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Generalization of best proximity points theorem for nonself proximal contractions of first kind
Fixed Point Theory and Applicationsvolume 2019, Article number: 7 (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.
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. [6,7,8,9,10,11,12,13,14,15,16,17,18,19]). 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].
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])
Given nonselfmappings $S:M\rightarrow N$ and $T:N\rightarrow M $. The pair $(S,T)$ is said to form a proximal cyclic contraction if there exists a nonnegative number $k<1$ such that
for all $u,a\in M$ and $v,b\in N$.
Definition 2.3
([21])
A nonselfmapping $S: M\rightarrow N$ is said to be a proximal contraction of the first kind if there exists a nonnegative number $\alpha<1 $ such that
for all $u_{1},u_{2},a_{1},a_{2} \in M$.
Definition 2.4
([24])
Let $\beta\in(0,+\infty)$. A βcomparison function is a map $\varphi:[0,+\infty)\rightarrow{}[0,+\infty)$ satisfying the following properties:
 $(P_{1})$ :

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])
Let $\beta\in(0,+\infty)$ and $\varphi\in \varPhi_{\beta}$. Then

(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])
A nonselfmapping $T:M\rightarrow N$ is said to be a proximal quasicontraction if there exists a number $q\in {}[ 0,1)$ such that
whenever $a,b,u,v\in M$ satisfy the condition that $d(u,Ta)=d(M,N)$ and $d(v,Tb)=d(M,N)$.
Main results and theorems
Now, we start this section by introducing the following concept.
Definition 3.1
Let $\beta\in(0,+\infty)$. A nonself mapping $T:M\rightarrow N$ is said to be a proximal βquasicontraction if and only if there exist $\varphi\in\varPhi_{\beta }$ and positive numbers $\alpha_{0},\ldots,\alpha_{4}$ such that
For all $a,b,u,v\in M$ satisfying, $d(u,Ta)=d(M,N)$ and $d(v,Tb)=d(M,N)$.
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
Let $(M,N)$ be a pair of nonempty closed subsets of a complete metric space $(X,d)$ such that $M_{0}$ and $N_{0}$ are nonempty. Let $S:M\longrightarrow N$ and $T:N\longrightarrow M$ be two mappings satisfying the following conditions:
 $(C_{1})$ :

$S(M_{0})\subset N_{0}$ 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
Since $M_{0}$ is a nonempty set, $M_{0}$ contains at least one element, say $a_{0}\in M_{0}$. Using the first hypothesis of the theorem, there exists $a_{1}\in M_{0}$ such that $d(a_{1},Sa_{0})=d(M,N)$. Again, since $S(M_{0})\subset N_{0}$, there exists $a_{2}\in M_{0}$ such that $d(a_{2},Sa_{1})=d(M,N)$. Continuing this process in a similar fashion to find $a_{n+1}\in M_{0}$ such that $d(a_{n+1},Sa_{n})=d(M,N)$. Since S is a proximal $\beta_{1}$quasicontraction mapping for $\psi\in\varPhi_{\beta_{1}}$ and since
then by Definition 3.1 we have
Now, if $\max\{ d(a_{n},a_{n1}), d(a_{n},a_{n+1})\}= d(a_{n},a_{n+1})$, then by Lemma 2.1 the above inequality becomes
which is a contradiction. Thus, $\max\{ d(a_{n},a_{n1}), d(a_{n},a_{n+1}) \}= d(a_{n},a_{n1})$, then the above inequality (2) becomes
By applying induction on n, the above inequality gives
Now, from the axioms of metric and Eq. (3), for positive integers $n< m$, we get
Hence, for every $\epsilon>0$ there exists $N>0$ such that
Therefore, $d(a_{n},a_{m})<\epsilon$ for all $m>n>N$. That is $\{ a_{n}\}$ is a Cauchy sequence in M. But M is a closed subset of the complete metric space X, then $\{a_{n}\}$ converges to some element $a_{\ast}\in M$.
Since $T(N_{0})\subset M_{0}$, by using a similar argument as above, there exists a sequence $\{b_{n}\}\subset N_{0}$ such that $d(b_{n+1},Tb_{n})=d(M,N)$ for each n. Since T is a proximal $\beta _{2}$quasicontraction mapping (say $\phi\in\varPhi_{\beta_{2}}$) and since $d(b_{n+1},Tb_{n})=d(b_{n},Tb_{n1})=d(M,N)$, we deduce from Definition 3.1 that
Using a similar argument as in the case of $\{a_{n}\}$, one can show that $\{b_{n}\}$ is a Cauchy sequence in the closed subset N of the complete space X. Thus $\{b_{n}\}$ converges to $b_{\ast}\in N$. Now we shall show that $a_{\ast}$ and $b_{\ast}$ are best proximal points of S and T, respectively. As the pair $(S,T)$ forms a proximal cyclic contraction, it follows that
Taking the limit as $n\longrightarrow+\infty$, in Eq. (4) we get $d(a_{\ast},b_{\ast})\leq kd(a_{\ast},b_{\ast})+(1k)d(M,N)$, and so, $(1k) d(a_{\ast},b_{\ast})\leq (1k)d(M,N)$. This implies
Using the fact that $d(M,N)\leq d(a_{\ast},b_{\ast})$ and (5), we get $d(a_{\ast},b_{\ast})=d(M,N)$. Therefore, we conclude that $a_{\ast }\in M_{0}$ and $b_{\ast}\in N_{0}$.
From one hand, since $S(M_{0})\subset N_{0}$ and $T(N_{0})\subset M_{0}$, there exist $u\in M$ and $v\in N$ such that
On the other hand, by (1), (6) and using the hypothesis of the theorem that S is a proximal $\beta_{1}$quasicontraction mapping, we deduce that
For simplicity, we denote
and
Thus,
Now, we show by contradiction that $\rho=0$. Suppose that $\rho>0$. First, we consider the case where the assertion (i) of $(C_{4})$ is satisfied, that is, ψ is continuous. Then, taking the limit as $n\rightarrow\infty$ in (7) and using (8) and Lemma 2.1, we obtain
which is a contradiction. Now, we assume the case where the assertion (ii) of $(C_{4})$ is satisfied, that is, $\beta_{1}>\max\{\alpha_{2},\alpha _{3}\}$. Then there exist $\epsilon>0$ and integer $N>0$ such that, for all $n>N$, we have
Therefore, the inequality (7) turns into the following inequality:
Since $\psi\in\varPhi_{\beta_{1}}$, by Lemma 2.1 we have
By letting $n\rightarrow\infty$, the above inequality yields
which is a contradiction as well. Thus, in both two cases we get $0=\rho =d(a_{\ast},u)$, which means that $u=a_{\ast}$ and so from equation (6) we get $d(a_{\ast},Sa_{\ast})=d(M,N)$. That is $a_{\ast}$ is a best proximity point for S.
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.
Now, we shall prove that the obtained best proximity points $a_{\ast}$ of S is unique. Assume to the contrary that there exists $x\in M $ such that $d(x,Sx)=d(M,N)$ and $x\neq a_{\ast}$. Since S is a proximal $\beta_{1}$quasicontractive mapping, we obtain
which is a contradiction. Similarly, using the same as above and the fact that T is a proximal $\beta_{2}$quasicontractive mapping, we see that the best proximity point $b_{\ast}$ of T is unique. □
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
Let $(M,N)$ be a pair of nonempty closed subsets of a complete metric space $(X,d)$ such that $M_{0}$ and $M_{0}$ are nonempty. Let $S:M\longrightarrow N$ and $T:N\longrightarrow M$ be mappings satisfy the following conditions:
 $(d_{1})$ :

