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Existence of best proximity points for controlled proximal contraction
Fixed Point Theory and Applicationsvolume 2015, Article number: 207 (2015)
Abstract
In this paper, we investigate the sufficient condition for the existence of best proximity points for nonselfmultivalued 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.
Introduction and preliminaries
Let $(X,d)$ be a metric space and A, B be subsets of X. We denote by $\operatorname{CL}(B)$, the set of all nonempty closed subsets of B. A point $x\in A$ is called a fixed point of a mapping $T:A\to \operatorname{CL}(B)$, if $x\in Tx$. The multivalued map T has no fixed point if $A\cap B=\emptyset$. In this case $d(x,Tx)>0$ for all $x\in A$. So, one can attempt to find the necessary condition so that the minimization problem
has at least one solution. A point $x^{\ast}\in X$ is said to be a best proximity point of the mapping $T:A\to B$ if $d(x^{\ast},Tx^{\ast})=\operatorname{dist}(A,B)$. When $A=B$, the best proximity point reduces to a fixed point of the mapping T. The following wellknown best approximation theorem is due to Fan.
Theorem 1.1
[1]
Let A be a nonempty compact convex subset of normed linear space X and $T:A\to X$ be a continuous function. Then there exists $x\in A$ such that
In this paper, we discuss sufficient conditions which ensure the existence of best proximity points for multivalued nonselfmappings 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]
Let $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with $A_{0}\neq\emptyset$. Then the pair $(A,B)$ is said to have the weak Pproperty if and only if for any $x_{1},x_{2}\in A$ and $y_{1},y_{2}\in B$,
Example 1.3
[14]
Let $X=\{(0,1),(1,0),(0,3),(3,0)\}$, endowed with a metric $d((x_{1},x_{2}), (y_{1},y_{2}))=x_{1}y_{1}+x_{2}y_{2}$. Let $A=\{(0,1),(1,0)\}$ and $B=\{(0,3),(3,0)\}$. Then, for
we have
Also, $A_{0}\neq\emptyset$. Thus the pair $(A,B)$ satisfies the weak Pproperty.
Definition 1.4
[3]
An element $x^{\ast}\in A$ is said to be a best proximity point of a multivalued nonselfmapping T, if $d(x^{\ast},Tx^{\ast})=\operatorname{dist}(A,B)$.
Theorem 1.5
[3]
Let A and B be two nonempty closed subsets of a complete metric space $(X,d)$ such that $A_{0}$ is nonempty. Let $T: A \to \operatorname{CB}(B)$ be a mapping satisfying the following conditions:

(i)
for each $x\in A_{0}$, we have $Tx \subseteq B_{0}$;

(ii)
the pair $(A,B)$ satisfies the Pproperty;

(iii)
there exists $\alpha\in(0,1)$ such that, for each $x,y\in A$, we have $H(Tx,Ty)\leq\alpha d(x,y)$.
Then there exists an element $x^{\ast}\in A_{0}$ such that $d(x^{\ast},Tx^{\ast})=\operatorname{dist}(A,B)$.
Best proximity theorems
We start this section by introducing the following definition.
Definition 2.1
Let A and B be nonempty subsets of a metric space $(X,d)$, $x_{0}\in A_{0}$, and $B(x_{0},r)$ is a closed ball in X. A mapping $T:A\to \operatorname{CL}(B)$ is said to be a proximal contraction on $B(x_{0},r)$, if there exists $\alpha\in(0,1)$ such that
Lemma 2.2
[16]
Let $(X,d)$ be a metric space, $B\in \operatorname{CL}(X)$, and $q>1$. Then, for each $x\in X$, there exists an element $b\in B$ such that
Now we are in a position to state and prove our first result.
Theorem 2.3
Let A and B be nonempty closed subsets of a complete metric space $(X,d)$. Assume that $A_{0}$ is nonempty and $T: A \to \operatorname{CL}(B)$ is a mapping satisfying the following conditions:

