# Existence of best proximity points for controlled proximal contraction

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## 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

Point Theorem Closed Subset Fixed Point Theorem Contraction Condition Contraction Mapping## 1 Introduction and preliminaries

*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

*T*. The following well-known 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 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, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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]

*P*-property if and only if for any \(x_{1},x_{2}\in A\) and \(y_{1},y_{2}\in B\),

### Example 1.3

[14]

*P*-property.

### 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]

*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**P*-*property*; - (iii)
*there exists*\(\alpha\in(0,1)\)*such that*,*for each*\(x,y\in A\),*we have*\(H(Tx,Ty)\leq\alpha d(x,y)\).

*Then there exists an element*\(x^{\ast}\in A_{0}\)

*such that*\(d(x^{\ast},Tx^{\ast})=\operatorname{dist}(A,B)\).

## 2 Best proximity theorems

We start this section by introducing the following definition.

### Definition 2.1

*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**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*\(d(x_{0},Tx_{0})+\operatorname{dist}(A,B)\leq(1-\sqrt{\alpha})r\).

*Then*

*T*

*has a best proximity point in*\(B(x_{0},r)\cap A_{0}\).

### Proof

*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

*P*-property. From (2.3) and (2.7), we have

*P*-property. From (2.7) and (2.12), we have

*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

*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

*P*-property. 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**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*,$$ d(Tx,Ty)\leq\alpha d(x,y)\quad \textit{for each }x,y\in B(x_{0},r) \cap A, $$(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*.

## 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

*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**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*,$$ H(T_{i}x,T_{i}y)\leq\alpha d(x,y)\quad \textit{for each } x,y\in B(a_{i},r_{i})\cap A, $$(3.1)*and*\(d(a_{i},T_{i}a_{i})+\operatorname{dist}(A,B)\leq(1-\sqrt{\alpha})r_{i}\).

*Then*

### Proof

*P*-property. From (3.2) and (3.6), we have

*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

### Example 3.2

*P*-property. 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*

*α*-

*contractions 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* *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* \(\lim_{i\to\infty} H(F_{T_{i}},F_{T_{0}})=0\).

## Notes

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