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. (1)

    Δ is associative and commutative;

  2. (2)

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

  3. (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 \(F_{x,y}\)):

  1. (PM-1)

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

  2. (PM-2)

    \(F_{x,y}(t)=F_{y,x}(t)\) for all \(t\in\mathbb{R}\);

  3. (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. (1)

    \(\varphi(t)>0\) for all \(t>0\);

  2. (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:

  1. (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.

  2. (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.