(i)
for each $x\in A_{0}$, we have $Tx \subseteq B_{0}$;

(ii)
the pair $(A,B)$ satisfies weak Pproperty;

(iii)
there exists $x_{0}\in A_{0}$ such that T is a proximal contraction on the closed ball $B(x_{0},r)$ and $d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1\sqrt{\alpha})r$.
Then T has a best proximity point in $B(x_{0},r)\cap A_{0}$.
Proof
By hypothesis (iii), we have $x_{0}\in A_{0}$ such that T is a proximal contraction on the closed ball $B(x_{0},r)$ and $d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1\sqrt{\alpha})r$. As $x_{0}\in A_{0}$. By (i), we have $y_{0}\in Tx_{0}\subseteq B_{0}$. Then there exists $x_{1}\in A_{0}$ such that
By using the triangular inequality, hypothesis (iii) and (2.3), we have
Since $x_{1}\in A_{0} \subseteq A$, $x_{1}\in B(x_{0},r)\cap A$. From (2.1), we have
As $\alpha>0$, by Lemma 2.2, we have $y_{1}\in Tx_{1}$ such that
Since $Tx_{1}\subseteq B_{0}$, there exists $x_{2}\in A_{0}$ such that
as $(A,B)$ satisfies the weak Pproperty. From (2.3) and (2.7), we have
Considering the triangular inequality, (2.4), and (2.9), we have
By construction, we have $x_{2}\in A_{0} \subseteq A$. Thus $x_{2}\in B(x_{0},r)\cap A$. Again from (2.1), we have
By using Lemma 2.2, we have $y_{2}\in Tx_{2}$ such that
Since $Tx_{2}\subseteq B_{0}$, there exists $x_{3}\in A_{0}$ such that
as $(A,B)$ satisfies the weak Pproperty. From (2.7) and (2.12), we have
From (2.11) and (2.13), we have
By considering the triangular inequality, (2.9), and (2.14), we have
as $x_{3}\in A_{0} \subseteq A$. Thus, $x_{3}\in B(x_{0},r)\cap A$. Continuing in the same way, we get two sequences $\{x_{n}\} \subseteq A_{0}$ with $x_{n}\in B(x_{0},r)\cap A$ and $\{y_{n}\}\subseteq B_{0}$ with $y_{n}\in Tx_{n}$ such that
Moreover,
For $n>m$, we have
Hence $\{x_{n}\}$ is a Cauchy sequence in $B(x_{0},r)\cap A \subseteq A$. A similar reasoning shows that $\{y_{n}\}$ is a Cauchy sequence in B. Since $B(x_{0},r)\cap A$ is closed in A, and A, B are closed subsets of a complete metric space, there exist $x^{\ast} \in B(x_{0},r)\cap A$ and $y^{\ast}\in B$ such that $x_{n} \to x^{\ast}$ and $y_{n} \to y^{\ast}$. By (2.15), we conclude that $d(x^{\ast},y^{\ast})=\operatorname{dist}(A,B)$ as $n\to\infty$. Clearly, $y^{\ast}\in Tx^{\ast}$, since $\lim_{n \to \infty}d(y_{n},Tx^{\ast})\leq\lim_{n\to \infty}H(Tx_{n},Tx^{\ast})=0$. Hence $\operatorname{dist}(A,B)\leq d(x^{\ast},Tx^{\ast})\leq d(x^{\ast},y^{\ast})=\operatorname{dist}(A,B)$. Therefore, $x^{\ast}$ is a best proximity point of the mapping T. □
Example 2.4
Let $X=\mathbb{R}^{2}$ be endowed with the metric $d((x_{1},y_{1}),(x_{2},y_{2}))=x_{1}x_{2}+{y_{1}y_{2}}$. Suppose that $A=\{(1,x): x\in\mathbb{R}\}$ and $B=\{(0,x): x\in\mathbb{R}\}$. Define $T:A\to \operatorname{CL}(B)$ by
Let us consider a ball $B(x_{0},r)$ with $x_{0}=(1,0.1)$ and $r=7.5$. Then it is easy to see that T is a proximal contraction on the closed ball $B((1,0.1),7.5)$ with $\alpha=\frac{1}{2}$. Also, we have $d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1\sqrt{\alpha})r$. Furthermore, $A_{0}=A$, $B_{0}=B$; for each $x\in A_{0}$ we have $Tx \subseteq B_{0}$ and the pair $(A,B)$ satisfies the weak Pproperty. Therefore, all the conditions of Theorem 2.3 hold and T has a best proximity point.
Corollary 2.5
Let A and B be nonempty closed subsets of a complete metric space $(X,d)$. Assume that $A_{0}$ is nonempty and $T: A \to B$ is a mapping satisfying the following conditions:

