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Best proximity point theorems for generalized αβproximal quasicontractive mappings
Fixed Point Theory and Applications volume 2017, Article number: 16 (2017)
Abstract
Herein, we search for some best proximity point results for a novel class of nonselfmappings \(T:A \longrightarrow B\) called generalized proximal αβquasicontractive. We illustrate our work by an example. Our results generalize and extend many recent results appearing in the literature. Several consequences are derived. As applications, we explore the existence of best proximity points for a metric space endowed with symmetric binary relation.
Introduction
Consider A and B two nonempty subsets of a metric space \((X,d)\). Let \(T:A\longrightarrow B\) be a nonselfmapping. The best proximity points of T are the points \(x\in A\) satisfying \(d(x,Tx)=d(A,B)\). Numerous works on best proximity point theory were studied by giving sufficient conditions assuring the existence and the uniqueness of such points. These theorems are a normal generalization of the contraction principle to the case of selfmappings. Several known results were derived. For additional information, see [1–7] and [8].
Recently, Samet et al. [9] introduced a novel class of contractive mappings called αψcontractive type mappings. They provided some interesting results to obtain the existence of fixed points for selfmappings. After that, Jleli et al. in [9] studied the existence and the uniqueness of best proximity points of nonselfmappings.
The main objective of this paper is to generalize the results of Jleli et al. [9] by introducing the proximal αβquasicontractive mappings on metric spaces involving βcomparison functions.
In fact, we have derived some theorems on best proximity points for a specific class of proximal generalized αβquasicontractive mappings. The presented results generalize the theorem of Jleli et al. [9] and many results existing in the literature. Moreover, we have shown that from our main theorems we are able to deduce various theorems of best proximity points for the case of metric spaces endowed with symmetric binary relations. Also, we have deduced some fixed point theorems already existing in the literature.
The paper is divided into five sections. Section 2 is dedicated to the notation adopted to provide definitions and evoking a compilation of pertinent results. Best proximity point theorems with their proofs are stated in Section 3, and we justify our results by a suitable example. Several consequences are obtained in Section 4. Finally, the existence of best proximity points and fixed point results are given in Section 5.
Preliminaries and definitions
Let \((A,B)\) be a pair of nonempty subsets of a metric space \((X,d)\). We adopt the following notations:
Definition 2.1
[10]
Let \(T: A\longrightarrow B\) 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
[11]
Let \(\beta\in(0,+\infty)\). A βcomparison function is a map \(\varphi:[0,+\infty)\to[0,+\infty)\) fulfilling the following properties:

(1)
φ is nondecreasing;

(2)
\(\lim_{n\to\infty}\varphi_{\beta}^{n}(t)=0\) for all \(t>0\), where \(\varphi_{\beta}^{n} \) denotes the nth iterate of \(\varphi _{\beta}\) and \(\varphi_{\beta}(t)=\varphi(\beta t)\);

(3)
there exists \(s\in(0,+\infty)\) such that \(\sum_{n=1}^{\infty}\varphi_{\beta}^{n}(s) < \infty\).
The set of all βcomparison functions φ satisfying (1)(3) will be denoted by \(\Phi_{\beta}\).
Remark 2.3
Let \(\alpha,\beta\in(0,+\infty)\). If \(\alpha<\beta\), then \(\Phi_{\beta }\subset\Phi_{\alpha}\).
A useful lemma concerning the comparison functions Φ was performed in [11].
Lemma 2.4
[11]
Let \(\beta\in(0,+\infty)\) and \(\varphi\in\Phi_{\beta}\). Then

(1)
\(\varphi_{\beta}\) is nondecreasing;

(2)
\(\varphi_{\beta} (t) < t\) for all \(t > 0\);

