New result on fixed point theorems for φ-contractions in Menger spaces

Abstract

Very recently, Fang (Fuzzy Sets Syst. 267:86-99, 2015) gave some fixed point theorems for probabilistic φ-contractions in Menger spaces. Fang’s results improve the one of Jachymski (Nonlinear Anal. 73:2199-2203, 2010) by relaxing the restriction on the gauge function φ. In this paper, inspired by the results of Fang, we prove a new fixed point theorem for a probabilistic φ-contraction in Menger spaces in which a weaker condition on the function φ is required. Our result improves the corresponding one of Fang and some others. Finally, an example is given to illustrate our result.

1 Introduction

Let $(X,F,{\varDelta })$ be a probabilistic metric space and $T: X\to X$ be a mapping. If there exists a gauge function $\varphi:\mathbb {R}^{+}\to\mathbb{R}^{+}$ such that

$$F_{Tx,Ty}\bigl(\varphi(t)\bigr)\geq F_{x,y}(t) \quad \mbox{for all } x,y\in X \mbox{ and } t>0, $$

then the mapping T is called a probabilistic φ-contraction. The probabilistic φ-contraction is a generalization of probabilistic k-contraction given by Sehgal and Bharucha-Reid [1]. In literature, many authors investigated fixed point theorems for probabilistic φ-contractions in Menger spaces; see [2–7]. On the fixed point theorems for other types of contractions in Menger or fuzzy metric spaces, please see [8–12]. Recently, Jachymski [13] proved a new fixed point theorem for a probabilistic φ-contraction in which the condition on the function φ is weakened. More precisely, the author gave the following result.

Theorem 1.1

([13])

Let $(X,F,{\varDelta })$ be a complete Menger probabilistic metric space with a continuous t-norm Δ of H-type, and let $\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a function satisfying conditions:

$$0< \varphi(t)< t \quad \textit{and}\quad \lim_{n\to\infty} \varphi^{n}(t)=0\quad \textit{for all }t>0. $$

If $T: X\to X$ is a probabilistic φ-contraction, then T has a unique fixed point $x^{*}\in X$, and $\{T^{n}x_{0}\}$ converges to $x^{*}$ for each $x_{0}\in X$.

Although Theorem 1.1 has been a very perfect result in which the condition on the gauge function φ is very simple, Fang [14] improves Theorem 1.1 by giving a new condition on φ recently. Let $\varphi: \mathbb{R}^{+}\to\mathbb{R}^{+}$ be a function satisfying the following condition:

$$ \mbox{for each }t>0 \mbox{ there exists } r\geq t \mbox{ such that } \lim _{n\to\infty}\varphi^{n}(t)=0. $$

(1.1)

Let $\boldsymbol{\Phi}_{\mathbf{w}}$ denote the set of all functions $\varphi: \mathbb {R}^{+}\to\mathbb{R}^{+}$ satisfying the condition (1.1) and let Φ denote the set of all functions $\varphi: \mathbb{R}^{+}\to\mathbb {R}^{+}$ satisfying the condition that $\lim_{n\to\infty}\varphi^{n}(t)=0$ for all $t>0$. In [14], Fang gave an example of $\varphi\in \boldsymbol{\Phi}_{\mathbf{w}}$ but $\varphi\notin\boldsymbol{\Phi}$.

By using the condition (1.1), Fang gave the following result.

Theorem 1.2

([14])

Let $(X,F,{\varDelta })$ be a complete Menger space with a t-norm Δ of H-type. If $T: X\to X$ is a probabilistic φ-contraction, where $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}$, then T has a unique fixed point $x^{*}\in X$, and $\{T^{n}x_{0}\}$ converges to $x^{*}$ for each $x_{0}\in X$.

Since the condition (1.1) is weaker than the one in Theorem 1.1, Theorem 1.2 improves Theorem 1.1. In [14], Fang asked the following question:

Can the condition (1.1) in Theorem 1.2 be replaced by a more weak condition?

In this paper, we give a positive answer to the question of Fang by proving a new fixed point theorem for a probabilistic φ-contraction in Menger spaces. In our result, the function φ is required to satisfy a more weak condition than (1.1) and the t-norm is not required to be of H-type. Our result improves the corresponding one of Fang [14] and some others. Finally, an example is given to illustrate our result.

2 Preliminaries

In the rest of this paper, let $\mathbb{R}=(-\infty,+\infty)$, $\mathbb {R}^{+}=[0,+\infty)$ and $\mathbb{N}$ denote the set of all natural numbers.

