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Best proximity point theorems for α-nonexpansive mappings in Banach spaces
- Dezhou Kong^{1, 2},
- Lishan Liu^{1, 3}Email author and
- Yonghong Wu^{3, 4}
https://doi.org/10.1186/s13663-015-0413-3
© Kong et al. 2015
Received: 25 January 2015
Accepted: 24 June 2015
Published: 4 September 2015
Abstract
In this paper, we discuss sufficient and necessary conditions for the existence of best proximity points for non-self-α-nonexpansive mappings in Banach spaces. We obtain convergence results under some assumptions, and we prove the existence of common best proximity points for a family of non-self-α-nonexpansive mappings.
Keywords
- best proximity point
- common best proximity point
- α-nonexpansive
- d-property
MSC
- 90C33
- 54H25
- 47H09
- 34B18
1 Introduction
Let \((X,d)\) be a metric space and A a nonempty subset of X. The mapping \(T:A\rightarrow X\) is said to have a fixed point in A if the fixed point equation \(Tx=x\) has at least one solution. In metric terminology, we say that \(x\in A\) is a fixed point of T if \(d(x,Tx)=0\). If the fixed point equation of a given mapping does not have a solution, then \(d(x,Tx)>0\) for all \(x\in A\). In such a situation, it is our aim to find an element \(x\in A\) such that \(d(x,Tx)\) is minimum in some sense, and the x is said to be the best approximation of the fixed point of T. In this paper, we will study the best approximation theory and best proximity pair theorems.
Consider the following well-known best approximation theorem due to Ky Fan [1].
Theorem 1.1
[1]
Let A be a nonempty compact convex set in a normed linear space X. If T is a continuous map from A into X, then there exists a point x in A such that \(\| x-Tx\|=\operatorname{dist}(Tx, A)\).
The point x in the theorem above is called a best approximation point of T in A. Note that if \(x\in A\) is a best approximation point, then \(\|x-Tx\|\) need not be the optimum. Best proximity point theorems have been explored to find sufficient conditions so that the minimization problem \(\min_{x\in A}\|x-Tx\|\) has at least one solution. To have a concrete lower bound, let us consider two nonempty subsets A, B of a metric space X and a mapping \(T:A\rightarrow B\). The natural question is whether one can find an element \(x_{0}\in A\) such that \(d(x_{0},Tx_{0})=\min\{d(x,Tx): x\in A\}\). Since \(d(x,Tx)\geq\operatorname{dist}(A, B)\), it is of interest to find a point \(x_{0}\in A\) such that \(d(x_{0},Tx_{0})=\operatorname{dist}(A, B)\). This situation motivates the researchers to develop the notion called best proximity point theory. It is worth to note that the best proximity point theorems can be viewed as a generalization of fixed point theorems, since most fixed point theorems can be derived as corollaries of best proximity point theorems. Some of the interesting results for best proximity points can be found in [2–18].
The following notion of weakly contractive mapping was introduced by Sankar Raj in [19].
Definition 1.1
[19]
Note that \(d(Tx,Ty)\leq d(x,y)-\psi(d(x,y))< d(x,y)\) if \(x,y\in A\) with \(x\neq y\). That is, T is a contractive map. In [19], Sankar Raj obtained the existence theorem of a best proximity point for weakly contractive mappings as follows.
Theorem 1.2
[19]
Let \((A, B)\) be a pair of two nonempty closed subsets of a complete metric space X such that \(A_{0}\) is nonempty. Let \(T: A\rightarrow B\) be a weakly contractive mapping such that \(T(A_{0})\subseteq B_{0}\). Assume that the pair \((A, B)\) has the d-property. Then there exists a unique point \(x^{*}\) in A such that \(d(x^{*},Tx^{*})=\operatorname{dist}(A,B)\).
In [20], Aoyama and Kohsaka introduced the self-α-nonexpansive mapping which is defined as follows.
Definition 1.2
[20]
It is obvious that every nonexpansive mapping is 0-nonexpansive. We denote the fixed point set of a mapping T by \(F(T)\). In [20], they proved the following result.
Theorem 1.3
[20]
Let X be a uniformly convex Banach space, let A be a nonempty, closed, and convex subset of X, and let \(T: A\rightarrow A\) be an α-nonexpansive mapping for some real number α such that \(\alpha< 1\). Then \(F(T)\) is nonempty if and only if there exists \(x\in A\) such that \(\{T^{n}(x)\}\) is bounded.
In [21] and [22], Abkar and Gabeleh studied the best proximity points of some non-self-mappings under appropriate conditions.
