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# Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces

Fixed Point Theory and Applications20122012:160

https://doi.org/10.1186/1687-1812-2012-160

• Received: 3 March 2012
• Accepted: 7 September 2012
• Published:

## Abstract

The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.

## Keywords

• Ishikawa iteration scheme with errors
• strictly hemicontractive operators
• Lipschitz operators
• Banach space

## 1 Preliminaries

Let K be a nonempty subset of an arbitrary Banach space E and ${E}^{\ast }$ be its dual space. The symbols $D\left(T\right)$, $R\left(T\right)$ and $F\left(T\right)$ stand for the domain, the range and the set of fixed points of T respectively (for a single-valued map $T:X\to X$, $x\in X$ is called a fixed point of T iff $T\left(x\right)=x$). We denote by J the normalized duality mapping from E to ${2}^{{E}^{\ast }}$ defined by
$J\left(x\right)=\left\{{f}^{\ast }\in {E}^{\ast }:〈x,{f}^{\ast }〉={\parallel x\parallel }^{2}={\parallel {f}^{\ast }\parallel }^{2}\right\}.$

Let T be a self-mapping of K.

Definition 1 Then T is called Lipshitzian if there exists $L>0$ such that
$\parallel Tx-Ty\parallel ⩽L\parallel x-y\parallel$
(1.1)

for all $x,y\in K$. If $L=1$, then T is called non-expansive, and if $0⩽L<1$, T is called contraction.

Definition 2[2, 3]
1. 1.
The mapping T is said to be pseudocontractive if the inequality
$\parallel x-y\parallel ⩽\parallel x-y+t\left(\left(I-T\right)x-\left(I-T\right)y\right)\parallel$
(1.2)

holds for each $x,y\in K$ and for all $t>0$. As a consequence of a result of Kato [4], it follows from the inequality (1.2) that T is pseudocontractive if and only if there exists $j\left(x-y\right)\in J\left(x-y\right)$ such that
$〈Tx-Ty,j\left(x-y\right)〉⩽{\parallel x-y\parallel }^{2}$
(1.3)
for all $x,y\in K$.
1. 2.
T is said to be strongly pseudocontractive if there exists a $t>1$ such that
$\parallel x-y\parallel \le \parallel \left(1+r\right)\left(x-y\right)-rt\left(Tx-Ty\right)\parallel$
(1.4)

for all $x,y\in D\left(T\right)$ and $r>0$.
1. 3.
T is said to be local strongly pseudocontractive if, for each $x\in D\left(T\right)$, there exists a ${t}_{x}>1$ such that
$\parallel x-y\parallel \le \parallel \left(1+r\right)\left(x-y\right)-r{t}_{x}\left(Tx-Ty\right)\parallel$
(1.5)

for all $y\in D\left(T\right)$ and $r>0$.
1. 4.
T is said to be strictly hemicontractive if $F\left(T\right)\ne \phi$ and if there exists a $t>1$ such that
$\parallel x-q\parallel \le \parallel \left(1+r\right)\left(x-q\right)-rt\left(Tx-q\right)\parallel$
(1.6)

for all $x\in D\left(T\right)$, $q\in F\left(T\right)$ and $r>0$.

It is easy to verify that an iteration scheme ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ which is T-stable on K is almost T-stable on K. Osilike [5] proved that an iteration scheme which is almost T-stable on X may fail to be T-stable on X.

Clearly, each strongly pseudocontractive operator is local strongly pseudocontractive.

Chidume [6] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset of ${L}_{p}$ (or ${l}_{p}$) into itself. Afterwards, several authors generalized this result of Chidume in various directions. Chidume [7] proved a similar result by removing the restriction ${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$. Tan and Xu [8] extended that result of Chidume to the Ishikawa iteration scheme in a p-uniformly smooth Banach space. Chidume and Osilike [2] improved the result of Chidume [6] to strictly hemicontractive mappings defined on a real uniformly smooth Banach space.

Recently, some researchers have generalized the results to real smooth Banach spaces, real uniformly smooth Banach spaces, real Banach spaces; or to the Mann iteration method, the Ishikawa iteration method; or to strongly pseudocontractive operators, local strongly pseudocontractive operators, strictly hemicontractive operators [919].

The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu [1] for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces. Our results extend, improve and unify the corresponding results in [2, 3, 10, 11, 1518, 2025].

## 2 Main results

We need the following results.

