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A best proximity point theorem for αproximal Geraghty nonself mappings
 Mohamed Iadh Ayari^{1}Email author
https://doi.org/10.1186/s1366301906618
© The Author(s) 2019
 Received: 15 November 2018
 Accepted: 21 May 2019
 Published: 3 June 2019
Abstract
In this paper, we search some best proximity point results for a new class of nonself mappings \(T:A \longrightarrow B\) called αproximal Geraghty mappings. Our results extend many recent results appearing in the literature. We suggest an example to support our result. Several consequences are derived. As applications, we investigate the existence of best proximity points for a metric space endowed with symmetric binary relation.
Keywords
 Best proximity points
 αProximal Geraghty nonself mappings on metric spaces
MSC
 47H10
 54H25
1 Introduction
One of the famous generalizations of the Banach contraction principle for the existence of fixed points for self mappings on metric spaces [5] is the theorem by Geraghty [8]. Consider A and B to be two nonempty subsets of a metric space \((X,d)\). Let \(T:A\longrightarrow B\) be a nonself mapping. Then the best proximity points of T are the points \(x\in A\) satisfying \(d(x,Tx)=d(A,B)\). Recently, several works on best proximity point theory were studied by giving sufficient conditions assuring the existence. Thus, several known results were derived. For additional information, see Refs. [2, 3, 7, 10, 12, 19–22], and [25].
Recently, Jleli, Karapinar, and Samet in [11] have introduced a new class of contractive mappings called α–ψcontractive type mappings. They have provided some results on the existence and uniqueness of best proximity points of such nonself mappings. There are many papers in the literature about αcontractions, see, for example, [1, 4, 13, 17], and [15, 16, 24]. Recently, Ayari in [9] proposed an extension for the case of α–βproximal quasicontractive mappings. We are interested in extending these works for the Geraghty functions by introducing the notion of αproximal Geraghty nonself mappings. The purpose of all of this is to provide a theorem on the existence and uniqueness of best proximity point for such mappings.
Kumam and Mongkolekeha in [14] proved new common best proximity point theorems for proximity commuting mappings using the concept of Geraghty theorem in complete metric spaces. Also Biligili, Karapinar, and Sadarangani in [6] also suggested a best proximity point theorem for a pair \((A,B)\) of subsets on a metric space X satisfying the Pproperty. This was accomplished by introducing the notion of generalized Geraghtycontraction.
In this work, we have established a new result on the existence and uniqueness of best proximity point for αproximal Geraghty nonself mappings defined on a closed subset of a complete metric space. Our result generalized results existing in the literature. Moreover, we have shown that from our main theorems we are able to deduce some other theorems of best proximity points for the case of metric spaces endowed with symmetric binary relations. We also have deduced the main fixed point theorem of Geraghty [8].
The paper is divided into five different sections as follows. Section 2 is dedicated to the notation adopted and to providing definitions. Moreover, best proximity point theorem is stated in Sect. 3 with its proof illustrated by an example. Then, several consequences are obtained in Sect. 4. Finally, the existence of best proximity points on metric spaces endowed with symmetric binary relations and a fixed point result are given in Sect. 5.
2 Preliminaries and definitions

\(d(A,B):=\inf \{d(a,b):a\in A,b\in B\}\);

\(A_{0}:=\{a\in A: \text{ there exists } b\in B \text{ such that } d(a,b)=d(A,B) \}\);

\(B_{0}:=\{b\in B: \text{ there exists } a\in A \text{ such that } d(a,b)=d(A,B) \}\).
Definition 2.1
([22])
Let \(T: A\rightarrow B\) be a mapping. An element \(x^{*}\) is said to be a best proximity point of T if \(d(x^{*},Tx ^{*})=d(A,B)\).
Definition 2.2
([18])
Let \((A,B)\) be a pair of nonempty subsets of a metric space \((X,d)\) such that \(A_{0}\) is nonempty. Then the pair \((A,B)\) is said to have the Pproperty iff \(d(x_{1},y_{1})=d(x_{2},y_{2})=d(A,B) \Longrightarrow d(x_{1},x_{2})=d(y_{1},y_{2})\), where \(x_{1},x_{2}\in A\) and \(y_{1},y_{2}\in B\).
Definition 2.3
([11])
Let \(T: A\rightarrow B\) and \(\alpha : A \times A\longrightarrow [0,+\infty )\). We say that T is said to be αproximal admissible if \(\alpha (x_{1},x_{2})\ge 1\) and \(d(u_{1},Tx_{1})=d(u_{2},Tx_{2})=d(A,B) \Longrightarrow \alpha (u_{1},u_{2})\ge 1\) for all \(x_{1},x_{2},u_{1},u _{2}\in A\).
Let us introduce the set F that is the class of all functions \(\beta :[0,\infty )\longrightarrow [0,1]\) such that, for any bounded sequence \(\{t_{n}\}\) of positive reals, \(\beta (t_{n})\longrightarrow 1\) implies \(t_{n}\longrightarrow 0\).
