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Some results on best proximity point on starshaped sets in probabilistic Banach (Menger) spaces
Fixed Point Theory and Applications volume 2016, Article number: 13 (2016)
Abstract
We first present the concepts of proximal contraction and proximal nonexpansive mappings on starshaped sets in probabilistic Banach (Menger) spaces. We derive some results about the best proximity points for these mappings in probabilistic Banach (Menger) spaces. Next, we bring some examples that defend our main results.
Introduction and preliminaries
The equation \(Tx = x\) for a mapping \(T: A \rightarrow B \) may have no solution whenever \(A\cap B =\emptyset\), where A, B are two nonempty subsets in a metric space \((X, d)\). Under this condition, it is beneficial to determine a point \(a_{0} \in A \) such that \(d(a_{0} , Ta_{0}) \) is minimal. If \(d(a_{0} , Ta_{0})\) is the global minimum value of \(\operatorname{dist}(A,B)\), i.e., \(d(a_{0} , Ta_{0}) = \operatorname{dist}(A,B) = \min\lbrace d(a, b) : a \in A, b\in B\rbrace\), then \(a_{0}\) is called best proximity point of T.
In 1969, Fan [1] proved one of the most classical theorems in best approximation theory. He showed that if \((V, \rho)\) is a topological vector space with seminorm p, \(W\subseteq V\), and \(T : W \rightarrow V\) is a mapping, then under certain conditions, there exists an element \(w_{0}\in W\) such that
Thereafter, this theorem has been generalized for continuous multivalued mappings by Reich [2, 3] and Sehgal and Singh [4].
Eldred et al. [5] showed that every relatively nonexpansive mapping has a proximal point under certain conditions. For further existence results of a best proximity point for several types of contractions, we refer to [6–25].
In 1942, a probabilistic metric (PM) space was introduced by Menger [26]. Schweizer and Sklar [27, 28] were two pioneers in the study of PM spaces.
PM spaces are very useful in probabilistic functional analysis, quantum particle physics, \(\epsilon^{\infty} \) theory, nonlinear analysis, and applications; see [29–33].
Indeed, the study of fixed point results in PM spaces is one of the most active research areas in fixed point theory. Sehgal and BharuchaReid [34] were two pioneers in this study. For further existence results of a fixed point and common fixed point in PM spaces, we refer, for example, to [35–37]. In 2014, Su and Zhang [38], proved some best proximity point theorems in PM spaces.
Let \(\Delta^{+}\) be the set of all distribution functions F (i.e., a nondecreasing and leftcontinuous function \(F : \mathbb{R} \rightarrow [0, 1]\) such that \(\inf_{t\in\mathbb{R}} F(t) = 0\) and \(\sup_{t\in \mathbb{R}} F(t) = 1\)) such that \(F(0) = 0\). Let X be a nonempty set, \(\epsilon_{0}=\chi_{(0,\infty)}\in\Delta ^{+}\), and \(F: X \times X\rightarrow\Delta^{+}\) (\(F(p,q)=F_{p,q}\)) be a mapping such that

(PM1)
\(F_{p,q}=\epsilon_{0}\) iff \(p=q\),

(PM2)
\(F_{ p,q}=F_{q,p}\), and

(PM3)
if \(F_{p,q}(t) = 1\) and \(F_{q,r}(s) = 1\), then \(F_{p,r}(t+s) = 1\)
for all \(p, q, r\in X\) and \(t,s\geq0\). Then \((X, F)\) is called a probabilistic metric space.
For wellknown definitions (such as tnorm, tnorm of Htype, probabilistic Menger space, complete probabilistic Menger space, probabilistic normed (PN) space, etc.) and known results, we refer to [27, 39].
First, we state some notation, definitions, and known results; afterward, we introduce concepts of proximal contraction, proximal nonexpansive, Pproperty, weak Pproperty, and semisharp proximinal pair in PM spaces. Throughout this paper, the minimum tnorm will be denoted by \(\Delta _{m}(a,b)=\min\{a,b\}\).
Lemma 1.1
([39])
Let \(( x_{n}) \) be a sequence in a probabilistic Menger space \((X, F,\Delta) \) such that Δ is a tnorm of Htype. If
for some \(k \in(0,1) \), then \(( x_{n}) \) is a Cauchy sequence.