(3)
\(\sum_{n=1}^{\infty}\varphi_{\beta}^{n}(t) < \infty\) for all \(t > 0\).
Definition 2.5
[7]
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 2.6
[9]
Let \(T: A\longrightarrow B\) and \(\alpha: A \times A\longrightarrow [0,+\infty)\). We say that T is αproximal admissible if \(\alpha(x_{1},x_{2})\ge1\) and \(d(u_{1},Tx_{1})=d(u_{2},Tx_{2})=d(A,B) \Longrightarrow\alpha(u_{1},u_{2})\ge1\) for all \(x_{1},x_{2},u_{1},u_{2}\in A\).
Definition 2.7
[9]
A nonselfmapping \(T: A\longrightarrow B\) is said to be a generalized αψproximal contraction, where \(\alpha: A \times A\longrightarrow[0,+\infty) \) and ψ is a \((c)\)comparison function if
where
Definition 2.8
[9]
A nonselfmapping \(T: A\longrightarrow B\) is said to be \((\alpha,d)\) regular, where \(\alpha: A \times A \longrightarrow[0,+,\infty)\), if for all \((x,y)\) such that \(0 \le\alpha(x,y)<1\), there exists \(u_{0} \in A_{0}\) such that
Main results and theorems
First, we introduce the following concept.
Definition 3.1
Let \((X,d)\) be a metric space and \((A,B)\) be a pair of nonempty subsets of X. Let \(\beta\in(0,+\infty)\). A nonself mapping \(T:A\to B\) is said to be a generalized αβproximal quasicontractive, where \(\alpha: A \times A\longrightarrow[0,+,\infty)\) iff there exist \(\varphi\in\Phi_{\beta }\) and positive numbers \(\alpha_{0},\ldots,\alpha_{4}\) such that
where
We propose the following best proximity point theorems.
Theorem 3.2
Let \((A,B)\) be a pair of nonempty closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let \(\alpha: A \times A\longrightarrow[0,+,\infty)\) and \(\varphi\in \Phi_{\beta}\). Consider a nonselfmapping \(T: A\longrightarrow B\) satisfying the following assertions:

(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 \(\alpha(x_{0},x_{1}) \ge1\);

(4)
if \(\{x_{n}\}\) 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 \(\alpha(x_{n(k)},x_{\ast})\ge1\) for all k;

(5)
there exists \(\beta\ge\max_{0\le k \le3}\{\alpha_{k},2\alpha_{4}\}\) such that T is generalized αβproximal quasicontractive.
Moreover, suppose that one of the following conditions holds:

φ is continuous;

\(\beta>\max\{\alpha_{2},\alpha_{4}\}\).
Then T has a best proximity point \(x_{\ast}\in A\) such that \(d(x_{\ast},Tx_{\ast})=d(A,B)\).
Theorem 3.3
In addition to the hypotheses of Theorem 3.2, suppose that T is \((\alpha,d)\) regular and \(\beta\ge\max\{\alpha_{0},2\alpha_{1},\alpha _{3},\alpha_{4}\}\). Then T has a unique best proximity point.
To prove the above theorems, we require the following lemma.
Lemma 3.4
Let \(T:A\longrightarrow B\) be a nonselfmapping and \(\alpha: A \times A\longrightarrow[0,+,\infty)\), satisfying the following conditions:

(1)
\(T(A_{0})\subset B_{0}\);