A mapping $F: \mathbb{R}\to[0,1]$ is called a distribution function if it is non-decreasing and left-continuous with $\inf_{t\in\mathbb {R}}F(t)=0$. If in addition $F(0)=0$, then F is called a distance distribution function. A distance distribution function F satisfying $\lim_{t\to\infty }F(t)=1$ is called a Menger distance distribution function.

The set of all Menger distance distribution functions is denoted by $\mathcal{D}^{+}$. It is known that $\mathcal{D}^{+}$ is partially ordered by the usual pointwise ordering of functions, that is, $F\leq G$ if and only if $F(t)\leq G(t)$ for all $t\geq0$. The maximal element in $\mathcal{D}^{+}$ on this order is the distance distribution function $\epsilon_{0}$ defined by

$$\epsilon_{0}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0,& t=0, \\ 1,& t>0. \end{array}\displaystyle \right . $$

Definition 2.1

([15])

A binary operation ${\varDelta } :[0,1]\times{}[0,1]\rightarrow{}[ 0,1]$ is a t-norm if Δ satisfies the following conditions:

• (1) Δ is associative and commutative;

• (2) ${\varDelta } (a,1)=a$ for all $a\in{}[0,1]$;

• (3) ${\varDelta }(a, b)\leq {\varDelta } (c,d)$ whenever $a\leq c$ and $b\leq d$ for all $a,b,c,d\in{}[0,1]$.

Two typical examples of the continuous t-norm are ${\varDelta } _{P}(a,b)=ab$ and ${\varDelta } _{M}(a,b)=\min\{a,b\}$ for all $a,b\in{}[0,1]$.

Definition 2.2

([16])

A t-norm Δ is said to be of Hadžić-type (for short H-type) if the family of functions $\{{\varDelta }^{m}(t)\}_{m=1}^{\infty}$ is equicontinuous at $t=1$, where

$${\varDelta }^{1}(t)={\varDelta }(t,t), \qquad {\varDelta }^{m+1}(t)={ \varDelta }\bigl(t,{\varDelta }^{m}(t)\bigr), \quad m=1,2,\ldots, t\in[0,1]. $$

It is easy to see that ${\varDelta }_{M}$ is a t-norm of H-type but ${\varDelta }_{P}$ is not of H-type. Here we give a new t-norm of H-type by ${\varDelta }_{M}$ and ${\varDelta }_{P}$.

Example 2.1

Let ${\varDelta }(x,1)={\varDelta }(1,x)=x$ for all $x\in[0,1]$, ${\varDelta }(x,y)={\varDelta }_{P}(x,y)$ for all $x,y\in[0,1]$ with $\max\{x,y\}\in[0,\frac{1}{2}]$ and ${\varDelta }(x,y)={\varDelta }_{M}(x,y)$ for all $x,y\in[0,1]$ with $\max\{x,y\} \in(\frac{1}{2},1]$. It is easy to check that Δ is a t-norm. Now we show that it is of H-type. For any given $\epsilon\in (0,\frac{1}{2})$, set $\delta=\epsilon$. Then $1-\delta=1-\epsilon> \frac{1}{2}$. Thus, for all $t\in(1-\delta,1)$, one has ${\varDelta }^{n}(t)=t>1-\delta=1-\epsilon$ for all $n\in\mathbb{N}$. For $\epsilon\in [\frac{1}{2},1)$, taking $\delta\in(0,\frac{1}{2})$ arbitrarily, then we have $1-\delta>\frac{1}{2}\geq1-\epsilon$. Thus for all $t\in (1-\delta,1)$, ${\varDelta }^{n}(t)=t>1-\delta>\frac{1}{2}\geq 1-\epsilon$ for all $n\in\mathbb{N}$. Therefore, Δ is a t-norm of H-type.

Example 2.2

Let $\delta\in(0,1]$ and let Δ be a t-norm. Define ${\varDelta }_{\delta}$ by ${\varDelta }_{\delta}(x,y)={\varDelta }(x,y)$, if $\max\{x,y\}\leq1-\delta$, and ${\varDelta }_{\delta}(x,y)=\min\{x,y\}$, if $\max\{x,y\}>1-\delta$. then ${\varDelta }_{\delta}$ is a t-norm of H-type; see [17]. However, if ${\varDelta }_{\delta}(x,1)={\varDelta }_{\delta}(1,x)=x$ for all $x\in[0,1]$, ${\varDelta }_{\delta}(x,y)=\delta$ for all $x,y\in[\delta,1)$ and ${\varDelta }_{\delta}(x,y)=0$ for all $x,y\in[0,1]$ with $\min\{x,y\}\in[0,\delta)$, then ${\varDelta }_{\delta}$ is a t-norm but not of H-type.

