallel {y}_{n}-{x}_{n}\parallel \to 0$ and ${u}_{n}-u\to 0$. We derive

$\underset{n\to \mathrm{\infty }}{lim sup}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{proj}_{Fix\left(T\right)}\left(u\right)-{y}_{n}〉\le 0.$

Finally, we prove that ${x}_{n}\to {proj}_{Fix\left(T\right)}\left(u\right)$. Note that

$\begin{array}{r}〈T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right),{x}_{n+1}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{1em}{0ex}}=〈T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right),{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉+〈T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right),{x}_{n+1}-{x}_{n}〉\\ \phantom{\rule{1em}{0ex}}\le {\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}+\parallel T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel \parallel {x}_{n+1}-{x}_{n}\parallel \end{array}$
(3.19)

and

$\begin{array}{r}{\parallel {y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}=〈{y}_{n}-{\varsigma }_{n}{u}_{n}-\left(1-{\varsigma }_{n}\right){x}_{n},{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{2em}{0ex}}+〈{\varsigma }_{n}{u}_{n}+\left(1-{\varsigma }_{n}\right){x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right),{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{1em}{0ex}}\le 〈{\varsigma }_{n}{u}_{n}+\left(1-{\varsigma }_{n}\right){x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right),{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{1em}{0ex}}=\left(1-{\varsigma }_{n}\right)〈{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right),{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{2em}{0ex}}-{\varsigma }_{n}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{1em}{0ex}}\le \frac{\left(1-{\varsigma }_{n}\right)}{2}{\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}+\frac{1}{2}{\parallel {y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}-{\varsigma }_{n}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉.\end{array}$

Then

$\begin{array}{rl}{\parallel {y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\le & \left(1-{\varsigma }_{n}\right){\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\\ -2{\varsigma }_{n}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉.\end{array}$
(3.20)

By (3.12), (3.16), and (3.20), we get

$\begin{array}{r}{\parallel {x}_{n+1}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}={\parallel \left(1-{\zeta }_{n}\right)\left({y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\right)+{\zeta }_{n}\left(T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\right)\parallel }^{2}\\ \phantom{\rule{1em}{0ex}}\le {\parallel \left(1-{\zeta }_{n}\right)\left({y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\right)\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}+2{\zeta }_{n}〈T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right),{x}_{n+1}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{1em}{0ex}}\le {\left(1-{\zeta }_{n}\right)}^{2}\left(1-{\varsigma }_{n}\right){\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}+2{\zeta }_{n}{\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}-2{\varsigma }_{n}{\left(1-{\zeta }_{n}\right)}^{2}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{y}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)〉\\ \phantom{\rule{2em}{0ex}}+2{\zeta }_{n}\parallel T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel \parallel {x}_{n+1}-{x}_{n}\parallel \\ \phantom{\rule{1em}{0ex}}\le \left[1-\left(1-2{\zeta }_{n}\right){\varsigma }_{n}\right]{\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}+{\zeta }_{n}^{2}{\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\\ \phantom{\rule{2em}{0ex}}+2{\varsigma }_{n}{\left(1-{\zeta }_{n}\right)}^{2}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{proj}_{Fix\left(T\right)}\left(u\right)-{y}_{n}〉\\ \phantom{\rule{2em}{0ex}}+2{\zeta }_{n}\parallel T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel \left(\kappa +2\right)\left({\varsigma }_{n}+{\zeta }_{n}\right){M}_{2}\\ \phantom{\rule{1em}{0ex}}=\left(1-{\gamma }_{n}\right){\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}+{\gamma }_{n}{\delta }_{n},\end{array}$
(3.21)

where ${\gamma }_{n}=\left(1-2{\zeta }_{n}\right){\varsigma }_{n}$ and

$\begin{array}{rl}{\delta }_{n}=& \frac{2{\left(1-{\zeta }_{n}\right)}^{2}}{1-2{\zeta }_{n}}〈{proj}_{Fix\left(T\right)}\left(u\right)-{u}_{n},{proj}_{Fix\left(T\right)}\left(u\right)-{y}_{n}〉+\frac{{\zeta }_{n}^{2}}{\left(1-2{\zeta }_{n}\right){\varsigma }_{n}}{\parallel {x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel }^{2}\\ +\frac{2{\zeta }_{n}}{1-2{\zeta }_{n}}\parallel T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel \left(\kappa +2\right){M}_{2}\\ +\frac{2{\zeta }_{n}^{2}}{\left(1-2{\zeta }_{n}\right){\varsigma }_{n}}\parallel T{x}_{n}-{proj}_{Fix\left(T\right)}\left(u\right)\parallel \left(\kappa +2\right){M}_{2}.\end{array}$

It is clear that ${\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_{n}=\mathrm{\infty }$ and ${lim sup}_{n\to \mathrm{\infty }}{\delta }_{n}\le 0$. We can therefore apply Lemma 2.2 to (3.21) and conclude that ${x}_{n}\to {proj}_{Fix\left(T\right)}\left(u\right)$ as $n\to \mathrm{\infty }$. This completes the proof. □

Corollary 3.6 Let $\left\{{\varsigma }_{n}\right\}$ and $\left\{{\zeta }_{n}\right\}$ be two real number sequences in $\left(0,1\right)$. For ${x}_{0}\in C$ arbitrarily, let the sequence $\left\{{x}_{n}\right\}$ be generated by

${x}_{n+1}=\left(1-{\zeta }_{n}\right){proj}_{C}\left[\left(1-{\varsigma }_{n}\right){x}_{n}\right]+{\zeta }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.$
(3.22)

Assume ${lim}_{n\to \mathrm{\infty }}{\varsigma }_{n}={lim}_{n\to \mathrm{\infty }}\frac{{\varsigma }_{n}}{{\zeta }_{n}}={lim}_{n\to \mathrm{\infty }}\frac{{\zeta }_{n}^{2}}{{\varsigma }_{n}}=0$. Then we have

1. (1)

the sequence $\left\{{x}_{n}\right\}$ is bounded;

2. (2)

the sequence $\left\{{x}_{n}\right\}$ is asymptotically regular, that is, ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n+1}-{x}_{n}\parallel =0$.

