# On solving Lipschitz pseudocontractive operator equations

## Abstract

We analyze the convergence of the Mann-type double sequence iteration process to the solution of a Lipschitz pseudocontractive operator equation on a bounded closed convex subset of arbitrary real Banach space into itself. Our results extend the result in (Moore in Comp. Math. Appl. 43: 1585-1589, 2002).

MSC:47H10, 54H25.

## 1 Introduction

Let E be a real Banach space and ${E}^{\ast }$ be the dual space of E. Let J be the normalized duality mapping from E to ${2}^{{E}^{\ast }}$ defined by

$J\left(x\right)=\left\{f\in {E}^{\ast }:〈x,f〉=\parallel x\parallel \parallel f\parallel ,\parallel f\parallel =\parallel x\parallel \right\}$

for all $x\in E$ where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. A single-valued duality map will be denoted by j.

An operator $T:E\to E$ is said to be

• pseudocontractive if there exists $j\left(x-y\right)\in J\left(x-y\right)$ such that

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

for any $x,y\in E$;

• accretive if for any $x,y\in E$, there exists $j\left(x-y\right)\in J\left(x-y\right)$ satisfying

$〈Tx-Ty,j\left(x-y\right)〉\ge 0;$
• strongly pseudocontractive if there exist $j\left(x-y\right)\in J\left(x-y\right)$ and a constant $\lambda \in \left(0,1\right)$ such that

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

for any $x,y\in E$;

• strongly accretive if for any $x,y\in E$, there exist $j\left(x-y\right)\in J\left(x-y\right)$ and a constant $t\in \left(0,1\right)$ satisfying

$〈Tx-Ty,j\left(x-y\right)〉\ge t{\parallel x-y\parallel }^{2}$

for all $x,y\in E$.

As a consequence of a result of Kato , the concept of pseudocontractive operators can equivalently be defined as follows:

T is strongly pseudocontractive if there exists $\lambda \in \left(0,1\right)$ such that the inequality

$\parallel x-y\parallel \le \parallel x-y+r\left[\left(I-T-\lambda I\right)x-\left(I-T-\lambda I\right)y\right]\parallel$
(1.1)

holds for all $x,y\in E$ and $r>0$. If $\lambda =0$ in the inequality (1.1), then T is pseudocontractive.

It is easy to see that T is pseudocontractive if and only if $I-T$ is accretive where I denotes the identity mapping on E.

Let C be a compact convex subset of a real Hilbert space and let $T:C\to C$ be a Lipschitz pseudocontraction. It remains as an open question whether the Mann iteration process always converges to a fixed point of T. In  it was proved that the Ishikawa iteration process converges strongly to a fixed point of T. In 2001, Mutangadura and Chidume  constructed the following example to demonstrate that the Mann iteration process is not guaranteed to converge to a fixed point of a Lipschitz pseudocontraction mapping a compact convex subset of a real Hilbert space H into itself.

Example 

Let $H={\mathrm{\Re }}^{2}$ with the usual Euclidean inner product, and for $x=\left(a,b\right)\in H$ define ${x}^{\perp }=\left(b,-a\right)$. Now, let $C={B}_{1}\left(o\right)$; the closed unit ball in H and let ${C}_{1}=\left\{x\in H:\parallel x\parallel \le \frac{1}{2}\right\}$, ${C}_{2}=\left\{x\in H:\frac{1}{2}\le \parallel x\parallel \le 1\right\}$. Define the map $T:C\to C$ by

Observe that T is pseudocontractive, Lipschitz continuous (with Lipschitz constant 5) and has the origin $\left(0,0\right)$ as its unique fixed point; C is compact and convex. However, for any $x\in {C}_{1}$, we have

${\parallel \left(1-\lambda \right)x+\lambda Tx\parallel }^{2}=\left(1+{\lambda }^{2}\right){\parallel x\parallel }^{2}>{\parallel x\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\lambda \in \left(0,1\right),$

while for any $x\in {C}_{2}$, we have

${\parallel \left(1-\lambda \right)x+\lambda Tx\parallel }^{2}\ge \frac{1}{2}{\parallel x\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }\lambda \in \left(0,1\right),$

and therefore no Mann sequence can converge to $\left(0,0\right)$, the unique fixed point of T, unless the initial guess is the fixed point itself.

Moore  introduced the concept of a Mann-type double sequence iteration process and proved that it converges strongly to a fixed point of a continuous pseudocontraction which maps a bounded closed convex nonempty subset of a real Hilbert space into itself.