Lemma 3[26]

Let${\left\{{\alpha }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{\beta }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{\gamma }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$and${\left\{{\omega }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$be nonnegative real sequences such that
${\alpha }_{n+1}\le \left(1-{\omega }_{n}\right){\alpha }_{n}+{\omega }_{n}{\beta }_{n}+{\gamma }_{n},\phantom{\rule{1em}{0ex}}n\ge 0,$

with${\left\{{\omega }_{n}\right\}}_{n=0}^{\mathrm{\infty }}\subset \left[0,1\right]$, ${\sum }_{n=0}^{\mathrm{\infty }}{\omega }_{n}=\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_{n}<\mathrm{\infty }$and${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0$. Then${lim}_{n\to \mathrm{\infty }}{\alpha }_{n}=0$.

Lemma 4[27]

Let${\left\{{a}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{b}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$be sequences of nonnegative real numbers and$0\le \theta <1$, so that
${a}_{n+1}\le \theta {a}_{n}+{b}_{n},\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}n\ge 0.$
1. (i)

If ${lim}_{n\to \mathrm{\infty }}{b}_{n}=0$, then ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$.

2. (ii)

If ${\sum }_{n=0}^{\mathrm{\infty }}{b}_{n}<\mathrm{\infty }$, then ${\sum }_{n=0}^{\mathrm{\infty }}{a}_{n}<\mathrm{\infty }$.

Lemma 5[4]

Let$x,y\in X$. Then$\parallel x\parallel \le \parallel x+ry\parallel$for every$r>0$if and only if there is$f\in J\left(x\right)$such that$Re〈y,f〉\ge 0$.

Lemma 6[2]

Let$T:D\left(T\right)\subseteq X\to X$be an operator with$F\left(T\right)\ne \mathrm{\varnothing }$. Then T is strictly hemicontractive if and only if there exists$t>1$such that for all$x\in D\left(T\right)$and$q\in F\left(T\right)$, there exists$j\in J\left(x-q\right)$satisfying
$Re〈x-Tx,j\left(x-q\right)〉\ge \left(1-\frac{1}{t}\right){\parallel x-q\parallel }^{2}.$

Lemma 7[24]

Let X be an arbitrary normed linear space and$T:D\left(T\right)\subseteq X\to X$be an operator.
1. (i)

If T is a local strongly pseudocontractive operator and $F\left(T\right)\ne \mathrm{\varnothing }$, then $F\left(T\right)$ is a singleton and T is strictly hemicontractive.

2. (ii)

If T is strictly hemicontractive, then $F\left(T\right)$ is a singleton.

In the sequel, let $k=\frac{t-1}{t}\in \left(0,1\right)$, where t is the constant appearing in (1.6). Further L denotes the common Lipschitz constant of T and S, and I denotes the identity mapping on an arbitrary Banach space X.

Definition 8 Let K be a nonempty convex subset of X and $T,S:K\to K$ be two operators. Assume that ${x}_{o}\in K$ and ${x}_{n+1}=f\left(T,S,{x}_{n}\right)$ defines an iteration scheme which produces a sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}\subset K$. Suppose, furthermore, that ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ converges strongly to $q\in F\left(T\right)\cap F\left(S\right)\ne \phi$. Let ${\left\{{y}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ be any bounded sequence in K and put ${\epsilon }_{n}=\parallel {y}_{n+1}-f\left(T,S,{y}_{n}\right)\parallel$.
1. (i)

The iteration scheme ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ defined by ${x}_{n+1}=f\left(T,S,{x}_{n}\right)$ is said to be common-stable on K if ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$ implies that ${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$.

2. (ii)

The iteration scheme ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ defined by ${x}_{n+1}=f\left(T,S,{x}_{n}\right)$ is said to be almost common-stable on K if ${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}<\mathrm{\infty }$ implies that ${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$.

We now establish our main results.

Theorem 9 Let K be a nonempty closed convex subset of an arbitrary Banach space X and$T,S:K\to K$be two Lipschitz strictly hemicontractive operators. Suppose that${\left\{{u}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{v}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$are arbitrary bounded sequences in K, and${\left\{{a}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{b}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{c}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{a}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{b}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$and${\left\{{c}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$are any sequences in$\left[0,1\right]$satisfying
1. (i)

${a}_{n}+{b}_{n}+{c}_{n}=1={a}_{n}^{\mathrm{\prime }}+{b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}$,

2. (ii)

${c}_{n}^{\mathrm{\prime }}=o\left({b}_{n}^{\mathrm{\prime }}\right)$,

3. (iii)

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

4. (iv)

${\sum }_{n=0}^{\mathrm{\infty }}{b}_{n}^{\mathrm{\prime }}=\mathrm{\infty }$,

5. (v)