(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 \(\alpha(x_{0},x_{1}) \ge1\);
then there exists a sequence \(\{x_{n}\}\subset A_{0}\) such that \(d(x_{n+1},Tx_{n})=d(A,B)\) and \({\alpha(x_{n},x_{n+1})}\geq1\). Such a sequence \(\{x_{n}\}\) is a Cauchy sequence.
Proof
Thanks to condition (3), there exist \(x_{0},x_{1} \in A\) such that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha(x_{0},x_{1}) \ge1\). 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 \(\alpha(x_{1},x_{2}) \ge1\).
In a similar fashion, by induction, we can build a sequence \(\{x_{n}\} \subset A_{0}\) such that
Our next step is to prove that the sequence \(\{x_{n}\}\) is a Cauchy sequence.
Using the Pproperty, we deduce from (3.2) that
Since T is generalized αβproximal quasicontractive, there exists a function \(\varphi\in\Phi_{\beta}\) such that
On the other hand, using (3.2), (3.3) and the triangular inequality, we get
Hence,
where \(\beta\ge\max_{0\le k \le3}\{\alpha_{k},2\alpha_{4}\}\). Using inequalities (3.3), (3.4) and (3.5) and taking into consideration the fact that φ is nondecreasing, we get that
Assume that, for some n, we have \(d(x_{n1},x_{n})\le d(x_{n},x_{n+1})\). It follows that \(d(x_{n+1},x_{n})\leq\varphi_{\beta}(d(x_{n+1},x_{n}))< d(x_{n+1},x_{n})\), which is a contradiction.
Therefore, for all \(n\ge0\), we have necessary the inequality \(d(x_{n1},x_{n})> d(x_{n},x_{n+1})\). It follows that
By induction, we obtain that
Using the triangular inequality and the above inequality (3.7), we get
since the series \({ \sum_{n=1}^{+\infty}\varphi_{\beta}^{n}(d(x_{1},x_{0}))}\) converges. Thus, the sequence is a Cauchy sequence in the metric space \((X,d)\). □
Proof of Theorem 3.2
The fact that \((X,d)\) is complete and A is closed assures that the sequence \(\{x_{n}\}\) converges to some element \(x_{\ast}\in A\).
Using hypothesis (4) of the theorem, there exists a subsequence \(\{ x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x_{\ast})\ge1\) for all k. Since T is generalized αβproximal quasicontractive, then we have
where
By the triangular inequality and (3.2), we have
We obtain that
Using (3.8) and (3.11), we get
In addition, by the triangular inequality and (3.2) on (3.9), we get
As φ is nondecreasing, combining inequalities (3.12) and (3.13), we obtain
Assume \(\rho=d(x_{\ast},Tx_{\ast})d(A,B)>0\).
We consider two separate cases as follows.
If φ is continuous, as \(k\longrightarrow+\infty\), we get
which is a contradiction.
If \(\beta>\max \{\alpha_{2},\alpha_{4} \}\), we claim also that \(\rho=0\). Suppose by contradiction that \(\rho>0\).
Letting \(k\longrightarrow+\infty\) in (3.9), we get \(M_{T}(x_{n(k)},x_{\ast}) \longrightarrow\max \{\alpha_{2},\alpha _{4} \}\rho\). Then there exists \(\varepsilon>0\) and \(N>0\) such that for all \(n>N\), we have
Therefore,
Consequently, by letting \(k\to\infty\), we get
which is a contradiction as well. Hence, our claim holds. Thus, we prove that \(x_{\ast}\) is a best proximity point of T, that is,
□
Proof of Theorem 3.3
For the uniqueness, suppose that \(x_{\ast}\) and \(y_{\ast}\) are two distinct best proximity points of T. Let \(s=d(x_{\ast},y_{\ast})>0\). Using the Pproperty, we obtain \(d(Tx_{\ast},Ty_{\ast})=d(x_{\ast},y_{\ast})=s\). We consider two cases.
If \(\alpha(x_{\ast},y_{\ast})\ge1\). Since T is a generalized αβproximal quasicontraction, this gives
where
Using the triangular inequality in (3.17), we obtain
Combining (3.16) and (3.18) and using the nondecreasing property of the function φ, we conclude that
