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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