$L\left[{\left(1+L\right)}^{2}{b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}+\left(1+L\right)\left({b}_{n}+{c}_{n}\right)\right]+\frac{{c}_{n}^{\mathrm{\prime }}}{{b}_{n}^{\mathrm{\prime }}}\le k\left(k-s\right)$, $n\ge 0$,

where s is a constant in$\left(0,k\right)$. Suppose that${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$is the sequence generated from an arbitrary${x}_{0}\in K$by
$\begin{array}{r}{x}_{n+1}={a}_{n}^{\mathrm{\prime }}{x}_{n}+{b}_{n}^{\mathrm{\prime }}T{z}_{n}+{c}_{n}^{\mathrm{\prime }}{v}_{n},\\ {z}_{n}={a}_{n}{x}_{n}+{b}_{n}S{x}_{n}+{c}_{n}{u}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.\end{array}$
(2.1)
Let ${\left\{{y}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ be any sequence in K and define ${\left\{{\epsilon }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ by
${\epsilon }_{n}=\parallel {y}_{n+1}-{p}_{n}\parallel ,\phantom{\rule{1em}{0ex}}n\ge 0,$
where
$\begin{array}{r}{p}_{n}={a}_{n}^{\mathrm{\prime }}{y}_{n}+{b}_{n}^{\mathrm{\prime }}T{w}_{n}+{c}_{n}^{\mathrm{\prime }}{v}_{n},\\ {w}_{n}={a}_{n}{y}_{n}+{b}_{n}S{y}_{n}+{c}_{n}{u}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.\end{array}$
(2.2)
Then
1. (a)
the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ converges strongly to the common fixed point q of T and S. Also,
$\begin{array}{rcl}\parallel {x}_{n+1}-q\parallel & \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n}-q\parallel \\ +L\left(1+L\right){k}^{-1}{b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel +\left(1+L\right){k}^{-1}{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel ,\phantom{\rule{1em}{0ex}}n\ge 0,\end{array}$
(b)
$\begin{array}{rcl}\parallel {y}_{n+1}-q\parallel & \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {y}_{n}-q\parallel \\ +L\left(1+L\right){k}^{-1}{b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel +\left(1+L\right){k}^{-1}{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +{\epsilon }_{n},\phantom{\rule{1em}{0ex}}n\ge 0,\end{array}$

2. (c)

${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}<\mathrm{\infty }$ implies that ${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$, so that ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ is almost common-stable on K,

3. (d)

${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$ implies that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$.