Definition 1.2
Suppose that A is a nonempty subset of a probabilistic Menger space \((X, F,\Delta)\). Then the probabilistic diameter of A is the mapping \(D_{A}\) defined on \([0,\infty]\) by \(D_{A}(\infty)=1\) and \(D_{A}(x)=\lim_{t\rightarrow x^{}}\varphi_{A}(t)\), where \(\varphi_{A} (t)=\inf\{F_{a,b}(t): a,b\in A\}\).
A nonempty set A in a probabilistic Menger space is bounded if \(\lim_{x\rightarrow\infty}D_{A}(x)=1\). It is easy to see that \(F_{a,b}(t)\geq D_{A}(t)\) for all \(a,b\in A\) and \(t\geq0\).
Definition 1.3
Let \((X, F,\Delta)\) be a probabilistic Menger space, \(A\subseteq X \), and \(T:A\rightarrow A\) be a mapping. The mapping T is said to be an isometry if
Definition 1.4
Let \((X, F,\Delta)\) be a probabilistic Menger space, and \(A, B\subseteq X \). A mapping \(T:A\rightarrow B\) is said to be continuous at \(x\in A\) if for every sequence \((x_{n})\) in A that converges to x, the sequence \((Tx_{n})\) in B converges to Tx.
Remark 1.5
If T is an isometry mapping on subset A of a probabilistic Menger space \((X,F, \Delta) \), then T is a continuous mapping because
Also, it is easy to see that T is an injective mapping.
An immediate consequence of the definition of a PN space ([27], Section 15.1) is the following lemma.
Lemma 1.6
([27])
Let \((X,\nu, \Delta) \) be a PN space, and \(F^{\nu}\) be the function from \(X\times X\) into \(\Delta^{+} \) defined by
Then \((X,F^{\nu}, \Delta) \) is a probabilistic Menger space.
We call this probabilistic metric \(F^{\nu}\) on X the probabilistic metric induced by the probabilistic norm ν.
Definition 1.7
A PN space \((X,\nu, \Delta) \) is said to be a probabilistic Banach space if \((X,F^{\nu}, \Delta) \) is a complete probabilistic Menger space.
Remark 1.8
Let A, B, C be a nonempty subsets of a PN space \((X,\nu, \Delta) \) such that Δ is continuous tnorm and \(x\in A\). If two mappings \(T:A\rightarrow B\) and \(S:A\rightarrow C\) are continuous at x, then \(T+S\) is continuous at x because
Definition 1.9
Let A be a nonempty subset of a PM space \((X,F)\). A mapping \(T:A\rightarrow X \) is called a contraction (nonexpansive) if \(F_{Tx,Ty}(t)\geq F_{x,y} (\frac{t}{\alpha} )\) (\(F_{Tx,Ty}(t)\geq F_{x,y} (t)\)) for some \(0<\alpha<1 \) and for all \(x,y\in A \) and \(t>0 \).
Definition 1.10
Suppose that A and B are nonempty subsets of a PM space \((X,F)\). Then the probabilistic distance of A, B is the mapping \(F_{A,B}\) defined on \([0,\infty]\) by
Also, if A and B are nonempty subsets of a PN space \((X,\nu, \Delta ) \), then \({F^{\nu}}_{A,B}(t)=\nu_{AB}(t)=\sup_{x\in A, y\in B}\nu _{xy}(t)\), where \(F^{\nu}\) is the probabilistic metric induced by the probabilistic norm ν.
Definition 1.11
Let \((X, F)\) be a PM space. For subsets A and B of X, define:
Clearly, if \(A_{0}\) (or \(B_{0}\)) is a nonempty subset, then A and B are nonempty subsets.
Definition 1.12
Let \((X, F) \) be a PM space, and \((A,B) \) be a pair of nonempty subsets of X. A mapping \(T : A\rightarrow B \) is called the proximal contraction (proximal nonexpansive) if there exists a real number \(0<\alpha<1 \) such that
for all \(u ,v,x,y\in A\) and \(t>0\).
Example 1.13
Let \(X=[0,2] \), and \(T:X\rightarrow X\) be the mapping defined by \(Tx=\frac{1}{8}x \). If \(F_{x,y}(t)=\frac{t}{t+xy} \), then it is easy to check that \(F_{X,X}(t)=1 \). If \(F_{u,Tx}(t)=1=F_{v,Ty}(t)\), then for \(\alpha=\frac{1}{8} \), we have \(F_{u,v}(t)= F_{x,y}(\frac{t}{\alpha})\), where \(u,v,x,y \in X\). Therefore, T is a proximal contraction.