Definition 1.1 

Let $\mathcal{N}$ denote the set of all nonnegative integers (the natural numbers) and let E be a normed linear space. By a double sequence in E is meant a function $f:\mathcal{N}×\mathcal{N}\to E$ defined by $f\left(n,m\right)={x}_{n,m}\in E$. A double sequence $\left\{{x}_{n,m}\right\}$ is said to converge strongly to ${x}^{\ast }$ if given any $ϵ>0$, there exist $N,M>0$ such that $\parallel {x}_{n,m}-{x}^{\ast }\parallel <ϵ$ for all $n\ge N$, $m\ge M$. If $\mathrm{\forall }n,r\ge N$, $\mathrm{\forall }m,t\ge M$, we have $\parallel {x}_{n,r}-{x}_{m,t}\parallel <ϵ$, then the double sequence is said to be Cauchy. Furthermore, if for each fixed n, ${x}_{n,m}\to {x}_{n}^{\ast }$ as $m\to \mathrm{\infty }$ and then ${x}_{n}^{\ast }\to {x}^{\ast }$ as $n\to \mathrm{\infty }$, then ${x}_{n,m}\to {x}^{\ast }$ as $n,m\to \mathrm{\infty }$.

Theorem 1.1 

Let C be a bounded closed convex nonempty subset of a (real) Hilbert space H, and let $T:C\to C$ be a continuous pseudocontractive map. Let ${\left\{{\alpha }_{n}\right\}}_{n\ge 0},{\left\{{a}_{k}\right\}}_{k\ge 0}\subset \left(0,1\right)$ be real sequences satisfying the following conditions:

1. (i)

${lim}_{k\to \mathrm{\infty }}{a}_{k}=1$,

2. (ii)

${lim}_{k,r\to \mathrm{\infty }}\left({a}_{k}-{a}_{r}\right)/\left(1-{a}_{k}\right)=0$, $\mathrm{\forall }0,

3. (iii)

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

4. (iv)

${\sum }_{n\ge 0}{\alpha }_{n}=\mathrm{\infty }$.

For an arbitrary but fixed $\omega \in C$, and for each $k\ge 0$, define ${T}_{k}:C\to C$ by ${T}_{k}x=\left(1-{a}_{k}\right)\omega +{a}_{k}Tx$, $\mathrm{\forall }x\in C$. Then the double sequence ${\left\{{x}_{k,n}\right\}}_{k\ge 0,n\ge 0}$ generated from an arbitrary ${x}_{0,0}\in C$ by

${x}_{k,n+1}=\left(1-{\alpha }_{n}\right){x}_{k,n}+{\alpha }_{n}{T}_{k}{x}_{k,n},\phantom{\rule{1em}{0ex}}k,n\ge 0,$

converges strongly to a fixed point ${x}_{\mathrm{\infty }}^{\ast }$ of T in C.

The following lemma will be useful in the sequel.

Lemma 1.2 

Let $\left\{{\delta }_{n}\right\}$ and $\left\{{\sigma }_{n}\right\}$ be two sequences of nonnegative real numbers satisfying the inequality

${\delta }_{n+1}\le \gamma {\delta }_{n}+{\sigma }_{n},\phantom{\rule{1em}{0ex}}n\ge 0.$

Here $\gamma \in \left[0,1\right)$. If ${lim}_{n\to \mathrm{\infty }}{\sigma }_{n}=0$, then ${lim}_{n\to \mathrm{\infty }}{\delta }_{n}=0$.

It is our purpose in this paper to extend Theorem 1.1 from Hilbert space to an arbitrary real Banach space with no further assumptions on the real sequences ${\left\{{\alpha }_{n}\right\}}_{n\ge 0}$, ${\left\{{a}_{k}\right\}}_{k\ge 0}$.

