Strong convergence of modified Halpern’s iterations for a k-strictly pseudocontractive mapping
© Li et al.; licensee Springer 2013
Received: 27 June 2012
Accepted: 20 February 2013
Published: 12 March 2013
In this paper, we discuss three modified Halpern iterations as follows:
and obtained the strong convergence results of the iterations (I)-(III) for a k-strictly pseudocontractive mapping, where satisfies the conditions: (C1) and (C2) , respectively. The results presented in this work improve the corresponding ones announced by many other authors.
Let H be a real Hilbert space with the inner product and the norm and let C be a nonempty closed convex subset of H.
and we can assume also that so that .
It is obvious that a k-strictly pseudocontractive mapping is Lipschitzian with . The class of nonexpansive mappings is a subclass of strictly pseudocontractive mappings in a Hilbert space, but the converse implication may be false. We remark that the class of strongly pseudo-contractive mappings is independent from the class of k-strict pseudo-contractions.
The strong convergence of Halpern’s iteration to a fixed point of T has also been proved in Banach spaces; see, e.g., [4–10]. Reich [4, 5] has showed the strong convergence of the sequence (1.4), where satisfies the conditions (C1), (C2) and (C5), is decreasing (noting that the condition (C5) is a special case of condition(C4)). In 1997, Shioji and Takahashi  extended Wittmann’s result to Banach spaces. In 2002, Xu  obtained a strong convergence theorem, where satisfies the following conditions: (C1), (C2) and (C6) . In particular, the canonical choice of satisfies the conditions (C1), (C2) and (C6).
However, is a real sequence satisfying the conditions (C1) and (C2) sufficient to guarantee the strong convergence of Halpern’s iteration (1.4) for nonexpansive mappings? It remains an open question, see .
where and satisfies the conditions (C1) and (C2).
where , , are three real sequences in , satisfying . He showed that the sequence satisfying the conditions (C1) and (C2) is sufficient to guarantee the strong convergence of the modified Halpern’s iterative sequence (1.7) for nonexpansive mappings.
The purpose of this paper is to present a significant answer to the above open question. We will show that the sequence satisfying the conditions (C1) and (C2) is sufficient to guarantee the strong convergence of the modified Halpern’s iterative sequences (1.5)-(1.7) for k-strictly pseudocontractive mappings, respectively. The results present in this paper improve and develop the corresponding results of [7, 10, 12, 13].
In what follows we will need the following:
Lemma 2.1 
for all and for all .
Lemma 2.2 
Let C be a nonempty closed convex subset of a real Hilbert space H, be a k-strictly pseudocontractive mapping. Then is demiclosed at zero.
Lemma 2.3 
3 Main results
In this section, proving the following theorems, we show that the conjunction of (C1) and (C2) is a sufficient condition on our iteration (I)-(III), respectively.
where , , are three real sequences in , satisfying and . Suppose that satisfies the conditions: (C1) ; (C2) . Then as , converges strongly to some fixed point of T and , where is the metric projection from H onto .
This proves the boundedness of the sequence , which leads to the boundedness of .
and hence the desired result is obtained by the condition (C1) and .
We may assume that since H is reflexive and is bounded. From (3.4), it follows from Lemma 2.2 that .
This completes the proof of Theorem 3.1. □
where , . Suppose that satisfies the conditions: (C1) ; (C2) . Then as , converges strongly to some fixed point of T and , where is the metric projection from H onto .
This proves the boundedness of the sequence , which implies that the sequence is bounded also.
This completes the proof of Theorem 3.2. □
where , , . Suppose that satisfies the conditions: (C1) ; (C2) . Then as , converges strongly to some fixed point of T, and , where is the metric projection from H onto .
This implies that is a quasi-firmly type nonexpansive mapping (see, for example, ). is also a strongly quasi-nonexpansive mapping (see, for example, ). Hence it follows from [11, 16] (see Theorem 3.1 and Remark 1 of  or Corollary 8 of ) that converges strongly to a point .
This completes the proof of Theorem 3.3. □
Remark 3.1 Theorems 3.1-3.3 improve the main results of [7, 10, 12, 13] from a nonexpansive mapping to a k-strictly pseudocontractive mapping, respectively. Theorems 3.1-3.3 show that the real sequence satisfying the two conditions (C1) and (C2) is sufficient for the strong convergence of the iterative sequences (I)-(III) for k-strictly pseudocontractive mappings, respectively. Therefore, our results give a significant partial answer to the open question.
The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article. The first author was supported by the Natural Science Foundational Committee of Hebei Province (Z2011113) and Hebei Normal University of Science and Technology (ZDJS 2009 and CXTD2010-05).
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