Definition 1.14
Let X be a vector space, and A be a nonempty subset of X. Then the subset A is called a pstarshaped set if there exists a point \(p \in A \) such that \(\alpha p + (1\alpha)x \in A\) for all \(x\in A\), \(\alpha\in [0,1] \), and p is called the center of A.
Clearly, each convex set C is a pstarshaped set for each \(p \in C \). Let \((X,\nu,\Delta_{m} )\) be a PN space, A be a pstarshaped set, B be a qstarshaped set, and \(\nu_{p  q}=\nu_{AB}\). If \(x\in A_{0}\), then there exists a point \(y\in B\) such that \(\nu _{xy}(t)=\nu_{AB}(t)\) for all \(t>0\). So we have
for all \(t>0\). Therefore, \(\nu_{(\alpha p+(1\alpha)x)(\alpha q +(1\alpha)y)}(t)=\nu_{AB}( t)\), which means that \(A_{0} \) is a pstarshaped set and, similarly, that \(B_{0} \) is a qstarshaped set.
Definition 1.15
Let \((X, F) \) be a PM space. A pair \((A,B) \) of nonempty subsets of X is said to have the Pproperty (weak Pproperty) if \(A_{0}\neq\emptyset \) and
for all \(u,v\in A_{0}\), \(x,y\in B_{0}\), and \(t>0\).
Example 1.16
Let \(X=\mathbb{R}^{2}\) and define
Clearly, \((X,F,\Delta_{m})\) is a complete probabilistic Menger space. Let
Then it is easy to check that \(A_{0}=A \), \(B_{0}=B \), and \(F_{A,B}(t)=\frac{t}{t+1} \). If
then \(x=y \) and \(u=v \), so that
Therefore, the pair \((A, B)\) has the Pproperty.
Example 1.17
Let \(X=\mathbb{R}^{2}\) and define
Let \(A=\lbrace(0,0)\rbrace\) and \(B=\lbrace(x,y)\in X : y=1+\sqrt {1x^{2}} \rbrace\). Clearly, \(A_{0}=\{(0,0)\} \) and \(B_{0}=\{ (1,1),(1,1)\} \). If
then
where \((x,y),(u,v)\in B_{0} \). Therefore, the pair \((A,B) \) has the weak Pproperty.
Definition 1.18
Let \((X, F) \) be a PM space. A pair \((A,B) \) of nonempty subsets of X is called a semisharp proximinal pair if there exists at most one \((x_{0}, y_{0}) \in A \times B\) such that \(F_{x, y_{0}}(t) = F_{A,B}(t)=F_{x_{0}, y}(t) \) for all \((x, y) \in A \times B \).
It is easy to check that if a pair \((A,B) \) has the Pproperty, then the pair \((A,B) \) is a semisharp proximinal pair. Clearly, a semisharp proximinal pair \((A,B) \) does not necessarily have the Pproperty.
Example 1.19
Suppose that \(X =\mathbb{R} \), \(A=\{10,10\} \), \(B=\{2,2\} \), and \(F_{x,y}(t)=\frac{t}{t+xy} \). It is easy to verify that \(F_{A,B}(t)=\frac{t}{t+8} \), \(A_{0}= A\), \(B_{0}= B\), and \(( A,B) \) is a semisharp proximinal pair but does not have the Pproperty.
Remark 1.20
It is easy to check that the Pproperty is stronger than the weak Pproperty. If a pair \((A,B) \) has the weak Pproperty and \(T : A\rightarrow B \) is a nonexpansive mapping, then for all \(u,v, x,y\in A\), we have
That is, T is a proximal nonexpansive mapping. Similarly, if a pair \((A,B) \) has the weak Pproperty and \(T : A\rightarrow B \) is a contraction mapping, then T is a proximal contraction mapping. Also, a pair \((A,B) \) has the Pproperty if and only if both pairs \((A,B) \) and \((B,A) \) have the weak Pproperty.
Definition 1.21
Let X and Y be vector spaces. A mapping \(T:X\rightarrow Y\) is affine if
for all \(n\in\mathbb{N}\), \(x_{1},\ldots,x_{n}\in X\), and \(\lambda_{1},\ldots , \lambda_{n} \in\mathbb{R}\) such that \(\sum_{i=1}^{n}\lambda_{i}=1\).