(i)
for each $x\in A_{0}$, we have $Tx \in B_{0}$;

(ii)
the pair $(A,B)$ satisfies the weak Pproperty;

(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,
$$ d(Tx,Ty)\leq\alpha d(x,y)\quad \textit{for each }x,y\in B(x_{0},r) \cap A, $$(2.18)and $d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1\sqrt{\alpha})r$.
Then T has a best proximity point in $B(x_{0},r)\cap A_{0}$.
If we assume that $X=A=B$, then Theorem 2.3 reduces to the following fixed point theorem.
Corollary 2.6
Let $(X,d)$ be a complete metric space and $T: X \to \operatorname{CL}(X)$ be a mapping. Assume that there exist $x_{0}\in X$ and $\alpha\in(0,1)$ satisfying
and $d(x_{0},Tx_{0})\leq(1\sqrt{\alpha})r$. Then T has a fixed point.
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 nonselfmultivalued 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
Let A and B be nonempty closed subsets of a complete metric space $(X,d)$. Assume that $A_{0}$ is nonempty and $T_{i}: A \to \operatorname{CL}(B)$, $i=1,2$ are mappings satisfying the following conditions:

(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 Pproperty;

(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,
$$ H(T_{i}x,T_{i}y)\leq\alpha d(x,y)\quad \textit{for each } x,y\in B(a_{i},r_{i})\cap A, $$(3.1)and $d(a_{i},T_{i}a_{i})+\operatorname{dist}(A,B)\leq(1\sqrt{\alpha})r_{i}$.
Then
Proof
Let $x_{0}\in B_{T_{1}}$, then we have $y_{0}\in T_{2}x_{0}$ such that
Since $y_{0}\in T_{2}x_{0}\subseteq B_{0}$, we have $x_{1}\in A_{0}$ such that
We know that $T_{2}$ is a proximal contraction for closed ball $B(a_{2},r_{2})$. Without loss of generality, we take $a_{2}=x_{0}$ and $r_{2}=r$ such that $d(x_{0},T_{2}x_{0})+\operatorname{dist}(A,B)\leq(1\sqrt{\alpha})r$. Clearly, $x_{1}\in B(x_{0},r)\cap A$, since $x_{1}\in A_{0}\subseteq A$ and
By hypothesis (iii), we have
As $\alpha>0$, by Lemma 2.2, we have $y_{1}\in T_{2}x_{1}$ such that
Since $T_{2}x_{1}\subseteq B_{0}$, there exists $x_{2}\in A_{0}$ such that
as $(A,B)$ satisfies the weak Pproperty. From (3.2) and (3.6), we have
Considering the triangular inequality, (3.3), and (3.8), we have
Also, $x_{2}\in A_{0}\subseteq A$. Thus, $x_{2}\in B(x_{0},r)\cap A$. Continuing in the same way, we get two sequences $\{x_{n}\} \subseteq A_{0}$ with $x_{n}\in B(x_{0},r)\cap A$ and $\{y_{n}\}\subseteq B_{0}$ with $y_{n}\in T_{2}x_{n}$ such that
Moreover,
For $n>m$, we have
