On Weak Probabilistic φ-Contractions in Menger Probabilistic Metric Spaces

Abstract

Two new concepts are introduced in this paper: ZH-set and weak probabilistic $\phi$-contraction. Firstly, a fixed-point theorem for weak probabilistic $\phi$-contraction is given in Menger probabilistic metric spaces, which generalizes and unifies several results in the existing literature. Then, a best proximity point theorem is obtained. Finally, two examples are provided to prove the effectiveness of our results.

1. Introduction

Fixed-point theory is a very important branch of modern mathematics. Since the publication of Banach fixed-point theorem, research on fixed-point theory has been extremely rich both in theory and in application. Banach first obtained fixed points on k-contraction in complete metric spaces. Since then, scholars have obtained abundant fixed-point theorems in different spaces with different contraction conditions.

Menger probabilistic metric spaces (abbreviated as MPM spaces), a core concept in the theory of probabilistic metric spaces, was proposed by Karl Menger in 1942 [1]. It aims to generalize the concept of traditional metric spaces to a probabilistic framework, enabling a more flexible description of distance relationships in uncertain environments.

Sehgal provided the Banach-type fixed point theorem in MPM spaces for the first time [2]. Since then, various researchers have delved into probabilistic $\phi$-contractions and derived several fixed-point theorems in MPM spaces. However, due to the complexity of MPM space, strong conditions are often attached in order to obtain fixed points, one of which is that $\phi$ fulfilled ${\sum }_{n=1}^{\infty }{\phi }^n\left(s\right)<\infty$ for all $s>0$. Weakening the additional conditions in probabilistic $\phi$-contraction, or more precisely reducing the requirements for the $\phi$ function, has become a direction of researchers’ efforts.

$\acute{\mathrm{C}}$iri$\acute{\mathrm{c}}$ [3] positively attempted to modify the assumptions regarding the function $\phi$. $\acute{\mathrm{C}}$iri$\acute{\mathrm{c}}$ obtained a fixed-point theorem under the condition: $\phi \left(0\right)=0,\phi \left(s\right)<s$ and ${lim}_{z\to s^{+}}sup\phi \left(z\right)<s$ for all $s>0$.

Later, Jachymski [4] obtained the same result under the condition: $0<\phi \left(s\right)<s$ and ${lim}_{n\to \infty }{\phi }^n\left(s\right)=0$ for all $s>0$, where ${\phi }^n\left(s\right)$ denotes the nth iteration of $\phi$.

Fang [5] relaxed restrictions on the function $\phi$ as well. Fang [5] obtained the fixed-point result under the condition that $\varphi$ satisfied the following: for $s>0$ there exists $z\ge s$, such that ${lim}_{n\to \infty }{\varphi }^n\left(z\right)=0$ holds.

In [6], Choudhury and Das discussed a new kind of contraction, and the functions they used were as follows $\varphi :[0,\infty )\to [0,\infty )$, which is a strictly increasing function satisfying the following conditions: it is left-continuous at every point in its domain, in particular, it is continuous at 0; its minimum and maximum function values can be taken to 0 and infinity, respectively. The concept of control functions has paved the way for establishing novel fixed point outcomes within MPM spaces. Literatures that utilize the $\varphi$-function include [7,8,9,10]. In the aftermath, a number of alternative control functions have been employed in issues concerning fixed points within PM spaces, with examples such as [3,5,11].

Mihet and Zaharia [12] investigated the existence of fixed points for several classes of probabilistic $\phi$-contractions under Fang-type conditions. They also raised several open questions, one of which is whether the result of Jachymski is equivalent to that of Fang?

Alegre and Romaguera [13], and Gregori et al. [14] proved the above equivalence using different methods. In their work, they used sets

$$A=\{s>0|\underset{n\to \infty }{lim}{\phi }^n\left(s\right)=0\}$$

or

$$A^{\prime }=\{s>0|\underset{n\to \infty }{lim}{\phi }^n\left(s\right)\ne 0\},$$

where $A^{\prime }$ is the complement of set A. It can be seen from their work that set A plays an important role.

Taking inspiration from set A, we propose a new concept in this paper, called ZH-set, which plays a vital role in the definition of weak probabilistic $\phi$-contraction.

We notice that a probabilistic $\phi$-contraction involves the following three aspects:

(i)

Two sets: X and $[0,\infty )$;

(ii)

Two mappings: $T:X\to X$ and $\phi :[0,\infty )\to [0,\infty )$;

(iii)

The bridge to connect the above two sets and two mappings:

$$F_{Tw,Tv}\left(\phi \left(s\right)\right)\ge F_{w,v}\left(s\right)\mathrm{for}\mathrm{all}w,v\in X\mathrm{and}s\in (0,\infty ).$$

In this paper, we introduce a “small” set, which we call ZH-set. The reason we say it is a “small” set is because sometimes the elements contained in the set are relatively little, and sometimes its Lebesgue measure may be 0.

We use the ZH-set—the set P—to replace $[0,\infty )$, and then define a weak probabilstic $\phi$-contraction, which involves the following three aspects:

(i${)}^{\prime }$

Two sets: X and P;

(ii${)}^{\prime }$

Two mappings: $T:X\to X$ and $\phi :P\to P$;

(iii${)}^{\prime }$

The bridge to connect the above two sets and two mappings:

$$F_{Tw,Tv}\left(\phi \left(s\right)\right)\ge F_{w,v}\left(s\right)\mathrm{for}\mathrm{all}w,v\in X\mathrm{and}s\in P-\left\{0\right\}.$$

Because P is relatively small, the mapping $\phi :P\to P$ is relatively simple. Thus, the relational formula $F_{Tw,Tv}\left(\phi \left(s\right)\right)\ge F_{w,v}\left(s\right)$ which is difficult to be verified and satisfied in the whole scope is relatively easy to be verified and satisfied. In fact, in this paper, the condition “$0<\phi \left(s\right)<s$ and ${lim}_{n\to \infty }{\phi }^n\left(s\right)=0$ for all $s>0$” can be weakened to the condition “${lim}_{n\to \infty }{\phi }^n\left(s\right)=0$ for $s\in P-\left\{0\right\}$, where $P\subset [0,\infty )$ is a ZH-set.”