In Section 2, we show some results on the best proximity points in probabilistic Banach (Menger) spaces. For example, if \((A,B) \) is a semisharp proximinal pair of a probabilistic Banach space \((X, \nu,\Delta_{m}) \) such that A is a pstarshaped set, \(A_{0}\) is a nonempty compact set, B is a qstarshaped set and \(\nu_{p  q}(t)=\nu_{AB}(t)\) for all \(t>0\), then every proximal nonexpansive mapping \(T : A\rightarrow B \) with \(T(A_{0} )\subseteq B_{0}\) has a best proximity point. We also prove that if A is a nonempty, compact, and convex subset of a probabilistic Banach space \((X, \nu,\Delta_{m}) \) and \(T:A\rightarrow A \) is a nonexpansive mapping, then T has a fixed point. Finally, we give some examples which defend our main results.
Proximity point for proximal contraction and proximal nonexpansive mappings
We first give the following lemma and then we state the main results of this paper. We recall that if \(A_{0}\) (or \(B_{0}\)) is a nonempty subset, then A and B are nonempty subsets.
Lemma 2.1
Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a tnorm of Htype, and \(A,B \subseteq X \) be such that \(A_{0} \) is a nonempty closed set. If \(T : A\rightarrow B \) is a proximal contraction mapping such that \(T(A_{0} )\subseteq B_{0}\), then there exists a unique \(x\in A_{0} \) such that \(F_{x,Tx}(t)=F_{A,B}(t) \) for all \(t>0\).
Proof
Since \(A_{0} \) is nonempty and \(T(A_{0} )\subseteq B_{0}\), there exist \(x_{1},x_{0}\in A_{0}\) such that \(F_{x_{1},Tx_{0}}(t)=F_{A,B}(t) \). Since \(Tx_{1}\in B_{0} \), there exists \(x_{2}\in A_{0}\) such that \(F_{x_{2},Tx_{1}}(t)=F_{A,B}(t) \). Continuing this process, we obtain a sequence \(( x_{n})\subseteq A_{0} \) such that \(F_{x_{n+1},Tx_{n}}(t)=F_{A,B}(t) \) for all \(n\in\mathbb {N} \) and \(t>0\). Since for all \(n\in\mathbb{N} \),
and T is a proximal contraction, we have
Therefore, by Lemma 1.1, \(( x_{n}) \) is a Cauchy sequence and so converges to some \(x\in A_{0} \). Again by the assumption \(T(A_{0} )\subseteq B_{0}\), \(Tx \in B_{0}\). Then there exists an element \(u\in A_{0} \) such that \(F_{u,Tx}(t)=F_{A,B}(t)\) for all \(t>0\). Since for all \(n\in\mathbb{N} \),
by the hypothesis we have
Letting \(n\rightarrow\infty\) shows that \(x_{n}\rightarrow u \) and thus \(x =u\), so \(F_{x,Tx}(t)=F_{A,B}(t) \). If there exists another element y such that \(F_{y,Ty}(t)=F_{A,B}(t) \), then by the hypothesis we have \(F_{x,y}(t)\geq F_{x,y}(\frac{t}{\alpha}) \), which means that \(x=y\). □
Proposition 2.2
Let \((X, F,\Delta) \) be a probabilistic Menger space, and \(A,B \subseteq X \) be such that \(A_{0} \) is a nonempty set. Suppose that \(T : A\rightarrow B \) is a proximal contraction mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \). Denote \(G = g(A)\) and
Then \(Tg^{1}\) is a proximal contraction, and \(G_{0}=A_{0} \).