Hence $\{x_{n}\}$ is a Cauchy sequence in $B(x_{0},r)\cap A\subseteq A$. A similar reasoning shows that $\{y_{n}\}$ is a Cauchy sequence in B. Since $B(x_{0},r)\cap A$ is closed in A, and A, B are closed subsets of a complete metric space, there exist $u^{\ast} \in B(x_{0},r)\cap A$ and $v^{\ast}\in B$ such that $x_{n} \to u^{\ast}$ and $y_{n} \to v^{\ast}$. By (3.9), we conclude that $d(u^{\ast},v^{\ast})=\operatorname{dist}(A,B)$ as $n\to\infty$. Clearly, $v^{\ast}\in T_{2}u^{\ast}$. Then we have $\operatorname{dist}(A,B)\leq d(u^{\ast},T_{2}u^{\ast})\leq d(u^{\ast},v^{\ast})=\operatorname{dist}(A,B)$. Therefore $u^{\ast}$ is a best proximity point of $T_{2}$. Now, we have
Similarly, if $\mathfrak{x}_{\mathfrak{0}}\in B_{T_{2}}$, then we have $\mathfrak{u}^{\ast}\in B_{T_{1}}$ such that
Thus, we have
□
Example 3.2
Let $X=\mathbb{R}^{2}$ be endowed with the metric $d((x_{1},y_{1}),(x_{2},y_{2}))=x_{1}x_{2}+{y_{1}y_{2}}$. Suppose that $A=\{(1,x): x\in\mathbb{R}\}$ and $B=\{(0,x): x\in\mathbb{R}\}$. Define $T_{1},T_{2}:A\to \operatorname{CL}(B)$ by
and
It is easy to see that $T_{1}$ is a proximal contraction on the closed ball $B(x_{0}=(1,0.1),r=7.5)$ with $\alpha=\frac{1}{2}$ and $d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1\sqrt{\alpha})r$. Further, $T_{2}$ is a proximal contraction on the closed ball $B(x_{1}=(1,1.25),r_{1}=8)$ with $\alpha=\frac{1}{2}$ and $d(x_{1},Tx_{1})+\operatorname{dist}(A,B)\leq (1\sqrt{\alpha})r_{1}$. Furthermore, it is easy to see that $A_{0}=A$, $B_{0}=B$, and for each $x\in A_{0}$ we have $T_{i}x \subseteq B_{0}$ for each $i=1,2$ and the pair $(A,B)$ satisfies the weak Pproperty. All the conditions of Theorem 3.1 hold. Thus the conclusion holds. That is,
If we assume that $X=A=B$, then Theorem 3.1 reduces to the following stability result.
Corollary 3.3
Let $(X,d)$ be a complete metric space and $T_{i}: X \to \operatorname{CL}(X)$, $i=1,2$ be mappings. Assume that there exist $\alpha\in(0,1)$ and $a_{1},a_{2}\in X$ such that, for each i, we have
and $d(a_{i},T_{i}a_{i})\leq(1\sqrt{\alpha})r_{i}$. Let $F_{T_{1}}$ and $F_{T_{2}}$ denote the sets of fixed points of $T_{1}$ and $T_{2}$ respectively. Then
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)
Let $(X,d)$ be a complete metric space and $T_{i}: X \to \operatorname{CL}(X)$, $i=1,2$ be αcontractions with the same α, that is,
where $\alpha\in(0,1)$. Then
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 αcontractions with the same α. If $\lim_{i\to\infty} H(T_{i}x,T_{0}x)=0$ uniformly for all $x\in X$, then $\lim_{i\to\infty} H(F_{T_{i}},F_{T_{0}})=0$.
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Keywords
 Point Theorem
 Closed Subset
 Fixed Point Theorem
 Contraction Condition
 Contraction Mapping