For other t-norms of H-type, the reader may refer to [16].

Definition 2.3

([18])

A triple $(X,F,{\varDelta })$ is called a Menger probabilistic metric space (for short, Menger space) if X is a nonempty set, Δ is a t-norm, and F is a mapping from $X\times X\to\mathcal{D}^{+}$ satisfying the following conditions (for $x,y\in X$, denote $F(x,y)$ by $F_{x,y}$):

• (PM-1) $F_{x,y}(t)=\epsilon_{0}(t)$ for all $t\in\mathbb{R}$ if and only if $x=y$;

• (PM-2) $F_{x,y}(t)=F_{y,x}(t)$ for all $t\in\mathbb{R}$;

• (PM-3) $F_{x,y}(t+s)\geq{ \varDelta }(F_{x,z}(t), F_{z,y}(s))$ for all $x,y,z\in X$ and $t,s>0$.

Definition 2.4

([15])

Let $(X,F,{\varDelta })$ be a Menger space and $\{x_{n}\}$ be a sequence in X. The sequence $\{x_{n}\}$ is said to be convergent to $x\in X$ if $\lim_{n\to\infty}F_{x_{n},x}(t)=1$ for all $t>0$; the sequence $\{x_{n}\}$ is said to be a Cauchy sequence if for any given $t>0$ and $\epsilon\in(0,1)$, there exists $N_{\epsilon,t}\in\mathbb {N}$ such that $F_{x_{n},x_{m}}(t)>1-\epsilon$ whenever $m,n>N_{t,\epsilon }$; the Menger space $(X,F,{\varDelta })$ is said to be complete, if each Cauchy sequence in X is convergent to some point in X.

3 Main results

In this section, let $\boldsymbol{\Phi}_{\mathbf{w}^{*}}$ denote the set of all functions $\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying the following condition:

$$\begin{aligned}& \mbox{for each } t_{1},t_{2}>0 \mbox{ there exists } r\geq \max\{ t_{1},t_{2}\} \mbox{ and } N\in\mathbb{N} \\& \quad \mbox{such that } \varphi ^{n}(r)< \min\{t_{1},t_{2} \} \mbox{ for all } n> N. \end{aligned}$$

(3.1)

Obviously, the condition (3.1) implies that

$$\begin{aligned} \begin{aligned}[b] &\mbox{for each } t>0 \mbox{ there exists } r\geq t \mbox{ and } N\in \mathbb{N} \\ &\quad \mbox{such that } \varphi^{n}(r)< t \mbox{ for all } n> N. \end{aligned} \end{aligned}$$

(3.2)

It is easy to see that for each $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}$, $\varphi\in \boldsymbol{\Phi}_{\mathbf{w}^{*}}$. In fact, if $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}$, then for each $t_{1},t_{2}>0$, there exist $r_{1}\geq t_{1}$ and $r_{2}\geq t_{2}$ such that $\lim_{n\to\infty}\varphi^{n}(r_{1})=\lim_{n\to\infty}\varphi^{n}(r_{2})=0$. Assume that $t_{1}\leq t_{2}$. Then there exists $N\in\mathbb{N}$ such that $\varphi^{n}(r_{2})< t_{1}$ for all $n>N$. Thus $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$.

However, if $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$, then it is unnecessary that $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}$.

Example 3.1

Let $\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ by $\varphi (t)=t$ for all $t\in[0,1]$, $\varphi(t)=t-1$ for all $t\in(1,\infty)$. Then $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$. In fact, for each $t_{1},t_{2}\in (0,\infty)$, there exists $N\in\mathbb{N}$ such that $r=1+N+\epsilon >\max\{t_{1},t_{2}\}$, where $\epsilon\in(0,\min\{t_{1},t_{2},1\})$. Then we have $\varphi^{n}(r)=\epsilon< \min\{t_{1},t_{2}\}$ for all $n> N+1$. So $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$. However, since $\lim_{n\to\infty }\varphi^{n}(r)\neq0$ for all $r>0$, $\varphi\notin\boldsymbol{\Phi}_{\mathbf{w}}$.