Proof From (ii), we have ${c}_{n}^{\mathrm{\prime }}={t}_{n}{b}_{n}^{\mathrm{\prime }}$, where ${t}_{n}\to 0$ as $n\to \mathrm{\infty }$. It follows from Lemma 7 that $F\left(T\right)\cap F\left(S\right)$ is a singleton; that is, $F\left(T\right)\cap F\left(S\right)=\left\{q\right\}$ for some $q\in K$. Set
$M=max\left\{\underset{n\ge 0}{sup}\left\{\parallel {u}_{n}-q\parallel \right\},\underset{n\ge 0}{sup}\left\{\parallel {v}_{n}-q\parallel \right\}\right\}.$
Since T is strictly hemicontractive, it follows form Lemma 6 that
$Re〈x-Tx,j\left(x-q\right)〉\ge k{\parallel x-q\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in K,$
which implies that
$Re〈\left(I-T-kI\right)x-\left(I-T-kI\right)q,j\left(x-q\right)〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in K.$
In view of Lemma 5, we have
$\parallel x-q\parallel \le \parallel x-q+r\left[\left(I-T-kI\right)x-\left(I-T-kI\right)q\right]\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in K,\mathrm{\forall }r>0.$
(2.3)
Also,
$\begin{array}{rcl}\left(1-{b}_{n}^{\mathrm{\prime }}\right){x}_{n}& =& \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right){x}_{n+1}+{b}_{n}^{\mathrm{\prime }}\left(I-T-kI\right){x}_{n+1}\\ +{b}_{n}^{\mathrm{\prime }}\left(T{x}_{n+1}-T{z}_{n}\right)-{c}_{n}^{\mathrm{\prime }}\left({v}_{n}-{x}_{n}\right),\end{array}$
(2.4)
and
$\left(1-{b}_{n}^{\mathrm{\prime }}\right)q=\left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right)q+{b}_{n}^{\mathrm{\prime }}\left(I-T-kI\right)q.$
(2.5)
From (2.4) and (2.5), we infer that for all $n\ge 0$,
$\begin{array}{rcl}\left(1-{b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n}-q\parallel & \ge & \parallel \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right)\left({x}_{n+1}-q\right)+{b}_{n}^{\mathrm{\prime }}\left(I-T-kI\right)\left({x}_{n+1}-q\right)\parallel \\ -{b}_{n}^{\mathrm{\prime }}\parallel T{x}_{n+1}-T{z}_{n}\parallel -{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-{x}_{n}\parallel \\ =& \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n+1}-q+\frac{{b}_{n}^{\mathrm{\prime }}}{1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}}\left(I-T-kI\right)\left({x}_{n+1}-q\right)\parallel \\ -{b}_{n}^{\mathrm{\prime }}\parallel T{x}_{n+1}-T{z}_{n}\parallel -{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-{x}_{n}\parallel \\ \ge & \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n+1}-q\parallel -{b}_{n}^{\mathrm{\prime }}\parallel T{x}_{n+1}-T{z}_{n}\parallel \\ -{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-{x}_{n}\parallel ,\end{array}$
which implies that for all $n\ge 0$,
(2.6)
(2.7)
(2.8)
Substituting (2.8) in (2.7), we have
$\begin{array}{rcl}\parallel {x}_{n+1}-{z}_{n}\parallel & \le & \left[{b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}+{c}_{n}\right]\parallel {x}_{n}-q\parallel \\ +L{b}_{n}^{\mathrm{\prime }}\left[\left[1+\left(1+L\right){b}_{n}+{c}_{n}\right]\parallel {x}_{n}-q\parallel \\ +{c}_{n}\parallel {u}_{n}-q\parallel \right]+{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +{c}_{n}\parallel {u}_{n}-q\parallel \\ =& \left[\left(1+L\right){b}_{n}^{\mathrm{\prime }}+L\left(1+L\right){b}_{n}{b}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}+{c}_{n}^{\mathrm{\prime }}\\ +\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\right]\parallel {x}_{n}-q\parallel \\ +{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\parallel {u}_{n}-q\parallel .\end{array}$
(2.9)
Substituting (2.9) in (2.6), we get
$\begin{array}{rcl}\parallel {x}_{n+1}-q\parallel & \le & \left(1-k{b}_{n}^{\mathrm{\prime }}+{k}^{-1}{c}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n}-q\parallel +{k}^{-1}L{b}_{n}^{\mathrm{\prime }}\left[\left[\left(1+L\right){b}_{n}^{\mathrm{\prime }}\\ +L\left(1+L\right){b}_{n}{b}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}+{c}_{n}^{\mathrm{\prime }}+\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\right]\parallel {x}_{n}-q\parallel \\ +{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\parallel {u}_{n}-q\parallel \right]+{k}^{-1}{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ =& \left[1-{b}_{n}^{\mathrm{\prime }}\left[k-{k}^{-1}L\left(\left(1+L\right){b}_{n}^{\mathrm{\prime }}+L\left(1+L\right){b}_{n}{b}_{n}^{\mathrm{\prime }}\\ +\left(1+L\right){b}_{n}+{c}_{n}^{\mathrm{\prime }}+\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\right)-{k}^{-1}{t}_{n}\right]\right]\parallel {x}_{n}-q\parallel \\ +{k}^{-1}L{b}_{n}^{\mathrm{\prime }}\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\parallel {u}_{n}-q\parallel +{k}^{-1}\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ \le & \left[1-{b}_{n}^{\mathrm{\prime }}\left[k-{k}^{-1}L\left({\left(1+L\right)}^{2}{b}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}\\ +{c}_{n}^{\mathrm{\prime }}+\left(1+L\right){c}_{n}\right)-{k}^{-1}{t}_{n}\right]\right]\parallel {x}_{n}-q\parallel \\ +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}M+{k}^{-1}\left(1+L\right){b}_{n}^{\mathrm{\prime }}{t}_{n}M\\ =& \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n}-q\parallel +{k}^{-1}\left(1+L\right)M{b}_{n}^{\mathrm{\prime }}\left(L{c}_{n}+{t}_{n}\right).\end{array}$
Put
we have
${\alpha }_{n+1}\le \left(1-{\omega }_{n}\right){\alpha }_{n}+{\omega }_{n}{\beta }_{n}+{\gamma }_{n},\phantom{\rule{1em}{0ex}}n\ge 0.$

Observe that ${\sum }_{n=0}^{\mathrm{\infty }}{\omega }_{n}=\mathrm{\infty }$, ${\omega }_{n}\in \left[0,1\right]$ and ${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0$. It follows from Lemma 3 that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-q\parallel =0$.

