Fixed point iteration for a countable family of multivalued strictly pseudocontractivetype mappings
 C. E. Chidume^{1, 2} and
 M. E. Okpala^{2}Email authorView ORCID ID profile
Received: 22 July 2015
Accepted: 27 August 2015
Published: 17 September 2015
Abstract
This paper introduces a new averaged algorithm for finding a common fixed point of a countably infinite family of generalized kstrictly pseudocontractive multivalued mappings. The new iterative sequence introduced is proved to be an approximating fixed point sequence for common fixed points of a countably infinite family of this class of mappings. Furthermore, under some mild assumptions, strong convergence theorems are also proved for this class of mappings. The method of proof used here is new and enables to overcome many strong restrictions appearing in contemporary literature. The stated theorems improve and generalize many recent works in iterative scheme for multivalued mappings.
Keywords
Mathematics Subject Classification
Background
Let (X, d) be a metric space, K a nonempty subset of X, and \(T:K\rightarrow 2^K\) be a multivalued mapping. A vector \(x\in K\) is a fixed point of T if \(x\in Tx\). For a single valued mapping T, a fixed point is any \(x\in K\) such that \(Tx=x\). We denote the collection of all fixed points of T by F(T). Many well known researchers like Brouwer (1912), Daffer and Kaneko (1995), Deimling (1992), and Kirk Downing and Kirk (1977), Geanakoplos (2003), Kakutani (1941), Markin (1973), Nadler (1969), Nash (1950, 1951) and Reich and Zaslavski (2002a, b, 2006), have studied fixed points for multivalued mappings.
Fixed point theory for multivalued mappings continues to attract a lot of attention because of its numerous real world applications in game theory and market economy, differential inclusions, and constrained optimization. They are also desirable in devising critical points in optimal control problems, energy management problems, signal processing, image reconstruction and a host of other problems.
Game theory and market economy is, perhaps, the most socially recognized application of multivalued mappings.
Though many theory for multivalued mappings in the literature have dealth with the existence of fixed points for such mappings, only very few have dealth with iterative algorithms for computing them. The problem of how to find such fixed points is part of what is addressed in this paper.
The first work on fixed points for multivalued (nonexpansive) mappings by the application of Hausdorff metric was done by Markin (1973), and followed by an extensive work by Nadler (1969). Since then, there are many results that have appeared in the literature and which have found novel applications in both pure and applied sciences. Notable among these results is the work of Browder (1967).
In studying the operator equation \(Au=0\) (when the mapping A is monotone), Browder (1967), introduced a new operator T defined by \(T:=IA,\) where I is the identity mapping on the Hilbert H. He called the operator a pseudocontractive mapping and the solutions of \(Au=0\), are exactly the fixed points of the pseudocontractive mapping T. An important proper subclass of the pseudocontractive mappings is the well know nonexpansive mappings.
Definition 1.1

pseudocontractive, in the terminology of Browder and Petryshyn (1967), if there exists \(k\in [0,1)\) such that$$\begin{aligned} \Vert TxTy\Vert ^2\le \Vert xy\Vert ^2+k\Vert (xTx)(yTy)\Vert ^2,\quad \forall x,y\in K. \end{aligned}$$(1)

monotone if$$\begin{aligned} \langle TxTy, xy\rangle \ge 0,\quad \forall x,y\in D(T). \end{aligned}$$

Definition of the mapping There is a problem of getting a right definition for the multivalued analogue which would be a generalization of the singlevalued case. There are several definitions available which will be a generalisation of the single valued case and one has to get the most natural among them to be able to establish some convergence theorems.

Identities In multivalued settings, the metric induced by the norm on X is not applicable and there is the need to develop new identities and other notions of distances which will be applicable. One notion of metric for sets that is readily applicable here is the Hausdorf metric.

Inference Many theorems and lemmas that are developed for single valued mappings cannot be carried over to multivalued cases and it is always difficult to make conclusions.
Chidume et al. (2013), introduced a multivalued analogue of Definition 1.2 as follows;
Definition 1.2
They proved a convergence theorem for this class of mapping as stated below:
Theorem 1.3
Very recently, Chidume and Okpala (2014) introduced a different class of multivalued strictly pseudocontractive mapping as given below:
Definition 1.4
The class of mapping introduced here is natural and has been proved to be a proper superset of the class introduced in Chidume et al. (2013).
They developed some new identities regarding Hausdorf metric and used a Krasnoselskii type algorithm and obtained the following theorem.
Theorem 1.5
We seek to prove strong convergence theorems, using a new averaged algorithm, for common fixed point of a countably infinite family of this general class of mappings in a real Hilbert space. Our theorem generalizes the results of Chidume et al. (2013), Chidume and Ezeora (2014), Panyanak (2007), Song and Wang (2008), among others and extends to a countable family the results of Chidume and Okpala (2014).
Preliminaries
We shall need the following definitions and notations in the sequel:
We casually denote \((D(A,B))^2\) by \(D^2(A,B)\) for all \(A, B\in CB(X)\) for simplicity of notation.
Definition 2.1