2. Preliminaries

Throughout this paper, let $R=(-\infty ,+\infty )$. We define $D_{+}$ as the space of all mappings $F:R\to [0,1]$, such that F is left-continuous, non-decreasing on R, satisfies $F\left(0\right)=0$, and $F(+\infty )={lim}_{w\to +\infty }F\left(w\right)=1$.

A triangular norm (for a short t-norm) is a binary operation $\diamond :[0,1]\times [0,1]\to [0,1]$, which is commutative, associative, and monotone and has 1 as the unit element. Basic examples are as follows:

• The Łukasiewicz t-norm: ${\diamond }_L(a,b)=max(a+b-1,0)$
• The product t-norm: ${\diamond }_P(a,b)=ab$
• The minimum t-norm: ${\diamond }_M(a,b)=min\{a,b\}$

The minimum t-norm ${\diamond }_M$ is the strongest t-norm, that is, ${\diamond }_M(w,v)\ge \diamond (w,v)$ for each $w,v\in [0,1]$ and each t-norm ⋄.

If ⋄ is a t-norm, then ${\diamond }^n\left(s\right)$ is defined for every $s\in [0,1]$ and $n\in N$ as 1 if $n=0$ and ${\diamond }^n\left(s\right)=\diamond ({\diamond }^{n-1}\left(s\right),s)$ if $n\ge 1$. A t-norm ⋄ is said to be of H-type if the family $\{{\diamond }^n\left(s\right):n\in N\}$ is equicontinuous at $s=1$. Obviously, the minimum ${\diamond }_M$ is a t-norm of H-type, in addition, there are a large number of t-norms of H-type (see [15] for details).

Definition 1

([15]). A Menger probabalistic metric space (MPM space) is a triple $(M,F,\diamond )$, where M is a nonempty set, ⋄ is a t-norm, and F is a mapping from $M\times M$ to $D_{+}$, where $\left(F\right(w,v)$ is denoted by $F_{w,v})$, satisfying the following conditions:

(PM1) $F_{w,v}\left(s\right)=1$ for all $s>0$ if $w=v$,

(PM2) $F_{w,v}\left(s\right)=F_{v,w}\left(s\right)$ for all $w,v\in M$,

(PM3) $F_{w,u}(s+t)\ge \diamond (F_{w,v}\left(s\right),F_{v,u}\left(t\right))$ for $w,v,u\in M,s,t\ge 0$.

Definition 2

([15]). Let $(M,F,\diamond )$ be an MPM space.

(1) 

A sequence $\left\{w_n\right\}\subset M$ converges to some point $w\in M$, if and only if ${lim}_{n\to \infty }{F}_{w_n,w}\left(s\right)=1$ for all $s>0$.

(2) 

A sequence $\left\{w_n\right\}\subset M$ is called Cauchy if and only if ${lim}_{n,m\to \infty }{F}_{w_n,w_m}\left(s\right)=1$ for all $s>0$.

(3) 

An MPM space $(M,F,\diamond )$ is said to be complete if every Cauchy sequence in M is convergent.

Definition 3

([15]). Let $(M,F,\diamond )$ be an MPM space. $T:M\to M$ is called a probabilistic φ-contraction if it satisfies

$$F_{Tw,Tv}\left(\phi \left(s\right)\right)\ge F_{w,v}\left(s\right)$$

for all $w,v\in M$ and $s>0,$ where $\phi :[0,+\infty )\to [0,+\infty )$ is a function satisfying certain conditions.

Definition 4

([16]). The distance of a point $w\in M$ from a non-empty set V for $s>0$ is defined as

$$F_{w,V}\left(s\right)=\underset{v\in V}{sup}{F}_{w,v}\left(s\right),$$

and the distance between two non-empty sets W and V for $s>0$ is defined as

$$F_{W,V}\left(s\right)=sup\{F_{a,b}\left(s\right):w\in W,v\in V\}.$$

Definition 5.

Let $(W,V)$ be a pair of non-empty subsets of an MPM space $(M,F,\diamond )$. Let P be a ZH-set. Then, the pair $(W,V)$ is said to have the S-property if and only if for $w_1,w_2\in W$ and $v_1,v_2\in V$, satisfying

$$F_{w_1,v_1}\left(s\right)=F_{W,V}\left(s\right)\mathit{and}{F}_{w_2,v_2}\left(s\right)=F_{W,V}\left(s\right)\mathit{for}\mathit{all}s\in P,$$

implies that $F_{w_1,w_2}\left(s\right)=F_{v_1,v_2}\left(s\right)$ for all $s\in P$.

3. Main Results

Definition 6.

A set $P\subset [0,+\infty )$ is said to be a ZH-set if for each $n\in N$ there exists $p_1,p_2\in P$, such that $p_1\ge n$ and $p_2\le 1/n$. Equivalently, P contains a pair of sequences, say ${\left\{w_n\right\}}_{n\in N}$ and ${\left\{v_n\right\}}_{n\in N}$, such that ${lim}_{n\to \infty }{w}_n=0$ and ${lim}_{n\to \infty }{v}_n=\infty$.

Example 1.

Let $P=[0,+\infty )$, then P is a ZH-set.

Example 2.

Let $P_1=N\bigcup \left\{0\right\}$, $P_2=N\bigcup \{{\displaystyle \frac{1}{n}}:n\in N\}$, $P_3=\{m+{\displaystyle \frac{1}{n}}:m,n\in N\}$, then $P_1,P_2,P_3$ are ZH-sets. It is obvious that $P_1,P_2,P_3$ are countable and $u\left(P_i\right)=0$ for $i=1,2,3$, where u is the Lebesgue measure.

Definition 7.

A function $\phi :P\to P$ is said to be a weak gauge function if $P\subset [0,+\infty )$ is a ZH-set and ${lim}_{n\to \infty }{\phi }^n\left(p\right)=0$ for each $p\in P$.

Example 3.

Let $P=P_1=N\bigcup \left\{0\right\}$ and $\phi :P\to P$ be defined by

$$\phi \left(p\right)=\left\{\begin{array}{cc}p-1,\hfill & if p>1\hfill \\ 0,\hfill & if p=1\hfill \end{array},\right.$$

then φ is a weak gauge function.

