1 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

$$\rho(w_{0}-Tw_{0})=d(Tw_{0}, W). $$

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 Bharucha-Reid [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 left-continuous 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

  1. (PM1)

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

  2. (PM2)

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

  3. (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 well-known definitions (such as t-norm, t-norm of H-type, 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, P-property, weak P-property, and semisharp proximinal pair in PM spaces. Throughout this paper, the minimum t-norm 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 t-norm of H-type. If

$$F_{x_{n},x_{n+1}}(kt)\geq F_{x_{n-1},x_{n}}(t) \quad (n\geq1, t>0) $$

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

$$F_{Tx,Ty}(t)=F_{x,y}(t) \quad \forall x,y\in X, \forall t\geq0. $$

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

$$F_{Tx_{n}, Tx}(t)=F_{x_{n},x}(t)\rightarrow1 \quad \forall t>0. $$

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

$$F^{\nu}(p,q)=\nu_{p-q}. $$

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 t-norm 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

$$\nu_{(T+S)(x)-(T+S)(x_{n})}(t)\geq\Delta\biggl(\nu_{T(x)-T(x_{n})}\biggl(\frac {t}{2} \biggr),\nu_{S(x)-S(x_{n})}\biggl(\frac{t}{2}\biggr)\biggr)\rightarrow1 \quad \forall t>0. $$

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

$$F_{A,B}(t)=\sup_{x\in A, y\in B}F_{x,y}(t) \quad \forall t\geq0 . $$

Also, if A and B are nonempty subsets of a PN space \((X,\nu, \Delta ) \), then \({F^{\nu}}_{A,B}(t)=\nu_{A-B}(t)=\sup_{x\in A, y\in B}\nu _{x-y}(t)\), 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:

$$\begin{aligned}& A_{0}=\bigl\lbrace x\in A :\exists y\in B \text{ s.t. } \forall t \geq0, F_{x,y}(t)=F_{A,B}(t)\bigr\rbrace , \\& B_{0}=\bigl\lbrace y\in B :\exists x\in A \text{ s.t. } \forall t \geq0, F_{x,y}(t)=F_{A,B}(t)\bigr\rbrace . \end{aligned}$$

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

$$\begin{aligned}& F_{u,Tx}(t)=F_{A,B}(t)=F_{v,Ty}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{x,y}\biggl( \frac{t}{\alpha}\biggr) \\& \bigl(F_{u,Tx}(t)=F_{A,B}(t)=F_{v,Ty}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{x,y}(t) \bigr) \end{aligned}$$

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 \(F_{x,y}(t)=\frac{t}{t+|x-y|} \), 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 p-star-shaped 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 p-star-shaped set for each \(p \in C \). Let \((X,\nu,\Delta_{m} )\) be a PN space, A be a p-star-shaped set, B be a q-star-shaped set, and \(\nu_{p - q}=\nu_{A-B}\). If \(x\in A_{0}\), then there exists a point \(y\in B\) such that \(\nu _{x-y}(t)=\nu_{A-B}(t)\) for all \(t>0\). So we have

$$\begin{aligned} \nu_{A-B}(t) \geq& \nu_{(\alpha p+(1-\alpha)x)-(\alpha q +(1-\alpha )y)}(t) \\ \geq& \Delta_{m}\bigl(\nu_{\alpha(p-q)}(\alpha t), \nu_{(1-\alpha )(x-y)}\bigl((1-\alpha)t\bigr)\bigr) \\ =& \Delta_{m}\bigl(\nu_{p-q}( t),\nu_{ x-y }(t) \bigr) \\ =& \Delta_{m}\bigl(\nu_{A-B}( t),\nu_{A-B}(t) \bigr) \\ =&\nu_{A-B}( t) \end{aligned}$$

for all \(t>0\). Therefore, \(\nu_{(\alpha p+(1-\alpha)x)-(\alpha q +(1-\alpha)y)}(t)=\nu_{A-B}( t)\), which means that \(A_{0} \) is a p-star-shaped set and, similarly, that \(B_{0} \) is a q-star-shaped 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 P-property (weak P-property) if \(A_{0}\neq\emptyset \) and

$$\begin{aligned}& F_{u,x}(t)=F_{A,B}(t)=F_{v,y}(t) \quad \Longrightarrow\quad F_{u,v}(t)= F_{x,y}(t) \\& \bigl(F_{u,x}(t)=F_{A,B}(t)=F_{v,y}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{x,y}(t) \bigr) \end{aligned}$$