From Example 3.1 we see that $\boldsymbol{\Phi}_{\mathbf{w}^{*}}$ is a proper subclass of $\boldsymbol{\Phi}_{\mathbf{w}}$. On $\boldsymbol{\Phi}_{\mathbf{w}^{*}}$, $\boldsymbol{\Phi}_{\mathbf{w}}$, and Φ, we have $\boldsymbol{\Phi} \subset \boldsymbol{\Phi} _{\mathbf{w}}\subset\boldsymbol{\Phi}_{\mathbf{w}^{*}}$.

Lemma 3.1

Let $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$. Then for each $t>0$, there exists $r\geq t$ such that $\varphi(r)< t$.

Proof

Suppose that there is $t_{0}>0$ such that $\varphi(r)\geq t_{0}$ for all $r\geq t_{0}$. By induction, we obtain $\varphi^{n}(r)\geq t_{0}$ for all $n\in\mathbb{N}$. From (3.2) it follows that there exist $r\geq t_{0}$ and $N\in\mathbb{N}$ such that $\varphi^{n}(r)< t_{0}$ for all $n>N$, which contradicts $\varphi^{n}(r)\geq t_{0}$ for all $r\geq t_{0}$ and $n\in\mathbb{N}$. Thus for each $t>0$, there exists $r\geq t$ such that $\varphi(r)\leq t$. This completes the proof. □

Lemma 3.2

Let $(X,F,{\varDelta })$ be a Menger space and $x,y\in X$. If there exists a function $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$ such that

$$ F_{x,y}\bigl(\varphi(t)\bigr)\geq F_{x,y}(t),\quad \forall t>0, $$

(3.3)

then $x=y$.

Proof

First by a similar proof with Lemma 2.2 of [14] we can show that for all $n\in\mathbb{N}$ and $t>0$, one has $\varphi ^{n}(t)>0$. By induction, from (3.3) it follows that

$$ F_{x,y}\bigl(\varphi^{n}(t)\bigr)\geq F_{x,y}(t) \quad \mbox{for all } n\in\mathbb {N} \mbox{ and } t>0. $$

(3.4)

Next we show that $F_{x,y}(t)=1$ for all $t>0$. In fact, if there exists $t_{0}>0$ such that $F_{x,y}(t_{0})<1$, then since $\lim_{t\to\infty }F_{x,y}(t)=1$ there is $t_{1}>t_{0}$ such that

$$ F_{x,y}(t)>F_{x,y}(t_{0}) \quad \mbox{for all } t \geq t_{1}. $$

(3.5)

Since $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$, there exist $t_{2}\geq\max\{ t_{1},t_{0}\}$ and $N\in\mathbb{N}$ such that $\varphi^{n}(t_{2})< \min\{ t_{0},t_{1}\}$ for all $n>N$. By the monotonicity of $F_{x,y}(\cdot)$, from (3.4) and (3.5) it follows that, for each $n> N$,

$$F_{x,y}(t_{0})\geq F_{x,y}\bigl( \varphi^{n}(t_{2})\bigr)\geq F_{x,y}(t_{2}) \geq F_{x,y}(t_{1})>F_{x,y}(t_{0}). $$

It is a contradiction. Therefore, $F_{x,y}(t)=1$ for all $t>0$, i.e., $x=y$. This completes the proof. □

Lemma 3.3

Let $(X,F,{\varDelta })$ be a Menger space where Δ is continuous at $(1,1)$ and let $\{x_{n}\}$ be a sequence in X. Suppose that there exists a function $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$ satisfying the following conditions:

• (1) $\varphi(t)>0$ for all $t>0$;

• (2) $F_{x_{n},x_{m}}(\varphi(t))\geq F_{x_{n-1},x_{m-1}}(t)$ for all $n,m\in\mathbb{N}$ and $t>0$.

Then $\lim_{n\to\infty}F_{x_{n},x_{n+k}}(t)=1$ for all $k\in\mathbb{N}$ and $t>0$.