Example 4.

Let $P=P_2=N\bigcup \{{\displaystyle \frac{1}{n}}:n\in N\}$ and $\phi :P\to P$ defined by

$$\phi \left(p\right)=\left\{\begin{array}{cc}n-1,\hfill & if p=n\ge 2\hfill \\ {\displaystyle \frac{1}{n+1}},\hfill & if p={\displaystyle \frac{1}{n}}\hfill \end{array},\right.$$

then φ is a weak gauge function.

Denote $\mathrm{\Psi }\left(P\right)$ as the class of weak gauge functions for a ZH-set P.

Definition 8.

Let $(M,F,\diamond )$ be an MPM space. A mapping $T:M\to M$ is called a weak probabilistic φ-contraction if there exists a ZH-set P and $\phi \in \mathrm{\Psi }\left(P\right)$, such that

$$F_{Tw,Tv}\left(\phi \left(s\right)\right)\ge F_{w,v}\left(s\right)$$

(1)

for $w,v\in M$ and $s\in P-\left\{0\right\}$.

The following lemma plays a crucial role in the proof of our main results.

Lemma 1.

Let $(M,F,\diamond )$ be an MPM space and $T:M\to M$ be a weak probabilistic φ-contraction for $\phi \in \mathrm{\Psi }\left(P\right)$, where P is a ZH-set. Then, for each $s>0$ and $s_0\in P-\left\{0\right\}$, there exists $n_0\in N$, such that

(i) 

${\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)<s$;

(ii) 

${\phi }^n\left(s_0\right)<s$ as $n\ge n_0$.

Proof. 

Firstly, we show that for $s_0\in P-\left\{0\right\}$ and $n\in N$, ${\phi }^n\left(s_0\right)>0$.

Conversely, suppose that there exists $n\in N$, such that ${lim}_{n\to \infty }{\phi }^n\left(s_0\right)=0$, then

$$0=F_{T^{n}w,T^{n}w}\left(0\right)=F_{T^{n}w,T^{n}w}\left({\phi }^n\left(s_0\right)\right)\ge F_{T^{n-1}w,T^{n-1}w}\left({\phi }^{n-1}\left(s_0\right)\right)\ge \cdots \ge F_{w,w}\left(s_0\right)=1,$$

which is a contradiction.

Secondly, since ${lim}_{n\to \infty }{\phi }^n\left(s_0\right)=0$, then for each $s>0$, there exists $n_1\in N$, such that when $n\ge n_1$, ${\phi }^n\left(s_0\right)<s$.

We claim there exists $n_0\ge n_1$, such that ${\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)$.

Suppose that for $n\ge n_1$, ${\phi }^{n+1}\left(s_0\right)\ge {\phi }^n\left(s_0\right)$. Then, we have

$$\underset{n\to \infty }{lim}{\phi }^n\left(s_0\right)\ge {\phi }^{n_1}\left(s_0\right)>0,$$

which is a contradiction. Thus, for each $s>0$ and $s_0\in P-\left\{0\right\}$, there exists $n_0\in N$, such that

(i)

${\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)<s$;

(ii)

${\phi }^n\left(s_0\right)<s$ as $n\ge n_0$.

□

Theorem 1.

Let $(M,F,\diamond )$ be a complete MPM-space with ⋄ of H-type and $T:M\to M$ is a weak probabilistic φ-contraction for some ZH-set P with $\phi \in \mathrm{\Psi }\left(P\right)$. Then, T is a Picard mapping.

Proof. 

Let $w_0\in M$ be an arbitrary point, we define the sequence $\left\{w_n\right\}$ in M by $w_n=T^n{w}_0$ for all $n\in N$. If $w_{n+1}=w_n$ for some $n\in N$, then T has a fixed point. Therefore, in the following proof, we can suppose $w_{n+1}\ne w_n$ for each $n\in N$.

We complete the proof by the following five steps.

Step 1. We show that for $\forall s>0$,

$$\underset{n\to \infty }{lim}{F}_{w_n,w_{n+1}}\left(s\right)=1.$$

(2)

Fix $s>0$ and let $ϵ\in (0,1]$. Since ${lim}_{s\to \infty }{F}_{w_0,w_1}\left(s\right)=1,$ there exists $s_0\in P$, such that $F_{w_0,w_1}\left(s_0\right)>1-ϵ.$

For $s>0$ and $s_0\in P$, by Lemma 1, there exists $n_0\in N$, such that when $n\ge n_0$, ${\phi }^n\left(s_0\right)<s$. Then

$$F_{w_n,w_{n+1}}\left(s\right)\ge F_{w_n,w_{n+1}}\left({\phi }^n\left(s_0\right)\right)\ge \cdots \ge F_{w_0,w_1}\left(s_0\right)>1-ϵ,$$

thus we have ${lim}_{n\to \infty }{F}_{w_n,w_{n+1}}\left(s\right)=1$.

Step 2.

$$F_{w_n,w_{n+k}}\left({\phi }^{n_0}\left(s_0\right)\right)\ge {\diamond }^{k-1}\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(t_0\right))\right)$$

(3)

for $k\in N$, where $s_0\in P,{\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)$.

It is obvious for $k=1$, since

$$F_{w_n,w_{n+1}}\left({\phi }^{n_0}\left(s_0\right)\right)\ge F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))={\diamond }^0\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))\right).$$

Assume that (3) holds for some k, then

$$\begin{array}{cc}\hfill F_{w_n,w_{n+k+1}}\left({\phi }^{n_0}\left(s_0\right)\right)& =F_{w_n,w_{n+k+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right)+{\phi }^{n_0+1}\left(s_0\right))\hfill \\ \hfill & \ge \diamond (F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right)),F_{w_{n+1},w_{n+k+1}}\left({\phi }^{n_0+1}\left(s_0\right)\right))\hfill \\ \hfill & \ge \diamond (F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right)),F_{w_n,w_{n+k}}\left({\phi }^{n_0}\left(s_0\right)\right))\hfill \\ \hfill & \ge \diamond (F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right)),{\diamond }^{k-1}\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))\right)\hfill \\ \hfill & ={\diamond }^k\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))\right).\hfill \end{array}$$

Step 3. $\left\{w_n\right\}$ is a Cauchy sequence in M.