Proof
Since \(G\subseteq A \), \(F_{G,B}(t)\leq F_{A,B} (t)\) for all \(t>0 \). Assume that \(x\in A_{0} \subseteq g(A_{0})\). Then \(x=g(x^{\prime}) \) for some \(x^{\prime} \in A_{0}\), and so there exists \(y\in B \) such that \(F_{A,B}(t)=F_{g(x^{\prime}),y}(t)\leq F_{G,B}(t) \) for all \(t>0 \). Thus, \(F_{A,B}(t)=F_{G,B}(t) \) for all \(t>0 \). Now we show that \(Tg^{1} \) is a proximal contraction. To this end, suppose that \(u,v,x,y\in G \) are such that
By the hypothesis we have
for some \(\alpha\in(0,1) \). Therefore, \(Tg^{1}\) is a proximal contraction. If \(x\in G_{0} \), then \(x\in G\subseteq A\), and there exists \(y\in B \) such that \(F_{x,y}(t)=F_{G,B}(t)=F_{A,B}(t)\) for all \(t>0 \), so that \(x\in A_{0} \). If \(x\in A_{0} \subseteq A\), then there exists \(y\in B \) such that \(F_{x,y}(t)=F_{A,B}(t)=F_{G,B}(t)\) for all \(t>0 \). On the other hand, by the hypothesis \(x\in G \), and therefore \(G_{0}=A_{0} \). □
Corollary 2.3
Let the hypotheses of Lemma 2.1 be satisfied. Suppose that \(T : A\rightarrow B \) is a proximal contraction mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g: A\rightarrow A\) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \). Then there exists a unique \(x\in A_{0} \) such that \(F_{gx,Tx}(t)=F_{A,B}(t) \).
Proof
By Proposition 2.2, \(Tg^{1}:G=g(A)\rightarrow B\) is proximal contraction, and \(Tg^{1}(G_{0})=Tg^{1}(A_{0})\subseteq T(A_{0})\subseteq B_{0}\). Now by Lemma 2.1 there exists a unique \(x'\in A_{0} \) such that \(F_{x',Tg^{1}x'}(t)=F_{A,B}(t) \). Since \(A_{0}\subseteq g(A_{0})\), there exists \(x\in A_{0} \) such that \(x'=g(x)\), so that \(F_{g(x),Tx}(t)=F_{A,B}(t) \). Note that g is an injective mapping, therefore, by Lemma 2.1, x is unique, and hence the result follows. □
Theorem 2.4
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, \(A,B \subseteq X \) be such that A is a convex set, \(A_{0}\) be a nonempty compact set, and B be a bounded convex set. Suppose that \(T : A\rightarrow B \) is a continuous affine and proximal nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \). Then there exists an element \(x\in A_{0} \) such that \(\nu _{gxTx}(t)=\nu_{AB}(t) \) for all \(t>0\).
Proof
Fix \(z\in A_{0} \) and \(i \in(0,1)\). We define the mapping \(T_{i}:A\rightarrow B \) by
We show that \(T_{i} \) is a proximal contraction. Let \(u,v,x,y\in A \) be such that
Since T is an affine mapping, we have
So by the hypothesis we have
Hence, \(T_{i} \) is a proximal contraction. Let \(x\in A_{0} \), so that \(Tx\in B_{0} \) and \(Tz\in B_{0} \). Therefore, there exist \(u,v\in A_{0} \) such that
Put \(y=iu+(1i)v\in A \). Then
and thus \(T_{i}(A_{0})\subseteq B_{0} \). By Corollary 2.3 there exists a unique \(x_{i}\in A_{0} \) such that \(\nu_{gx_{i}T_{i}x_{i}}(t)=\nu _{AB}(t) \) for all \(t>0 \). Fix \(j\in(0,1) \). Then
Now letting \(i\rightarrow1 \), we obtain
Then letting \(j\rightarrow1 \), we have
So we can create a sequence \((x_{n})\) in \(A_{0}\) such that
Since \(A_{0}\) is compact, the sequence \((x_{n})\) has a subsequence \((x_{n_{k}})\) such that \(x_{n_{k}}\rightarrow x\in A_{0}\). By Remark 1.5, g is continuous mapping, and so \(gT\) is a continuous mapping by Remark 1.8. Indeed, since \(\Delta_{m}\) is a continuous tnorm, \(p\rightarrow \nu_{P}\) is continuous ([27], Chapter 12), and we get
as required. □
Theorem 2.5
Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a tnorm of Htype, and \((A,B) \) be a pair of subsets of X with the weak Pproperty such that \(A_{0}\) is a nonempty closed set. If \(T:A\rightarrow B \) is a contraction mapping such that \(T(A_{0} )\subseteq B_{0}\), then there exists a unique x in A such that \(F_{x,Tx}(t)=F_{A,B}(t) \) for all \(t>0 \).
Proof
It is a direct consequence of Remark 1.20 and Lemma 2.1. □
Clearly, the pair \((A, A) \) has the Pproperty, so we have the following result.
Corollary 2.6
Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a tnorm of Htype. Then every contraction selfmapping from each nonempty closed subset of X has a unique fixed point.