$S(A_{0})\subset M_{0}$ 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])
Let X be a nonempty set. A mapping $T:X\longrightarrow X$ is called βquasicontractive, if there exist $\beta>0$ and $\varphi \in\varPhi_{\beta}$ such that
where
with $\alpha_{i}\geq0$ for $i=0, 1,2,3,4$.
Corollary 3.2
Let $(X,d)$ be a complete metric space. Let $S,T:X\longrightarrow X$ be two selfmappings satisfying the following conditions:
 $(E_{1})$ :

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 $v=b=2 \in N$.
Now, since $d(u,Sa)=d(1,S(1))= d(1,2)=1=d(M,N)$ and $d(v,Tb)=d(2,T(2))= d(2,1)=1=d(M,N)$. Therefore,
So, $(S,T)$ are proximal cyclic contraction for any $0\leq k<1$. Now we shall show that S is proximal $\beta_{1}$quasicontraction mapping with $\psi(t)=\frac{1}{7}t,\beta_{1}=2$ and $\alpha_{i}=\frac{1}{5}$ for$i=0,1,2,3$ and $\alpha_{4} = \frac{1}{100}$. Note that $\psi(t)= \frac{1}{7}t \in\varPhi_{2} $ since $\psi_{\beta_{1}}t= \psi_{2}t= \frac{2}{7} t $. As above the only $a,b,u,v\in M$ such that $d(u,Sa)=d(M,N)=1=d(v,Sb)$ is $a=b=u=v =1 \in M$. But
So, S is a proximal $\beta_{1}$quasicontraction mapping. We deduce using our Theorem 3.1, that S has a unique best proximity point which is $a_{\ast} =1$ in this example.
Similarly, by using the same argument as above, we can show that T is proximal $\beta_{2}$quasicontraction mapping with $\phi(t)=\frac{1}{8}t,\beta_{2}=3$ and $\alpha_{i}=\frac{1}{6}$ for$i=0,1,2,3$ and $\alpha_{4} = \frac{1}{100}$. Note that $\phi(t)= \frac{1}{8}t \in\varPhi_{3} $ since $\phi_{\beta_{2}}t= \phi_{3}(t)= \frac{3}{8} t $. As above the only $a,b,u,v\in N$ such that $d(u,Ta)=d(M,N)=1=d(v,Tb)$ is $a=b=u=v =2 \in M$. But
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 $b_{\ast} =2$.
Finally, $\psi(t)$ and $\phi(t)$ are continuous mappings as well as $\beta_{1}, \beta_{2} > \max_{0\leq i \leq3}\{\alpha_{i} \} $. Therefore
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].
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MSC
 47H10
 54H25
Keywords
 Best proximity points
 Proximal βquasicontractive mappings on metric spaces and proximal cyclic contraction