Proof

It is easy to see that the condition (1) implies that $\varphi^{n}(t)>0$ for all $t>0$ and the condition (2) implies that

$$ F_{x_{n},x_{n+1}}\bigl(\varphi^{n}(t)\bigr)\geq F_{x_{0},x_{1}}(t), \quad \forall n\in \mathbb{N} \mbox{ and } \forall t>0. $$

(3.6)

We first prove that

$$ \lim_{n\to\infty}F_{x_{n},x_{n+1}}(t)=1, \quad \forall t>0. $$

(3.7)

Since $\lim_{t\to\infty}F_{x_{0},x_{1}}(t)=1$, for any $\epsilon\in(0,1)$, there exists $t_{0}>0$ such that $F_{x_{0},x_{1}}(t_{0})>1-\epsilon$. For each $t>0$, since $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$, there exist $t_{1}\geq\max \{t,t_{0}\}$ and $N\in\mathbb{N}$ such that $\varphi^{n}(t_{1})< \min\{ t,t_{0}\}$ for all $n\geq N$. By the monotonicity of $F_{x,y}(\cdot)$, from (3.6) we have

$$\begin{aligned} F_{x_{n},x_{n+1}}(t)&\geq F_{x_{n},x_{n+1}}\bigl(\varphi^{n}(t_{1}) \bigr) \\ &\geq F_{x_{0},x_{1}}(t_{1})\geq F_{x_{0},x_{n}}(t_{0}) \\ &>1-\epsilon\quad \mbox{for all }n\geq N, \end{aligned}$$

which implies that (3.7) holds. Assume that $\lim_{n\to\infty }F_{x_{n},x_{n+k}}(t)=1$ for each $k\in\mathbb{N}$ and $t>0$. Since Δ is continuous at $(1,1)$, we have

$$F_{x_{n},x_{n+k+1}}(t)\geq{ \varDelta }\bigl(F_{x_{n},x_{n+k}}(t/2),F_{x_{n+k},x_{n+k+1}}(t/2) \bigr)\to{ \varDelta }(1,1)=1\quad \mbox{as } n\to\infty. $$

By induction we conclude that

$$\lim_{n\to\infty}F_{x_{n},x_{n+k}}(t)=1, \quad \forall k\in\mathbb{N} \mbox{ and } \forall t>0. $$

This completes the proof. □

Lemma 3.4

Let $(X,F,{\varDelta })$ be a Menger space where Δ is of H-type and continuous at $(1,1)$ and let $\{x_{n}\}$ be a sequence in X. Suppose that there exists a function $\varphi\in\boldsymbol {\Phi}_{\mathbf{w}^{*}}$ satisfying the conditions (1) and (2) in Lemma  3.3. Then $\{x_{n}\}$ is a Cauchy sequence.

Proof

Let $t>0$. By Lemma 3.1 there is $r\geq t$ such that $\varphi(r)< t$. We show by induction that

$$ F_{x_{n},x_{n+k}}(t)\geq{ \varDelta }^{k}\bigl(F_{x_{n},x_{n+1}}\bigl(t- \varphi (r)\bigr)\bigr), \quad \forall k\in\mathbb{N}. $$

(3.8)

Obviously, (3.8) holds for $k=1$. Assume that (3.8) holds for some $k\in \mathbb{N}$. By (2) in Lemma 3.3 we have

$$\begin{aligned} F_{x_{n},x_{n+k+1}}(t)&\geq{ \varDelta }\bigl(F_{x_{n},x_{n+1}}\bigl(t-\varphi (r) \bigr),F_{x_{n+1},x_{n+k+1}}\bigl(\varphi(r)\bigr)\bigr) \\ &\geq{ \varDelta }\bigl(F_{x_{n},x_{n+1}}\bigl(t-\varphi(r)\bigr), F_{x_{n},x_{n+k}}(r) \bigr) \\ &\geq{ \varDelta }\bigl(F_{x_{n},x_{n+1}}\bigl(t-\varphi (r)\bigr),F_{x_{n},x_{n+k}}(t) \bigr) \\ &\geq{ \varDelta }\bigl(F_{x_{n},x_{n+1}}\bigl(t-\varphi(r)\bigr),{\varDelta }^{k}\bigl(F_{x_{n},x_{n+1}}\bigl(t-\varphi(r)\bigr)\bigr)\bigr) \\ &={\varDelta }^{k+1}\bigl(F_{x_{n},x_{n+1}}\bigl(t-\varphi(r)\bigr)\bigr). \end{aligned}$$

It follows that (3.8) holds for $k+1$. So (3.8) holds for all $k\in \mathbb{N}$.