We also have
$\begin{array}{rcl}\left(1-{b}_{n}^{\mathrm{\prime }}\right){y}_{n}& =& \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right){p}_{n}+{b}_{n}^{\mathrm{\prime }}\left(I-T-kI\right){p}_{n}\\ +{b}_{n}^{\mathrm{\prime }}\left(T{p}_{n}-T{w}_{n}\right)-{c}_{n}^{\mathrm{\prime }}\left({v}_{n}-{y}_{n}\right).\end{array}$
(2.10)
From (2.5) and (2.10), it follows that for all $n\ge 0$,
$\begin{array}{rcl}\left(1-{b}_{n}^{\mathrm{\prime }}\right)\parallel {y}_{n}-q\parallel & \ge & \parallel \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right)\left({p}_{n}-q\right)+{b}_{n}^{\mathrm{\prime }}\left(I-T-kI\right)\left({p}_{n}-q\right)\parallel \\ -{b}_{n}^{\mathrm{\prime }}\parallel T{p}_{n}-T{w}_{n}\parallel -{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-{y}_{n}\parallel \\ =& \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right)\parallel {p}_{n}-q\\ +\frac{{b}_{n}^{\mathrm{\prime }}}{1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}}\left(I-T-kI\right)\left({p}_{n}-q\right)\parallel \\ -{b}_{n}^{\mathrm{\prime }}\parallel T{p}_{n}-T{w}_{n}\parallel -{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-{y}_{n}\parallel \\ \ge & \left(1-\left(1-k\right){b}_{n}^{\mathrm{\prime }}\right)\parallel {p}_{n}-q\parallel -{b}_{n}^{\mathrm{\prime }}\parallel T{p}_{n}-T{w}_{n}\parallel \\ -{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-{y}_{n}\parallel ,\end{array}$
which implies that for all $n\ge 0$,
(2.11)
(2.12)
(2.13)
Substituting (2.13) in (2.12), we have
$\begin{array}{rcl}\parallel {p}_{n}-{w}_{n}\parallel & \le & \left[{b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}+{c}_{n}\right]\parallel {y}_{n}-q\parallel \\ +L{b}_{n}^{\mathrm{\prime }}\left[\left[1+\left(1+L\right){b}_{n}+{c}_{n}\right]\parallel {y}_{n}-q\parallel \\ +{c}_{n}\parallel {u}_{n}-q\parallel \right]+{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +{c}_{n}\parallel {u}_{n}-q\parallel \\ =& \left[\left(1+L\right){b}_{n}^{\mathrm{\prime }}+L\left(1+L\right){b}_{n}{b}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}+{c}_{n}^{\mathrm{\prime }}\\ +\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\right]\parallel {y}_{n}-q\parallel \\ +{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\parallel {u}_{n}-q\parallel .\end{array}$
(2.14)
Substituting (2.14) in (2.11), we get
$\begin{array}{rcl}\parallel {p}_{n}-q\parallel & \le & \left(1-k{b}_{n}^{\mathrm{\prime }}+{k}^{-1}{c}_{n}^{\mathrm{\prime }}\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L{b}_{n}^{\mathrm{\prime }}\left[\left[\left(1+L\right){b}_{n}^{\mathrm{\prime }}\\ +L\left(1+L\right){b}_{n}{b}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}+{c}_{n}^{\mathrm{\prime }}+\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\right]\parallel {y}_{n}-q\parallel \\ +{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\parallel {u}_{n}-q\parallel \right]+{k}^{-1}{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ =& \left[1-{b}_{n}^{\mathrm{\prime }}\left[k-{k}^{-1}L\left(\left(1+L\right){b}_{n}^{\mathrm{\prime }}+L\left(1+L\right){b}_{n}{b}_{n}^{\mathrm{\prime }}\\ +\left(1+L\right){b}_{n}+{c}_{n}^{\mathrm{\prime }}+\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\right)-{k}^{-1}{t}_{n}\right]\right]\parallel {y}_{n}-q\parallel \\ +{k}^{-1}L{b}_{n}^{\mathrm{\prime }}\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}\parallel {u}_{n}-q\parallel +{k}^{-1}\left(1+L{b}_{n}^{\mathrm{\prime }}\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ \le & \left[1-{b}_{n}^{\mathrm{\prime }}\left[k-{k}^{-1}L\left({\left(1+L\right)}^{2}{b}_{n}^{\mathrm{\prime }}+\left(1+L\right){b}_{n}\\ +{c}_{n}^{\mathrm{\prime }}+\left(1+L\right){c}_{n}\right)-{k}^{-1}{t}_{n}\right]\right]\parallel {y}_{n}-q\parallel \\ +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \end{array}$
(2.15)
for any $n\ge 0$. Thus (2.15) implies that
$\begin{array}{rcl}\parallel {y}_{n+1}-q\parallel & \le & \parallel {y}_{n+1}-{p}_{n}\parallel +\parallel {p}_{n}-q\parallel \\ \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +{\epsilon }_{n}\\ =& \left(1-{\omega }_{n}\right)\parallel {y}_{n}-q\parallel +{\omega }_{n}{\beta }_{n}+{\gamma }_{n}.\end{array}$
(2.16)
With
we have
${\alpha }_{n+1}\le \left(1-{\omega }_{n}\right){\alpha }_{n}+{\omega }_{n}{\beta }_{n}+{\gamma }_{n},\phantom{\rule{1em}{0ex}}n\ge 0.$

Observe that ${\sum }_{n=0}^{\mathrm{\infty }}{\omega }_{n}=\mathrm{\infty }$, ${\omega }_{n}\in \left[0,1\right]$ and ${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0$. It follows from Lemma 3 that ${lim}_{n\to \mathrm{\infty }}\parallel {y}_{n}-q\parallel =0$.

