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Best proximity point theorems for generalized αβproximal quasicontractive mappings
 Mohamed Iadh Ayari^{1}Email author
https://doi.org/10.1186/s1366301706121
© The Author(s) 2017
 Received: 27 May 2017
 Accepted: 4 September 2017
 Published: 1 November 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.
Keywords
 best proximity points
 αβproximal quasicontractive mappings on metric spaces
MSC
 47H10
 54H25
1 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.
2 Preliminaries and definitions
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]
 (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 \(\sum_{n=1}^{\infty}\varphi_{\beta}^{n}(s) < \infty\).
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]
 (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]
Definition 2.8
[9]
3 Main results and theorems
First, we introduce the following concept.
Definition 3.1
We propose the following best proximity point theorems.
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 \(\alpha(x_{0},x_{1}) \ge1\);
 (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.

φ is continuous;

\(\beta>\max\{\alpha_{2},\alpha_{4}\}\).
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
 (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 \(\alpha(x_{0},x_{1}) \ge1\);
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 \(\alpha(x_{1},x_{2}) \ge1\).
Our next step is to prove that the sequence \(\{x_{n}\}\) is a Cauchy sequence.
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 \(x_{\ast}\in A\).
We consider two separate cases as follows.
If \(\beta>\max \{\alpha_{2},\alpha_{4} \}\), we claim also that \(\rho=0\). Suppose by contradiction that \(\rho>0\).
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 \(\alpha(x_{\ast},y_{\ast})<1\). 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 \(\alpha (x_{\ast},u_{1})\ge1\).
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 \(i=0,1,2,3,4\).
It is easy to see that the pair \((A,B)\) satisfies the Pproperty.
4 Consequences
Several consequences of the main theorems are suggested in this section.
Corollary 4.1
[9]
 (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.
Proof
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
 (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 \(x,y\in A\).
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. □
5 Applications
5.1 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]
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 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.

φ is continuous;

\(\beta>\max\{\alpha_{2},\alpha_{4}\}\).
Proof
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. □
5.2 Application to fixed point results
Let us recall the following definition.
Definition 5.7
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
 (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.

φ is continuous;

\(\beta>\max\{\alpha_{2},\alpha_{4}\}\).
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.
6 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.
Declarations
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