## 2 Main results

Theorem 2.1 Let C be a bounded closed convex subset of a Banach space E and $T:C\to C$ be a Lipschitz pseudocontraction with $F\left(T\right)\ne \mathrm{\varnothing }$. Let ${\left\{{\alpha }_{n}\right\}}_{n\ge 0},{\left\{{a}_{k}\right\}}_{k\ge 0}\subset \left(0,1\right)$ be real sequences satisfying the following conditions:

1. (i)

${lim}_{k\to \mathrm{\infty }}{a}_{k}=1$,

2. (ii)

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

For an arbitrary but fixed $\omega \in C$, and for each $k\ge 0$, define ${T}_{k}:C\to C$ by ${T}_{k}x=\left(1-{a}_{k}\right)\omega +{a}_{k}Tx$, $\mathrm{\forall }x\in C$. Then the double sequence ${\left\{{x}_{k,n}\right\}}_{k\ge 0,n\ge 0}$ generated from an arbitrary ${x}_{0,0}\in C$ by

${x}_{k,n+1}=\left(1-{\alpha }_{n}\right){x}_{k,n}+{\alpha }_{n}{T}_{k}{x}_{k,n},\phantom{\rule{1em}{0ex}}k,n\ge 0$
(2.1)

converges strongly to a fixed point ${x}^{\ast }$ of T in C.

Proof Since T is Lipschitzian, there exists $L>0$ such that

Since T is pseudocontractive, for each $k\ge 0$, we have

$〈{T}_{k}x-{T}_{k}y,j\left(x-y\right)〉={a}_{k}〈Tx-Ty,j\left(x-y\right)〉\le {a}_{k}{\parallel x-y\parallel }^{2}.$

Hence, ${T}_{k}$ is Lipschitz and strongly pseudocontractive. Also, C is invariant under ${T}_{k}$ for all $k\ge 0$, by convexity. Thus, for each $k\ge 0$, ${T}_{k}$ has a unique fixed point ${x}_{k}^{\ast }$, say, in C.

Now, we proceed in the following steps.

1. (I)

for each $k\ge 0$, ${x}_{k,n}\to {x}_{k}^{\ast }\in C$ as $n\to \mathrm{\infty }$.

2. (II)

${x}_{k}^{\ast }\to {x}^{\ast }\in C$ as $k\to \mathrm{\infty }$.

3. (III)

${x}^{\ast }\in F\left(T\right)$.

Proof of (I). In fact, it follows from (2.1) that

$\begin{array}{rcl}{x}_{k,n}& =& {x}_{k,n+1}+{\alpha }_{n}{x}_{k,n}-{\alpha }_{n}{T}_{k}{x}_{k,n}\\ =& \left(1+{\alpha }_{n}\right){x}_{k,n+1}+{\alpha }_{n}\left(I-{T}_{k}-\lambda I\right){x}_{k,n+1}-\left(2-\lambda \right){\alpha }_{n}{x}_{k,n+1}+{\alpha }_{n}{x}_{k,n}\\ +{\alpha }_{n}\left({T}_{k}{x}_{k,n+1}-{T}_{k}{x}_{k,n}\right)\\ =& \left(1+{\alpha }_{n}\right){x}_{k,n+1}+{\alpha }_{n}\left(I-{T}_{k}-\lambda I\right){x}_{k,n+1}-\left(2-\lambda \right){\alpha }_{n}\left[\left(1-{\alpha }_{n}\right){x}_{k,n}+{\alpha }_{n}{T}_{k}{x}_{k,n}\right]\\ +{\alpha }_{n}{x}_{k,n}+{\alpha }_{n}\left({T}_{k}{x}_{k,n+1}-{T}_{k}{x}_{k,n}\right)\\ =& \left(1+{\alpha }_{n}\right){x}_{k,n+1}+{\alpha }_{n}\left(I-{T}_{k}-\lambda I\right){x}_{k,n+1}-\left(1-\lambda \right){\alpha }_{n}{x}_{k,n}\\ +\left(2-\lambda \right){\alpha }_{n}^{2}\left({x}_{k,n}-{T}_{k}{x}_{k,n}\right)+{\alpha }_{n}\left({T}_{k}{x}_{k,n+1}-{T}_{k}{x}_{k,n}\right).\end{array}$

Thus, if ${x}_{k}^{\ast }$ is a fixed point of ${T}_{k}$, $k\ge 0$, then

$\begin{array}{rcl}{x}_{k,n+1}-{x}_{k}^{\ast }& =& \left(1+{\alpha }_{n}\right)\left({x}_{k,n+1}-{x}_{k}^{\ast }\right)+{\alpha }_{n}\left(I-{T}_{k}-\lambda I\right)\left({x}_{k,n+1}-{x}_{k}^{\ast }\right)\\ -\left(1-\lambda \right){\alpha }_{n}\left({x}_{k,n}-{x}_{k}^{\ast }\right)+\left(2-\lambda \right){\alpha }_{n}^{2}\left({x}_{k,n}-{T}_{k}{x}_{k,n}\right)+{\alpha }_{n}\left({T}_{k}{x}_{k,n+1}-{T}_{k}{x}_{k,n}\right).\end{array}$

