On approximation of fixed points of multivalued pseudocontractive mappings in Hilbert spaces
- Felicia Obiageli Isiogugu^{1, 2}Email author
https://doi.org/10.1186/s13663-016-0548-x
© Isiogugu 2016
Received: 24 August 2015
Accepted: 22 April 2016
Published: 4 May 2016
Abstract
Keywords
MSC
1 Introduction
The approximation of fixed points of multivalued mappings with respect to the Hausdorff metric, using Mann [2] or Ishikawa [3] iteration scheme, has never been successful without imposing the condition that either the fixed point set of S is strict or that S is a multivalued mapping for which \(P_{S}\) satisfies some contractive conditions (see, e.g., [1, 4–9], and references therein). Among these recent studies, Chidume et al. [1] introduced the class of multivalued pseudocontractive mappings as follows.
Definition 1.1
([1])
Theorem 1.1
([1])
Theorem 1.2
([1])
Recently, Isiogugu et al. [10] observed that there are multivalued Lipschitzian maps with nonempty fixed point set for which neither the fixed point set of S is strict nor \(P_{S}\) satisfies any of the contractive conditions studied so far by authors. They also noted that these classes of maps have another interesting property of some existing maps considered recently by authors (see, e.g., [1, 4–9, 11, 12], and references therein). They therefore suggested to approximate the fixed points of multivalued mappings S directly instead of \(P_{S}\) and without imposing the strict fixed point set condition on S. This suggestion was also due to the fact that it has not been established that if a multivalued map S belongs to a class of maps, then \(P_{S}\) necessarily belongs to the same class of maps and that the fixed point set of S need not be strict. Consequently, they introduced the ‘type-one’ condition, which guarantees the weak convergence of the Mann sequence \(\{r_{n}\}\) without imposing the condition that the fixed point set of S is strict to a fixed point of a multivalued quasi-nonexpansive mapping S in a real Hilbert space. They also proved that under this condition, if S is nonexpansive, then \(I-S\) is demiclosed at zero. They obtained the following results.
Proposition 1.1
([10])
Let H be a real Hilbert space, and C a nonempty weakly closed subset of H. Let \(S:C\subseteq H\rightarrow P(H)\) be a multivalued mapping from C into the family of all proximinal subsets of H. Suppose that S is a nonexpansive mapping of type one. Suppose that \(\{r_{n}\}_{n=1}^{\infty }\subseteq C\) is such that \(\{r_{n}\}\) weakly converges to p and a sequence \(\{g_{n}\}\) with \(g_{n}\in Sr_{n}\) and \(\|r_{n}-g_{n}\|=d(r_{n},Sr_{n})\) for all \(n\in\mathbb{N}\) is such that \(\{ r_{n}-g_{n}\}\) strongly converges to 0. Then \(0\in(I-S)p\) (i.e., \(p=v\) for some \(v\in Sp\)).
Theorem 1.3
([10])
Theorem 1.4
([10])
Therefore, our purpose in this work is to establish that the type-one condition introduced by Isiogugu et al. [10] guarantees the weak (respectively, strong) convergence of the Mann (respectively, Ishikawa) sequence \(\{r_{n}\}\) to a fixed point of k-strictly pseudocontractive (respectively, pseudocontractive) mapping S of Chidume et al. [1] without the condition that the fixed point set of S is strict in a real Hilbert space. It is also proved that, under this condition, \(I-S\) is demiclosed at zero if S is k-strictly pseudocontractive. The results obtained extend, complement, and improve the results on multivalued and single-valued mappings and also give a partial solution to the problem of removing the strict fixed point set condition usually imposed on S.
2 Preliminaries
In the sequel, we shall need the following definitions and lemmas.
Definition 2.1
Let X be a nonempty set, and let \(S:X\rightarrow X\) be a map. A point \(r\in X\) is called a fixed point of S if \(r=Sr\). If \(S:X\rightarrow2^{X}\) is a multivalued map, then r is a fixed point of S if \(r\in Sr\). If \(Sr=\{r\}\), then r is called a strict fixed point of S. The set \(F(S)=\{r\in D(S):r\in Sr\} \) (respectively, \(F(S)=\{r\in D(S):r= Sr\}\)) is called the fixed point set of a multivalued (respectively, single-valued) map S, whereas the set \(F_{s}(S)=\{r\in D(S):Sr=\{r\}\}\) is called the strict fixed point set of S.
