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

## 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.

## 1 Introduction

Let 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

$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H.$

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

The mapping $T:H\to H$ is said to be pseudocontractive (see, for example, [1, 2]) if

${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H$
(1.1)

and it is said to be strongly pseudocontractive if there exists $k\in \left(0,1\right)$ such that

${\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+k{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H.$
(1.2)

Let $F\left(T\right):=\left\{x\in H:Tx=x\right\}$, and let K be a nonempty subset of H. A mapping $T:K\to K$ is called hemicontractive if $F\left(T\right)\ne \mathrm{\varnothing }$ and

${\parallel Tx-{x}^{\ast }\parallel }^{2}\le {\parallel x-{x}^{\ast }\parallel }^{2}+{\parallel x-Tx\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in H,{x}^{\ast }\in F\left(T\right).$

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 .

In 1974, Ishikawa  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 $\left\{{x}_{n}\right\}$ be a sequence defined iteratively by the Ishikawa iterative scheme

$\left\{\begin{array}{c}{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{y}_{n},\hfill \\ {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}$
(1.3)

where $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ are sequences satisfying the conditions

1. (i)
$0\le {\alpha }_{n}\le {\beta }_{n}\le 1$

;

2. (ii)
${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0$

;

3. (iii)
${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}{\beta }_{n}=\mathrm{\infty }$

.

Then the sequence $\left\{{x}_{n}\right\}$ 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  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 

For all $x,y\in H$ and $\lambda \in \left[0,1\right]$, the following well-known identity holds:

${\parallel \left(1-\lambda \right)x+\lambda y\parallel }^{2}=\left(1-\lambda \right){\parallel x\parallel }^{2}+\lambda {\parallel y\parallel }^{2}-\lambda \left(1-\lambda \right){\parallel x-y\parallel }^{2}.$

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 $\left\{{\beta }_{n}\right\}$ be a sequence in $\left[0,1\right]$ satisfying

1. (iv)
${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_{n}=\mathrm{\infty }$

;

2. (v)
${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0$

.

For arbitrary ${x}_{1}\in K$, let $\left\{{x}_{n}\right\}$ be a sequence defined iteratively by the S-iterative scheme

$\left\{\begin{array}{c}{x}_{n+1}=T{y}_{n},\hfill \\ {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1.\hfill \end{array}$
(2.1)

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

Proof From Schauder’s fixed point theorem, $F\left(T\right)$ is nonempty since K is a compact convex set and T is continuous. Let ${x}^{\ast }\in F\left(T\right)$. Using the fact that T is hemicontractive, we obtain

${\parallel T{x}_{n}-{x}^{\ast }\parallel }^{2}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {x}_{n}-T{x}_{n}\parallel }^{2}$
(2.2)

and

${\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}\le {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {y}_{n}-T{y}_{n}\parallel }^{2}.$
(2.3)

With the help of (2.1), (2.2) and Lemma 2.1, we obtain the following estimates: (2.4) (2.5)

Substituting (2.4) and (2.5) in (2.3) we obtain

$\begin{array}{rcl}{\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}& \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left(1-{\beta }_{n}\right){\parallel {x}_{n}-T{y}_{n}\parallel }^{2}+{\beta }_{n}{\parallel T{x}_{n}-T{y}_{n}\parallel }^{2}\\ -{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}.\end{array}$
(2.6)

Also, with the help of condition (C) and (2.6), we have

$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& =& {\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left(1-{\beta }_{n}\right){\parallel {x}_{n}-T{y}_{n}\parallel }^{2}+{\beta }_{n}{\parallel T{x}_{n}-T{y}_{n}\parallel }^{2}\\ -{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel T{x}_{n}-T{y}_{n}\parallel }^{2}-{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{L}^{2}{\parallel {x}_{n}-{y}_{n}\parallel }^{2}-{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ =& {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{L}^{2}{\beta }_{n}^{2}{\parallel {x}_{n}-T{x}_{n}\parallel }^{2}-{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ =& {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\beta }_{n}\left(1-\left(2+{L}^{2}\right){\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}.\end{array}$
(2.7)

Now, by ${lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0$, there exists ${n}_{0}\in \mathbb{N}$ such that for all $n\ge {n}_{0}$,

${\beta }_{n}\le \frac{1}{2\left(2+{L}^{2}\right)},$
(2.8)

and with the help of (2.8), (2.7) yields

${\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-\frac{1}{2}{\beta }_{n}{\parallel {x}_{n}-T{x}_{n}\parallel }^{2},$

which implies

$\frac{1}{2}{\beta }_{n}{\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2},$

so that

$\frac{1}{2}\sum _{j=N}^{n}{\beta }_{j}{\parallel {x}_{j}-T{x}_{j}\parallel }^{2}\le {\parallel {x}_{N}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}.$

The rest of the argument follows exactly as in the proof of theorem of . 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 $\left\{{\beta }_{n}\right\}$ be a sequence in $\left[0,1\right]$ satisfying conditions (iv) and (v).

Assume that ${P}_{K}:H\to K$ is the projection operator of H onto K. Let $\left\{{x}_{n}\right\}$ be a sequence defined iteratively by

$\left\{\begin{array}{c}{x}_{n+1}={P}_{K}\left(T{y}_{n}\right),\hfill \\ {y}_{n}={P}_{K}\left(\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n}\right),\phantom{\rule{1em}{0ex}}n\ge 1.\hfill \end{array}$

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

Proof The operator ${P}_{K}$ is nonexpansive (see, e.g., ). K is a Chebyshev subset of H so that ${P}_{K}$ is a single-valued mapping. Hence, we have the following estimate:

$\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& =& {\parallel {P}_{K}\left(T{y}_{n}\right)-{P}_{K}{x}^{\ast }\parallel }^{2}\\ \le & {\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\beta }_{n}\left(1-\left(2+{L}^{2}\right){\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}.\end{array}$

The set $K=K\cup T\left(K\right)$ is compact, and so the sequence $\left\{\parallel {x}_{n}-T{x}_{n}\parallel \right\}$ 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. .

## References

1. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis. Am. Math. Soc., Providence; 1976.

2. 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)90085-6

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

4. 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.2011.05.078

5. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

6. 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.

## Acknowledgements

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

## Author information

Authors

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Correspondence to Sunhong Lee.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors read and approved the final manuscript.

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Kang, S.M., Rafiq, A. & Lee, S. Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces. J Inequal Appl 2013, 132 (2013). https://doi.org/10.1186/1029-242X-2013-132

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

### Keywords

• iterative scheme
• Lipschitzian mappings
• hemicontractive mappings
• Hilbert spaces 