Motivated and inspired by the above mentioned work, in this article, we consider a map \(T: A\rightarrow B\), where A and B are nonempty subsets of a Banach space X, which is non-self-α-nonexpansive in the sense of Definition 1.2. We attempt to study the sufficient and necessary conditions for the existence of a best proximity point for non-self-α-nonexpansive mappings and convergence results. Moreover, we discuss the existence of common best proximity points for a family of non-self-α-nonexpansive mappings. When the map T is considered to be a self-map, then our result reduces to the fixed point theorem of Aoyama and Kohsaka for α-nonexpansive mappings. Our results are generalization and improvement of the recent results obtained by many authors.
2 Preliminaries
In [23], Kirk et al. proved the following lemma which guarantees the nonemptiness of \(A_{0}\) and \(B_{0}\).
Lemma 2.1
[23]
Let X be a reflexive Banach space and A be a nonempty, closed, bounded, and convex subset of X, and B be a nonempty, closed, and convex subset of X. Then \(A_{0}\) and \(B_{0}\) are nonempty and satisfy \(P_{B}(A_{0})\subseteq B_{0}\), \(P_{A}(B_{0})\subseteq A_{0}\).
Also, in [24], Sadiq Basha and Veeramani proved that \(A_{0}\) is contained in the boundary of A. It is easy to verify that \(A_{0}\) and \(B_{0}\) are closed convex subsets of A and B, respectively, if A and B are closed and convex.
The notion called the d-property was introduced in [25].
Definition 2.1
[25]
Definition 2.2
[25]
A normed linear space X is said to have the d-property if and only if every pair \((A, B)\) of nonempty and closed convex subsets of X has the d-property.
Lemma 2.3
[25]
Let A, B be nonempty, closed, and convex subsets of a strictly convex space X such that \(A_{0}\) is nonempty. Then the restriction of the metric projection mapping \(P_{A_{0}}\) to \(B_{0}\) is an isometry. That is, \(P_{A_{0}}: B_{0}\rightarrow A_{0}\) is an isometry.
Let us define the notion of non-self-α-nonexpansive maps as follows.
Definition 2.3
Remark 2.1
We note that a non-self-nonexpansive mapping (\(\| Tx-Ty\|\leq\|x-y\|\), \(x,y\in A\)) \(T:A \rightarrow B\) is a non-self-0-nonexpansive mapping.
The following example shows that there is a discontinuous non-self-α-nonexpansive mapping.
Example 2.1
Proof
Lemma 2.4
Let A, B be nonempty, closed, and convex subsets of a strictly convex space X such that \(A_{0}\) is nonempty. Then \(P^{2}(x)=x\) for all \(x\in A_{0}\cup B_{0}\).
Proof
Remark 2.2
We note that there exists \(x\in A_{0}\) such that \(\|x-T(x)\|=\operatorname{dist}(A,B)\) if and only if x is the fixed point of \(P_{A_{0}} T\).
3 Best proximity point theorems
Now let us use the above characterization of strictly convex spaces to prove the following best proximity point theorem of non-self-α-nonexpansive mappings.
Theorem 3.1
Let X be a uniformly convex Banach space and A, B be nonempty, closed, and convex subsets of X. Suppose that \(A_{0}\) is nonempty and \(T: A\rightarrow B\) is an α-nonexpansive mapping on \(A_{0}\) for some real number α such that \(0<\alpha< 1\) and \(T(A_{0})\subseteq B_{0}\). Then T has at least one best proximity point if and only if there exists \(x\in A_{0}\) such that \(\{(PT)^{n}(x)\}\) is bounded. Moreover, if T is continuous and \(\|(PT)^{2}(x)-x\|\leq r\|(PT)(x)-x\|\) for all \(x\in A_{0}\), where \(0< r<\sqrt{2}\), then \((PT)^{n}(x)\) converges to a proximity point for all \(x\in A_{0}\).
Proof
Remark 3.1
If \(A=B\), Theorem 3.1 is the fixed point theorem of Aoyamma and Kohsaka [20] with convergence result.
Remark 3.2
If T is a weakly contractive mapping, by Theorem 3.1, we can obtain the best proximity point theorem of Sankar Raj [19] with convergence result.
Corollary 3.1
Let X be a uniformly convex Banach space and A, B be nonempty, closed, and convex subsets of X. Suppose that A is bounded and \(T: A\rightarrow B\) is an α-nonexpansive mapping for some real number \(\alpha< 1\) such that \(T(A_{0})\subseteq B_{0}\). Then T has at least one best proximity point.