Let $t>0$. Define $a_{n}=\inf_{k\geq1}F_{x_{n},x_{n+k}}(t)$. Since $\varphi \in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$, by Lemma 3.1 there exists $t_{0}\geq t$ such that $\varphi(t_{0})< t$. So by the condition (2) we have

$$\begin{aligned} a_{n}&=\inf_{k\geq1}F_{x_{n},x_{n+k}}(t) \\ &\geq\inf_{k\geq1}F_{x_{n},x_{n+k}}\bigl(\varphi(t_{0}) \bigr) \\ &\geq\inf_{k\geq1}F_{x_{n-1},x_{n-1+k}}(t_{0}) \\ &\geq\inf_{k\geq1}F_{x_{n-1},x_{n-1+k}}(t) \\ &=a_{n-1}\quad \mbox{for all } n\in\mathbb{N}. \end{aligned}$$

So $\{a_{n}\}$ is non-decreasing. Since $\{a_{n}\}$ is bounded, there exists $a\in[0,1]$ such that $a_{n}\to a$ as $n\to\infty$. Assume that $a<1$. Then there exists $\eta\in(0,1)$ such that $a+\eta<1$. For any given $\epsilon\in(0,1/2)$, by the definition of $a_{n}$ there exists $k=k(\epsilon,n)\in\mathbb{N}$ such that

$$ a_{n}\geq F_{x_{n},x_{n+k}}(t)-\epsilon/2. $$

(3.9)

By Lemma 3.3 one has $\lim_{n\to\infty}F_{x_{n},x_{n+1}}(t-\varphi (r))=1$. Therefore there exist $\delta\in(0,1)$ and $N\in\mathbb{N}$ such that $F_{x_{n},x_{n+1}}(t-\varphi(r))\in(1-\delta, 1)$ for all $n>N$. Since Δ is of H-type, ${\varDelta }^{k}(F_{x_{n},x_{n+1}}(t-\varphi(r)))>1-\epsilon/2$ for all $n>N$ and all $k\in\mathbb{N}$. Further combing (3.8) and (3.9) we get

$$1>a+\eta>a_{n}\geq1-\epsilon $$

for all $n>N$, which implies that

$$1>a+\delta>a\geq1. $$

It is a contradiction. So $a=1$. Since $a_{n}\to1$ as $n\to\infty$, there exists $N'\in\mathbb{N}$ such that $a_{n}>1-\epsilon$ for all $n>N$. Then by the definition of $\{a_{n}\}$, we have

$$F_{x_{n},x_{n}+k}(t)>1-\epsilon $$

for all $n\in\mathbb{N}$ with $n>N'$ and all $k\in\mathbb{N}$. Thus $\{ x_{n}\}$ is a Cauchy sequence. This completes the proof. □

Theorem 3.1

Let $(X,F,{\varDelta })$ be a complete Menger space where Δ is of H-type and continuous at $(1,1)$. Let $T: X\to X$ be a probabilistic φ-contraction, where $\varphi\in\boldsymbol{\Phi} _{\mathbf{w}^{*}}$ satisfies $\varphi(t)>0$ for all $t>0$. Then T has a unique fixed point $x^{*}\in X$, and $\{T^{n}x_{0}\}$ converges to $x^{*}$ for each $x_{0}\in X$.

Proof

Take $x_{0}\in X$ arbitrarily and define the sequence $\{ x_{n}\}$ by $x_{n}=Tx_{n-1}$ for each $n\in\mathbb{N}$. Since T is a probabilistic φ-contraction, we have

$$F_{x_{n},x_{m}}\bigl(\varphi(t)\bigr)=F_{Tx_{n-1},Tx_{m-1}}\bigl(\varphi(t)\bigr) \geq F_{x_{n-1},x_{m-1}}(t),\quad \forall m,n\in\mathbb{N} \mbox{ and } \forall t>0. $$

So, from Lemma 3.4 it follows that $\{x_{n}\}$ is a Cauchy sequence. Since X is complete, there exists $x^{*}\in X$ such that $x_{n}\to x^{*}$ as $n\to\infty$.

Next we show that $x^{*}$ is a fixed point of T. For any $t>0$, Lemma 3.1 shows that there exists $r\geq t$ such that $\varphi(r)< t$. By the monotonicity of Δ we get

$$\begin{aligned} F_{x^{*},Tx^{*}}(t)&\geq{ \varDelta }\bigl(F_{x^{*},x_{n+1}}\bigl(t-\varphi (r) \bigr),F_{x_{n+1},Tx^{*}}\bigl(\varphi(r)\bigr)\bigr) \\ &= {\varDelta }\bigl(F_{x^{*},x_{n+1}}\bigl(t-\varphi(r)\bigr),F_{Tx_{n},Tx^{*}}\bigl( \varphi (r)\bigr)\bigr) \\ &\geq{ \varDelta }\bigl(F_{x^{*},x_{n+1}}\bigl(t-\varphi(r)\bigr),F_{x_{n},x^{*}}(r) \bigr) \\ &\geq{ \varDelta }(c_{n},c_{n}), \end{aligned}$$

