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Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces
Journal of Inequalities and Applicationsvolume 2013, Article number: 132 (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.
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
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
and it is said to be strongly pseudocontractive if there exists $k\in (0,1)$ such that
Let $F(T):=\{x\in H:Tx=x\}$, and let 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 Siterative scheme introduced by Sahu and Petruşel [4] in 2011.
In this paper, we establish strong convergence for the Siterative 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 wellknown 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 Siterative 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
and
With the help of (2.1), (2.2) and Lemma 2.1, we obtain the following estimates:
Substituting (2.4) and (2.5) in (2.3) we obtain
Also, with the help of condition (C) and (2.6), we have
Now, by ${lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0$, there exists ${n}_{0}\in \mathbb{N}$ such that for all $n\ge {n}_{0}$,
and with the help of (2.8), (2.7) yields
which implies
so that
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 singlevalued 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 Siterative 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].
References
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Ishikawa S: Fixed point by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S00029939197403364695
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Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362546X(91)90200K
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Acknowledgements
The authors would like to thank the referees for useful comments and suggestions.
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Keywords
 iterative scheme
 Lipschitzian mappings
 hemicontractive mappings
 Hilbert spaces