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Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces
Journal of Inequalities and Applications volume 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
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
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The authors would like to thank the referees for useful comments and suggestions.
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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/10.1186/1029242X2013132
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DOI: https://doi.org/10.1186/1029242X2013132
Keywords
 iterative scheme
 Lipschitzian mappings
 hemicontractive mappings
 Hilbert spaces