For $s>0$, as proved in Lemma 1, we can find $s_0\in P$ and $n_0$, such that,

$${\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)<s.$$

Then,

$$F_{w_n,w_{n+k}}\left(s\right)\ge F_{w_n,w_{n+k}}\left({\phi }^{n_0}\left(s_0\right)\right)\ge {\diamond }^{k-1}\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))\right).$$

Let $ϵ\in (0,1]$, since $\left\{{\diamond }^n\left(s\right)\right\}$ is equicontinuous at $s=1$ and ${\diamond }^n\left(1\right)=1$, so there exists $\delta >0$, such that

$${\diamond }^n\left(s\right)>1-ϵ for all s\in (1-\delta ,1]and n\in N.$$

(4)

By step 1, we have that

$$\underset{n\to \infty }{lim}{F}_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))=1.$$

Thus, there exists $n_1>n_0$, such that

$$F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s\right)-{\phi }^{n_0+1}\left(s\right))>1-\delta$$

for all $n>n_1$. It follows from (4) that

$$F_{w_n,w_{n+k}}\left(s\right)\ge {\diamond }^{k-1}\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s\right)-{\phi }^{n_0+1}\left(s\right))\right)>1-ϵ$$

for all $n>n_1$ and $k\in N$.

Thus, $\left\{w_n\right\}$ is a Cauchy sequence in M.

Step 4. T has a fixed point.

Since M is complete and $\left\{w_n\right\}$ is a Cauchy sequence in M, there exists a $w^{*}\in M$, such that ${lim}_{n\to \infty }{F}_{w_n,w^{*}}\left(s\right)=1$ for $\forall s>0.$ For $s>0$, as proved in Lemma 1, we can find $s_0\in P$ and $n_0$, such that

$${\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)<s.$$

Then

$$\begin{array}{cc}\hfill F_{w^{*},T{w}^{*}}\left(s\right)& \ge F_{w^{*},T{w}^{*}}\left({\phi }^{n_0}\left(s_0\right)\right)\hfill \\ \hfill & \ge \diamond (F_{w^{*},w_{n+1}}\left({\phi }^{n_0}\left(s_0\right)\right)-{\phi }^{n_0+1}\left(s_0\right)),F_{w_{n+1},T{w}^{*}}\left({\phi }^{n_0+1}\left(s_0\right)\right))\hfill \\ \hfill & \ge \diamond (F_{w^{*},w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right)),F_{w_n,w^{*}}\left({\phi }^{n_0}\left(s_0\right)\right))\hfill \\ \hfill & \ge \diamond (b_n,b_n),\hfill \end{array}$$

where $b_n=\min \{F_{w^{*},w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right)),F_{w_n,w^{*}}\left({\phi }^{n_0}\left(s_0\right)\right)\}$.

Since ${lim}_{n\to \infty }{b}_n=1$ and ⋄ is continuous, passing $n\to \infty$, we obtained $F_{w^{*},T{w}^{*}}\left(s\right)=1$ for all $s>0$.

Thus, $T{w}^{*}=w^{*}$.

Step 5. T has at most one fixed point.

Suppose, on the contrary, that there exists another fixed point $v^{*}$ of T, such that $T{w}^{*}=w^{*}\ne T{v}^{*}=v^{*}$. Then, there exists $s_1>0$ such that $F_{w^{*},v^{*}}\left(s_1\right)<1.$ Since ${lim}_{t\to \infty }{F}_{w^{*},v^{*}}\left(s\right)=1$, there exists $s_0\in P$ and $s_0>s_1$, such that

$$F_{w^{*},v^{*}}\left(s_0\right)>F_{w^{*},v^{*}}\left(s_1\right).$$

By Lemma 1, there exists $n_0\in N$, such that ${\phi }^{n_0}\left(s_0\right)<s_1$.

Then, we have

$$F_{w^{*},v^{*}}\left(s_1\right)\ge F_{w^{*},v^{*}}\left({\phi }^{n_0}\left(s_0\right)\right)\ge F_{w^{*},v^{*}}\left(s_0\right)>F_{w^{*},v^{*}}\left(s_1\right),$$

which is a contradiction. Therefore, the fixed point of T is unique. □

Next, we will demonstrate the generality and validity of our results. First of all, we prove that many existing results can be obtained by our theorem.

Theorem 2

([4]). Let $(M,F,\diamond )$ be a complete MPM space with ⋄ of H-type and $\phi :[0,\infty )\to [0,\infty )$ be a function such that $0<\phi \left(s\right)<s$ and ${lim}_{n\to \infty }{\phi }^n\left(s\right)=0$ for all $s>0$. If T is a probabilistic φ-contraction, then T is a Picard mapping.

Proof. 

Let P = $[0,\infty )$, It clearly satisfies all the conditions in Theorem 1. □

Theorem 3

([5]). Let $(M,F,\diamond )$ be a complete MPM space with ⋄ of H-type and $\varphi :[0,\infty )\to [0,\infty )$ satisfying: for $s>0$ there exists $z\ge s$, such that

$$\underset{n\to \infty }{lim}{\varphi }^n\left(z\right)=0$$

(5)

holds. If T is a probabilistic ϕ-contraction, then T is a Picard mapping.

Proof. 

Let $P=\{r:{lim}_{n\to \infty }{\varphi }^n\left(r\right)=0\}$, then P is a ZH-set by the property (5) of $\varphi$.

Define $\phi :P\to P$ by $\phi \left(p\right)=\varphi \left(p\right)$ for $p\in P$, then $\phi$ is well-defined and $\phi \in \mathrm{\Psi }\left(P\right)$. In fact, for $p\in P$, by the definition of P, ${lim}_{n\to \infty }{\varphi }^n\left(p\right)=0$, then

$$\underset{n\to \infty }{lim}{\varphi }^n\left(\phi \left(p\right)\right)=\underset{n\to \infty }{lim}{\varphi }^n\left(\varphi \left(p\right)\right)=\underset{n\to \infty }{lim}{\varphi }^{n+1}\left(p\right)=0.$$

Thus, $\phi \left(p\right)\in P$,

$${\phi }^2\left(p\right)=\phi \left(\phi \left(p\right)\right)=\varphi \left(\phi \left(p\right)\right)=\varphi \left(\varphi \left(p\right)\right)={\varphi }^2\left(p\right).$$

We can obtain ${\phi }^n\left(p\right)={\varphi }^n\left(p\right)$ for $n\in N$ in a similar way.