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

$$F_{(x,y),(u,v)}(t)=\frac{t}{t+\sqrt{(x-u)^{2}+(y-v)^{2}}}. $$

Clearly, \((X,F,\Delta_{m})\) is a complete probabilistic Menger space. Let

$$\begin{aligned}& A=\biggl\{ \biggl(0,\frac{1}{n}\biggr) : n\in\mathbb{N}\biggr\} \cup\bigl\{ (0,0)\bigr\} , \\& B=\biggl\{ \biggl(1,\frac{1}{n}\biggr) : n\in \mathbb{N}\biggr\} \cup\bigl\{ (1,0)\bigr\} . \end{aligned}$$

Then it is easy to check that \(A_{0}=A \), \(B_{0}=B \), and \(F_{A,B}(t)=\frac{t}{t+1} \). If

$$F_{(0,x),(1,y)}(t)= F_{A,B}(t)=\frac{t}{t+1}=F_{(0,u),(1,v)}(t), $$

then \(x=y \) and \(u=v \), so that

$$F_{(0,x),(0,u)}(t)=\frac{t}{t+|x-u|}=\frac{t}{t+|y-v|}=F_{(1,y),(1,v)}(t). $$

Therefore, the pair \((A, B)\) has the P-property.

Example 1.17

Let \(X=\mathbb{R}^{2}\) and define

$$F_{(x,y),(u,v)}(t)=\frac{t}{t+\sqrt{(x-u)^{2}+(y-v)^{2}}}. $$

Let \(A=\lbrace(0,0)\rbrace\) and \(B=\lbrace(x,y)\in X : y=1+\sqrt {1-x^{2}} \rbrace\). Clearly, \(A_{0}=\{(0,0)\} \) and \(B_{0}=\{ (-1,1),(1,1)\} \). If

$$F_{(0,0),(x,y)}(t)=F_{A,B}(t)=\frac{t}{t+\sqrt{2}}=F_{(0,0),(u,v)}(t), $$

then

$$1=F_{(0,0),(0,0)}(t)\geq F_{(x,y),(u,v)}(t), $$

where \((x,y),(u,v)\in B_{0} \). Therefore, the pair \((A,B) \) has the weak P-property.

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 P-property, then the pair \((A,B) \) is a semisharp proximinal pair. Clearly, a semisharp proximinal pair \((A,B) \) does not necessarily have the P-property.

Example 1.19

Suppose that \(X =\mathbb{R} \), \(A=\{-10,10\} \), \(B=\{-2,2\} \), and \(F_{x,y}(t)=\frac{t}{t+|x-y|} \). 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 P-property.

Remark 1.20

It is easy to check that the P-property is stronger than the weak P-property. If a pair \((A,B) \) has the weak P-property and \(T : A\rightarrow B \) is a nonexpansive mapping, then for all \(u,v, x,y\in A\), we have

$$ F_{u,Tx}(t)=F_{A,B}(t)=F_{v,Ty}(t) \quad \Longrightarrow\quad F_{u,v}(t)\geq F_{Tx,Ty}(t)\geq F_{x,y}(t). $$

That is, T is a proximal nonexpansive mapping. Similarly, if a pair \((A,B) \) has the weak P-property and \(T : A\rightarrow B \) is a contraction mapping, then T is a proximal contraction mapping. Also, a pair \((A,B) \) has the P-property if and only if both pairs \((A,B) \) and \((B,A) \) have the weak P-property.

