# Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces

- Nawab Hussain
^{1}, - DR Sahu
^{2}and - Arif Rafiq
^{3}Email author

**2013**:247

https://doi.org/10.1186/1687-1812-2013-247

© Hussain et al.; licensee Springer. 2013

**Received: **7 May 2013

**Accepted: **5 September 2013

**Published: **7 November 2013

## Abstract

The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal *et al.* (J. Nonlinear Convex. Anal. 8(1): 61-79, 2007), associated with nonexpansive and *ϕ*-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

## Keywords

*ϕ*-hemicontractive mappingsBanach spaces

## Dedication

Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday

## 1 Preliminaries

*K*be a nonempty subset of an arbitrary Banach space

*X*, and let ${X}^{\ast}$ be its dual space. Let $T:X\to X$ be an operator. The symbols $D(T)$ and $R(T)$ stand for the domain and the range of

*T*, respectively. We denote $F(T)$ by the set of fixed points of a single-valued mapping $T:K\to K$. We denote by

*J*the normalized duality mapping from

*X*to ${2}^{{X}^{\ast}}$ defined by

Let $T:D(T)\subseteq X\to X$ be an operator.

**Definition 1**

*T*is called

*L*-

*Lipschitzian*if there exists $L\ge 0$ such that

for all $x,y\in D(T)$. If $L=1$, then *T* is called *non-expansive*, and if $0\u2a7dL<1$, *T* is called *contraction*.

- (i)
*T*is said to be strongly pseudocontractive if there exists a $t>1$ such that for each $x,y\in D(T)$, there exists $j(x-y)\in J(x-y)$ satisfying$Re\u3008Tx-Ty,j(x-y)\u3009\le \frac{1}{t}{\parallel x-y\parallel}^{2}.$ - (ii)
*T*is said to be strictly hemicontractive if $F(T)\ne \mathrm{\varnothing}$ and if there exists a $t>1$ such that for each $x\in D(T)$ and $q\in F(T)$, there exists $j(x-y)\in J(x-y)$ satisfying$Re\u3008Tx-q,j(x-q)\u3009\le \frac{1}{t}{\parallel x-q\parallel}^{2}.$ - (iii)
*T*is said to be*ϕ*-strongly pseudocontractive if there exists a strictly increasing function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\varphi (0)=0$ such that for each $x,y\in D(T)$, there exists $j(x-y)\in J(x-y)$ satisfying$Re\u3008Tx-Ty,j(x-y)\u3009\le {\parallel x-y\parallel}^{2}-\varphi (\parallel x-y\parallel )\parallel x-y\parallel .$ - (iv)
*T*is said to be*ϕ*-hemicontractive if $F(T)\ne \mathrm{\varnothing}$ and if there exists a strictly increasing function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\varphi (0)=0$ such that for each $x\in D(T)$ and $q\in F(T)$, there exists $j(x-y)\in J(x-y)$ satisfying$Re\u3008Tx-q,j(x-q)\u3009\le {\parallel x-q\parallel}^{2}-\varphi (\parallel x-q\parallel )\parallel x-q\parallel .$

Clearly, each strictly hemicontractive operator is *ϕ*-hemicontractive.

*K*of a normed space $X,S:K\to K$ and $T:K\to K$,

- (a)
where $\{{b}_{n}\}$ is a sequence in $[0,1]$;

- (b)
where $\{{b}_{n}\}$, $\{{b}_{n}^{\prime}\}$ are sequences in $[0,1]$ is known as the Ishikawa [2] iteration scheme;

- (c)
where $\{{b}_{n}\}$, $\{{b}_{n}^{\prime}\}$ are sequences in $[0,1]$, is known as the Agarwal-O’Regan-Sahu [5] iteration scheme;

- (d)
where $\{{b}_{n}\}$, $\{{b}_{n}^{\prime}\}$ are sequences in $[0,1]$, is known as the modified Agarwal-O’Regan-Sahu iteration scheme.

Chidume [1] 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 [3, 6–12].

The purpose of this paper is to characterize conditions for the convergence of the iterative scheme in the sense of Agarwal *et al.* [5] associated with nonexpansive and *ϕ*-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Our results improve and generalize most results in recent literature [1, 3, 5, 6, 8, 9, 11, 12].

## 2 Main result

The following result is now well known.

**Lemma 3** [13]

*For all*$x,y\in X$

*and*$j(x+y)\in J(x+y)$,

Now, we prove our main result.

**Theorem 4**

*Let*

*K*

*be a nonempty closed and convex subset of an arbitrary Banach space*

*X*,

*let*$S:K\to K$

*be nonexpansive*,

*and let*$T:K\to K$

*be a uniformly continuous*

*ϕ*-

*hemicontractive mapping such that*

*S*

*and*

*T*

*have the common fixed point*.

*Suppose that*${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*and*${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$

*are sequences in*$[0,1]$

*satisfying conditions*

- (i)
${lim}_{n\to \mathrm{\infty}}(1-{b}_{n})={lim}_{n\to \mathrm{\infty}}{b}_{n}^{\prime}=0$,

- (ii)
${\sum}_{n=1}^{\mathrm{\infty}}(1-{b}_{n})=\mathrm{\infty}$.

