- Research
- Open Access
Convergence theorems for finite family of a general class of multi-valued strictly pseudo-contractive mappings
- Charles E Chidume^{1},
- Mmaduabuchi E Okpala^{1}Email author,
- Abdulmalik U Bello^{1} and
- Patrice Ndambomve^{1}
https://doi.org/10.1186/s13663-015-0365-7
© Chidume et al. 2015
- Received: 7 November 2014
- Accepted: 24 June 2015
- Published: 16 July 2015
Abstract
Let H be a real Hilbert space, K a nonempty subset of H, and \(T:K\rightarrow \mathit{CB}(K)\) a multi-valued mapping. Then T is called a generalized k-strictly pseudo-contractive multi-valued mapping if there exists \(k\in[0,1)\) such that, for all \(x,y\in D(T)\), we have \(D^{2}(Tx,Ty)\leq\|x-y\|^{2}+kD^{2}(Ax,Ay)\), where \(A:=I-T\), and I is the identity operator on K. A Krasnoselskii-type algorithm is constructed and proved to be an approximate fixed point sequence for a common fixed point of a finite family of this class of maps. Furthermore, assuming existence, strong convergence to a common fixed point of the family is proved under appropriate additional assumptions.
Keywords
- generalized k-strictly pseudo-contractive multi-valued mappings
- multi-valued maps
MSC
- 47H04
- 47H09
- 47H10
1 Introduction
Let \(T : D(T) \subseteq H \rightarrow \mathit{CB}(H)\) be a multi-valued mapping on H. A point \(x\in D(T)\) is called a fixed point of T if \(x \in Tx\). The fixed point set of T is denoted by \(F(T) := \{x \in D(T) : x \in Tx\}\).
The study of fixed points for multi-valued nonexpansive mappings using Hausdorff metric was introduced by Markin [1], and studied extensively by Nadler [2]. Since then, many results have appeared in the literature (see, e.g., Nadler [2] and Panyanak [3], and the references contained in them). Many of these results have found nontrivial applications in pure and applied sciences. Examples of such applications are, in control theory, convex optimization, differential inclusions, and economics (especially in game theory and market economy). For early results involving fixed points of multi-valued mappings and their applications see, for example, Brouwer [4], Daffer and Kaneko [5], Downing and Kirk [6], Geanakoplos [7], Kakutani [8], Nash [9, 10]. For details on the applications of this type of mappings in nonsmooth differential equations, one may consult Chang [11], Chidume et al. [12], Deimling [13], Khan et al. [14, 15], Reich et al. [16–18], Song and Wnag [19] and the references therein.
In studying the equation \(Au=0\), where A is a monotone operator defined on a real Hilbert space, Browder [20], introduced an operator T defined by \(T:=I-A\), where I is the identity mapping on H. He called such an operator a pseudo-contractive mapping. It is easily seen that the zeros of A are precisely the fixed points of the pseudo-contractive mapping T. It is well known that every nonexpansive mapping is pseudo-contractive but the converse is not true. In fact, in general, pseudo-contractive mappings are not necessarily continuous.
Moreover, Browder and Petryshyn [21] introduced the subclass of single-valued pseudo-contractive maps given below.
Definition 1.1
Since then, several extensions of this class of mappings to multi-valued cases have been defined and studied. For results on the approximation of common fixed points of families of multi-valued nonexpansive mappings (see, for example, Abbas et al. [22]), and for multi-valued strictly pseudo-contractive mappings see, Chidume et al. [12], Chidume and Ezeora [23], Ofoedu and Zegeye [24], Panyanak [3], Shahzad and Zegeye [25] and the references therein.
The class of multi-valued pseudo-contractive mappings introduced by Chidume et al. [12] is as follows.
Definition 1.2
Remark 1.3
Using the above definition, Chidume et al. [12] proved the following theorem, which extends the result of Browder and Petryshyn [21].