Definition 2.4
([8])
3 Main results and theorems
First, we introduce the following concept which is a natural generalization of the definition of Geraghty.
Definition 3.1
We propose the following best proximity point theorem.
Theorem 3.2
 (1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;
 (2)
T is αproximal admissible;
 (3)
There exist elements \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha (x_{0},x_{1}) \ge 1\);
 (4)
If \(\{x_{n}\}\) is a sequence in A such that \(\alpha (x_{n},x_{n+1}) \ge 1\) and \({\lim_{n\longrightarrow +\infty }x_{n}=x_{\ast }\in A}\), then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha (x_{n(k)},x_{\ast })\ge 1\) for all k.
Proof
Thanks to condition (3), there exist \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha (x_{0},x_{1}) \ge 1\). As \(T(A_{0})\subset B_{0}\), there exists \(x_{2}\in A_{0}\) such that \(d(x_{2},Tx_{1})=d(A,B)\).
As T is αproximal admissible and using \(\alpha (x_{0},x _{1}) \ge 1\), \(d(x_{1},Tx_{0})=d(x_{2},Tx_{1})=d(A,B)\), this implies that \(\alpha (x_{1},x_{2}) \ge 1\).
Thus, the sequence \(\{x_{n}\}\) is a Cauchy sequence in the closed subset A of the space \((X,d)\).
The fact that \((X,d)\) is complete and A is closed assures that the sequence \(\{x_{n}\}\) converges to some element \(x_{\ast }\in A\).
For the uniqueness, suppose that there are two distinct best proximity points for T such that \(x_{\ast }\neq y_{\ast }\). Thus \(r=d(x_{ \ast },y_{\ast })>0\). Since \(d(Tx_{\ast },x_{\ast })=d(Ty_{\ast },y _{\ast })=d(A,B)\), using the Pproperty, we conclude that \(r=d(Tx _{\ast },Ty_{\ast })\).
Since T is an αproximal Geraghty nonself mapping, we obtain \(r\le \beta (r)r\). Thus \(\beta (r)\ge 1\). Since \(\beta (r)\le 1\), we conclude that \(\beta (r)=1\); and therefore \(r=0\), which is a contradiction. □
Example
Consider the complete Euclidian space \(X= \mathbb{R}^{2}\) with the metrics \(d((x_{1},y_{1}), (x_{2},y_{2}))=x _{1}x_{2}+y_{1}y_{2}\). Let \(A=\{(a,0), a\in [2,3]\}\) and \(B=\{(b,0), b\in [\frac{1}{3},\frac{1}{2}]\}\).
Consider the nonself mapping \(T:A\longrightarrow B\) such that \(T(a,0)=(\frac{1}{a},0)\). \(T(A)=B\) and A and B are closed subsets on the complete space \((\mathbb{R}^{2},d)\). It is easy to see that the couple (A,B) satisfies the Pproperty. The function \(\alpha (x,y)=1\) for all \(x,y\in A\). We have \(d(T(2,0),(2,0))=\frac{1}{2}2= \frac{3}{2}=d(A,B)\). So hypotheses (2), (3), and (4) of the theorem are satisfied. What is remaining is to prove hypothesis (5) of Theorem 3.2 that T is a proximal Geraghty mapping.
If \(a\neq a'\), we have for all \(a,a' \in A=[2,3]\), \(aa'\ge 4\); meanwhile \(1+aa'\le 2\). Thus hypothesis (5) of our Theorem 3.2 is satisfied for the function \(\beta :[0,+\infty )\longrightarrow [0,1]\) such that \(\beta (u)=\frac{1}{1+u}\). So the conclusion is the existence and uniqueness of best proximity point of the mapping T which is \((2,0)\).
4 Consequences
For the case \(\alpha =1\), the definition of Geraghty is the following.
Definition 4.1
Several consequences of our main theorem are suggested in this section.
Corollary 4.2
 (1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;
 (2)
There exist elements \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\);
 (3)
T is a proximal Geraghty mapping.
Proof
This is an immediate consequence of our main Theorem 3.2 by taking \(\alpha (x,y)=1\) for all \(x,y\in A\). □
We can also suggest some corollary for the cases \(\beta (u)=e^{ku}\), where \(k>0\), \(\beta (u)=\frac{1}{u+1}\), \(\beta (u)= \bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} 1;&u=0, \\ \frac{\ln (1+u)+1}{u} ; &u>0 \end{array}}\) and \(\beta (u)=\frac{1}{2\frac{2}{\pi }\arctan (\frac{1}{u^{\alpha }} )}\) where \(0<\alpha <1\), \(u>0\).