(3.10)

where $c_{n}=\min\{F_{x^{*},x_{n+1}}(t-\varphi(r)),F_{x_{n},x^{*}}(r)\}$. Since $c_{n}\to1$ as $n\to\infty$ and Δ is continuous at $(1,1)$, from (3.10) we have

$$F_{x^{*},Tx^{*}}(t)\geq{ \varDelta }(c_{n},c_{n})\to{ \varDelta }(1,1)=1, $$

which implies that $x^{*}=Tx^{*}$.

Finally, we prove that $x^{*}$ is the unique fixed point of T. Suppose that T has another fixed point $x'\in X$. Then we have

$$F_{x^{*},x'}\bigl(\varphi(t)\bigr)=F_{Tx^{*},Tx'}\bigl(\varphi(t)\bigr) \geq F_{ x^{*}, x'}(t), \quad \forall t>0. $$

From Lemma 3.2 it follows that $x^{*}=x'$. Thus $x^{*}$ is the unique fixed point of T. This completes the proof. □

Corollary 3.1

Let $(X,F,{\varDelta })$ be a complete Menger space where Δ is of H-type and continuous at $(1,1)$. Let $T_{0},T_{1}: X\to X$ be two mappings such that

$$ F_{T_{0}x,T_{0}y}\bigl(\varphi(t)\bigr)\geq F_{x,y}(t) \quad \textit{and}\quad F_{T_{1}x,T_{1}y}(t)\geq F_{x,y}(t) \quad \textit{for all }x,y\in X\textit{ and }t>0, $$

(3.11)

where $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$ satisfies $\varphi(t)>0$ for all $t>0$. If $T_{0}$ commutes with $T_{1}$, then $T_{0}$ and $T_{1}$ have a unique common fixed point in X.

Proof

Let $T=T_{0}T_{1}$. Then (3.11) implies that T is a probabilistic φ-contraction. From Theorem 3.1 it follows that T has a unique fixed point $x^{*}\in X$. Since $T_{0}$ commutes with $T_{1}$, we have $T_{0}T_{1}x^{*}=T_{1}T_{0}x^{*}$. Further we have $T(T_{0}x^{*})=(T_{0}T_{1})(T_{0}x^{*})=T_{0}(T_{0}T_{1}x^{*})=T_{0}(Tx^{*})=T_{0}x^{*}$, which implies that $T_{0}x^{*}$ is a fixed point of T. Since T has a unique fixed point $x^{*}$, one has $T_{0}x^{*}=x^{*}$. Similarly, we have $T_{1}x^{*}=x^{*}$. Thus $x^{*}$ is the common fixed point of $T_{0}$ and $T_{1}$. Assume that $x'\in X$ is another common fixed point of $T_{0}$ and $T_{1}$. Since $T_{0}$ commutes with $T_{1}$, we have $T(T_{0}x')=(T_{0} T_{1})(T_{0}x')=T_{0}(T_{0}T_{1}x')=T_{0}(T_{1}T_{0}x')=T_{0}x'$, which implies that $T_{0}x'$ is the fixed point of T. Since $x^{*}$ is a unique fixed point of T, one has $x'=T_{0}x'=x^{*}$. Thus $x^{*}$ is the unique common fixed point of $T_{0}$ and $T_{1}$. This completes the proof. □

Finally, we give an example to illustrate Theorem 3.1.

Example 3.2

Let $X=\{3^{n+2}:n\in\mathbb{N}\}\cup\{0,3\}$ and define the mapping $F: X\times X\to\mathcal{D}^{+}$ by $F_{x,y}(0)=0$ for all $x,y\in X$, $F_{x,x}(t)=1$ for all $x\in X$ and $t>0$,

$$\begin{aligned}& F_{0,3}(t)=F_{3,0}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{3}{5},&0< t\leq3, \\ 1,& t>3 \end{array}\displaystyle \right . \quad \mbox{and}\quad F_{x,y}(t)=F_{y,x}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{2}, &0< t\leq|x-y|, \\ 1,& t>|x-y| \end{array}\displaystyle \right . \end{aligned}$$

for all $x,y\in X$ with $x\neq y$ and $\{x,y\}\neq\{0,3\}$. It is easy to see that $(X,F,{\varDelta }_{M})$ is a complete Menger space.