So, ${lim}_{n\to \infty }{\phi }^n\left(p\right)={lim}_{n\to \infty }{\varphi }^n\left(p\right)=0$. Thus, $\phi \in \mathrm{\Psi }\left(P\right)$.

Since $F_{Tw,Tv}\left(\varphi \left(s\right)\right)\ge F_{w,v}\left(s\right)$ for all $s>0$, in particular,

$$F_{Tw,Tv}\left(\varphi \left(s\right)\right)\ge F_{w,v}\left(s\right)\mathrm{for}\mathrm{all}s\in P-\left\{0\right\}.$$

That is $F_{Tw,Tv}\left(\phi \left(s\right)\right)\ge F_{w,v}\left(s\right)$ for all $s\in P-\left\{0\right\}$. So, T is a weak probabilistic $\phi$-contraction. Therefore, we can apply Theorem 1 and thus T is a Picard mapping. □

In [6], Choudhury and Das defined a function $\varphi$ as follows:

Definition 9

([6]). A function $\varphi :[0,\infty )\to [0,\infty )$ is said to be a type of special function if it satisfies the following conditions:

(i) 

$\varphi \left(t\right)=0$ if and only if $t=0$.

(ii) 

ϕ is strictly increasing and $\varphi \left(t\right)\to \infty$ as $t\to \infty$.

(iii) 

ϕ is left-continuous in $(0,\infty )$.

(iv) 

ϕ is continuous at 0.

The main result in [6] is described as follows: Let $(M,F,{\diamond }_M)$ be a complete MPM space with ${\diamond }_M=\min \{a,b\}$ and $T:M\to M$ satisfies the following conditions:

$$F_{Tw,Tv}\left(\varphi \left(cs\right)\right)\ge F_{w,v}\left(\varphi \left(s\right)\right)$$

for $w,v\in M$ and $s>0$, $c\in (0,1)$, where $\varphi :[0,\infty )\to [0,\infty )$ is a special function satisfying Definition 9. Then, T has a unique fixed point.

At the end of the paper [6], Choudhury and Das put forward an open question: Is the above result true for other t-norms? We provide a more general theorem than the above result by Theorem 1, thus providing at least a partial answer to this question.

Theorem 4.

Suppose $(M,F,\diamond )$ is a complete MPM-space, ⋄ is a t-norm of H-type. Suppose that $T:M\to M$ satisfies:

$$F_{Tw,Tv}\left(\varphi \left(cs\right)\right)\ge F_{w,v}\left(\varphi \left(s\right)\right)$$

for $w,v\in M$, $s>0$, where ϕ is as described in Definition 9, $c\in (0,1)$, then, T is a Picard mapping.

Proof. 

Let $P=\left\{\varphi \right(s):s\in [0,+\infty \left)\right\}$, then P is a ZH-set by the properties of $\varphi$ in Definition 9.

Define $\phi :P\to P$ by $\phi \left(p\right)=\varphi \left(c{\varphi }^{-1}\left(p\right)\right)$ for $p\in P$, then $\phi$ is well-defined and $\phi \in \mathrm{\Psi }\left(P\right)$. In fact, for $p\in P$, by the definition of P, $\phi \left(p\right)\in P$,

$${\phi }^2\left(p\right)=\phi \left(\phi \left(p\right)\right)=\varphi \left(c{\varphi }^{-1}\left(\phi \left(p\right)\right)\right)=\varphi \left(c{\varphi }^{-1}\left(\varphi \left(c{\varphi }^{-1}\left(p\right)\right)\right)\right)=\varphi \left(c^2{\varphi }^{-1}\left(p\right)\right).$$

We can obtain ${\phi }^n\left(p\right)=\varphi \left(c^n{\varphi }^{-1}\left(p\right)\right)$ for $n\in N$ in a similar way.

By the properties of $\varphi$, ${lim}_{n\to \infty }\varphi \left(c^n{\varphi }^{-1}\left(p\right)\right)=0$. So,

$$\underset{n\to \infty }{lim}{\phi }^n\left(p\right)=\underset{n\to \infty }{lim}{\varphi }^n\left(p\right)=0.$$

Thus, $\phi \in \mathrm{\Psi }\left(P\right)$.

Since $F_{Tw,Tv}\left(\varphi \left(cs\right)\right)\ge F_{w,v}\left(\varphi \left(s\right)\right)$ for all $s>0$ is equivalent to $F_{Tw,Tv}(\varphi \left(c{\varphi }^{-1}\left(t\right)\right)\ge F_{w,v}\left(t\right)$ for $t=\varphi \left(s\right)\in P$, that is

$$F_{Tw,Tv}\left(\phi \left(t\right)\right)\ge F_{w,v}\left(t\right)\mathrm{for}\mathrm{all}t\in P-\left\{0\right\}.$$

So, T is a weak probabilistic $\phi$-contraction. Therefore, we can apply Theorem 1 and thus T is a Picard mapping. □

Example 5.

Let $M=[0,\infty )$ and define $F:M\times M\to D_{+}$ as follows:

$$F(w,v)\left(s\right)=F_{w,v}\left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{s}{s+\sigma (w,v)}},\hfill & \mathit{if}\sigma (w,v)\ge s,\hfill \\ 1,\hfill & \mathit{if}\sigma (w,v)<s.\hfill \end{array}\right.$$

(6)

Then $(M,F,{\diamond }_M)$ is a complete MPM space (see Example in [5]), where $\sigma (w,v)=|w-v|$.

Let $T:M\to M$ be defined by $Tw={\displaystyle \frac{w}{1+w}}$. Firstly, we show that T is not the Sehgal contraction. Suppose, on the contrary, that there exists $k\in (0,1)$, such that $F_{Tw,Tv}\left(ks\right)\ge F_{w,v}\left(s\right)$ for all $w,v\in M$ and $s>0$.