Definition 1.21

Let X and Y be vector spaces. A mapping \(T:X\rightarrow Y\) is affine if

$$T\Biggl(\sum_{i=1}^{n}\lambda_{i} x_{i}\Biggr)=\sum_{i=1}^{n} \lambda_{i} T(x_{i}) $$

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 p-star-shaped set, \(A_{0}\) is a nonempty compact set, B is a q-star-shaped set and \(\nu_{p - q}(t)=\nu_{A-B}(t)\) 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.

2 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 t-norm of H-type, 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 \(n\in\mathbb{N} \),

$$F_{x_{n},Tx_{n-1}}(t)=F_{A,B}(t)= F_{x_{n+1},Tx_{n}}(t) \quad (t>0) $$

and T is a proximal contraction, we have

$$F_{x_{n+1},x_{n}}(t)\geq F_{x_{n},x_{n-1}}\biggl(\frac{t}{\alpha}\biggr)\quad (0< \alpha< 1, t>0). $$

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 \(n\in\mathbb{N} \),

$$F_{u,Tx}(t)=F_{A,B}(t)=F_{x_{n+1},Tx_{n}}(t)\quad (t>0), $$

by the hypothesis we have

$$F_{u,x_{n+1}}(t)\geq F_{x,x_{n}}\biggl(\frac{t}{\alpha}\biggr)\geq F_{x,x_{n}}(t) \quad (t>0). $$

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

$$G_{0} =\bigl\lbrace z\in G : \exists y\in B \textit{ s.t. } \forall t \geq0, F_{z,y}(t)=F_{G,B}(t)\bigr\rbrace . $$

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

$$F_{u,Tg^{-1}x}(t)=F_{G,B}(t)=F_{A,B}(t)=F_{v,Tg^{-1}y}(t) \quad (t>0). $$

By the hypothesis we have

$$F_{u,v}(t)\geq F_{g^{-1}x,g^{-1}y}\biggl(\frac{t}{\alpha } \biggr)=F_{gg^{-1}x,gg^{-1}y}\biggl(\frac{t}{\alpha}\biggr)=F_{x,y}\biggl( \frac{t}{\alpha }\biggr) \quad (t>0) $$

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 \(\nu _{gx-Tx}(t)=\nu_{A-B}(t) \) 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

$$T_{i}x=(1-i)Tz+iTx. $$

We show that \(T_{i} \) is a proximal contraction. Let \(u,v,x,y\in A \) be such that

$$\nu_{u-T_{i}x}(t)=\nu_{A-B}(t)=\nu_{v-T_{i}y}(t)\quad (t>0). $$

Since T is an affine mapping, we have

$$\nu_{u-T((1-i)z+ix)}(t)=\nu_{A-B}(t)=\nu_{v-T((1-i)z+iy)}(t)\quad (t>0). $$

So by the hypothesis we have

$$\begin{aligned} \nu_{u-v}(t) & \geq\nu_{(1-i)z+ix-(1-i)z-iy}(t) \\ &= \nu_{i(x-y)}(t)=\nu_{x-y}\biggl(\frac{t}{i}\biggr)\quad (t>0). \end{aligned}$$

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

$$\nu_{u-Tx}(t)=\nu_{A-B}(t)=\nu_{v-Tz}(t)\quad (t>0). $$

Put \(y=iu+(1-i)v\in A \). Then

$$\begin{aligned} \nu_{y-T_{i}x}(t)&= \nu_{iu+(1-i)v-(1-i)Tz-iTx}(t) \\ &= \nu_{i(u-Tx)+(1-i)(v-Tz)}(t) \\ &\geq \Delta_{m}\bigl(\nu_{i(u-Tx)} (it ),\nu_{(1-i)(v-Tz)} \bigl((1-i)t \bigr)\bigr) \\ &= \Delta_{m}\bigl( \nu_{u-Tx}(t),\nu_{v-Tz}(t)\bigr) \\ &= \Delta_{m}\bigl(\nu_{A-B}(t),\nu_{A-B}(t)\bigr)= \nu_{A-B}(t) \quad (t>0), \end{aligned}$$

