 Research
 Open Access
Generalization of best proximity points theorem for nonself proximal contractions of first kind
 Mohamed Iadh Ayari^{1}Email author,
 Zead Mustafa^{2} and
 Mohammed Mahmoud Jaradat^{2}
https://doi.org/10.1186/s1366301906574
© The Author(s) 2019
 Received: 12 November 2018
 Accepted: 30 January 2019
 Published: 25 February 2019
Abstract
The primary objective of this paper is the study of the generalization of some results given by Basha (Numer. Funct. Anal. Optim. 31:569–576, 2010). We present a new theorem on the existence and uniqueness of best proximity points for proximal βquasicontractive mappings for nonselfmappings \(S:M\rightarrow N\) and \(T:N\rightarrow M\). Furthermore, as a consequence, we give a new result on the existence and uniqueness of a common fixed point of two self mappings.
Keywords
 Best proximity points
 Proximal βquasicontractive mappings on metric spaces and proximal cyclic contraction
MSC
 47H10
 54H25
1 Introduction
In 1969, Fan in [2] proposed the concept best proximity point result for nonself continuous mappings \(T:A\longrightarrow X\) where A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space X. He showed that there exists a such that \(d(a,Ta)=d(Ta,A)\). Many extensions of Fan’s theorems were established in the literature, such as in work by Reich [3], Sehgal and Singh [4] and Prolla [5].
In 2010, [1], Basha introduce the concept of best proximity point of a nonself mapping. Furthermore he introduced an extension of the Banach contraction principle by a best proximity theorem. Later on, several best proximity points results were derived (see e.g. [6–19]). Best proximity point theorems for nonself set valued mappings have been obtained in [20] by Jleli and Samet, in the context of proximal orbital completeness condition which is weaker than the compactness condition.
The aim of this article is to generalize the results of Basha [21] by introducing proximal βquasicontractive mappings which involve suitable comparison functions. As a consequence of our theorem, we obtain the result of Basha in [21] and an analogous result on proximal quasicontractions is obtained which was first introduced by Jleli and Samet in [20].
2 Preliminaries and definitions
Let \((M,N)\) be a pair of nonempty subsets of a metric space \((X,d)\). The following notations will be used throughout this paper: \(d(M,N):=\inf\{d(m,n):m\in M, n\in N\}\); \(d(x,N):=\inf\{d(x,n):n\in N\}\).
Definition 2.1
([1])
Let \(T:M\rightarrow N\) be a nonselfmapping. An element \(a_{\ast}\in M\) is said to be a best proximity point of T if \(d(a_{\ast },Ta_{\ast})=d(M,N)\).
Note that in the case of selfmapping, a best proximal point is the normal fixed point, see [22, 23].
Definition 2.2
([21])
Definition 2.3
([21])
Definition 2.4
([24])
 \((P_{1})\) :

φ is nondecreasing.
 \((P_{2})\) :

\(\lim_{n\rightarrow\infty}\varphi _{\beta }^{n}(t)=0\) for all \(t>0\), where \(\varphi_{\beta}^{n}\) denote the nth iteration of \(\varphi_{\beta}\) and \(\varphi_{\beta}(t)=\varphi (\beta t)\).
 \((P_{3})\) :

There exists \(s\in(0,+\infty)\) such that \(\sum_{n=1}^{\infty}\varphi_{\beta}^{n}(s)<\infty\).
 \((P_{4})\) :

\((\mathrm{id}\varphi_{\beta} ) \circ\varphi _{\beta}(t) \leq\varphi_{\beta} \circ(\mathrm{id}\varphi_{\beta})(t) \mbox{ for all } t \geq0\), where \(\mathrm{id}: [0,\infty) \longrightarrow[0,\infty) \) is the identity function.
Throughout this work, the set of all functions φ satisfying \((P_{1}), (P_{2})\) and \((P_{3})\) will be denoted by \(\varPhi_{\beta}\).
Remark 2.1
Let \(\alpha,\beta\in(0,+\infty)\). If \(\alpha<\beta\), then \(\varPhi_{\beta}\subset\varPhi_{\alpha}\).
We recall the following useful lemma concerning the comparison functions \(\varPhi_{\beta}\).
Lemma 2.1
([24])
 (i)
\(\varphi_{\beta}\) is nondecreasing;
 (ii)
\(\varphi_{\beta} (t) < t\) for all \(t > 0\);
 (iii)
\(\sum_{n=1}^{\infty}\varphi_{\beta}^{n}(t) < \infty\) for all \(t > 0 \).
Definition 2.5
([20])
3 Main results and theorems
Now, we start this section by introducing the following concept.
Definition 3.1
Let \((M,N)\) be a pair of nonempty subsets of a metric space \((X,d)\). The following notations will be used throughout this paper: \(M_{0}:=\{u\in M:\text{ there exists }v\in N\text{ with }d(u,v)=d(M,N)\} \);\(N_{0}:=\{v\in N:\text{ there exists }u\in M\text{ with }d(u,v)=d(M,N)\} \).
Our main result is giving by the following best proximity point theorems.
Theorem 3.1
 \((C_{1})\) :