Further, if ${\sum }_{n=0}^{\mathrm{\infty }}{\varsigma }_{n}=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}\frac{\parallel {x}_{n+1}-{x}_{n}\parallel }{{\zeta }_{n}}=0$, then the sequence $\left\{{x}_{n}\right\}$ converges strongly to ${proj}_{Fix\left(T\right)}\left(0\right)$, which is the minimum-norm fixed point of T.

Proof Letting ${u}_{n}=u=0$ in (3.12), we obtain (3.22). Consequently, by Theorem 3.5, we find that the sequence $\left\{{x}_{n}\right\}$ generated by (3.22) converges strongly to ${proj}_{Fix\left(T\right)}\left(0\right)$, which is the minimum-norm fixed point of T. □

## References

1. Ceng LC, Petruşel A, Yao JC: Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings. Appl. Math. Comput. 2009, 209: 162-176. 10.1016/j.amc.2008.10.062

2. Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschitz pseudo-contractions. Proc. Am. Math. Soc. 2001, 129: 2359-2363. 10.1090/S0002-9939-01-06009-9

3. Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudo-contractive maps. Proc. Am. Math. Soc. 2004, 132: 831-840. 10.1090/S0002-9939-03-07101-6

4. Cho SY, Qin X, Kang SM: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854-857. 10.1016/j.aml.2011.10.031

5. Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147-150. 10.1090/S0002-9939-1974-0336469-5

6. Kang SM, Cho SY, Qin X: Hybrid projection algorithms for approximating fixed points of asymptotically quasi-pseudocontractive mappings. J. Nonlinear Sci. Appl. 2012, 5: 466-474.

7. Kang SM, Rafiq A: On convergence results for Lipschitz pseudocontractive mappings. J. Appl. Math. 2012., 2012: Article ID 902601 10.1155/2012/902601

8. Kang SM, Rafiq A, Lee S: Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 132 10.1186/1029-242X-2013-132

9. Li XS, Kim JK, Huang NJ: Viscosity approximation of common fixed points for L -Lipschitzian semigroup of pseudocontractive mappings in Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 936121 10.1155/2009/936121

10. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506-510. 10.1090/S0002-9939-1953-0054846-3

11. Morales CH, Jung JS: Convergence of paths for pseudocontractive mappings in Banach spaces. Proc. Am. Math. Soc. 2000, 128: 3411-3419. 10.1090/S0002-9939-00-05573-8

12. Ofoedu EU, Zegeye H: Further investigation on iteration processes for pseudocontractive mappings with application. Nonlinear Anal. 2012, 75: 153-162. 10.1016/j.na.2011.08.015

13. Qin X, Cho YJ, Kang SM, Zhou H: Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions. Nonlinear Anal. 2009, 71: 685-690. 10.1016/j.na.2008.10.102

14. Song YS, Chen R: An approximation method for continuous pseudocontractive mappings. J. Inequal. Appl. 2006., 2006: Article ID 28950 10.1155/JIA/2006/28950

15. Udomene A: Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces. Nonlinear Anal. 2007, 67: 2403-2414. 10.1016/j.na.2006.09.001

16. Wen DJ, Chen YA: General iterative method for generalized equilibrium problems and fixed point problems of k -strict pseudo-contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 125 10.1186/1687-1812-2012-125

17. Yao Y, Colao V, Marino G, Xu HK: Implicit and explicit algorithms for minimum-norm fixed points of pseudocontractions in Hilbert spaces. Taiwan. J. Math. 2012, 16: 1489-1506.

18. Yao Y, Liou YC: Strong convergence of an implicit iteration algorithm for a finite family of pseudocontractive mappings. J. Inequal. Appl. 2008., 2008: Article ID 280908 10.1155/2008/280908

19. Yao Y, Liou YC, Chen R: Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces. Nonlinear Anal. 2007, 67: 3311-3317. 10.1016/j.na.2006.10.013

20. Yao Y, Liou YC, Marino G: A hybrid algorithm for pseudo-contractive mappings. Nonlinear Anal. 2009, 71: 4997-5002. 10.1016/j.na.2009.03.075

21. Yao Y, Marino G, Xu HK, Liou YC: Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces. J. Inequal. Appl. 2014., 2014: Article ID 206 10.1186/1029-242X-2014-206

22. Zegeye H, Shahzad N, Alghamdi MA: Convergence of Ishikawa’s iteration method for pseudocontractive mappings. Nonlinear Anal. 2011, 74: 7304-7311. 10.1016/j.na.2011.07.048

23. Zegeye H, Shahzad N, Alghamdi MA: Minimum-norm fixed point of pseudocontractive mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 926017 10.1155/2012/926017

24. Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal. 2009, 70: 4039-4046. 10.1016/j.na.2008.08.012

25. Yao Y, Shahzad N: New methods with perturbations for non-expansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 79 10.1186/1687-1812-2011-79

26. Yu ZT, Lin LJ, Chuang CS: Mathematical programming with multiple sets split monotone variational inclusion constraints. Fixed Point Theory Appl. 2014., 2014: Article ID 20 10.1186/1687-1812-2014-20

27. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240-256. 10.1112/S0024610702003332

## Acknowledgements

The authors are very grateful to the reviewers for their valuable comments and suggestions.

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Correspondence to Shin Min Kang or Chahn Yong Jung.

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