and thus \(T_{i}(A_{0})\subseteq B_{0} \). By Corollary 2.3 there exists a unique \(x_{i}\in A_{0} \) such that \(\nu_{gx_{i}-T_{i}x_{i}}(t)=\nu _{A-B}(t) \) for all \(t>0 \). Fix \(j\in(0,1) \). Then

$$\begin{aligned} \nu_{gx_{i}-Tx_{i}}(t) & \geq\Delta_{m}\bigl(\nu_{gx_{i}-T_{i}x_{i}}(jt), \nu _{T_{i}x_{i}-Tx_{i}}\bigl((1-j)t\bigr)\bigr) \\ & =\Delta_{m}\bigl(\nu_{A-B}(jt),\nu_{(1-i)(Tz-Tx_{i})} \bigl((1-j)t\bigr)\bigr) \\ &= \Delta_{m} \biggl(\nu_{A-B}(jt),\nu_{Tz-Tx_{i}} \biggl( \frac {(1-j)t}{1-i} \biggr) \biggr) \\ &\geq\Delta_{m} \biggl(\nu_{A-B}(jt),D_{B} \biggl( \frac{(1-j)t}{1-i} \biggr) \biggr)\quad (t>0). \end{aligned}$$

Now letting \(i\rightarrow1 \), we obtain

$$\lim_{i\rightarrow1}\nu_{gx_{i}-Tx_{i}}(t)\geq\Delta_{m} \bigl(\nu_{A-B}(jt),1\bigr)=\nu _{A-B}(jt)\quad \bigl(\forall j \in(0,1), t>0\bigr). $$

Then letting \(j\rightarrow1 \), we have

$$\lim_{i\rightarrow1}\nu_{gx_{i}-Tx_{i}}(t)=\nu_{A-B}(t) \quad (t>0). $$

So we can create a sequence \((x_{n})\) in \(A_{0}\) such that

$$\nu_{gx_{n}-Tx_{n}}(t)\rightarrow\nu_{A-B}(t) \quad (t>0). $$

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 \(g-T\) is a continuous mapping by Remark 1.8. Indeed, since \(\Delta_{m}\) is a continuous t-norm, \(p\rightarrow \nu_{P}\) is continuous ([27], Chapter 12), and we get

$$\nu_{gx -Tx }(t)=\lim_{k\rightarrow\infty}\nu _{gx_{n_{k}}-Tx_{n_{k}}}(t)= \nu_{A-B}(t), $$

as required. □

Theorem 2.5

Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a t-norm of H-type, and \((A,B) \) be a pair of subsets of X with the weak P-property 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 P-property, so we have the following result.

Corollary 2.6

Let \((X, F,\Delta) \) be a complete probabilistic Menger space such that Δ is a t-norm of H-type. Then every contraction self-mapping 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 p-star-shaped set, \(A_{0}\) be a nonempty compact set, B be a q-star-shaped set, and let \(\nu_{p - q}(t)=\nu_{A-B}(t)\) 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 \(\nu_{x - Tx}(t)=\nu_{A-B}(t)\) for all \(t>0\).

Proof

For each integer \(i\geq1 \), define \(T_{i}:A_{0}\rightarrow B_{0} \) by

$$T_{i}(x)= \biggl(1-\frac{1}{i} \biggr) Tx +\frac{1}{i}q \quad ( x\in A_{0}). $$

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

$$\nu_{u-T_{i}x}(t)=\nu_{v-T_{i}y}(t)=\nu_{A_{0}-B_{0}}(t)= \nu_{A-B}(t)=\nu _{s-Tx}(t)=\nu_{r-Ty}(t). $$