Using inequality (1.1), it follows that

$\begin{array}{rcl}\parallel {x}_{k,n+1}-{x}_{k}^{\ast }\parallel & \ge & \left(1+{\alpha }_{n}\right)\parallel {x}_{k,n+1}-{x}_{k}^{\ast }\parallel -\left(1-\lambda \right){\alpha }_{n}\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel \\ -\left(2-\lambda \right){\alpha }_{n}^{2}\parallel {x}_{k,n}-{T}_{k}{x}_{k,n}\parallel -{\alpha }_{n}\parallel {T}_{k}{x}_{k,n+1}-{T}_{k}{x}_{k,n}\parallel .\end{array}$
(2.2)

On the other hand, by (2.1) we obtain

$\begin{array}{rcl}\parallel {x}_{k,n+1}-{x}_{k,n}\parallel & =& {\alpha }_{n}\parallel {T}_{k}{x}_{k,n}-{x}_{k,n}\parallel \\ \le & {\alpha }_{n}\left(\parallel {T}_{k}{x}_{k,n}-{x}_{k}^{\ast }\parallel +\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel \right)\\ =& {\alpha }_{n}\left({a}_{k}\parallel T{x}_{k,n}-{x}_{k}^{\ast }\parallel +\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel \right)\\ \le & {\alpha }_{n}\left({a}_{k}L\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel +\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel \right)\\ \le & {\alpha }_{n}\left(L+1\right)\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel .\end{array}$

Therefore,

$\begin{array}{rcl}\parallel {T}_{k}{x}_{k,n+1}-{T}_{k}{x}_{k,n}\parallel & =& {a}_{k}\parallel T{x}_{k,n+1}-T{x}_{k,n}\parallel \\ \le & {a}_{k}L\parallel {x}_{k,n+1}-{x}_{k,n}\parallel \\ \le & L\parallel {x}_{k,n+1}-{x}_{k,n}\parallel \\ \le & {\alpha }_{n}L\left(L+1\right)\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel .\end{array}$
(2.3)

Substituting (2.3) into (2.2), we arrive at

$\begin{array}{rcl}\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel & \ge & \left(1+{\alpha }_{n}\right)\parallel {x}_{k,n+1}-{x}_{k}^{\ast }\parallel -\left(1-\lambda \right){\alpha }_{n}\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel \\ -\left(2-\lambda \right){\alpha }_{n}^{2}\parallel {x}_{k,n}-{T}_{k}{x}_{k,n}\parallel -L\left(L+1\right){\alpha }_{n}^{2}\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel ,\end{array}$

which implies that

$\begin{array}{rcl}{\alpha }_{n}\parallel {x}_{k,n+1}-{x}_{k}^{\ast }\parallel & \le & \left(1-\lambda \right){\alpha }_{n}\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel +{\alpha }_{n}^{2}\left[L\left(L+1\right)\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel \\ +\left(2-\lambda \right)\parallel {x}_{k,n}-{T}_{k}{x}_{k,n}\parallel \right],\end{array}$

and so

$\begin{array}{rcl}\parallel {x}_{k,n+1}-{x}_{k}^{\ast }\parallel & \le & \left(1-\lambda \right)\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel +{\alpha }_{n}\left[L\left(L+1\right)\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel \\ +\left(2-\lambda \right)\parallel {x}_{k,n}-{T}_{k}{x}_{k,n}\parallel \right].\end{array}$
(2.4)

Since C is bounded, there exists $M>0$ such that

$M=max\left\{L\left(L+1\right)\underset{n\ge 0}{sup}\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel ,\left(2-\lambda \right)\underset{n\ge 0}{sup}\parallel {x}_{k,n}-{T}_{k}{x}_{k,n}\parallel \right\}.$

Hence, it follows from (2.4) that

$\parallel {x}_{k,n+1}-{x}_{k}^{\ast }\parallel \le \left(1-\lambda \right)\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel +{\alpha }_{n}M.$

Since $\lambda \in \left(0,1\right)$ and ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$, it follows from Lemma 1.2 that

$\underset{n\to \mathrm{\infty }}{lim}\parallel {x}_{k,n}-{x}_{k}^{\ast }\parallel =0,$

i.e., ${x}_{k,n}\to {x}_{k}^{\ast }$ as $n\to \mathrm{\infty }$.