\(S(M_{0})\subset N_{0}\) and \(T(N_{0})\subset M_{0}\);
 \((C_{2})\) :

there exist \(\beta_{1}, \beta_{2}\geq\max\{ \alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}, 2\alpha_{4}\}\) such that S is a proximal \(\beta_{1}\)quasicontraction mapping (say, \(\psi\in\varPhi_{\beta_{1}}\)) and T is a proximal \(\beta_{2} \)quasicontraction mapping (say, \(\phi\in\varPhi_{\beta_{2}}\)).
 \((C_{3})\) :

The pair \((S,T)\) forms a proximal cyclic contraction.
 \((C_{4})\) :

Moreover, one of the following two assertions holds:
 (i)
ψ and ϕ are continuous;
 (ii)
\(\beta_{1},\beta_{2}>\max\{\alpha_{2},\alpha _{3}\} \).
 (i)
Proof
Similarly, by using word by word the above argument after replacing u by v, S by T, \(\beta_{1}\) by \(\beta_{2}\) and ψ by ϕ, we get that \(v=b_{\ast}\) and hence by (6) \(b_{\ast}\) is a best proximity point for the nonself mapping T.
In Theorem 3.1 by taking \(\alpha_{0}=\alpha_{1}=\alpha_{2}=\alpha_{3}=0 , \alpha _{4}=1,\beta_{1}=\beta_{2}=1\) and \(\psi(t)=\phi(t)=qt\) which is a continuous function and belongs to \(\varPhi_{1}\), we obtain Corollary 3.3 in [21].
Corollary 3.1
 \((d_{1})\) :

\(S(A_{0})\subset M_{0}\) and \(T(M_{0})\subset N_{0}\).
 \((d_{2})\) :

S and T are proximal quasicontractions.
 \((d_{3})\) :

The pair \((S,T)\) form a proximal cyclic contraction.
Then S has a unique best proximity point \(a_{\ast }\in M\) such that \(d(a_{\ast},Sa_{\ast})=d(M,N)\) and T has a unique best proximity point \(b_{\ast}\in N\) such that \(d(b_{\ast},Tb_{\ast })=d(M,N)\). Also, these best proximity points satisfies \(d(a_{\ast },b_{\ast })=d(M,N)\).
Proof
The result follows immediately from Theorem 3.1 by taking \(\alpha _{0} = \alpha_{1} =\alpha_{2} =\alpha_{3} = 1 \) and \(\alpha_{4} = \frac {1}{2} \), \(\beta_{1}=\beta_{2} = 1\) and \(\psi(t)=\phi(t)=qt\). □
The following definition, which was introduced in [24], is needed to derive a fixed point result as a consequence of our main theorem.
Definition 3.2
([24])
Corollary 3.2
 \((E_{1})\) :

S is \(\beta_{1}\)quasicontractive ( say, \(\psi\in\varPhi_{\beta_{1}}\)) and T is \(\beta_{2}\)quasicontractive (say, \(\phi\in\varPhi_{\beta_{2}}\)).
 \((E_{2})\) :

For all \(a,b\in X,d(Sa,Tb)\leq kd(a,b)\) for some \(k\in(0,1)\).
 \((E_{3})\) :

Moreover, one of the following assertions holds:
 (i)
ψ and ϕ are continuous;
 (ii)
\(\beta_{1},\beta_{2}>\max\{\alpha _{2},\alpha_{3}\}\).
 (i)
Then S and T have a common unique fixed point.
Proof
This result follows from Theorem 3.1 by taking \(M=N=X\) and noticing that the hypotheses \((E_{1})\) and \((E_{2})\) of the corollary coincide with the first, second and the third conditions of Theorem 3.1. □
Example 3.1
Let \(X=\mathbb{R}\) with the metric \(d(x,y)=xy\), then \((X,d)\) is complete metric space. Let \(M=[0,1]\) and \(N=[2,3]\). Also, let \(S:M\longrightarrow N\) and \(T:N\longrightarrow M\) be defined by \(S(x)=3x\) and \(T(y)=3y\). Then it is easy to see that \(d(M,N)=1\), \(M_{0}=\{1\}\) and \(N_{0}=\{2\}\). Thus, \(S(M_{0}) = S(\{1\}) = \{2\} = N_{0}\) and \(T(M_{0}) = T(\{2\}) = \{1\} = M_{0}\).
Now we show that the pair \((S,T)\) forms a proximal cyclic contraction. \(d(u,Sa) = d(M,N) =1\) implies that \(u=a=1 \in M\) and \(d(v,Tb = d(M,N) =1\) implies that \(v=b=2 \in N\).
So, T is a proximal \(\beta_{2}\)quasicontraction mapping. We deduce, using Theorem 3.1, that T has a unique best proximity point which is \(b_{\ast} =2\).
4 Conclusion
Improvements to some best proximity point theorems are proposed. In particular, the result due to Basha [21] for proximal contractions of first kind is generalized. Furthermore, we propose a similar result on existence and uniqueness of best proximity point of proximal quasicontractions introduced by Jleli and Samet in [20]. This has been achieved by introducing βquasicontractions involving βcomparison functions introduced in [24].
Declarations
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Authors’ contributions
The authors contributed equally to the preparation of the paper. The authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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