Now we define

$$u'= \biggl(1-\frac{1}{i} \biggr) s +\frac{1}{i}p\in A_{0},\qquad v'= \biggl(1-\frac{1}{i} \biggr) r + \frac{1}{i}p\in A_{0}, $$

so we have

$$\begin{aligned} \nu_{A-B}(t)&\geq\nu_{u'-T_{i}x}(t) =\nu_{ (1-\frac{1}{i} ) s +\frac{1}{i}p- (1-\frac{1}{i} ) Tx -\frac{1}{i}q}(t) \\ & =\nu_{ (1-\frac{1}{i} )(s-Tx)+\frac{1}{i}(p-q)}(t) \\ & \geq\Delta_{m} \biggl(\nu_{ (1-\frac{1}{i} )(s-Tx)} \biggl( t \biggl(1- \frac{1}{i} \biggr) \biggr) ,\nu_{\frac{1}{i}(p-q)} \biggl( t \biggl( \frac {1}{i} \biggr) \biggr) \biggr) \\ & =\Delta_{m}\bigl( \nu_{s-Tx}(t),\nu_{p-q}(t) \bigr) \\ & =\Delta_{m}\bigl(\nu_{A-B}(t),\nu_{A-B}(t)\bigr)= \nu_{A-B}(t). \end{aligned}$$

Hence, \(\nu_{u'-T_{i}x}(t)= \nu_{A-B}(t)\). Since \(\nu _{u-T_{i}x}(t)= \nu_{A-B}(t)\) 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

$$\begin{aligned} \nu_{u-v}(t)&=\nu_{u'-v'}(t) =\nu_{ (1-\frac{1}{i} )(s-r)}(t) \\ &= \nu_{s-r} \biggl(\frac{t}{1-\frac{1}{i}} \biggr)\geq\nu_{x-y} \biggl(\frac {t}{1-\frac{1}{i}} \biggr). \end{aligned}$$

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 \(\nu_{u_{i}-T_{i}u_{i}}(t)=\nu _{A_{0}-B_{0}}(t)= \nu_{A-B}(t)\). 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 \(\nu_{v_{i}-Tu_{i}}(t)= \nu _{A-B}(t)\). So we have

$$\begin{aligned} \nu_{A-B}(t)&\geq \nu_{ (1-\frac{1}{i} )v_{i} +\frac {1}{i}p-T_{i}u_{i}}(t) \\ &= \nu_{ (1-\frac{1}{i} ) v_{i} +\frac{1}{i}p- (1-\frac {1}{i} ) Tu_{i} -\frac{1}{i}q}(t) \\ &\geq\Delta_{m} \biggl(\nu_{(1-\frac{1}{i})( v_{i}- Tu_{i})}\biggl( t\biggl(1- \frac {1}{i}\biggr)\biggr) ,\nu_{\frac{1}{i}(p-q)}\biggl( t\biggl( \frac{1}{i}\biggr) \biggr) \biggr) \\ &=\Delta_{m}\bigl( \nu_{v_{i}- Tu_{i}}(t),\nu_{p-q}(t)\bigr) \\ &= \Delta_{m}\bigl(\nu_{A-B}(t), \nu_{A-B}(t) \bigr)=\nu_{A-B}(t). \end{aligned}$$

Thus, \(\nu_{A-B}(t)= \nu_{ (1-\frac{1}{i} )v_{i} +\frac {1}{i}p-T_{i}u_{i}}(t)\). Since \((A,B) \) is a semisharp proximinal pair and \(\nu_{A-B}(t)=\nu _{u_{i}-T_{i}u_{i}}(t) \), we have \(u_{i}=(1-\frac{1}{i}) v_{i} +\frac{1}{i}p\), and so

$$\nu_{u_{i}-v_{i}}(t)=\nu_{\frac{1}{i}(v_{i}-p)}(t)=\nu_{v_{i}-p}(it) . $$

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

$$\nu_{u_{i}-v_{i}}(t)=\nu_{v_{i}-p}(it)\geq\nu_{v_{i}-p}(jt)\geq \Delta _{m} \biggl(\nu_{v_{i}-z}\biggl(\frac{j}{2}t \biggr),\nu_{z-p}\biggl(\frac{j}{2}t\biggr)\biggr). $$

Letting \(i\rightarrow\infty\), we have

$$\lim_{i\rightarrow\infty}\nu_{u_{i}-v_{i}}(t)\geq\nu_{z-p} \biggl(\frac {j}{2}t\biggr) \quad (\forall j\geq1). $$