Let $w={\displaystyle \frac{1}{n}},v={\displaystyle \frac{1}{2n}},s={\displaystyle \frac{1}{4n}}$, then when $n>1$

$$\sigma (w,v)={\displaystyle \frac{1}{2n}}>{\displaystyle \frac{1}{4n}}=s,\sigma (Tw,Tv)={\displaystyle \frac{{\displaystyle \frac{1}{2n}}}{(1+{\displaystyle \frac{1}{n}})(1+{\displaystyle \frac{1}{2n}})}}>{\displaystyle \frac{k}{4n}}=ks.$$

By (6), we have

$$F_{w,v}\left(s\right)={\displaystyle \frac{s}{s+\sigma (w,v)}}={\displaystyle \frac{{\displaystyle \frac{1}{4n}}}{{\displaystyle \frac{1}{4n}}+{\displaystyle \frac{1}{2n}}}}$$

$$F_{Tw,Tv}\left(ks\right))={\displaystyle \frac{ks}{ks+\sigma (Tw,Tv)}}={\displaystyle \frac{{\displaystyle \frac{k}{4n}}}{{\displaystyle \frac{k}{4n}}+{\displaystyle \frac{{\displaystyle \frac{1}{2n}}}{(1+{\displaystyle \frac{1}{n}})(1+{\displaystyle \frac{1}{2n}})}}}}.$$

Since $F_{Tw,Tv}\left(ks\right)\ge F_{w,v}\left(s\right)$, we have

$${\displaystyle \frac{{\displaystyle \frac{k}{4n}}}{{\displaystyle \frac{k}{4n}}+{\displaystyle \frac{{\displaystyle \frac{1}{2n}}}{(1+{\displaystyle \frac{1}{n}})(1+{\displaystyle \frac{1}{2n}})}}}}\ge {\displaystyle \frac{{\displaystyle \frac{1}{4n}}}{{\displaystyle \frac{1}{4n}}+{\displaystyle \frac{1}{2n}}}}.$$

After a simple calculation, we obtain

$$k\ge {\displaystyle \frac{1}{(1+{\displaystyle \frac{1}{n}})(1+{\displaystyle \frac{1}{2n}})}}.$$

Passing limit as $n\to \infty$, we have $k\ge 1$, which is a contradiction. Thus, T is not the Sehgal contraction.

Let $P=[0,+\infty )\bigcap Q$, where Q is the set of all rational numbers. Let $\phi :P\to P$ be defined by

$$\phi \left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{s}{1+s}},\hfill & s\in [0,1)\bigcap Q,\hfill \\ {\displaystyle \frac{13}{6}},\hfill & s\in [1,2]\bigcap Q,\hfill \\ s-{\displaystyle \frac{4}{3}},\hfill & s\in [2,+\infty )\bigcap Q.\hfill \end{array}\right.$$

(7)

We now show that T is a weak probabilistic φ-contraction, i.e., T satisfies (1).

Let $s\in P$. If $\sigma (Tw,Tv)=|Tw-Tv|<\phi \left(s\right)$, then we have $F_{Tw,Tv}\left(\phi \left(s\right)\right)=1\ge F_{w,v}\left(s\right)$, (1) holds. Suppose that $\sigma (Tw,Tv)=|Tw-Tv|\ge \phi \left(s\right)$. From (7), it is easy to see that

$$\phi \left(s\right)\ge {\displaystyle \frac{s}{1+s}}\mathit{for}\mathit{all}s\in [0,+\infty )\bigcap Q,$$

thus $\sigma (Tw,Tv)\ge {\displaystyle \frac{s}{1+s}}$. Note that

$$\sigma (Tw,Tv)={\displaystyle \frac{|w-v|}{1+w+v+wv}}\le {\displaystyle \frac{\sigma (w,v)}{1+\sigma (w,v)}}.$$

We obtain ${\displaystyle \frac{s}{1+s}}\le {\displaystyle \frac{\sigma (w,v)}{1+\sigma (w,v)}}$, which implies that $\sigma (w,v)\ge s$ since the function $h\left(t\right)={\displaystyle \frac{t}{1+t}}$ is strictly increasing on $[0,\infty )$. By (6), we have $F_{w,v}\left(s\right)={\displaystyle \frac{s}{s+\sigma (w,v)}}$. Thus,

$$F_{Tw,Tv}\left(\phi \left(s\right)\right)={\displaystyle \frac{\phi \left(s\right)}{\phi \left(s\right)+\sigma (Tw,Tv)}}\ge {\displaystyle \frac{{\displaystyle \frac{s}{1+s}}}{{\displaystyle \frac{s}{1+s}}+{\displaystyle \frac{\sigma (w,v)}{1+\sigma (w,v)}}}}\ge {\displaystyle \frac{s}{s+\sigma (w,v)}}=F_{w,v}\left(s\right),$$

i.e., (1) holds. This shows that all the conditions of Theorem 1 are satisfied. By Theorem 1, we conclude that T has a unique fixed point $w^{*}$ in M. Indeed, $w=0$ is a unique fixed point of T. However, Theorem 2 in [4] cannot be applied to this example because the φ defined by (7) does not meet the conditions $\phi \left(s\right)<s$ for all $s>0$ since $\phi \left(1\right)={\displaystyle \frac{13}{6}}>1$.

Lemma 2

([17]). Let W and V be two non-empty subsets of a PM-space $(M,F,\diamond )$, then $F_{W,V}\left(s\right)$ is a distribution function.

Then, we define a novel weak contraction.

Definition 10.

Let $(M,F,\diamond )$ be a MPM-space and $W,V$ be two disjoint non-empty subsets of M. Let P be a ZH-set and $\varphi \in \mathrm{\Psi }\left(P\right)$. A non-self mapping $T:W\to V$ is called weak ϕ-proximal contraction if

$$F_{Tw,Tv}\left(\varphi \left(t\right)\right)\ge F_{w,v}\left(t\right),$$

(8)

where $w,v\in W$, $s\in P$.

Definition 11.