Definition 2.2
Definition 2.3
Definition 2.4
Definition 2.5
Let X be a Banach space. Let \(S:D(S)\subseteq X\rightarrow2^{X}\) be a multivalued mapping. Then \(I-S\) is said to be weakly demiclosed at zero if for any sequence \(\{r_{n}\}_{n=1}^{\infty}\subseteq D(S)\) such that \(\{r_{n}\}\) converges weakly to z and a sequence \(\{g_{n}\}\) with \(g_{n}\in Sr_{n}\) for all \(n\in\mathbb{N}\) such that \(\{r_{n}-g_{n}\}\) strongly converges to zero, we have \(z\in Sz\) (i.e., \(0\in (I-S)z\)).
Definition 2.6
([15])
Definition 2.7
Let X be a Banach space, and \(S:D(S)\subseteq X\rightarrow2^{X}\) a multivalued mapping. The graph of \(I-S\) is said to be closed in \(\sigma(X,X^{*})\times(X,\|\cdot\| )\) (i.e., \(I-S\) is weakly demiclosed or demiclosed) if for any sequence \(\{r_{n}\}_{n=1}^{\infty}\subseteq D(S)\) such that \(\{r_{n}\}\) weakly converges to z and a sequence \(\{g_{n}\}\) with \(g_{n}\in Sr_{n}\) for all \(n\in\mathbb{N}\) such that \(\{r_{n}-g_{n}\}\) strongly converges to g, we have \(g\in(I-S)z\) (i.e., \(g=z-v\) for some \(v\in Sz\)). Here I denotes the identity on X, \(\sigma(X,X^{*})\) is the weak topology of X, and \((X,\|\cdot\|)\) is the norm (or strong) topology of X.
Definition 2.8
([6])
Definition 2.9
([10])
Lemma 2.1
Lemma 2.2
([16])
Lemma 2.3
([4])
3 Main results
We now obtain a demiclosedness property in the sense that if \(\{r_{n}\}_{n=1}^{\infty}\subseteq C\) is such that \(\{r_{n}\}\) weakly converges to z and a sequence \(\{g_{n}\}\) with \(g_{n}\in Sr_{n}\) and \(\|r_{n}-g_{n}\|=d(r_{n},Sr_{n})\) for all \(n\in\mathbb{N}\) is such that \(\{ r_{n}-g_{n}\}\) strongly converges to 0, then \(0\in(I-S)z\) (i.e., \(z=v\) for some \(v\in Sz\)).
Proposition 3.1
Let H be a real Hilbert space, and Ca nonempty weakly closed subset of H. Let \(S:C\subseteq H\rightarrow P(H)\) be a multivalued mapping from C into the family of all proximinal subsets of H. If S is a k-strictly pseudocontractive mapping of type one, then \((I-S)\) is demiclosed at zero (i.e., the graph of \(I-S\) is closed at zero in \(\sigma(H,H^{*})\times(H,\|\cdot\|)\) or weakly demiclosed at zero).
Proof
Therefore, \(z=q\in Sz\). □
We now obtain some strong and weak convergence results for the class of pseudocontractive mappings and k-strictly pseudocontractive mappings of Chidume et al. [1], respectively, in Hilbert spaces.
Theorem 3.1
Proof
Theorem 3.2
Proof
We now have the following corollaries with proofs easily following from the definition and Theorems 3.1 and 3.2, respectively.
Corollary 3.1
([10])
Corollary 3.2
([10])
Concluding Remark
It is not clear if Theorem 3.1 and Theorem 3.2 will still hold if the classes of pseudocontractive and k-strictly pseudocontractive mappings considered are replaced with the classes k-strictly pseudocontractive-type and pseudocontractive-type mappings, respectively, considered by Isiogugu [5].
Declarations
Acknowledgements
The author is a post doctoral fellow at the School of Mathematics, Statistics and Computer Sciences, University of Kwazulu Natal, South Africa. She is grateful to the University for the scholarship, making their facilities available, and for hospitality. She is also grateful to the University of Nigeria Nsukka for granting her study leave.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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