Suppose that ${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$. It follows from equation (2.15) that
$\begin{array}{rcl}{\epsilon }_{n}& \le & \parallel {y}_{n+1}-q\parallel +\parallel {p}_{n}-q\parallel \\ \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +\parallel {y}_{n+1}-q\parallel \to 0,\end{array}$

as $n\to \mathrm{\infty }$; that is, ${\epsilon }_{n}\to 0$ as $n\to \mathrm{\infty }$. □

Using the techniques in the proof of Theorem 9, we have the following results.

Theorem 10 Let X, K, T, S, s, ${\left\{{u}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{v}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{z}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{w}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{y}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$and${\left\{{p}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$be as in Theorem  9. Suppose that${\left\{{a}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{b}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{c}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{a}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{b}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$and${\left\{{c}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$are sequences in$\left[0,1\right]$satisfying conditions (i), (iii)-(v) of Theorem  9 with
$\sum _{n=0}^{\mathrm{\infty }}{c}_{n}^{\mathrm{\prime }}<\mathrm{\infty }.$

Then the conclusions of Theorem  9 hold.

Theorem 11 Let X, K, T, S, s, ${\left\{{u}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{v}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{z}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{w}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{y}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$and${\left\{{p}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$be as in Theorem  9. Suppose that${\left\{{a}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{b}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{c}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{a}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{b}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$and${\left\{{c}_{n}^{\mathrm{\prime }}\right\}}_{n=0}^{\mathrm{\infty }}$are sequences in$\left[0,1\right]$satisfying condition (i), (iii) and (v) of Theorem  9 with
where m is a constant. Then
1. (a)
the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ converges strongly to the common fixed point q of T and S. Also,
$\parallel {x}_{n+1}-q\parallel \le \left(1-sm\right)\parallel {x}_{n}-q\parallel +C,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,$

where
$C={k}^{-1}\left(1+L\right)\left[\underset{n\ge 0}{Lsup}\left\{{c}_{n}\parallel {u}_{n}-q\parallel \right\}+\underset{n\ge 0}{sup}\left\{{c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \right\}\right],$
(b)
$\begin{array}{rcl}\parallel {y}_{n+1}-q\parallel & \le & \left(1-sm\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +{\epsilon }_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,\end{array}$
1. (c)

${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$ implies that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$.

Proof As in the proof of Theorem 9, we conclude that $F\left(T\right)\cap F\left(S\right)=\left\{q\right\}$ and
$\begin{array}{rcl}\parallel {x}_{n+1}-q\parallel & \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {x}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ \le & \left(1-sm\right)\parallel {x}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel \\ \le & \left(1-sm\right)\parallel {x}_{n}-q\parallel +C,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0.\end{array}$
Let

Observe that $0\le \theta <1$ and ${lim}_{n\to \mathrm{\infty }}{b}_{n}=0$. It follows from Lemma 4 that ${lim}_{n\to \mathrm{\infty }}\parallel {x}_{n}-q\parallel =0$.

Also, from (2.15), we have
$\begin{array}{rcl}\parallel {y}_{n+1}-q\parallel & \le & \left(1-s{b}_{n}^{\mathrm{\prime }}\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){b}_{n}^{\mathrm{\prime }}{c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +{\epsilon }_{n}\\ \le & \left(1-sm\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +{\epsilon }_{n}.\end{array}$
Suppose that ${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$. It follows from equation (2.15) that
$\begin{array}{rcl}{\epsilon }_{n}& \le & \parallel {y}_{n+1}-q\parallel +\parallel {p}_{n}-q\parallel \\ \le & \left(1-sm\right)\parallel {y}_{n}-q\parallel +{k}^{-1}L\left(1+L\right){c}_{n}\parallel {u}_{n}-q\parallel \\ +{k}^{-1}\left(1+L\right){c}_{n}^{\mathrm{\prime }}\parallel {v}_{n}-q\parallel +\parallel {y}_{n+1}-q\parallel \to 0,\end{array}$

as $n\to \mathrm{\infty }$; that is, ${\epsilon }_{n}\to 0$ as $n\to \mathrm{\infty }$.