Proof of (II). We prove that ${\left\{{x}_{k}^{\ast }\right\}}_{k=0}^{\mathrm{\infty }}={\left\{{T}_{k}{x}_{k}^{\ast }\right\}}_{k=0}^{\mathrm{\infty }}$ converges to some ${x}^{\ast }\in C$. For this purpose, we need only to prove that ${\left\{{x}_{k}^{\ast }\right\}}_{0}^{\mathrm{\infty }}$ is a Cauchy sequence.

In fact, we have

$\begin{array}{rcl}{\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel }^{2}& =& 〈{x}_{l}^{\ast }-{x}_{m}^{\ast },j\left({x}_{l}^{\ast }-{x}_{m}^{\ast }\right)〉\\ =& 〈{T}_{l}{x}_{l}^{\ast }-{T}_{m}{x}_{m}^{\ast },j\left({x}_{l}^{\ast }-{x}_{m}^{\ast }\right)〉\\ =& 〈\left(1-{a}_{l}\right)\omega +{a}_{l}T{x}_{l}^{\ast }-\left(1-{a}_{m}\right)\omega -{a}_{m}T{x}_{m}^{\ast },j\left({x}_{l}^{\ast }-{x}_{m}^{\ast }\right)〉\\ =& \left({a}_{m}-{a}_{l}\right)〈\omega ,j\left({x}_{l}^{\ast }-{x}_{m}^{\ast }\right)〉+{a}_{l}〈T{x}_{l}^{\ast }-T{x}_{m}^{\ast },j\left({x}_{l}^{\ast }-{x}_{m}^{\ast }\right)〉\\ +\left({a}_{l}-{a}_{m}\right)〈T{x}_{m}^{\ast },j\left({x}_{l}^{\ast }-{x}_{m}^{\ast }\right)〉\\ \le & |{a}_{l}-{a}_{m}|\left(\parallel \omega \parallel \parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel +\parallel T{x}_{m}^{\ast }\parallel \parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel \right)\\ +{a}_{l}〈T{x}_{l}^{\ast }-T{x}_{m}^{\ast },j\left({x}_{l}^{\ast }-{x}_{m}^{\ast }\right)〉\\ \le & |{a}_{l}-{a}_{m}|\left(\parallel \omega \parallel +\parallel T{x}_{m}^{\ast }\parallel \right)\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel +{a}_{l}\lambda {\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel }^{2}\\ \le & |{a}_{l}-{a}_{m}|\left(\parallel \omega \parallel +\parallel T{x}_{m}^{\ast }\parallel \right)\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel +\lambda {\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel }^{2},\end{array}$

that is,

$\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel \le \left[|{a}_{l}-{a}_{m}|\left(\parallel \omega \parallel +\parallel T{x}_{m}^{\ast }\parallel \right)+\lambda \parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel \right],$

hence

$\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel \le 2\frac{|{a}_{l}-{a}_{m}|}{1-\lambda }d,$

where $d=diamC$. If follows from condition (i) that

$\underset{l,m\to \mathrm{\infty }}{lim}\parallel {x}_{l}^{\ast }-{x}_{m}^{\ast }\parallel =0.$

This completes step (II) of the proof.

Proof of (III). In order to accomplish step (III), we first have to prove that ${\left\{{x}_{k}^{\ast }\right\}}_{k=0}^{\mathrm{\infty }}$ is an approximate fixed point sequence for T. In fact, from ${T}_{k}{x}_{k}^{\ast }=\left(1-{a}_{k}\right)\omega +{a}_{k}T{x}_{k}^{\ast }$, we have

$\begin{array}{rcl}\parallel {x}_{k}^{\ast }-T{x}_{k}^{\ast }\parallel & =& \parallel {x}_{k}^{\ast }-\frac{1}{{a}_{k}}{T}_{k}{x}_{k}^{\ast }+\frac{1-{a}_{k}}{{a}_{k}}\omega \parallel \\ =& \parallel {x}_{k}^{\ast }-\frac{1}{{a}_{k}}{x}_{k}^{\ast }+\frac{1-{a}_{k}}{{a}_{k}}\omega \parallel \\ =& \parallel \frac{1-{a}_{k}}{{a}_{k}}\left(\omega -{x}_{k}^{\ast }\right)\parallel \\ \le & \frac{1-{a}_{k}}{{a}_{k}}\left(\parallel \omega \parallel +\parallel {x}_{k}^{\ast }\parallel \right)\\ \le & \frac{1-{a}_{k}}{{a}_{k}}\cdot 2d,\end{array}$