Theorem 1.4
([12])
Chidume and Ezeora [23] extended the theorems obtained in Chidume et al. [12] to a finite family \(\{T_{i}, i=1,2,\ldots ,m\}\) of multi-valued \(k_{i}\)-strictly pseudo-contractive mappings using a Krasnoselskii-type algorithm.
More precisely, they obtained the following theorems.
Theorem 1.5
([23])
Let K be a nonempty, closed, and convex subset of a real Hilbert space H and \(T_{i}:K \rightarrow \mathit{CB}(K)\) be a finite family of multi-valued \(k_{i}\)-strictly pseudo-contractive mappings, \(k_{i} \in (0,1)\), \(i = 1, \ldots ,m\), such that \(\bigcap_{i=1}^{m}F(T_{i})\neq\emptyset\). Assume that, for \(p\in\bigcap_{i=1}^{m}F(T_{i})\), \(T_{i}p=\{p\}\).
Theorem 1.6
([23])
Chidume and Okpala [26] introduced the following definition for multi-valued k-strictly pseudo-contractive mappings.
Definition 1.7
([26])
Remark 1.8
It is shown in [26] that every multi-valued k-strictly pseudo-contractive map as defined in [12] is a generalized k-strictly pseudo-contractive multi-valued mapping. An example is given in [26] of a generalized k-strictly pseudo-contractive mapping that is not a multi-valued k-strictly pseudo-copseudo-contractive mapping. The definition given here appears to be more natural than that given in [12].
It is our purpose in this paper to prove that a Krasnoselskii-type algorithm, under appropriate conditions, converges strongly to a common fixed point of a finite family \(\{T_{i}, i=1,2,\ldots,m\}\) of generalized \(k_{i}\)-strictly pseudo-contractive multi-valued mappings in a real Hilbert space. In the setting where the algorithms agree, our theorems generalize the results of Chidume et al. [12], and Chidume and Ezeora [23]. Moreover, they improve and extend to a finite family the results of Chidume and Okpala [26]. Also, several assumptions in the results of Chidume and Ezeora [23] (e.g., \(\lambda_{i}\in(k,1)\), for all i and \(T_{i}\) is continuous (see, e.g., [23], Theorem 2.4) and hemicompact for each i), are significantly weakened.
The rest of this paper is organized as follows. Some known results and useful lemmas are listed in Section 2. In Section 3, we state and prove our main theorem and the corollaries that follow from the theorem. In the last section, we show an illustrative example where our theorem is applicable.
2 Preliminaries
- (i)
\(x_{n}\rightarrow x\): \(\{x_{n}\}\) converges strongly to x as \(n\rightarrow\infty\).
- (ii)
H: a real Hilbert space with an induced norm \(\|\cdot\|\).
- (iii)
\(F(T):=\{x\in K: x\in Tx\}\).
- (iv)
\(\mathit{CB}(K)\), is the collection of nonempty, closed, and bounded subsets of K.
Definition 2.1
for each \(x,y\in K\). If \(L<1\) in inequality (2.1), the mapping T is called a contraction, and if \(L=1\), it is called nonexpansive.
We recall the following proposition.
Proposition 2.2
([26])
Let K be a nonempty subset of a real Hilbert space H and \(T:K\rightarrow \mathit{CB}(K)\) be a generalized k-strictly pseudo-contractive multi-valued 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 \(T:K\rightarrow \mathit{CB}(K)\) is said to be hemicompact if, for any sequence \(\{x_{n}\}\) such that \(\lim_{n\rightarrow\infty}d(x_{n}, Tx_{n})=0\), there exists a subsequence, say, \(\{x_{n_{k}}\} \) of \(\{x_{n}\}\) such that \(x_{n_{k}}\rightarrow p\in K\).
Remark 2.5
If K is compact, then every multi-valued mapping \(T:K\rightarrow \mathit{CB}(K)\) is hemicompact.
The following lemma will also be used in the sequel.
