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# On fixed point theorems involving altering distances in Menger probabilistic metric spaces

Journal of Inequalities and Applications20132013:576

https://doi.org/10.1186/1029-242X-2013-576

• Accepted: 11 November 2013
• Published:

## Abstract

In this paper, we show by means of an example that the results of Babačev (Appl. Anal. Discrete Math. 6:257-264, 2012) do not hold for the class of t-norms $T\le {T}_{p}$. Further, we prove a fixed point theorem for quasi-type contraction involving altering distance functions which is weaker than that proposed by Babačev but for any continuous t-norm in a complete Menger space.

MSC:47H10, 54H25.

## Keywords

• fixed point
• Menger PM-spaces
• continuous t-norm

## 1 Introduction

Probabilistic metric spaces (in short PM-spaces) are a probabilistic generalization of metric spaces which are appropriate to carry out the study of those situations wherein distances are measured in terms of distribution functions rather than non-negative real numbers. The study of PM-spaces was initiated by Menger . Schweizer and Sklar  further enriched this concept and provided a new impetus by proving some fundamental results on this theme.

The first result on fixed point theory in PM-spaces was given by Sehgal and Bharueha-Reid  wherein the notion of probabilistic B-contraction was introduced and a generalization of the classical Banach fixed point principle to complete Menger PM-spaces was given. In , it was proved that any B-contraction on a complete Menger space $\left(S,F,{T}_{M}\right)$, where t-norm ${T}_{M}$ is defined by ${T}_{M}\left(x,y\right)=min\left\{x,y\right\}$, has a unique fixed point. In 1982, Hadžić  extended the result contained in  for a more general class of t-norms called H-type t-norms (see also ).

After that several types of contractions and associated fixed point theorems have been established in PM-spaces by various authors, e.g.,  (see also ). We also refer to a nice book on this topic by Hadžić and Pap . In this continuation, Choudhury and Das  extended the classical metric fixed point result of Khan et al.  by introducing the idea of altering distance functions in PM-spaces. In , it was proved that any probabilistic ϕ-contraction on a complete Menger space $\left(S,F,{T}_{M}\right)$ has a unique fixed point.

An open problem that remains to be investigated is whether the results are valid in the cases of any arbitrary continuous t-norm. (However, in  Miheţ gave an affirmative answer to the question raised in  using the idea of probabilistic boundedness and H-type t-norms along with some additional conditions.) Very recently, Babačev in  extended and improved the results of Choudhury and Das  for a nonlinear generalized contraction wherein she used the associated t-norm as a min norm.

In this paper, we show by means of an example that Babačev’s  results do not hold for the class of t-norms $T\le {T}_{p}$. Further, we prove a fixed point theorem for quasi-type contraction involving altering distance functions in a complete Menger space for any continuous t-norm T.

## 2 Preliminaries

Consistent with Choudhury and Das , Choudhury et al.  and Babačev , the following definitions and results will be needed in the sequel.

In the standard notation, let ${\mathcal{D}}^{+}$ be the set of all distribution functions $F:\mathbb{R}\to \left[0,1\right]$ such that F is a nondecreasing, left-continuous mapping which satisfies $F\left(0\right)=0$ and ${sup}_{x\in \mathbb{R}}F\left(x\right)=1$. The space ${\mathcal{D}}^{+}$ is partially ordered by the usual point-wise ordering of functions, i.e., $F\le G$ if and only if $F\left(t\right)\le G\left(t\right)$ for all $t\in \mathbb{R}$. The maximal element for ${\mathcal{D}}^{+}$ in this order is the distribution function given by
${ϵ}_{0}\left(t\right)=\left\{\begin{array}{ll}0,& t\le 0,\\ 1,& t>0.\end{array}$

Definition 2.1 

A binary operation $T:\left[0,1\right]×\left[0,1\right]\to \left[0,1\right]$ is a continuous t-norm if it satisfies the following conditions:
1. (a)

T is commutative and associative,

2. (b)

T is continuous,

3. (c)

$T\left(a,1\right)=a$ for all $a\in \left[0,1\right]$,

4. (d)

$T\left(a,b\right)\le T\left(c,d\right)$ whenever $a\le c$ and $b\le d$, and $a,b,c,d\in \left[0,1\right]$.

The following are the three basic continuous t-norms:
1. (i)

The minimum t-norm, ${T}_{M}$, is defined by ${T}_{M}\left(a,b\right)=min\left\{a,b\right\}$.

2. (ii)

The product t-norm, ${T}_{p}$, is defined by ${T}_{p}\left(a,b\right)=a\cdot b$.

3. (iii)

The Lukasiewicz t-norm, ${T}_{L}$, is defined by ${T}_{L}\left(a,b\right)=max\left\{a+b-1,0\right\}$.