Now letting \(j \rightarrow\infty\), we have

$$\lim_{i\rightarrow\infty}\nu_{u_{i}-v_{i}}(t)\geq\lim_{j \rightarrow\infty} \nu_{z-p}\biggl(\frac{j}{2}t\biggr)=1. $$

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 \(\nu _{A-B}(t)=\nu_{u-Tx}(t)\). Since we know that \(\nu_{A-B}(t)=\nu _{v_{i}-Tu_{i}}(t)\) 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 \(\nu_{A-B}(t)=\nu _{x-Tx}(t) \), 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 P-property such that A is a p-star-shaped set, \(A_{0} \) be a nonempty compact set, B be a q-star-shaped set, and let \(\nu_{p - q}(t)=\nu_{A-B}(t)\) 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

$$G_{0} =\bigl\lbrace z\in G : \exists y\in B \textit{ s.t. } \forall t \geq0, F_{z,y}(t)=F_{G,B}(t)\bigr\rbrace . $$

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 p-star-shaped set, \(A_{0}\) be a nonempty compact set, B be a q-star-shaped set, and let \(\nu_{p - q}(t)=\nu_{A-B}(t)\) 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 \(\nu_{gx - Tx}(t)=\nu_{A-B}(t)\) 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 P-property 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 \(\nu _{gx-Tx}(t)=\nu_{A-B}(t) \) 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 \(F_{(x,x'),(y,y')}(t)=\frac{t}{t+|x-y|+|x'-y'|} \). 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

$$F_{g(0,x),g(0,y)}(t)=F_{(0,-x),(0,-y)}(t)=\frac{t}{t+|x-y|}=F_{(0,x),(0,y)}(t). $$

If \((0,u),(0,x), (0,v),(0,y)\in A \) are such that

$$\frac{t}{t+1+|u-\frac{x}{4}|} = F_{(0,u),T(0,x)}(t)=F_{A,B}(t)= F_{(0,v),T(0,y)}(t)=\frac{t}{t+1+|v-\frac{y}{4}|}, $$

then \(u= \frac{x}{4}\) and \(v= \frac{y}{4}\), so that

$$F_{(0,u),(0,v)}(t)=F_{(0,\frac{x}{4}),(0, \frac{y}{4})}(t)=\frac {t}{t+\frac{1}{4}|x-y|}=F_{(0,x),(0,y)} \biggl(\frac{t}{\frac{1}{4}} \biggr). $$

Therefore, all the hypothesis of Corollary 2.3 are satisfied, and we also have

$$F_{(0,0),T(0,0)}(t)=F_{(0,0),(1,0)}(t)=\frac{t}{t+1}=F_{A,B}(t). $$

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, \(\nu_{A-B}(t)=\frac{t}{t+1} \), \(A_{0}=\lbrace2\rbrace\), and \(B_{0}=\lbrace3\rbrace\). For every \(x\in A \), define \(T:A\rightarrow B \) by \(Tx=5-x \) 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

$$\nu_{2-T2}(t)=\nu_{2-3}(t)=\frac{t}{t+1}= \nu_{A-B}(t). $$

The following example shows that the weak P-property of the pair \((A, B) \) cannot be removed from Theorem 2.5.

Example 2.16

Let \(X =\mathbb{R} \), \(A =\lbrace-10,10\rbrace\), \(B = \lbrace-2, 2\rbrace\), and \(F_{p,q}(t)=\frac{t}{t+|p-q|} \). 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 P-property.