Proof
Since X is a uniformly convex Banach space and A is bounded, we see that \(A_{0}\) is nonempty and \(\{(PT)^{n}(x)\}\) is bounded for any \(x\in A_{0}\). By Theorem 3.1, T has at least one best proximity point. □
Corollary 3.2
Let X be a uniformly convex Banach space and A, B be nonempty, closed, and convex subsets of X, and let \(A_{0}\) be nonempty. Suppose that \(T(A_{0})\subseteq B_{0}\) and \(T: A\rightarrow B\) is a nonexpansive mapping on \(A_{0}\), i.e. \(\|T(x)-T(y)\|\leq\|x-y\|\), for all \(x, y\in A_{0}\). Then T has at least one best proximity point if and only if there exists \(x\in A_{0}\) such that \(\{(PT)^{n}(x)\}\) is bounded.
Proof
It is well known that nonexpansive mappings are 0-nonexpansive. By Theorem 3.1, the assertion is proved. □
Let A, B be nonempty convex subsets of a normed linear space. A mapping \(T: A\rightarrow B\) is said to be affine if \(T(\lambda x+(1-\lambda)y)=\lambda T(x)+(1-\lambda)T(y)\), for all \(x, y\in A\) and \(\lambda\in(0,1)\). For convenience, we define \(F_{A}(T)=\{x\in A: \|x-T(x)\|=\operatorname{dist}(A, B)\}\).
Theorem 3.2
- (i)
T, S are, respectively, non-self-α-nonexpansive and β-nonexpansive mappings on \(A_{0}\) such that \(\alpha, \beta< 1\);
- (ii)
T is an affine and continuous mapping, and \(T(A_{0})\subseteq B_{0}\);
- (iii)
\(P_{B_{0}}S(x)=TS(x)\) for all \(x\in F_{A}(T)\).
Proof
Remark 3.3
In Theorem 3.2, S does not need to be continuous.
Example 3.1
4 Common best proximity points
In this section, we discuss sufficient conditions for the existence of common best proximity points for α-nonexpansive mappings.
Theorem 4.1
- (i)
T, S are respectively α-nonexpansive and β-nonexpansive mappings on \(A_{0}\) such that \(\alpha, \beta< 1\);
- (ii)
T is an affine and continuous mapping, and \(T(A_{0})\subseteq B_{0}\), \(S(A_{0})\subseteq B_{0}\);
- (iii)
for any \(x\in F_{A}(T)\), there exists \(y\in F_{A}(T)\) such that \(S(x)=T(y)\).
Proof
The following theorem guarantees the existence of a common best proximity point for a finite family \(\Im=\{T_{1}, T_{2}, \ldots, T_{n}\} \) of affine, α-nonexpansive mappings.
Theorem 4.2
- (i)
\(T_{i}\) are respectively \(\alpha_{i}\)-nonexpansive mappings on \(A_{0}\) such that \(\alpha_{i}< 1\), for all \(i=1, 2, \ldots, n\);
- (ii)
\(T_{i}\) are an affine and continuous mappings, and \(T_{i}(A_{0})\subseteq B_{0}\) for all \(i=1, 2, \ldots, n\);
- (iii)
for any \(x\in\bigcap_{k=1} ^{i-1} F_{A}(T_{k})\), there exists \(y\in\bigcap_{k=1} ^{i-1} F_{A}(T_{k})\) such that \(T_{i}(x)=T_{1}(y)\) for all \(i=1, 2, \ldots, n\).
Proof
From the proof of Theorem 3.2, we know that \(F_{A}(T_{i})\) is a nonempty, closed, and convex subset of \(A_{0}\) for all \(i=1, 2, \ldots, n\). Theorem 4.1 implies that \(F_{A}(T_{1})\cap F_{A}(T_{2})\) is a nonempty, closed, and convex set. Following a similar argument as Theorem 4.1, we can prove that \(T_{3}\) on \(F_{A}(T_{1})\cap F_{A}(T_{2})\) is an \(\alpha_{3}\)-nonexpansive mapping. Thus there exists a best proximity point \(z^{*}\in F_{A}(T_{1})\cap F_{A}(T_{2})\). By repeating the argument, we can prove that there exists \(x^{*}\in\bigcap_{i=1} ^{n} F_{A}(T_{i})\) such that \(\|x^{*}-T_{i}(x^{*})\|=\operatorname{dist}(A, B)\) for all \(i=1, 2, \ldots, n\). □
Example 4.1
Declarations
Acknowledgements
The first and second authors were supported financially by the National Natural Science Foundation of China (11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
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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