Regarding the pointwise ordering, the following inequalities hold:
${T}_{L}<{T}_{p}<{T}_{M}.$

Definition 2.2 A Menger probabilistic metric space (briefly, Menger PM-space) is a triple $\left(X,F,T\right)$, where X is a nonempty set, T is a continuous t-norm, and F is a mapping from $X×X$ into ${\mathcal{D}}^{+}$ such that if ${F}_{xy}$ denotes the value of F at the pair $\left(x,y\right)$, the following conditions hold:

(PM1) ${F}_{xy}\left(t\right)={ϵ}_{0}\left(t\right)$ if and only if $x=y$,

(PM2) ${F}_{xy}\left(t\right)={F}_{yx}\left(t\right)$,

(PM3) ${F}_{xy}\left(t+s\right)\ge T\left({F}_{xz}\left(t\right),{F}_{zy}\left(s\right)\right)$ for all $x,y,z\in X$ and $s,t\ge 0$.

Remark 2.1 

Every metric space is a PM-space. Let $\left(X,d\right)$ be a metric space and $T\left(a,b\right)=min\left\{a,b\right\}$ be a continuous t-norm. Define ${F}_{xy}\left(t\right)={ϵ}_{0}\left(t-d\left(x,y\right)\right)$ for all $x,y\in X$ and $t>0$. The triple $\left(X,F,T\right)$ is a PM-space induced by the metric d.

Definition 2.3 Let $\left(X,F,T\right)$ be a Menger PM-space.
1. (1)

A sequence ${\left\{{x}_{n}\right\}}_{n}$ in X is said to be convergent to x in X if for every $ϵ>0$ and $\lambda >0$, there exists a positive integer N such that ${F}_{{x}_{n}x}\left(ϵ\right)>1-\lambda$ whenever $n\ge N$.

2. (2)

A sequence ${\left\{{x}_{n}\right\}}_{n}$ in X is called a Cauchy sequence if for every $ϵ>0$ and $\lambda >0$, there exists a positive integer N such that ${F}_{{x}_{n}{x}_{m}}\left(ϵ\right)>1-\lambda$ whenever $n,m\ge N$.

3. (3)

The space X is said to be complete if every Cauchy sequence in X is convergent to a point in X.

The $\left(ϵ,\lambda \right)$-topology  in a Menger space $\left(X,F,T\right)$ is introduced by the family of neighborhoods ${N}_{x}$ of a point $x\in X$ given by
${N}_{x}=\left\{{N}_{x}\left(ϵ,\lambda \right):ϵ>0,\lambda \in \left(0,1\right)\right\},$
where
${N}_{x}\left(ϵ,\lambda \right)=\left\{y\in X:{F}_{xy}\left(ϵ\right)>1-\lambda \right\}.$

The $\left(ϵ,\lambda \right)$-topology is a Hausdorff topology. In this topology the function f is continuous in ${x}_{0}\in X$ if and only if for every sequence ${x}_{n}\to {x}_{0}$ it holds that $f\left({x}_{n}\right)\to f\left({x}_{0}\right)$.

Definition 2.4 (Altering distance function )

The control function $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is called an altering distance function if it has the following properties:
1. (i)

ψ is monotone increasing and continuous,

2. (ii)

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

The following category of functions was introduced in .

Definition 2.5 

A function $\varphi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is said to be a Φ-function if it satisfies the following conditions:
1. (i)

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

2. (ii)

$\varphi \left(t\right)$ is strictly monotone increasing and $\varphi \left(t\right)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$,

3. (iii)

ϕ is left continuous in $\left(0,\mathrm{\infty }\right)$,

4. (iv)

ϕ is continuous at 0.

The class of all Φ-functions will be denoted by Φ.

Lemma 2.1 

Let $\left(X,F,T\right)$ be a Menger PM-space. Let $\varphi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be a Φ-function. Then the following statement holds.

If for $x,y\in X$, $0, we have ${F}_{xy}\left(\varphi \left(t\right)\right)\ge {F}_{xy}\left(\varphi \left(t/c\right)\right)$ for all $t>0$, then $x=y$.

Theorem 2.1 

Let $\left(X,F,T\right)$ be a complete Menger PM-space with a continuous t-norm T which satisfies $T\left(a,a\right)\ge a$ for every $a\in \left[0,1\right]$. Let $c\in \left(0,1\right)$ be fixed. If for a Φ-function ϕ and a self-mapping f on X,
$\begin{array}{rl}{F}_{fxfy}\left(\varphi \left(t\right)\right)\ge & min\left\{{F}_{xy}\left(\varphi \left(\frac{t}{c}\right)\right),{F}_{xfx}\left(\varphi \left(\frac{t}{c}\right)\right),{F}_{yfy}\left(\varphi \left(\frac{t}{c}\right)\right),\\ {F}_{xfy}\left(2\varphi \left(\frac{t}{c}\right)\right),{F}_{yfx}\left(2\varphi \left(\frac{t}{c}\right)\right)\right\}\end{array}$
(2.1)

holds for every $x,y\in X$ and all $t>0$, then f has a unique fixed point in X.

## 3 Main results

We begin with the following example.