Lemma 2.6
([27])
Lemma 2.7
([23])
The following properties of the Hausdorf distance were established in [26].
Lemma 2.8
([26])
- (a)
\(D(B_{1}, B_{2})=D(x+B_{1}, x+B_{2})\),
- (b)
\(D(B_{1}, B_{2})=D(-B_{1},-B_{2})\),
- (c)
\(D(x+B_{1}, y+B_{2})\leq\|x-y\|+D(B_{1},B_{2})\),
- (d)
\(D(\{x\},B_{1})=\sup_{b_{1}\in B_{1}}\|x-b_{1}\|\),
- (e)
\(D(\{x\}, B_{1})=D(0,x-B_{1})\).
3 Main results
In this section, we prove strong convergence theorems for a common fixed point of a finite family of generalized k-strictly pseudo-contractive multi-valued mappings in a real Hilbert space.
Theorem 3.1
Proof
Corollary 3.2
Let K be nonempty, closed, and convex subset of a real Hilbert space H. For \(i=1,2,\ldots,m\), let \(T_{i}:K\rightarrow \mathit{CB}(K)\) be a family of generalized \(k_{i}\)-strictly pseudo-contractive multi-valued mapping with \(\bigcap_{i=1}^{m} F(T_{i})\neq\emptyset\). Assume that, for \(p\in\bigcap_{i=1}^{m} F(T_{i})\), \(T_{i}p=\{p\}\), and that \(T_{i_{0}}\) is hemicompact for some \(i_{0}\). Then the sequence \(\{x_{n}\}\) defined by (3.1) converges strongly to a common fixed point of \(\{T_{i}, i=1,2,\ldots,m\}\).
Proof
Remark 3.3
- (i)
The theorem is proved for the much larger class of generalized k-strictly pseudo-contractive multi-valued mappings.
- (ii)
No continuity assumption is imposed on our maps.
- (iii)
Only one arbitrary map is required to be hemicompact.
- (iv)
The condition \(\lambda_{i}\in(k,1)\), for all i is replaced by the weaker condition \(\lambda_{0}\in(k,1)\).
Corollary 3.4
Let K be nonempty, compact and convex subset of a real Hilbert space H, and for \(i=1,\ldots,m\), let \(T_{i}:K\rightarrow \mathit{CB}(K)\) be a family of generalized \(k_{i}\)-strictly pseudo-contractive multi-valued mappings with \(\bigcap_{i=1}^{m} F(T_{i})\neq\emptyset\). Assume that, for \(p\in\bigcap_{i=1}^{m} F(T_{i})\), \(T_{i}p=\{p\}\). Then the sequence \(\{x_{n}\}\) defined by (3.1) converges strongly to a common fixed point of the maps \(T_{i}\).
Proof
Since every multi-valued map defined on a compact set is necessarily hemicompact, \(T_{i}:K\rightarrow \mathit{CB}(K)\) is hemicompact for each i. Thus, by Corollary 3.2, \(\{x_{n}\}\) converges strongly to some \(p\in\bigcap_{i=1}^{m} F(T_{i})\). □
Remark 3.5
- (i)
The class of mappings considered in this paper contains the class of multi-valued k-strictly pseudo-contractive mappings as special case, which itself properly contains the class of multi-valued nonexpansive maps.
- (ii)
The algorithm here is of Krasnoselskii-type, which is well known to have a geometric order of convergence.
- (iii)
The method of proof used here is of independent interest as it does not assume that Tx is weakly closed for each \(x\in K\), or proximinal subset of K, as imposed in [12] and [23].
- (iv)
In the case where we have only one map, \(m=1\), we recover all the results of Chidume and Okpala [26].
4 An example
We finally give an example where the theorems are applicable. For the example, we shall need the following lemma.
Lemma 4.1
Proof
Remark 4.2
If we take \(c=4\) in this lemma, we recover Lemma 3.5 of [26].
Example 4.3
Declarations
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Authors’ Affiliations
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