Corollary 4.3
 (1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;
 (2)
There exist elements \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\);
 (3)The nonself mapping T satisfiesfor all \(x,y \in A\), for some constant \(k>0\).$$ d(Tx,Ty)\le \exp \bigl(kd(x,y)\bigr)d(x,y) $$
Corollary 4.4
 (1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;
 (2)
There exist elements \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\);
 (3)The nonself mappingfor all \(x,y \in A\).$$ d(Tx,Ty)\le \frac{d(x,y)}{1+d(x,y)} $$
Corollary 4.5
 (1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;
 (2)
There exist elements \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\);
 (3)The nonself mapping T satisfiesfor all \(x,y \in A\).$$ d(Tx,Ty)\le \ln \bigl(1+d(x,y)\bigr) $$
Corollary 4.6
 (1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;
 (2)
There exist elements \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\);
 (3)The nonself mapping T satisfiesfor all \(x,y \in A\).$$ d(Tx,Ty)\le \frac{d(x,y)}{2\frac{2}{\pi }\arctan (\frac{1}{d(x,y)^{ \alpha }} )} ,\quad 0< \alpha < 1 $$
5 Applications
Our first consequence is the theorem of Geraghty for the existence of fixed point.
Corollary 5.1
([8])
Let \((X,d)\) be a metric space and T be an operator. Suppose that there exist \(\beta :[0,\infty )\longrightarrow [0,1]\) such that, for any bounded sequence \(\{t_{n}\}\) of positive reals, \(\beta (t_{n})\longrightarrow 1\) implies \(t_{n}\longrightarrow 0\).
Proof
By considering \(A=B=X\) and the function \(\alpha (x,y)=1\) in Theorem 3.2, we guarantee the existence and uniqueness of a fixed point of such a self mapping T. □
In order to apply our results on best proximity points on a metric space endowed with symmetric binary relation, we need some preliminaries.
Let \((X,d)\) be a metric space and \(\mathcal{R}\) be a symmetric binary relation over X.
Definition 5.2
([11])
A nonself mapping \(T:A\longrightarrow B\) is a proximal comparative mapping if \(x \mathcal{R} y\) and \(d(u_{1},Tx)=d(u_{2},Ty)=d(A,B)\) for all \(x,y,u_{1},u_{2} \in A\), then \(u_{1} \mathcal{R} u_{2}\).
Definition 5.3
([23])
We say that \((X,d,\mathcal{R})\) is regular if, for a sequence \(\{x_{n}\}\) in X, we have \(x_{n}\mathcal{R}x_{n+1}\) for all \(n\in \mathbb{N}_{0}\) and \(\lim_{n\rightarrow \infty }d(x_{n},x)=0\) for some \(x\in X\), then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(x_{n(k)}\mathcal{R}x\) for all \(k\in \mathbb{N}_{0}\).
Definition 5.4
We have the following best proximity point result.
Corollary 5.5
 (1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;
 (2)
T is proximal comparative mapping;
 (3)
There exist elements \(x_{0},x_{1} \in A_{0}\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(x_{0} \mathcal{R} x_{1}\);
 (4)
If \((A,d,\mathcal{R})\) is regular;
 (5)
There exists \(\beta \in F\) such that \(T:A\to B\) is Geraghty contractive.
Then T has a unique best proximity point \(x_{\ast }\in A\) such that \(d(x_{\ast },Tx_{\ast })=d(A,B)\).
Proof
Assume that \(\alpha (x,y) \ge 1\), and \(d(u_{1},Tx)=d(u_{2},Tx)=d(A,B)\), for some \(x,y,u_{1},u_{2} \in A\). By the definition of α, we get \(x \mathcal{R} y\) and \(d(u_{1},Tx)=d(u_{2},Tx)=d(A,B)\). Condition (2) of Corollary implies \(u_{1} \mathcal{R} u_{2}\), which gives us \(\alpha (u_{1},u_{2})\ge 1\).
Condition (3) means that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha (x _{0},x_{1})\ge 1\).
The condition \(T:A\to B\) is Geraghty contractive means that T is an αproximal Geraghty mapping. Also the condition \((A,d, \mathcal{R})\) is regular implies that if \(\{x_{n}\}\) is a sequence in A such that \(\alpha (x_{n},x_{n+1}) \ge 1\) and \({\lim_{n\longrightarrow +\infty }x_{n}=x_{\ast }\in A}\), then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha (x_{n(k)},x_{\ast })\ge 1\) for all k.
Now all the hypotheses of Theorem 3.2 are satisfied, which implies the existence and uniqueness of a proximity point for the nonself mapping T. □
6 Conclusion
We recall that we managed in this paper to propose a new best proximity point for αproximal Geraghty nonself mappings. This was achieved by introducing the notion of αproximal Geraghty nonself mappings which is an extension of the definition of Geraghty for the case of self mappings. As applications, we have established not only the existence but also the uniqueness of best proximity point results for the case of nonselfmappings on metric spaces endowed with symmetric binary relations.
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