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 1-star-shaped set, B is 2-star-shaped set,

$$\nu_{A-B}(t)=\sup_{x\in A, y\in B}\nu_{x-y}(t)= \frac{t}{t+1},\qquad A_{0}=\lbrace1\rbrace,\qquad B_{0}= \lbrace2\rbrace, $$

and

$$\nu_{1-2}(t)=\frac{t}{t+|1-2|}=\frac{t}{t+1}= \nu_{A-B}(t). $$

Also, \((A,B) \) is a semisharp proximinal pair. Now for each \(x\in A \), define \(T:A\rightarrow B \) by \(Tx=3-x \). If \(u,v,x,y\in A \), then

$$\frac{t}{t+|u-3+x|} =\nu_{u-Tx}(t)=\nu_{A-B}(t)= \nu_{v-Ty}(t)=\frac {t}{t+|v-3+y|}, $$

so that \(u= x=1 \) and \(v= y=1 \). Thus,

$$\nu_{u-v}(t)=1=\nu_{x-y}(t). $$

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

$$\nu_{1-T1}(t)=\nu_{1-2}(t)=\frac{t}{t+1}= \nu_{A-B}(t). $$

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, \(\nu_{A-B}(t)= \frac{t}{t+1} \), B is not convex but is a \((0,1) \)-star-shaped set, and A is \((0,0)\)-star-shaped set. Clearly, \(A_{0}=A\) and \(B_{0}=B_{2} \). So

$$\nu_{(0,0)-(0,1)}(t)=\frac{t}{t+ |0|+|1|}=\frac{t}{t+1}= \nu_{A-B}(t), $$

and \((A,B) \) is a semisharp proximinal pair. Suppose that \(T:A\rightarrow B \) is defined by

$$ T(x,0)= \left\{ \textstyle\begin{array}{l@{\quad}l} (0,1), & x= 0, \\ (\sin x,1), & x\neq0, \end{array}\displaystyle \right. $$

and \((u,0),(v,0),(x,0),(y,0)\in A \) are such that

$$\nu_{(u,0)-T(x,0)}(t)=\nu_{A-B}(t)=\frac{t}{t+1}= \nu_{(v,0)-T(y,0)}(t). $$

If \(x=y=0 \), then \(u=v=0 \), and therefore

$$\nu_{(u,0)-(v,0)}(t)=\nu_{(0,0)-(0,0)}(t)=1=\nu_{(x,0)-(y,0)}(t). $$

If \(x,y\neq0\), then \(u=\sin x\), \(v=\sin y \), and therefore

$$\begin{aligned} \nu_{(u,0)-(v,0)}(t)&=\nu_{(\sin x,0)-( \sin y,0)}(t)=\frac{t}{t+|\sin x-\sin y|} \\ &\geq\frac{t}{t+| x- y|} \\ &=\nu_{(x,0)-(y,0)}(t). \end{aligned}$$

If \(x=0\) and \(y\neq0\), then \(u=0 \) and \(v=\sin y \), and therefore

$$\nu_{(u,0)-(v,0)}(t)=\nu_{(0,0)-(\sin y,0)}(t)=\frac{t}{t+|\sin y|}\geq \frac{t}{t+| y|}\geq\nu_{(0,0)-(y,0)}(t). $$

If \(x\neq0\) and \(y= 0\), then \(u=\sin x \) and \(v=0 \), and therefore

$$\nu_{(u,0)-(v,0)}(t)=\nu_{(\sin x,0)-(0,0)}(t)=\frac{t}{t+|\sin x|}\geq \frac{t}{t+| x|}\geq\nu_{(x,0)-(0,0)}(t). $$

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

$$\nu_{(0,0)-T(0,0)}(t)=\nu_{(0,0)-(0,1)}(t)=\frac{t}{t+1}= \nu_{A-B}(t). $$

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, \(\nu_{A-B}(t)=\frac{t}{t+\frac{7}{8}} \), the pair \((A,B)\) has the P-property, \(A_{0}=\lbrace1\rbrace\), 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

$$\nu_{Tx-Ty}(t)=\nu_{-\frac{1}{8}(x-y)}(t)=\nu_{x-y}(8t)\geq \nu_{x-y}(t). $$

Therefore, all the hypotheses of Corollary 2.12 are satisfied, and hence T has a best proximity point, and we also have

$$\nu_{1-T1}(t)=\nu_{1-\frac{15}{8}}(t)=\frac{t}{t+\frac{7}{8}}= \nu_{A-B}(t). $$