where $d=diamC$. Hence ${lim}_{k\to \mathrm{\infty }}\parallel {x}_{k}^{\ast }-T{x}_{k}^{\ast }\parallel =0$. Since ${x}_{k}^{\ast }\to {x}^{\ast }$ as $k\to \mathrm{\infty }$, T is continuous and using continuity of the norm, we get ${lim}_{k\to \mathrm{\infty }}\parallel {x}^{\ast }-T{x}^{\ast }\parallel =0$, i.e., ${x}^{\ast }=T{x}^{\ast }$. This completes the proof. □

Corollary 2.2 Let C be a bounded closed convex subset of a Banach space E and $T:C\to C$ be a nonexpansive mapping with $F\left(T\right)\ne \mathrm{\varnothing }$. Let ${\left\{{\alpha }_{n}\right\}}_{n\ge 0},{\left\{{a}_{k}\right\}}_{k\ge 0}\subset \left(0,1\right)$ be real sequences satisfying conditions (i)-(ii) in Theorem  2.1. For an arbitrary but fixed $\omega \in C$, and for each $k\ge 0$, define ${T}_{k}:C\to C$ by ${T}_{k}x=\left(1-{a}_{k}\right)\omega +{a}_{k}Tx$, $\mathrm{\forall }x\in C$. Then the double sequence ${\left\{{x}_{k,n}\right\}}_{k\ge 0,n\ge 0}$ generated from an arbitrary ${x}_{0,0}\in C$ by

${x}_{k,n+1}=\left(1-{\alpha }_{n}\right){x}_{k,n}+{\alpha }_{n}{T}_{k}{x}_{k,n},\phantom{\rule{1em}{0ex}}k,n\ge 0,$

converges strongly to a fixed point of T in C.

Proof Obvious, observing the fact that every nonexpansive mapping is Lipschitz and pseudocontractive. □

The following corollary follows from Theorem 2.1 on setting $\omega =0\in C$.

Corollary 2.3 Let C, E, T, ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{a}_{k}\right\}}_{k=0}^{\mathrm{\infty }}$ be as in Theorem  2.1. For an arbitrary but fixed $\omega \in C$, and for each $k\ge 0$, define ${T}_{k}:C\to C$ by ${T}_{k}x={a}_{k}Tx$ for all $x\in C$. Then the double sequence ${\left\{{x}_{k,n}\right\}}_{k\ge 0,n\ge 0}$ generated from an arbitrary ${x}_{0,0}\in C$ by

${x}_{k,n+1}=\left(1-{\alpha }_{n}\right){x}_{k,n}+{\alpha }_{n}{T}_{k}{x}_{k,n},\phantom{\rule{1em}{0ex}}k,n\ge 0,$

converges strongly to a fixed point of T in C.

Remark 2.1 Theorem 2.1 improves and extends Theorem 3.1 of Moore  in three respects:

1. (1)

It abolishes the condition that ${lim}_{r,k\to \mathrm{\infty }}\frac{{a}_{k}-{a}_{r}}{1-{a}_{k}}=0$.

2. (2)

It abolishes the condition that ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$.

3. (3)

The ambient space is no longer required to be a Hilbert space and is taken to be the more general Banach space instead.

Remark 2.2

1. (1)

Whereas the Ishikawa iteration process was proved to converge to a fixed point of a Lipschitz pseudocontractive mapping in compact convex subsets of a Hilbert space, we imposed no compactness conditions to obtain the strong convergence of the double sequence iteration process to a fixed point of a Lipschitz pseudocontraction.

2. (2)

Our results may easily be extended to the slightly more general classes of Lipschitz hemicontractive and Lipschitz quasi-nonexpansive mappings.

3. (3)

Prototypes of the sequences ${\left\{{a}_{k}\right\}}_{k=0}^{\mathrm{\infty }}$ and ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ are

${a}_{k}=\frac{k}{1+k}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\alpha }_{n}=\frac{1}{{\left(n+1\right)}^{2}}.$

## References

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

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

4. Moore C: A double sequence iteration process for fixed points of continuous pseudocontractions. Comput. Math. Appl. 2002, 43: 1585-1589. 10.1016/S0898-1221(02)00121-9

5. Liu QH: A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings. J. Math. Anal. Appl. 1990, 146: 302-305.

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