Theorem 2.7
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, and \((A,B) \) be a semisharp proximinal pair of X such that A is a pstarshaped set, \(A_{0}\) be a nonempty compact set, B be a qstarshaped set, and let \(\nu_{p  q}(t)=\nu_{AB}(t)\) for all \(t>0\). If \(T : A\rightarrow B \) is a proximal nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\), then there exists an element \(x\in A_{0} \) such that \(\nu_{x  Tx}(t)=\nu_{AB}(t)\) for all \(t>0\).
Proof
For each integer \(i\geq1 \), define \(T_{i}:A_{0}\rightarrow B_{0} \) by
Then by the hypothesis we have \(T_{i}(A_{0} )\subseteq B_{0}\). Next, we show that for each i, \(T_{i} \) is a proximal contraction with \(\alpha=1\frac{1}{i}< 1 \). To do this, suppose that \(x,y,u,v,s,r\in A_{0} \) and \(t>0\) are such that
Now we define
so we have
Hence, \(\nu_{u'T_{i}x}(t)= \nu_{AB}(t)\). Since \(\nu _{uT_{i}x}(t)= \nu_{AB}(t)\) and \((A,B) \) is a semisharp proximinal pair, we have \(u'=u\). By the same method we also have \(v'=v \). Since T is a proximal nonexpansive mapping, we have
Therefore, \(T_{i} \) is a proximal contraction with \(\alpha=1\frac {1}{i}< 1 \). By Lemma 2.1, for each \(i\geq1 \), there exists a unique \(u_{i}\in A_{0} \) such that \(\nu_{u_{i}T_{i}u_{i}}(t)=\nu _{A_{0}B_{0}}(t)= \nu_{AB}(t)\). Since \(A_{0} \) is compact and \(( u_{i})\subseteq A_{0} \), without loss of generality, we can assume that \(u_{i} \) is a convergent sequence and \(u_{i}\rightarrow x \in A_{0} \).
For each \(i\geq1\), since \(T(u_{i})\in T(A_{0})\subseteq B_{0}\), there exists \(v_{i}\in A_{0} \) such that \(\nu_{v_{i}Tu_{i}}(t)= \nu _{AB}(t)\). So we have
Thus, \(\nu_{AB}(t)= \nu_{ (1\frac{1}{i} )v_{i} +\frac {1}{i}pT_{i}u_{i}}(t)\). Since \((A,B) \) is a semisharp proximinal pair and \(\nu_{AB}(t)=\nu _{u_{i}T_{i}u_{i}}(t) \), we have \(u_{i}=(1\frac{1}{i}) v_{i} +\frac{1}{i}p\), and so
Since \(A_{0} \) is compact and \(( v_{i})\subseteq A_{0} \), without loss of generality, we can assume that \(v_{i} \) is a convergent sequence and \(v_{i}\rightarrow z \in A_{0} \). For every \(j\leq i\), we have
Letting \(i\rightarrow\infty\), we have
Now letting \(j \rightarrow\infty\), we have
Therefore, \(\nu_{u_{i}v_{i}}(t)\rightarrow1 \), so that \(z=\lim_{i\rightarrow\infty}v_{i}= \lim_{i\rightarrow\infty}u_{i}=x\). Since \(Tx\in B_{0} \), there must exist \(u\in A_{0} \) such that \(\nu _{AB}(t)=\nu_{uTx}(t)\). Since we know that \(\nu_{AB}(t)=\nu _{v_{i}Tu_{i}}(t)\) and T is a proximal nonexpansive mapping, it follows that \(\nu _{v_{i}u}(t)\geq\nu_{u_{i}x}(t) \rightarrow1\). This implies that \(u= \lim_{i\rightarrow\infty}v_{i}=x\) and then \(\nu_{AB}(t)=\nu _{xTx}(t) \), as required. □
Theorem 2.8
Let \((X,\nu,\Delta_{m}) \) be a probabilistic Banach space, \((A,B) \) be a semisharp proximinal pair of X with the weak Pproperty such that A is a pstarshaped set, \(A_{0} \) be a nonempty compact set, B be a qstarshaped set, and let \(\nu_{p  q}(t)=\nu_{AB}(t)\) for all \(t>0\). If \(T:A\rightarrow B \) is a nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\), then T has a best proximity point in \(A_{0} \).