which is a contradiction. So, \(s=0\) and therefore \(x_{\ast}=y_{\ast}\).
If \(\alpha(x_{\ast},y_{\ast})<1\). Since T is \((\alpha,d)\) regular, there exists \(u_{0}\in A_{0}\) such that \(\alpha(x_{\ast},u_{0})\ge1 \) and \(\alpha (y_{\ast},u_{0})\ge1\). Since \(T(A_{0})\subset B_{0}\), there exists \(u_{1}\in A_{0} \) such that \(d(u_{1},Tu_{0})=d(A,B)\).
We have \(d(x_{\ast},Tx_{\ast})=d(u_{1},Tu_{0})=d(A,B)\) and \(\alpha(x_{\ast},u_{0})\ge1\).
Using the fact that T is αproximal admissible, we get \(\alpha (x_{\ast},u_{1})\ge1\).
One can proceed further in a similar fashion to find \(\{u_{n}\}\in A_{0}\) such that
Using the Pproperty and (3.19), we have
As T is generalized αβproximal quasicontractive, then we get
Using (3.19) and (3.21), we get
Therefore, from (3.19), we conclude that
On the other hand, using (3.15), for all \(n\in\mathbb {N}\cup\{0\}\), we obtain
Using the triangular inequality and (3.20) in the above expression (3.24), and taking into consideration (3.15), we get
Since \(\alpha(u_{n+1},x_{\ast})\ge1\), combining (3.25) and (3.23), we get that
where \(\beta\ge\max\{\alpha_{0},2\alpha_{1},\alpha_{3},\alpha_{4}\}\). Assume that, for some n, we have \(d(u_{n},x_{\ast})\le d(u_{n+1},x_{\ast})\).
We have from (3.26)
which is a contradiction.
Therefore, for all \(n\geq0\), we have \(d(u_{n+1},x_{\ast})< d(u_{n},x_{\ast})\). Using (3.26), we have
By induction, we obtain
Hence, by letting \(n\longrightarrow+\infty\) in the above inequality, we obtain that \(\{u_{n}\}\) converges to \(x_{\ast}\).
Analogously, we can prove that \(\{u_{n}\}\) converges to \(y_{\ast}\). Using the uniqueness of limit, we conclude that \(x_{\ast}=y_{\ast}\). □
Example
Consider the complete Euclidian space \(X=\mathbb{R}^{2}\) with the metric \(d((x_{1},x_{2}),(y_{1},y_{2}))=x_{1}x_{2}+y_{1}y_{2}\). Let \(A=\{(\gamma,0):\gamma\in{}[0,1]\}\) and \(B=\{(\delta,1): \delta \in{}[0,1]\}\).
Also, let \(T:A\longrightarrow B\) be defined by \(T(\gamma,0)=(\frac{\gamma}{4},1)\). Then it is easy to see that \(d(A,B)=1\) and \(A_{0}=A\), \(B_{0}=B\). Now, we shall show that T is an αβproximal quasicontractive mapping with \(\varphi(t)=\frac{3}{4}t\), \(\alpha\equiv1\), and \(\beta_{1}=\frac{3}{4}\) and \(\alpha_{i}=\frac{1}{3^{i+1}}\) for \(i=0,1,2,3,4\).
Let \(x,y\in A\), where \(x=(\gamma_{1},0)\) and \(y=(\gamma_{2},0)\).
So, T is an αβproximal quasicontractive mapping with \(\alpha(x,y)=1\) for all \(x,y\in A\) and \(\varphi(t)=\frac{3}{4}t\), \(\beta=\frac{3}{4}\) and \(\alpha_{i}=\frac{1}{3^{i+1}}\) for \(i=0,1,2,3,4\). Since \(\beta=\frac{3}{4}\ge\max_{0\le k \le3}\{\alpha_{k},2\alpha_{4}\}\).
It is easy to see that the pair \((A,B)\) satisfies the Pproperty.
Since \(\alpha(x,y)=1\) for all \(x,y\in A\), then the mapping T is αadmissible. Also the fact that \(\beta=\frac{3}{4}\ge\max\{\frac{1}{3},\frac {2}{9},\frac{1}{81},\frac{1}{243}\}=\max\{\alpha_{0},2\alpha_{1},\alpha _{3},\alpha_{4}\}=\frac{1}{3}\) and T is \((\alpha,d)\) regular since \(\alpha\equiv1\) assures the uniqueness of the proximity point of T. Therefore, all the conditions of Theorems 3.2 and 3.3 are satisfied, and so T has a unique proximity point which is \(x_{\ast }=(0,0)\in A\).
Consequences
Several consequences of the main theorems are suggested in this section.
Corollary 4.1
[9]
Let A and B be nonempty closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let \(\alpha: A\times A\longrightarrow[0, +\infty)\) and \(\psi\in\Psi\). Suppose that \(T:A\longrightarrow B\) is a nonselfmapping satisfying the following conditions:

(1)
\(T(A_{0})\subset B_{0}\), and \((A,B)\) satisfies the Pproperty;

(2)
T is αproximal admissible;

(3)
there exist elements \(x_{0}\) and \(x_{1}\) such that
$$ d(x_{1},Tx_{0})=d(A,B),\qquad \alpha(x_{0},x_{1}) \ge1; $$ 
(4)
T is a generalized αψ proximal contraction;

(5)
if \(\{x_{n}\}\) is a sequence in A such that \(\alpha(x_{n},x_{n+1}) \ge1\) and \({\lim_{n\longrightarrow+\infty}x_{n}=x_{\ast}\in A}\), then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x_{\ast})\ge1\) for all k.
Then there exists an element \(x_{\ast}\in A_{0}\) such that \(d(x_{\ast},Tx_{\ast})=d(A,B)\).
Proof
First, we notice that using \(M(x,y)\) appearing in (2.1), we have the following inequality:
The existing best proximity point result follows immediately from Theorem 3.2 by taking \(\psi=\varphi\in\Phi_{2}\) and \(\beta\ge 2>\max\{1,1\}=1\). □
Corollary 4.2
[9]
In addition to the hypotheses of Corollary 4.1, suppose that T is \((\alpha,d)\) regular. Then T has a unique best proximity point.
Proof
Also it is an immediate consequence of Theorem 3.3 since the assertion \(\beta\ge\max\{\alpha_{0},2\alpha_{1},\alpha_{3},\alpha_{4}\}=2\) is satisfied, and therefore \(\psi=\varphi\in\Phi_{2}\). □
Corollary 4.3
Let A and B be nonempty closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Suppose that \(T: A\longrightarrow B\) is a nonselfmapping satisfying the following conditions:

(1)
\(T(A_{0})\subset B_{0}\), and \((A,B)\) satisfies the Pproperty;

(2)
there exists \(k\in(0,1)\) such that \(d(Tx,Ty)\le k d(x,y)\) for all \(x,y\in A\).
Then there exists a unique element \(x_{\ast}\in A_{0}\) such that \(d(x_{\ast},Tx_{\ast})=d(A,B)\).
Proof
This follows immediately from Theorem 3.2 by taking \(\alpha (x,y)=1\) for all \(x,y\in A\) and \(\varphi(t)=kt \) which is continuous, where \(k\in(0,1)\). Since \(\alpha(x,y)=1\) for all \(x,y\in A\), then condition (4) of our main Theorem 3.2 occurs. So, there exists a best proximity point for T.
The fact that \(\alpha(x,y)=1\) for all \(x,y\in A\) guarantees that T is \((\alpha,d)\) regular, which implies, by Theorem 3.3, that such a best proximity point for T is unique. □
Applications
Best proximity points for metric spaces endowed with symmetric binary relations
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 5.1
[9]
A nonselfmapping \(T:A\longrightarrow B\) 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 5.2
[12]
A subset A of X is called \(\mathcal{R}\)directed if, for every \(x, y \in A\), there exists \(z \in X\) such that \(x\mathcal{R}z\) and \(y\mathcal{R}z\).
Definition 5.3
[13]
We say that \((X,d,\mathcal{R})\) is regular if, for a sequence \(\{x_{n}\} \) in X, if we have \(x_{n}\mathcal{R}x_{n+1}\) for all \(n\in\mathbb {N}_{0}\) and \(\lim_{n\longrightarrow\infty}d(x_{n},x)=0\) 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
[11]
Let X be a nonempty set. A nonselfmapping \(T:A\to B\) is called βquasicontractive if there exist \(\beta>0\) and \(\varphi\in\Phi _{\beta}\) such that
where
with \(\alpha_{k}\ge0\) for \({k=0,\ldots,4}\).
We have the following best proximity point result.
Corollary 5.5
Let \((A,B)\) be a pair of nonempty closed subsets of a complete metric space \((X,d)\) such that \(A_{0}\) is nonempty. Let \(\mathcal{R}\) be a symmetric binary relation over X. Consider a nonselfmapping \(T: A\longrightarrow B\) satisfying the following assertions:

(1)
\(T(A_{0})\subset B_{0}\) and the pair \((A,B)\) satisfies the Pproperty;

(2)
T is a 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 \(\beta\ge\max_{0\le k \le3}\{\alpha_{k},2\alpha_{4}\} \) such that \(T:A\to B\) is βquasicontractive.
Moreover, assume that one of the following conditions holds:

φ is continuous;

