- Research
- Open Access

# 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

- modified iterative scheme
- nonexpansive mappings
*ϕ*-hemicontractive mappings- Banach 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

## References

- Chidume CE: Iterative approximation of fixed point of Lipschitz strictly pseudocontractive mappings.
*Proc. Am. Math. Soc.*1987, 99: 283–288.MathSciNetMATHGoogle Scholar - Ishikawa S: Fixed point by a new iteration method.
*Proc. Am. Math. Soc.*1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Liu LW: Approximation of fixed points of a strictly pseudocontractive mapping.
*Proc. Am. Math. Soc.*1997, 125: 1363–1366. 10.1090/S0002-9939-97-03858-6View ArticleMathSciNetMATHGoogle Scholar - Mann WR: Mean value methods in iteration.
*Proc. Am. Math. Soc.*1953, 26: 506–510.View ArticleMathSciNetMATHGoogle Scholar - Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings.
*J. Nonlinear Convex Anal.*2007, 8(1):61–79.MathSciNetMATHGoogle Scholar - Ciric LB, Ume JS: Ishikawa iterative process for strongly pseudocontractive operators in Banach spaces.
*Math. Commun.*2003, 8: 43–48.MathSciNetMATHGoogle Scholar - Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces.
*J. Math. Anal. Appl.*1995, 194: 114–125. 10.1006/jmaa.1995.1289MathSciNetView ArticleMATHGoogle Scholar - Liu Z, Kim JK, Kang SM: Necessary and sufficient conditions for convergence of Ishikawa iterative schemes with errors to
*ϕ*-hemicontractive mappings.*Commun. Korean Math. Soc.*2003, 18(2):251–261.MathSciNetView ArticleMATHGoogle Scholar - Liu Z, Xu Y, Kang SM: Almost stable iteration schemes for local strongly pseudocontractive and local strongly accretive operators in real uniformly smooth Banach spaces.
*Acta Math. Univ. Comen.*2008, LXXVII(2):285–298.MathSciNetMATHGoogle Scholar - Schu J: On a theorem of C.E. Chidume concerning the iterative approximation of fixed points.
*Math. Nachr.*1991, 153: 313–319. 10.1002/mana.19911530127MathSciNetView ArticleMATHGoogle Scholar - Xue Z: Iterative approximation of fixed point for
*ϕ*-hemicontractive mapping without Lipschitz assumption.*Int. J. Math. Math. Sci.*2005, 17: 2711–2718.MathSciNetMATHGoogle Scholar - Zhou HY, Cho YJ: Ishikawa and Mann iterative processes with errors for nonlinear
*ϕ*-strongly quasi-accretive mappings in normed linear spaces.*J. Korean Math. Soc.*1999, 36: 1061–1073.MathSciNetMATHGoogle Scholar - Xu HK: Inequality in Banach spaces with applications.
*Nonlinear Anal.*1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar - Kato T: Nonlinear semigroups and evolution equations.
*J. Math. Soc. Jpn.*1967, 19: 508–520. 10.2969/jmsj/01940508View ArticleMathSciNetMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis - Fixed Point Theory and Its Applications*. Yokohama Publishers, Yokohama; 2000.MATHGoogle Scholar - Takahashi W, Yao J-C: Weak and strong convergence theorems for positively homogeneous nonexpansive mappings in Banach spaces.
*Taiwan. J. Math.*2011, 15: 961–980.MathSciNetMATHGoogle Scholar - 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.1993.1287MathSciNetView ArticleMATHGoogle Scholar - 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.1998.5987MathSciNetView ArticleMATHGoogle Scholar - Hussain N, Rafiq A: On modified implicit Mann iteration method involving strictly hemicontractive mappings in smooth Banach spaces.
*J. Comput. Anal. Appl.*2013, 15(5):892–902.MathSciNetMATHGoogle Scholar - Hussain N, Rafiq A, Ciric LB: Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 160Google Scholar - Khan SH, Rafiq A, Hussain N: A three-step iterative scheme for solving nonlinear
*ϕ*-strongly accretive operator equations in Banach spaces.*Fixed Point Theory Appl.*2012., 2012: Article ID 149Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.