Proof
It is a direct consequence of Remark 1.20 and Theorem 2.7. □
Proposition 2.9
Let \((X, F,\Delta) \) be a probabilistic Menger space, and \(A,B \subseteq X \) be such that \(A_{0} \) is a nonempty set. Suppose that \(T : A\rightarrow B \) is a proximal nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \). Denote \(G = g(A)\) and
Then \(Tg^{1}\) is a proximal nonexpansive, and \(G_{0}=A_{0} \).
Proof
The result follows by using a similar argument as in the proof of Proposition 2.2. □
The following theorem is an immediate consequence of Theorem 2.7 and Proposition 2.9.
Theorem 2.10
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, \((A,B) \) be a semisharp proximinal pair of X such that A is a pstarshaped set, \(A_{0}\) be a nonempty compact set, B be a qstarshaped set, and let \(\nu_{p  q}(t)=\nu_{AB}(t)\) for all \(t>0\). If \(T : A\rightarrow B \) is a proximal nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \), then there exists an element \(x\in A_{0} \) such that \(\nu_{gx  Tx}(t)=\nu_{AB}(t)\) for all \(t>0\).
Corollary 2.11
Let \((X, \nu,\Delta_{m}) \) be a probabilistic Banach space, and let \((A,B) \) be a pair of convex subsets of X with the Pproperty such that \(A_{0}\) is a nonempty compact set. If \(T : A\rightarrow B \) is a nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\) and \(g : A\rightarrow A \) is an isometry mapping such that \(A_{0}\subseteq g(A_{0}) \), then there exists an element \(x\in A_{0} \) such that \(\nu _{gxTx}(t)=\nu_{AB}(t) \) for all \(t>0\).
In Corollary 2.11, if \(g(x)=x\), then we have the following corollary.
Corollary 2.12
With the hypotheses of the previous corollary, if \(T:A\rightarrow B \) is a nonexpansive mapping such that \(T(A_{0} )\subseteq B_{0}\), then T has a best proximity point.
In Corollary 2.12, if \(A=B\), then we have the following corollary.
Corollary 2.13
If A is a nonempty, compact, and convex subset of a probabilistic Banach space \((X, \nu,\Delta_{m}) \) and \(T:A\rightarrow A \) is a nonexpansive mapping, then T has a fixed point.
In the following, we give some examples that defend our main results.
Example 2.14
Let \(X=\mathbb{R}^{2} \), \(A=\lbrace(0,y) : y\in\mathbb{R}\rbrace\) and \(B=\lbrace(1,y): y\in\mathbb{R}\rbrace\). Suppose that \(T :A\rightarrow B\) is defined by \(T(0,y)= (1,\frac {y}{4} ) \), \(g :A\rightarrow A\) is defined by \(g(0,y)=(0,y) \), and \(F_{(x,x'),(y,y')}(t)=\frac{t}{t+xy+x'y'} \). It is easy to see that \((X,F,\Delta_{m})\) is a complete probabilistic Menger space, \(F_{A,B}(t)=\frac{t}{t+1} \), \(A_{0} =A\), \(B_{0}=B \), \(T(A_{0})\subseteq B_{0} \), and
If \((0,u),(0,x), (0,v),(0,y)\in A \) are such that
then \(u= \frac{x}{4}\) and \(v= \frac{y}{4}\), so that
Therefore, all the hypothesis of Corollary 2.3 are satisfied, and we also have
Example 2.15
Let \(X=\mathbb{R} \), \(A=[0,2] \) and \(B=[3,5] \). For every \(x\in X \), define \(\nu_{x} (t)=\frac{t}{t+x}\). It is easy to see that \((X, \nu,\Delta_{m}) \) is a probabilistic Banach space, \(\nu_{AB}(t)=\frac{t}{t+1} \), \(A_{0}=\lbrace2\rbrace\), and \(B_{0}=\lbrace3\rbrace\). For every \(x\in A \), define \(T:A\rightarrow B \) by \(Tx=5x \) and let g be the identity mapping. Clearly, T is a continuous affine and proximal nonexpansive mapping, and \(T(A_{0})=\{T(2)\}=\{3\}=B_{0} \). Therefore, all the hypotheses of Theorem 2.4 are satisfied, and also we have
The following example shows that the weak Pproperty of the pair \((A, B) \) cannot be removed from Theorem 2.5.