Lipschitzian if there exists \(L>0\) such that for each \(x,y\in K\),$$\begin{aligned} D(Tx,Ty)\le L \Vert xy\Vert , \end{aligned}$$(7)

nonexpansive if there exist \(L\le 1\) such that T is Lipschitchitzian.
Proposition 2.2
(Chidume and Okpala (2014)) Let K be a nonempty subset of a real Hilbert space H and \(T:K\rightarrow CB(K)\) be a generalized kstrictly pseudocontractive multivalued mapping. Then T is Lipschitzian.
Remark 2.3
Since every Lipschitz map is continuous, we would not make any continuity assumption on our mapping T throughout this paper.
Definition 2.4
A map \(T:K\rightarrow CB(K)\) is said to be hemicompact if, for any sequence \(\{x_n\}\) such that \(\lim \limits _{n\rightarrow \infty }d(x_n, Tx_n)=0, \) there exists a subsequence, say, \(\{x_{n_k}\} \) of \(\{x_n\}\) such that \(x_{n_k}\rightarrow p\in K\).
Remark 2.5
Trivial example of hemicompact mappings are mapping with compact domains.
Definition 2.6
Let H be a real Hilbert space and let T be a multivalued mapping. The multivalued mapping \(IT\) is said to be strongly demiclosed at 0 (see, e.g., Garcí aFalset et al. (2011)) if for any sequence \(\{x_n\}\subseteq D(T)\) such that \(x_n\rightarrow p\) and \(d(x_n, Tx_n)\) converges strongly to 0, then \(d(p, Tp)=0\).
Proposition 2.7
(Chidume and Okpala (2014)) Let K be a nonempty and closed subset of a real Hilbert space H and let \(T:K\rightarrow CB(K)\) be a generalized kstrictly pseudocontractive multivalued mapping. Then, \((IT)\) is strongly demiclosed at zero.
The following recurrent inequality will be used to make estimates in the sequel.
Lemma 2.8
Lemma 2.9
The following characterizations of the Hausdorf metric can be found in Chidume and Okpala (2014).
Lemma 2.10
 (a)
\(D(B_1, B_2)=D(x+B_1, x+B_2).\) Translation Invariance.
 (b)
\( D(B_1, B_2)=D(B_1,B_2).\)
 (c)
\( D(x+B_1, y+B_2)\le \Vert xy\Vert +D(B_1,B_2).\) Triangle inequality.
 (d)
\(D(\{x\},B_1)=\sup \limits _{b_1\in B_1}\Vert xb_1\Vert .\)
 (e)
\(D(\{x\}, B_1)=D(0,xB_1).\)
Fixed point iterations
The example given below shows that this general class of kstrictly pseudocontractive mappings actually exists and properly contains the class studied by Chidume et al. (2013), Osilike and Isiogugu (2011), Panyanak (2007), and a host of other authors. For the example, we shall need the following lemma, which is easy to verify.
Lemma 3.1
Remark 3.2
By setting \(c=4\) in the lemma above, we will recover Lemma (3.5) of Chidume and Okpala (2014).
Example 3.3
The following Lemma would be used in the sequel.
Lemma 3.4
Proof
Obviously, \(\Gamma _n^i\) is closed, convex and nonempty for each \(n\ge 1\) due to Lemma (2.10)(d).
Based upon these analyses, we now prove our main theorem. We will assume henceforth that K is a nonempty, closed and convex subset of a real Hilbert space H.
Theorem 3.5
Proof
Corollary 3.6
Let \(T_i:K\rightarrow CB(K)\) be a countably infinite family of generalized \(\kappa _i\) strictly pseudocontractive multivalued mappings such that for some \(\kappa \in (0,1), \,\,\kappa _i\in (0,\kappa ]\). Assume that \(\cap _{i=1}^\infty F(T_i)\ne \emptyset \) and suppose that for \(p\in \cap _{i=1}^\infty F(T_i),\) \(T_ip=\{p\}\). Assume \(T_{i_0}\) is hemicompact for some \(i_0.\) Then, the sequence \(\{x_n\}\) defined by algorithm (11) converges strongly to a fixed point of T.
Proof
Corollary 3.7
Let \(T_i:K\rightarrow CB(K)\) be a countably infinite family of generalized \(\kappa _i\) strictly pseudocontractive multivalued mapping, with \(\cap _{i=1}^\infty F(T_i)\ne \emptyset \) and assume that for \(p\in \cap _{i=1}^\infty F(T_i)\), \(T_ip=\{p\}\). Then, the sequence \(\{x_n\}\) defined by Eq. (11) converges strongly to a fixed point of T.
Proof
Since K is compact, the mappings \(T_i:K\rightarrow CB(K)\) is hemicompact. Thus, by Corollary (3.6), we have that \(\{x_n\}\) converges strongly to some \(p\in F(T)\). \(\square \)
Remark 3.8
 (i)
We proved the theorem for a countably infinite family of a much larger class of mapping which is the generalized kstrictly pseudocontractive multivalued mappings.
 (ii)
We only needed just one of the maps to be hemicompact and not all of them.
 (iii)
We replaced the ‘strong condition’ \(\delta _i\in (k,1)\) by a weaker condition \(\delta _0\in (k,1)\).
 (iv)
The condition \(\zeta _n^i\in \Gamma _n^i\) is more readily applicable than requiring that Tx is proximinal and weakly closed for each x, and then, computing \(\zeta _n=P_{Tx_n}x_n\) at each iterative step.
Conclusion
 (i)
The class of mappings considered in this paper contains the class of multivalued kstrictly pseudocontractive mappings as a special case, which itself properly contain the class of multivalued nonexpansive maps.
 (ii)
The algorithm here is of Krasnoselkii type, which is known to have a geometric order of convergence.
 (iii)
The condition that Tx be weakly closed for each \(x\in K\) as can be found, for example, in Chidume et al. (2013) and Chidume and Ezeora (2014) is dispensed with here.
Declarations
Authors’ contributions
CE proposed the problem. ME worked out the intricacies and drafted the first version of the manuscript. Both authors read and approved the final manuscript.
Acknowledgements
The authors would like to acknowledge funding from the African Capacity Building Foundation (ACBF) for the publication of this manuscript.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis 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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