Let W and V be two non-empty subsets of PM-space $(M,F,\diamond )$. Let P be a ZH-set. We define $W_P$ and $V_P$ as follows:

$$W_P=\left\{w\in W:\exists v\in V\mathit{such}\mathit{that}{F}_{w,v}\left(s\right)=F_{W,V}\left(s\right)\mathit{for}\mathit{all}s\in P\right\},$$

$$V_P=\left\{v\in V:\exists w\in W\mathit{such}\mathit{that}{F}_{w,v}\left(s\right)=F_{W,V}\left(s\right)\mathit{for}\mathit{all}s\in P\right\}.$$

Theorem 5.

Let $(M,F,\diamond )$ be a complete MPM-space with ⋄ of H-type and $W,V$ be two non-empty disjoint subsets of M where W is closed. Let P be a ZH-set and $\varphi \in \mathrm{\Psi }\left(P\right)$. Let $T:W\to V$ be a weak ϕ-proximal contraction mapping which satisfies the following conditions:

(i) 

$T\left(W_P\right)\subseteq V_P$ and $(W,V)$ satisfies the S-property,

(ii) 

there exist $w_0,w_1\in W_P$, such that $F_{w_1,T{w}_0}\left(s\right)=F_{W,V}\left(s\right)$ for all $s\in P$.

Then, there exists an element $w^{*}\in W$ such that $F_{w^{*},T{w}^{*}}\left(s\right)=F_{W,V}\left(t\right)$ for all $s>0$, that is, T has a best proximity point.

Proof. 

From the hypothesis, it can be known that the existence of $w_0,w_1\in W_P$, such that

$$F_{w_1,T{w}_0}\left(s\right)=F_{W,V}\left(s\right)\mathrm{for}\mathrm{all}s\in P$$

holds. Due to $T\left(W_P\right)\subseteq V_P$, there exists $w_2\in W_P$, such that

$$F_{w_2,T{w}_1}\left(s\right)=F_{W,V}\left(s\right)$$

holds. So, for all $s\in P$,

$$F_{w_1,T{w}_0}\left(s\right)=F_{W,V}\left(s\right)\mathrm{and}{F}_{w_2,T{w}_1}\left(s\right)=F_{W,V}\left(s\right).$$

Due to $T\left(W_P\right)\subseteq V_P$, there exists $w_3\in W_P$, such that

$$F_{w_3,T{w}_2}\left(s\right)=F_{W,V}\left(s\right).$$

holds. So, for all $s\in P$,

$$F_{w_2,T{w}_1}\left(s\right)=F_{W,V}\left(s\right)\mathrm{and}{F}_{w_3,T{w}_2}\left(s\right)=F_{W,V}\left(s\right).$$

After performing n steps in this way, we have for all $s\in P$,

$$F_{w_n,T{w}_{n-1}}\left(s\right)=F_{W,V}\left(s\right),$$

(9)

and

$$F_{w_{n+1},T{w}_n}\left(s\right)=F_{W,V}\left(s\right)\mathrm{for}\mathrm{all}s\in P.$$

(10)

Due to the fact that $(W,V)$ has the S-property, we obtain from (9) and (10), for all $s\in P$,

$$F_{w_n,w_{n+1}}\left(s\right)=F_{T{w}_{n-1},T{w}_n}\left(s\right)\mathrm{for}\mathrm{all}s\in P.$$

Since T is a weak $\varphi$-proximal contraction, we have for all $s\in P$,

$$F_{w_n,w_{n+1}}\left(\varphi \left(s\right)\right)=F_{T{w}_{n-1},T{w}_n}\left(\varphi \left(s\right)\right)\ge F_{w_{n-1},w_n}\left(s)\right).$$

(11)

We claim that $\left\{w_n\right\}$ is a Cauchy sequence. Firstly, we prove that for $s>0$,

$$\underset{n\to \infty }{lim}{F}_{w_n,w_{n+1}}\left(s\right)=1.$$

Fix $s>0$ and let $ϵ\in (0,1]$. Since ${lim}_{s\to \infty }{F}_{w_0,w_1}\left(s\right)=1,$ there exists $s_0\in P$, such that $F_{w_0,w_1}\left(s_0\right)>1-ϵ.$

For $s>0$ and $s_0\in P$, by Lemma 1, the existence of $n_0\in N$, such that ${\phi }^n\left(s_0\right)<s$ holds, when $n\ge n_0$. Then,

$$F_{w_n,w_{n+1}}\left(s\right)\ge F_{w_n,w_{n+1}}\left({\phi }^n\left(s_0\right)\right)\ge \cdots \ge F_{w_0,w_1}\left(s_0\right)>1-ϵ,$$

thus we have

$$\underset{n\to \infty }{lim}{F}_{w_n,w_{n+1}}\left(s\right)=1.$$

Secondly, as proof in Theorem 1,

$$F_{w_n,w_{n+k}}\left({\phi }^{n_0}\left(s_0\right)\right)\ge {\diamond }^{k-1}\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))\right)$$

for $k\in N$, where $s_0\in P,{\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)$.

Thirdly, for $s>0$, as proved in Lemma 1, we can find $s_0\in P$ and $n_0$, such that, ${\phi }^{n_0+1}\left(s_0\right)<{\phi }^{n_0}\left(s_0\right)<s$. Then

$$F_{w_n,w_{n+k}}\left(s\right)\ge F_{w_n,w_{n+k}}\left({\phi }^{n_0}\left(s_0\right)\right)\ge {\diamond }^{k-1}\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))\right).$$

Let $ϵ\in (0,1]$, since $\left\{{\diamond }^n\left(s\right)\right\}$ is equicontinuous at $s=1$ and ${\diamond }^n\left(1\right)=1$, so there exists $\delta >0$, such that

$${\diamond }^n\left(s\right)>1-ϵ \mathrm{for} \mathrm{all} s\in (1-\delta ,1] \mathrm{and} n\in N.$$

(12)

From the foregoing description, we have that

$$\underset{n\to \infty }{lim}{F}_{w_n,w_{n+1}}({\phi }^{n_0}\left(s_0\right)-{\phi }^{n_0+1}\left(s_0\right))=1.$$