Example 2.16
Let \(X =\mathbb{R} \), \(A =\lbrace10,10\rbrace\), \(B = \lbrace2, 2\rbrace\), and \(F_{p,q}(t)=\frac{t}{t+pq} \). Clearly, \((X, F,\Delta_{m}) \) is a complete probabilistic Menger space. Then \(A_{0} = A \), \(B_{0} = B \), and \(F_{A,B}(t)=\frac{t}{t+8} \). Let \(T : A \rightarrow B\) be a mapping given by \(T (10) = 2 \) and \(T (10) = 2 \). It is easy to see that for \(\alpha=\frac{1}{5} \), T is a contraction mapping with \(T (A_{0})\subseteq B_{0}\). The mapping T does not have any best proximity point because \(F_{x,Tx}(t)=\frac{t}{t+12} < \frac{t}{t+8}= F_{A,B}(t)\) for all \(x \in A \). It should be noted that the pair \((A, B) \) does not have the weak Pproperty.
Example 2.17
Let \(X=\mathbb{R} \), \(A=[0,1] \), and \(B=[2,3] \). For every \(x\in X \), define \(\nu_{x}(t)=\frac{t}{t+x} \). It is easy to see that \((X, \nu,\Delta_{m}) \) is a probabilistic Banach space, A is 1starshaped set, B is 2starshaped set,
and
Also, \((A,B) \) is a semisharp proximinal pair. Now for each \(x\in A \), define \(T:A\rightarrow B \) by \(Tx=3x \). If \(u,v,x,y\in A \), then
so that \(u= x=1 \) and \(v= y=1 \). Thus,
So T is a proximal nonexpansive, and \(T(A_{0})=B_{0}\). Therefore, all the hypotheses of Theorem 2.7 are satisfied, and we also have
Example 2.18
Let \(X=\mathbb{R}^{2} \), \(A=\lbrace(x,0) : 0\leq x \leq1\rbrace\), \(B_{1}=\lbrace(x,y): x+y=1, 1\leq x \leq0\rbrace\), \(B_{2}=\lbrace(x,1): 0\leq x \leq1\rbrace\), \(B=B_{1}\cup B_{2}\), and \(\nu_{(x,x')}(t)=\frac{t}{t+ x+x'} \). It is easy to see that \((X, \nu,\Delta_{m}) \) is a probabilistic Banach space, \(\nu_{AB}(t)= \frac{t}{t+1} \), B is not convex but is a \((0,1) \)starshaped set, and A is \((0,0)\)starshaped set. Clearly, \(A_{0}=A\) and \(B_{0}=B_{2} \). So
and \((A,B) \) is a semisharp proximinal pair. Suppose that \(T:A\rightarrow B \) is defined by
and \((u,0),(v,0),(x,0),(y,0)\in A \) are such that
If \(x=y=0 \), then \(u=v=0 \), and therefore
If \(x,y\neq0\), then \(u=\sin x\), \(v=\sin y \), and therefore
If \(x=0\) and \(y\neq0\), then \(u=0 \) and \(v=\sin y \), and therefore
If \(x\neq0\) and \(y= 0\), then \(u=\sin x \) and \(v=0 \), and therefore
Hence, T is proximal nonexpansive, and \(T(A_{0})\subseteq B_{2}=B_{0}\), so all the hypotheses of Theorem 2.7 are satisfied, and we also have
Example 2.19
Let \(X=\mathbb{R} \), \(A=[0,1] \), \(B=[\frac{15}{8},2] \), and \(\nu _{x}(t)=\frac{t}{t+x} \). Clearly, \((X,\nu,\Delta_{m})\) is a probabilistic Banach space, \(\nu_{AB}(t)=\frac{t}{t+\frac{7}{8}} \), the pair \((A,B)\) has the Pproperty, \(A_{0}=\lbrace1\rbrace\), and \(B_{0}=\lbrace\frac{15}{8}\rbrace\). If \(Tx=\frac{1}{8}x+2 \), then \(T(A_{0})=\{T(1)\}=\{\frac{15}{8}\}=B_{0} \). Let \(x,y\in A\). Then we have
Therefore, all the hypotheses of Corollary 2.12 are satisfied, and hence T has a best proximity point, and we also have
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Acknowledgements
The authors would like to express their sincere appreciation to the Shahrekord University and the Center of Excellence for Mathematics for financial support.
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MSC
 41A65
 47H10
 47H09
 46S50
Keywords
 proximal contraction
 proximal nonexpansive mappings
 best proximity point
 Pproperty
 starshaped