*For any*${x}_{1}\in K$,

*define the sequence*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*inductively as follows*:

*Then the following conditions are equivalent*:

- (a)
${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*converges strongly to the common fixed point**q**of**S**and**T*. - (b)
${\{S{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*and*${\{T{y}_{n}\}}_{n=1}^{\mathrm{\infty}}$*are bounded*.

*Proof* First, we prove that (a) implies (b).

Since *T* is *ϕ*-hemicontractive, it follows that $F(T)$ is a singleton. Let $F(S)\cap F(T)=\{q\}$ for some $q\in K$.

*S*and

*T*yields that

Thus, ${lim}_{n\to \mathrm{\infty}}T{y}_{n}=q$. Therefore, ${\{S{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{T{y}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are bounded.

Second, we need to show that (b) implies (a). Suppose that ${\{S{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{T{y}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are bounded.

It is clear that $\parallel {x}_{1}-q\parallel \le {M}_{1}$. Let $\parallel {x}_{n}-q\parallel \le {M}_{1}$. Next, we will prove that $\parallel {x}_{n+1}-q\parallel \le {M}_{1}$.

*T*ensures that

where the second inequality holds by the convexity of ${\parallel \cdot \parallel}^{2}$.

as $n\to \mathrm{\infty}$.

which is impossible. Hence, $\parallel {x}_{p+m+1}-q\parallel <\u03f5$. That is, (2.11) holds for all $m\ge 1$. Thus, (2.11) ensures that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=q$. This completes the proof. □

Taking $S=I$ in Theorem 4, we get the following.

**Corollary 5**

*Let*

*K*

*be a nonempty closed and convex subset of an arbitrary Banach space*

*X*,

*and let*$T:K\to K$

*be a uniformly continuous*

*ϕ*-

*hemicontractive mapping*.

*Suppose that*${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*and*${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$

*are sequences in*$[0,1]$

*satisfying conditions*(i)-(ii)

*of Theorem*4.

*For any*${x}_{1}\in K$,

*define the sequence*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*inductively as follows*:

*Then the following conditions are equivalent*:

- (a)
${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*converges strongly to the unique fixed point**q**of**T*. - (b)
${\{T{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*is bounded*.

**Remark 6**

- 1.
All the results can also be proved for the same iterative scheme with error terms.

- 2.
The known results for strongly pseudocontractive mappings are weakened by the

*ϕ*-hemicontractive mappings. - 3.
Our results hold in arbitrary Banach spaces, where as other known results are restricted for ${L}_{p}$ (or ${l}_{p}$) spaces and

*q*-uniformly smooth Banach spaces. - 4.
Theorem 4 is more general in comparison to the results of Agarwal

*et al.*[5] in the context of the class of*ϕ*-hemicontractive mappings. Theorem 4 extends convergence results coercing*ϕ*-hemicontractive mappings in the literature in the framework of Agarwal-O’Regan-Sahu iteration process (see also [14–21]).

## 3 Applications

**Theorem 7**

*Let*

*X*

*be an arbitrary real Banach space*, $S:X\to X$

*be nonexpansive*,

*and let*$T:X\to X$

*be uniformly continuous*

*ϕ*-

*strongly accretive operators*,

*respectively*.

*Suppose that*${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*and*${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$

*are sequences in*$[0,1]$

*satisfying conditions*(i)-(ii)

*of Theorem*4.

*For any*${x}_{1}\in X$,

*define the sequence*${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*inductively as follows*:

*where*$f\in X$,

*and*

*I*

*is the identity operator*.

*Then the following conditions are equivalent*:

- (a)
${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*converges strongly to the solution of the system*$Sx=f=Tx$. - (b)
${\{(I-S){x}_{n}\}}_{n=1}^{\mathrm{\infty}}$, ${\{(I-T){x}_{n}\}}_{n=1}^{\mathrm{\infty}}$

*and*${\{(I-T){y}_{n}\}}_{n=1}^{\mathrm{\infty}}$*are bounded*.

*Proof* Suppose that ${x}^{\ast}$ is the solution of the system $Sx=f=Tx$. Define $G,{G}^{\prime}:X\to X$ by $Gx=f+(I-S)x$ and ${G}^{\prime}x=f+(I-T)x$, respectively. Since *S* and *T* are nonexpansive and uniformly continuous *ϕ*-strongly accretive operators, respectively, so are *G* and ${G}^{\prime}$, then ${x}^{\ast}$ is the common fixed point of *G* and ${G}^{\prime}$. Thus, Theorem 7 follows from Theorem 4. □

**Example 8**Let $X=\mathbb{R}$ be the reals with the usual norm and $K=[0,1]$. Define $S:K\to K$ by

*T*is

*ϕ*-hemicontractive mapping with $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by $\varphi (t)=tan(t)$ for all $t\in [0,\mathrm{\infty})$. Moreover, 0 is the common fixed point of

*S*and

*T*. Let ${\{{b}_{n}\}}_{n=1}^{\mathrm{\infty}}$ and ${\{{b}_{n}^{\prime}\}}_{n=1}^{\mathrm{\infty}}$ be sequences in $[0,1]$ defined by

Then ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ defined by (2.1) in Theorem 4 converges to 0, which is the common fixed point of *S* and *T*.

## Declarations

### Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.

## Authors’ Affiliations

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