Thus, there exists $n_1>n_0$, such that

$$F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s\right)-{\phi }^{n_0+1}\left(s\right))>1-\delta$$

for all $n>n_1$. It follows from (12) that

$$F_{w_n,w_{n+k}}\left(s\right)\ge {\diamond }^{k-1}\left(F_{w_n,w_{n+1}}({\phi }^{n_0}\left(s\right)-{\phi }^{n_0+1}\left(s\right))\right)>1-ϵ$$

for all $n>n_1$ and $k\in N$. Thus, $\left\{w_n\right\}$ is a Cauchy sequence in M. Since $(M,F,\diamond )$ is complete and W is a closed subset of M, there exists $w^{*}\in W$, such that

$$\underset{n\to \infty }{lim}{F}_{w_n,w^{*}}\left(s\right)=1.$$

(13)

Since $\varphi \in \mathrm{\Psi }\left(P\right)$, we can pick $s_1>0$ such that $s>\varphi \left(s_1\right)$ holds. In fact, let $s_1={\phi }^{n_0}\left(s_0\right)$ as in Lemma 1.

According to the non-decreasing property of distribution function, we obtain,

$$\begin{array}{c}\hfill F_{T{w}_n,T{w}^{*}}\left(s\right)\ge F_{T{w}_n,T{w}^{*}}\left(\varphi \left(s_1\right)\right)\ge F_{w_n,w^{*}}\left(s_1\right).\end{array}$$

Passing limit as $n\to \infty$,

$$\underset{n\to \infty }{lim}{F}_{T{w}_n,T{w}^{*}}\left(s\right)=1.$$

(14)

Let us choose $\lambda >0$ be arbitrary, such that $t>2\lambda$, then

$$\begin{array}{cc}\hfill F_{w^{*},T{w}^{*}}\left(s\right)& \ge \diamond (F_{w^{*},w_{n+1}}\left(\lambda \right),F_{w_{n+1},T{w}^{*}}(s-\lambda ))\hfill \\ & \ge \diamond (F_{w^{*},w_{n+1}}\left(\lambda \right),\diamond (F_{w_{n+1},T{w}_n}(s-2\lambda ),F_{T{w}_n,T{w}^{*}}\left(\lambda \right)))\hfill \\ & =\diamond (F_{w^{*},w_{n+1}}\left(\lambda \right),\diamond (F_{W,V}(s-2\lambda ),F_{T{w}_n,T{w}^{*}}\left(\lambda \right)))\left(\mathrm{using}\right(10\left)\right).\hfill \end{array}$$

On the above inequality, taking the limit as $n\to \infty$ and using the results of (13), (14) we have,

$$F_{w^{*},T{w}^{*}}\left(s\right)\ge \diamond (1,\diamond (F_{W,V}(s-2\lambda ),1))=F_{W,V}(s-2\lambda ).$$

Due to the arbitrariness of $\lambda$ and the left continuity of $F_{W,V}\left(s\right)$, we have

$$F_{w^{*},T{w}^{*}}\left(s\right)\ge F_{W,V}\left(s\right),$$

this means

$$F_{w^{*},T{w}^{*}}\left(s\right)=F_{W,V}\left(s\right).$$

□

Example 6.

Let $M=[0,\infty )$ and define $F:M\times M\to D_{+}$ as follows:

$$F(w,v)\left(s\right)=F_{w,v}\left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{s}{s+\sigma (w,v)}},\hfill & \mathit{if}\sigma (w,v)\ge s,\hfill \\ 1,\hfill & \mathit{if}\sigma (w,v)<s.\hfill \end{array}\right.$$

(15)

Then, $(M,F,{\diamond }_M)$ is a complete MPM-space, where $\sigma (w,v)=|w-v|$.

We consider the following disjoint closed subsets:

$$W=[3,4],V=[0,{\displaystyle \frac{3}{4}}].$$

Let $T:W\to V$ be defined by $Tw={\displaystyle \frac{w}{1+w}}$. Then, $F_{W,V}\left(s\right)={\displaystyle \frac{s}{s+\frac{9}{4}}}$ for all $s>0$.

Note that $W_P=\left\{3\right\},V_P=\left\{{\displaystyle \frac{3}{4}}\right\}$ and $T\left(W_P\right)\subseteq V_P$.

For the only point $3\in W$ and the only point ${\displaystyle \frac{3}{4}}\in V$, $F_{3,{\displaystyle \frac{3}{4}}}\left(s\right)={\displaystyle \frac{s}{s+\frac{9}{4}}}=F_{W,V}\left(s\right)$ for all $s>0$.

Therefore, $(W,V)$ satisfies the S-property.

Let $P=[0,+\infty )\bigcap Q$, where Q is the set of all rational numbers. Let $\phi :P\to P$ be defined by

$$\phi \left(s\right)=\left\{\begin{array}{cc}{\displaystyle \frac{s}{1+s}},\hfill & s\in [0,1)\bigcap Q,\hfill \\ {\displaystyle \frac{13}{6}},\hfill & s\in [1,2]\bigcap Q,\hfill \\ s-{\displaystyle \frac{4}{3}},\hfill & s\in [2,+\infty )\bigcap Q.\hfill \end{array}\right.$$

(16)

By the same method as in Example 5, We can show that T is a weak ϕ-proximal contraction, that is,

$$F_{Tw,Tv}\left(\varphi \left(s\right)\right)\ge F_{w,v}\left(s\right),$$

for $w,v\in W$, $s\in P$. That means all the conditions of Theorem 5 are satisfied. By Theorem 5, it is concluded that T has a best proximity point. Indeed, $3\in W$ is the best proximity point.

4. Conclusions

In this paper, we introduce the concept of the ZH-set, and based on it, we give the concept of weak probabilistic $\phi$-contraction. We have proved a new fixed-point theorem for weak probabilistic contraction in Menger spaces with a t-norm of H-type, namely Theorem 1. It is worth noting that in the theorem, the contractive gauge function $\phi$ only needs to meet a quite weak condition. Therefore, this theorem improves, generalizes, and unifies some important fixed-point theorems, including the results of $\acute{\mathrm{C}}$iri$\acute{\mathrm{c}}$ [3], Jachymski [4], Fang [5], and Choudhury and Das [6]. What is more, we obtain a best proximity point theorem.