Existence of best proximity points for controlled proximal contraction
- Quanita Kiran^{1},
- Muhammad Usman Ali^{2},
- Tayyab Kamran^{2, 3} and
- Erdal Karapınar^{4, 5}Email author
https://doi.org/10.1186/s13663-015-0455-6
© Kiran et al. 2015
Received: 6 July 2015
Accepted: 4 November 2015
Published: 14 November 2015
Abstract
In this paper, we investigate the sufficient condition for the existence of best proximity points for non-self-multivalued mappings. Additionally, we discuss the stability theorem for such mappings. Our results improve and generalize some existing results on the topic in the literature, in particular, the results of Lim and of Abkar and Gabeleh.
Keywords
1 Introduction and preliminaries
Theorem 1.1
[1]
In this paper, we discuss sufficient conditions which ensure the existence of best proximity points for multivalued non-self-mappings satisfying contraction condition on the closed ball of a complete metric space. Moreover, we discuss the stability of the best proximity points. Our results extend and generalize some results by Lim [2], and Abkar and Gabeleh [3]. Some important best proximity theorems can be found in [4–15].
Now we recollect some notions, definitions, and results, for easy reference. \(\operatorname{dist}(A,B)= \inf\{d(a,b): a\in A, b\in B\}\), \(d(x,B)=\inf\{d(x,b): b\in B\}\), \(A_{0}=\{a\in A: d(a,b)=\operatorname{dist}(A,B)\ \text{for some }b\in B\}\), \(B_{0}=\{b\in B: d(a,b)=\operatorname{dist}(A,B)\text{ for some } a\in A\}\), \(\operatorname{CB}(B)\) is the set of all nonempty closed and bounded subsets of B and \(B(x_{0},r)=\{x\in X: d(x_{0},x)\leq r\}\).
Definition 1.2
[13]
Example 1.3
[14]
Definition 1.4
[3]
An element \(x^{\ast}\in A\) is said to be a best proximity point of a multivalued non-self-mapping T, if \(d(x^{\ast},Tx^{\ast})=\operatorname{dist}(A,B)\).
Theorem 1.5
[3]
- (i)
for each \(x\in A_{0}\), we have \(Tx \subseteq B_{0}\);
- (ii)
the pair \((A,B)\) satisfies the P-property;
- (iii)
there exists \(\alpha\in(0,1)\) such that, for each \(x,y\in A\), we have \(H(Tx,Ty)\leq\alpha d(x,y)\).
2 Best proximity theorems
We start this section by introducing the following definition.
Definition 2.1
Lemma 2.2
[16]
Now we are in a position to state and prove our first result.
Theorem 2.3
- (i)
for each \(x\in A_{0}\), we have \(Tx \subseteq B_{0}\);
- (ii)
the pair \((A,B)\) satisfies weak P-property;
- (iii)
there exists \(x_{0}\in A_{0}\) such that T is a proximal contraction on the closed ball \(B(x_{0},r)\) and \(d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1-\sqrt{\alpha})r\).
Proof
Example 2.4
Corollary 2.5
- (i)
for each \(x\in A_{0}\), we have \(Tx \in B_{0}\);
- (ii)
the pair \((A,B)\) satisfies the weak P-property;
- (iii)there exists \(x_{0}\in A_{0}\) such that T is a proximal contraction on the closed ball \(B(x_{0},r)\), that is,and \(d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1-\sqrt{\alpha})r\).$$ d(Tx,Ty)\leq\alpha d(x,y)\quad \textit{for each }x,y\in B(x_{0},r) \cap A, $$(2.18)
If we assume that \(X=A=B\), then Theorem 2.3 reduces to the following fixed point theorem.
Corollary 2.6
3 Stability of best proximity points
Stability of fixed point sets of multivalued mappings was initially investigated by Markin [15] and Nadler [16] with some strong conditions. Lim [2] proved the stability theorem for fixed point sets of multivalued contraction mappings by relaxing the condition assumed by Markin [15]. Abkar and Gabeleh [3] discussed the stability of best proximity point sets of non-self-multivalued mappings. In this section, we extend and generalize the stability theorems due to Abkar and Gabeleh [3], and Lim [2].
In this section, by \(B_{T_{1}}\) and \(B_{T_{2}}\) we denote the sets of best proximity points of \(T_{1}\) and \(T_{2}\), respectively.
Theorem 3.1
- (i)
for each \(x\in A_{0}\), we have \(T_{i}x \subseteq B_{0}\), \(i=1,2\);
- (ii)
the pair \((A,B)\) satisfies the weak P-property;
- (iii)for each \(i=1,2\), there exists \(a_{i}\) such that \(T_{i}\) is proximal contraction on the closed ball \(B(a_{i},r_{i})\) with the same α as a contraction constant, that is,and \(d(a_{i},T_{i}a_{i})+\operatorname{dist}(A,B)\leq(1-\sqrt{\alpha})r_{i}\).$$ H(T_{i}x,T_{i}y)\leq\alpha d(x,y)\quad \textit{for each } x,y\in B(a_{i},r_{i})\cap A, $$(3.1)
Proof
Example 3.2
If we assume that \(X=A=B\), then Theorem 3.1 reduces to the following stability result.
Corollary 3.3
Note that in this theorem \(B(a_{i},r_{i})\) are closed balls.
Remark 3.4
If \(r_{1}\), \(r_{2}\) are sufficiently large, then \(B(a_{1},r_{1})\) and \(B(a_{2},r_{2})\) are equal to X. In this case, from Corollary 3.3, we get the following result.
Corollary 3.5
(Lim [2], Lemma 1)
Corollary 3.6
(Lim [2], Theorem 1)
Let \((X,d)\) be a complete metric space and \(T_{i}: X \to \operatorname{CL}(X)\), \(i=1,2,\ldots \) , be α-contractions with the same α. If \(\lim_{i\to\infty} H(T_{i}x,T_{0}x)=0\) uniformly for all \(x\in X\), then \(\lim_{i\to\infty} H(F_{T_{i}},F_{T_{0}})=0\).
Declarations
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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