# Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces

- Shin Min Kang
^{1}, - Arif Rafiq
^{2}and - Sunhong Lee
^{1}Email author

**2013**:132

https://doi.org/10.1186/1029-242X-2013-132

© Kang et al.; licensee Springer 2013

**Received: **24 October 2012

**Accepted: **9 March 2013

**Published: **27 March 2013

## Abstract

In this paper, we establish strong convergence for the iterative scheme introduced by Sahu and Petruşel associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

**MSC:**47H10, 47J25.

## Keywords

## 1 Introduction

*H*be a Hilbert space, and let $T:H\to H$ be a mapping. The mapping

*T*is called

*Lipshitzian*if there exists $L>0$ such that

If $L=1$, then *T* is called *nonexpansive* and if $0\le L<1$, then *T* is called *contractive*.

*pseudocontractive*(see, for example, [1, 2]) if

*strongly pseudocontractive*if there exists $k\in (0,1)$ such that

*K*be a nonempty subset of

*H*. A mapping $T:K\to K$ is called

*hemicontractive*if $F(T)\ne \mathrm{\varnothing}$ and

It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractions. For the importance of fixed points of pseudocontractions, the reader may consult [1].

In 1974, Ishikawa [3] proved the following result.

**Theorem 1.1** *Let* *K* *be a compact convex subset of a Hilbert space* *H*, *and let* $T:K\to K$ *be a Lipschitzian pseudocontractive mapping*.

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

*let*$\{{x}_{n}\}$

*be a sequence defined iteratively by the Ishikawa iterative scheme*

*where*$\{{\alpha}_{n}\}$

*and*$\{{\beta}_{n}\}$

*are sequences satisfying the conditions*

- (i);$0\le {\alpha}_{n}\le {\beta}_{n}\le 1$
- (ii);${lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0$
- (iii).${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}{\beta}_{n}=\mathrm{\infty}$

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

Another iterative scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings is the *S*-iterative scheme introduced by Sahu and Petruşel [4] in 2011.

In this paper, we establish strong convergence for the *S*-iterative scheme associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

## 2 Main results

We need the following lemma.

**Lemma 2.1** [5]

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

*and*$\lambda \in [0,1]$,

*the following well*-

*known identity holds*:

Now we prove our main results.

**Theorem 2.2**

*Let*

*K*

*be a compact convex subset of a real Hilbert space*

*H*,

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

*be a Lipschitzian hemicontractive mapping satisfying*

*Let*$\{{\beta}_{n}\}$

*be a sequence in*$[0,1]$

*satisfying*

- (iv);${\sum}_{n=1}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty}$
- (v).${lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0$

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

*let*$\{{x}_{n}\}$

*be a sequence defined iteratively by the*

*S*-

*iterative scheme*

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to the fixed point of* *T*.

*Proof*From Schauder’s fixed point theorem, $F(T)$ is nonempty since

*K*is a compact convex set and

*T*is continuous. Let ${x}^{\ast}\in F(T)$. Using the fact that

*T*is hemicontractive, we obtain

*C*) and (2.6), we have

The rest of the argument follows exactly as in the proof of theorem of [3]. This completes the proof. □

**Theorem 2.3** *Let* *K* *be a compact convex subset of a real Hilbert space* *H*, *and let* $T:K\to K$ *be a Lipschitzian hemicontractive mapping satisfying condition* (*C*). *Let* $\{{\beta}_{n}\}$ *be a sequence in* $[0,1]$ *satisfying conditions* (iv) *and* (v).

*Assume that*${P}_{K}:H\to K$

*is the projection operator of*

*H*

*onto*

*K*.

*Let*$\{{x}_{n}\}$

*be a sequence defined iteratively by*

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

*Proof*The operator ${P}_{K}$ is nonexpansive (see,

*e.g.*, [2]).

*K*is a Chebyshev subset of

*H*so that ${P}_{K}$ is a single-valued mapping. Hence, we have the following estimate:

The set $K=K\cup T(K)$ is compact, and so the sequence $\{\parallel {x}_{n}-T{x}_{n}\parallel \}$ is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.2. This completes the proof. □

**Remark 2.4** In Theorem 1.1, putting ${\alpha}_{n}=1$, $0\le {\alpha}_{n}\le {\beta}_{n}\le 1$ implies ${\beta}_{n}=1$, which contradicts ${lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0$. Hence the *S*-iterative scheme is not the special case of Ishikawa iterative scheme.

**Remark 2.5** In Theorems 2.2 and 2.3, condition (*C*) is not new; it is due to Liu *et al.* [6].

## Declarations

### Acknowledgements

The authors would like to thank the referees for useful comments and suggestions.

## Authors’ Affiliations

## References

- Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In
*Nonlinear Functional Analysis*. Am. Math. Soc., Providence; 1976.Google Scholar - Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces.
*J. Math. Anal. Appl.*1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleGoogle 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 ArticleGoogle Scholar - Sahu DR, Petruşel A: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces.
*Nonlinear Anal.*2011, 74: 6012–6023. 10.1016/j.na.2011.05.078MathSciNetView ArticleGoogle Scholar - Xu HK: Inequalities in Banach spaces with applications.
*Nonlinear Anal.*1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleGoogle Scholar - Liu Z, Feng C, Ume JS, Kang SM: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings.
*Taiwan. J. Math.*2007, 11: 27–42.MathSciNetGoogle Scholar

## Copyright

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