Conversely, suppose that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$. Put

Observe that $0\le \theta <1$ and ${lim}_{n\to \mathrm{\infty }}{b}_{n}=0$. It follows from Lemma 4 that ${lim}_{n\to \mathrm{\infty }}\parallel {y}_{n}-q\parallel =0$. □

As an immediate consequence of Theorems 9 and 11, we have the following:

Corollary 12 Let K be a nonempty closed convex subset of an arbitrary Banach space X and$T,S:K\to K$be two Lipschitz strictly hemicontractive operators. Suppose that${\left\{{\alpha }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{\beta }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$are any sequences in$\left[0,1\right]$satisfying
1. (vi)

${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}=\mathrm{\infty }$,

2. (vii)

$L\left[{\left(1+L\right)}^{2}{\alpha }_{n}+\left(1+L\right){\beta }_{n}\right]\le k\left(k-s\right)$, $n\ge 0$,

where s is a constant in$\left(0,k\right)$. Suppose that${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$is the sequence generated from an arbitrary${x}_{0}\in K$by
Let ${\left\{{y}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ be any sequence in K and define ${\left\{{\epsilon }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ by
${\epsilon }_{n}=\parallel {y}_{n+1}-{p}_{n}\parallel ,\phantom{\rule{1em}{0ex}}n\ge 0,$
where
${p}_{n}=\left(1-{\alpha }_{n}\right){y}_{n}+{\alpha }_{n}T{w}_{n},$
and
${w}_{n}=\left(1-{\beta }_{n}\right){y}_{n}+{\beta }_{n}S{y}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.$
Then
1. (a)

the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ converges strongly to the common fixed point q of T and S,

2. (b)

${\sum }_{n=0}^{\mathrm{\infty }}{\epsilon }_{n}<\mathrm{\infty }$ implies that ${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$, so that ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ is almost common-stable on K,

3. (c)

${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$ implies that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$.

Corollary 13 Let X, K, T, S, s, ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{z}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{w}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{y}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$and${\left\{{p}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$be as in Theorem  9. Suppose that${\left\{{\alpha }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$, ${\left\{{\beta }_{n}\right\}}_{n=0}^{\mathrm{\infty }}$are sequences in$\left[0,1\right]$satisfying conditions (vi)-(vii) and (iii) of Theorem  9 with
${\alpha }_{n}\ge m>0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,$
where m is a constant. Then
1. (a)
the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ converges strongly to the common fixed point q of T and S. Also,
$\parallel {x}_{n+1}-q\parallel \le \left(1-sm\right)\parallel {x}_{n}-q\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,$
(b)
$\parallel {y}_{n+1}-q\parallel \le \left(1-sm\right)\parallel {y}_{n}-q\parallel +{\epsilon }_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,$

2. (c)

${lim}_{n\to \mathrm{\infty }}{y}_{n}=q$ implies that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=0$.

Example 14 Let $\mathbb{R}$ denote the set of real numbers with the usual norm, $K=\mathbb{R}$, and define $T,S:\mathbb{R}\to \mathbb{R}$ by
$Tx=\frac{2}{5}{sin}^{2}x,\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Sx=\frac{4}{5}x.$
Set $L=\frac{4}{5}$, $t=\frac{5}{4}$, $s=\frac{1}{400}$. Clearly, $F\left(T\right)\cap F\left(S\right)=\left\{0\right\}$ and
$|Tx-Ty|\le \frac{2}{5}|sinx-siny||sinx+siny|\le L|x-y|,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in \mathbb{R}.$

Clearly both T and S are Lipschitz operators on $\mathbb{R}$.

Also, it follows from (1.1) that
$\begin{array}{rcl}|\left(1+r\right)\left(x-y\right)-rt\left(Tx-Ty\right)|& \ge & \left(1+r\right)|x-y|-rt|Tx-Ty|\\ =& |x-y|+r\left(|x-y|-t|Tx-Ty|\right)\\ \ge & |x-y|\end{array}$
for any $x,y\in \mathbb{R}$ and $r>0$. Thus T is strongly pseudocontractive and Lemma 7 ensures that T is strictly hemicontractive. Put
then it can be easily seen that
$L\left[{\left(1+L\right)}^{2}{b}_{n}^{\mathrm{\prime }}+{c}_{n}^{\mathrm{\prime }}+\left(1+L\right)\left({b}_{n}+{c}_{n}\right)\right]+\frac{{c}_{n}^{\mathrm{\prime }}}{{b}_{n}^{\mathrm{\prime }}}\le 0.456\le 0.049375,\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0.$

It follows from Theorem 9 that the sequence ${\left\{{x}_{n}\right\}}_{n=0}^{\mathrm{\infty }}$ defined by (2.1) converges strongly to the common fixed point 0 of T and S in K and the iterative scheme defined by (2.1) is T-stable.