\(\beta>\max\{\alpha_{2},\alpha_{4}\}\).
Then T has a best proximity point \(x_{\ast}\in A\) such that \(d(x_{\ast},Tx_{\ast})=d(A,B)\).
Proof
Let us define the mapping \(\alpha: A\times A \longrightarrow[0, +\infty )\) by:
In order to apply our Theorem 3.2, we have to prove that T is αadmissible.
Assume that \(\alpha(x,y) \ge1\), 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 the corollary implies \(u_{1} \mathcal{R} u_{2}\), which gives us \(\alpha (u_{1},u_{2})\ge1\).
Condition (3) means that \(d(x_{1},Tx_{0})=d(A,B)\) and \(\alpha (x_{0},x_{1})\ge1\).
The condition \(T:A\to B\) is βquasicontractive means that T is generalized αβproximal quasicontractive. Also the condition \((A,d,\mathcal{R})\) is regular implies if \(\{x_{n}\}\) is a sequence in A such that \(\alpha(x_{n},x_{n+1}) \ge1\) and \({\lim_{n\longrightarrow +\infty}x_{n}=x_{\ast}\in A}\), then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x_{\ast})\ge1\) for all k.
Now all the hypotheses of Theorem 3.2 are satisfied, which implies the existence of a proximity point for the nonselfmapping T. □
Corollary 5.6
In addition to the hypotheses of Corollary 5.5, suppose that A is \(\mathcal{R}\)directed and \(\beta\ge\max\{\alpha_{0},2\alpha _{1},\alpha_{3},\alpha_{4}\}\). Then T has a unique best proximity point.
Proof
The fact that A is \(\mathcal{R}\)directed implies that the nonselfmapping \(T: A\longrightarrow B\) is \((\alpha,d)\) regular. So, by Theorem 3.3, we deduce the uniqueness of a best proximity point for T. □
Application to fixed point results
Let us recall the following definition.
Definition 5.7
Let A be a nonempty set of a metric space \((X,d)\). A selfmapping \(T:A\to A\) is called a generalized αβquasicontractive if there exist two functions \(\alpha: A\times A\longrightarrow[0, +\infty)\) and \(\varphi \in\Phi_{\beta}\), where \(\beta>0\), such that, for all \(x,y\in A\), we have
where
with \(\alpha_{k}\ge0\) for \({k=0,\ldots,4}\).
By considering the particular case, \(A=B\) in Theorems 3.2 and 3.3, the fixed point results were deduced as follows.
Corollary 5.8
Let A be a nonempty closed subset of a complete metric space \((X,d)\). Let \(T:A\to A\) be an αβquasicontractive mapping, where \(\beta\ge\max_{0\le k \le3}\{\alpha_{k},2\alpha_{4}\}\), satisfying the following assertions:

(1)
T is αproximal admissible;

(2)
there exist elements \(x_{0},x_{1} \in A\) such that \(\alpha(x_{0},x_{1}) \ge1\);

(3)
if \(\{x_{n}\}\) is a sequence in A such that \(\alpha(x_{n},x_{n+1}) \ge1\) for all n and \({\lim_{n\longrightarrow+\infty}x_{n}=x_{\ast}\in A}\), then there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x_{\ast})\ge1\) for all k.
Moreover, suppose that one of the following conditions holds:

φ is continuous;

\(\beta>\max\{\alpha_{2},\alpha_{4}\}\).
Then T has a fixed point.
Corollary 5.9
In addition to the hypotheses of Corollary 5.8, suppose that T is \((\alpha,d)\) regular and \(\beta\ge\max\{\alpha_{0},2\alpha_{1},\alpha _{3},\alpha_{4}\}\). Then T has a unique fixed point.
Conclusion
We recall that we have given in this paper some improvements to the best proximity point theorems previously made by JM, KE and SB in [9] for αψproximal contractive mappings. This improvement was obtained by introducing the proximal αβquasicontractive mappings on metric spaces involving βcomparison functions. As applications, we have established not only the existence but the uniqueness of best proximity point results for the case of nonselfmappings on metric spaces endowed with symmetric binary relations.
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Ayari, M.I. Best proximity point theorems for generalized αβproximal quasicontractive mappings. Fixed Point Theory Appl 2017, 16 (2017) doi:10.1186/s1366301706121
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MSC
 47H10
 54H25
Keywords
 best proximity points
 αβproximal quasicontractive mappings on metric spaces