Let $T: X\to X$ be a mapping defined by $T0=T3= T27=0$ and $T3^{n+3}=3^{n+2}$ for each $n\in\mathbb{N}$. Let $\varphi: \mathbb {R}^{+}\to\mathbb{R}^{+}$ be a function defined by

$$\varphi(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} t,&\mbox{if } 0\leq t\leq1, \\ t-1,&\mbox{if } t>1. \end{array}\displaystyle \right . $$

Then $\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}$, but $\varphi\notin\boldsymbol{\Phi} _{\mathbf{w}}$; see Example 3.1.

Next we show that T is a probabilistic φ-contraction, i.e., T satisfies the following condition:

$$ F_{Tx,Ty}\bigl(\varphi(t)\bigr)\geq F_{x,y}(t)\quad \mbox{for all } x,y\in X \mbox{ and } t>0. $$

(3.12)

First, it is easy to see that for $x,y\in\{0,3,27\}$, (3.12) holds for all $t>0$ since $T0=T3=T27=0$. Next we show that (3.12) holds for all $x,y\in X$ with $x\neq y$ and $\{x,y\}\nsubseteq\{0,3,27\}$ and $t>0$. Obviously, if $|Tx-Ty|<\varphi(t)$, then $F_{Tx,Ty}(\varphi(t))=1\geq F_{x,y}(t)$. So (3.12) holds. Now we consider all $x,y\in X$ with $x\neq y$ and $\{x,y\}\nsubseteq\{0,3,27\}$ and $t>0$ with $|Tx-Ty|\geq\varphi(t)$ by the following cases:

• (a) For $(x,y)\in\{ (0,3^{n+3}),(3,3^{n+3}), (27,3^{n+3}):n\in\mathbb {N}\}$, it is easy to conclude that $\varphi(t)\leq|Tx-Ty|$ implies that $t\leq|x-y|$ for all $t>0$. Thus if $\varphi(t)\leq|Tx-Ty|$, then

  $$F_{Tx,Ty}\bigl(\varphi(t)\bigr)=\frac{1}{2}=F_{x,y}(t) \quad \mbox{for all } t>0. $$

  Therefore (3.12) holds.

• (b) For $(x,y)\in\{(3^{n+3},3^{m+3}): m,n\in\mathbb{N} \mbox{ with } m>n\}$, we have $\varphi(t)\leq |Tx-Ty|=3^{m+2}-3^{n+2}<3(3^{m+2}-3^{n+2})=|y-x|$ for $t\in(0,1]$. For $t>1$, from $\varphi(t)=t-1\leq|Tx-Ty|=3^{m+2}-3^{n+2}$, we have $t\leq3^{m+2}-3^{n+2}+1< 3^{m+3}-3^{n+3}= |x-y|$ since $3^{m+3}-3^{n+3}-3^{m+2}+3^{n+2}=2(3^{m+2}-3^{n+2})>1$. So $\varphi (t)\leq|Tx-Ty|$ implies that $t\leq|x-y|$ for all $t>0$. Thus if $\varphi(t)\leq|Tx-Ty|$, then

  $$F_{Tx,Ty}\bigl(\varphi(t)\bigr)=\frac{1}{2}=F_{x,y}(t) \quad \mbox{for all } t>0. $$

  Therefore (3.12) holds.

By the discussion above, (3.12) holds for all $x,y\in X$ and $t>0$. Therefore, T is a probabilistic φ-contraction. All the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, T has a unique fixed point $x^{*}\in X$. Obviously, $x^{*}=0$ is the unique fixed point of T. However, since $\varphi\notin\boldsymbol{\Phi}_{\mathbf{w}}$, Theorem 1.2, i.e., Theorem 3.1 of [14] cannot be applied to this example.

4 Conclusion

In this paper, we prove a new fixed point theorems for a probabilistic φ-contraction in Menger spaces. In the theorem, a more weak condition on the gauge function φ is required. Thus our result improves Theorem 1.2 of Fang [14] and some others, such as Jachymski [13], Ćirić [2], and Xiao et al. [19]. By using Theorem 3.1, it is easy to prove some fixed point theorems for φ-contraction in fuzzy metric spaces like Theorems 4.1-4.4 in [14]. For shortening the length of this paper, we omit the proofs of these theorems.