## Declarations

### Acknowledgements

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

## Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan
(3)
Faculty of Mechanical Engineering, University in Belgrade, Al. Rudara 12-35, Belgrade, 11 070, Serbia

## References

1. Xu Y: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224: 91–101. 10.1006/jmaa.1998.5987
2. Chidume CE, Osilike MO: Fixed point iterations for strictly hemicontractive maps in uniformly smooth Banach spaces. Numer. Funct. Anal. Optim. 1994, 15: 779–790. 10.1080/01630569408816593
3. Weng X: Fixed point iteration for local strictly pseudo-contractive mapping. Proc. Am. Math. Soc. 1991, 113(3):727–731. 10.1090/S0002-9939-1991-1086345-8
4. Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508–520. 10.2969/jmsj/01940508
5. Osilike MO: Stable iteration procedures for strong pseudocontractions and nonlinear operator equations of the accretive type. J. Math. Anal. Appl. 1996, 204: 677–692. 10.1006/jmaa.1996.0461
6. Chidume CE:An iterative process for nonlinear Lipschitzian strongly accretive mappings in ${L}_{p}$ spaces. J. Math. Anal. Appl. 1990, 151: 453–461. 10.1016/0022-247X(90)90160-H
7. Chidume CE: Iterative solutions of nonlinear equations with strongly accretive operators. J. Math. Anal. Appl. 1995, 192: 502–518. 10.1006/jmaa.1995.1185
8. Tan KK, Xu HK: Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces. J. Math. Anal. Appl. 1993, 178: 9–21. 10.1006/jmaa.1993.1287
9. Chang SS: Some problems and results in the study of nonlinear analysis. Nonlinear Anal. TMA 1997, 30(7):4197–4208. 10.1016/S0362-546X(97)00388-X
10. Chang SS, Cho YJ, Lee BS, Kang SM: Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudocontractive mappings in Banach spaces. J. Math. Anal. Appl. 1998, 224: 149–165. 10.1006/jmaa.1998.5993
11. Chidume CE: Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings. Proc. Am. Math. Soc. 1987, 99(2):283–288.
12. Chidume CE: Approximation of fixed points of strongly pseudocontractive mappings. Proc. Am. Math. Soc. 1994, 120: 545–551. 10.1090/S0002-9939-1994-1165050-6
13. Chidume CE: Iterative solution of nonlinear equations in smooth Banach spaces. Nonlinear Anal. TMA 1996, 26(11):1823–1834. 10.1016/0362-546X(94)00368-R
14. Chidume CE, Osilike MO: Nonlinear accretive and pseudocontractive operator equations in Banach spaces. Nonlinear Anal. 1998, 31: 779–789. 10.1016/S0362-546X(97)00439-2
15. Deng L: On Chidume’s open questions. J. Math. Anal. Appl. 1993, 174(2):441–449. 10.1006/jmaa.1993.1129
16. Deng L: An iterative process for nonlinear Lipschitz and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces. Acta Appl. Math. 1993, 32: 183–196. 10.1007/BF00998152
17. Deng L:Iteration processes for nonlinear Lipschitz strongly accretive mappings in ${L}_{p}$ spaces. J. Math. Anal. Appl. 1994, 188(1):128–140. 10.1006/jmaa.1994.1416
18. Deng L, Ding XP: Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces. Nonlinear Anal., Theory Methods Appl. 1995, 24(7):981–987. 10.1016/0362-546X(94)00115-X
19. Zeng LC: Iterative approximation of solutions to nonlinear equations of strongly accretive operators in Banach spaces. Nonlinear Anal. TMA 1998, 31: 589–598. 10.1016/S0362-546X(97)00425-2
20. Harder AM, Hicks TL: A stable iteration procedure for nonexpansive mappings. Math. Jpn. 1988, 33: 687–692.
21. Harder AM, Hicks TL: Stability results for fixed point iteration procedures. Math. Jpn. 1988, 33: 693–706.
22. Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5
23. Liu LW: Approximation of fixed points of a strictly pseudocontractive mapping. Proc. Am. Math. Soc. 1997, 125(5):1363–1366. 10.1090/S0002-9939-97-03858-6
24. Liu Z, Kang SM, Shim SH: Almost stability of the Mann iteration method with errors for strictly hemicontractive operators in smooth Banach spaces. J. Korean Math. Soc. 2003, 40(1):29–40.
25. Park JA: Mann iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces. J. Korean Math. Soc. 1994, 31: 333–337.
26. Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194(1):114–125. 10.1006/jmaa.1995.1289
27. Berinde V: Generalized contractions and applications (Romanian). Editura Cub Press 22, Baia Mare; 1997.Google Scholar