Example 3.1 Let $X=\left[0,\mathrm{\infty }\right)$ and $T={T}_{p}$. For each $t\in \left(0,\mathrm{\infty }\right)$, define
${F}_{xy}\left(t\right)=\left\{\begin{array}{ll}\frac{min\left\{x,y\right\}}{max\left\{x,y\right\}},& \mathrm{\forall }t>0,x\ne y,\\ 1,& \mathrm{\forall }t>0,x=y.\end{array}$
It is clear that $\left(X,F,{T}_{p}\right)$ is a complete Menger PM-space (see ) (but here $\left(X,F,{T}_{M}\right)$ is not a PM-space). Let us consider the function
$f:X\to X,\phantom{\rule{2em}{0ex}}fx=x+1,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in X.$

Then it can be easily seen that the above example satisfies Theorem 2.1 for ${T}_{p}$. Indeed,

• Case I. If $x=y$, then inequality (2.1) is obviously true.

• Case II. If $x\ne y$ and $x, then we have $x+1, $x, $y, $x.

For any function $\varphi \in \mathrm{\Phi }$ and $c\in \left(0,1\right)$, inequality (2.1) becomes
$\frac{x+1}{y+1}\ge min\left\{\frac{x}{y},\frac{x}{x+1},\frac{y}{y+1},\frac{x}{y+1},\frac{min\left\{y,x+1\right\}}{max\left\{y,x+1\right\}}\right\}.$
If $x+1, then we have
$\frac{x+1}{y+1}\ge min\left\{\frac{x}{y},\frac{x}{x+1},\frac{y}{y+1},\frac{x}{y+1},\frac{x+1}{y}\right\}.$

Clearly, the inequality holds with the minimum value $x/\left(y+1\right)$.

If $x+1=y$, then we have
$\frac{x+1}{y+1}\ge min\left\{\frac{x}{y},\frac{x}{x+1},\frac{y}{y+1},\frac{x}{y+1},1\right\}.$

Then the inequality holds with the minimum value $x/\left(y+1\right)$.

And if $x+1>y$, then we have
$\frac{x+1}{y+1}\ge min\left\{\frac{x}{y},\frac{x}{x+1},\frac{y}{y+1},\frac{x}{y+1},\frac{y}{x+1}\right\}.$

The inequality holds with the minimum value $x/\left(y+1\right)$.

• Case III. If $x\ne y$ and $x>y$, then we have $x+1>y+1$, $x+1>y$.

By inequality (2.1), we have
$\frac{y+1}{x+1}\ge min\left\{\frac{y}{x},\frac{x}{x+1},\frac{y}{y+1},\frac{min\left\{x,y+1\right\}}{max\left\{x,y+1\right\}},\frac{y}{x+1}\right\}.$
If $x>y+1$, then we have
$\frac{y+1}{x+1}\ge min\left\{\frac{y}{x},\frac{x}{x+1},\frac{y}{y+1},\frac{y+1}{x},\frac{y}{x+1}\right\}.$

The inequality holds with the minimum value $y/\left(x+1\right)$.

If $x=y+1$, then we have
$\frac{y+1}{x+1}\ge min\left\{\frac{y}{x},\frac{x}{x+1},\frac{y}{y+1},1,\frac{y}{x+1}\right\}.$

The inequality holds with the minimum value $y/\left(x+1\right)$.

Finally, if $x, then we have
$\frac{y+1}{x+1}\ge min\left\{\frac{y}{x},\frac{x}{x+1},\frac{y}{y+1},\frac{x}{y+1},\frac{y}{x+1}\right\}.$

Again inequality holds with the minimum value $y/\left(x+1\right)$.

Thus the above example satisfies all the conditions of Theorem 2.1 with the t-norm ${T}_{p}$, but here the mapping f has no fixed point. Therefore, Theorem 2.1 cannot be generalized for $T\le {T}_{p}$.

Now, we are motivated to introduce our result.

Theorem 3.1 Let $\left(X,F,T\right)$ be a complete Menger PM-space with a continuous t-norm, and let $c\in \left(0,1\right)$ be fixed. If for a Φ-function ϕ and a self-mapping f on X,
$\begin{array}{rl}{F}_{fxfy}\left(\varphi \left(t\right)\right)\ge & min\left\{{F}_{xy}\left(\varphi \left(\frac{t}{c}\right)\right),{F}_{xfx}\left(\varphi \left(\frac{t}{c}\right)\right),\\ {F}_{yfy}\left(\varphi \left(\frac{t}{c}\right)\right),{F}_{yfx}\left(\varphi \left(\frac{t}{c}\right)\right)\right\}\end{array}$
(3.1)

holds for every $x,y\in X$ and all $t>0$, then f has a unique fixed point in X.

Proof Let ${x}_{0}\in X$. Now, construct a sequence $\left\{{x}_{n}\right\}$ in X as follows:
${x}_{n}=f{x}_{n-1},\phantom{\rule{1em}{0ex}}n=1,2,\dots .$
Applying (3.1) for $x={x}_{n-1}$ and $y={x}_{n}$, we have
$\begin{array}{rl}{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(t\right)\right)& ={F}_{f{x}_{n-1}f{x}_{n}}\left(\varphi \left(t\right)\right)\\ \ge min\left\{{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(t/c\right)\right),{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(t/c\right)\right),{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(t/c\right)\right),{F}_{{x}_{n}{x}_{n}}\left(\varphi \left(t/c\right)\right)\right\}\\ =min\left\{{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(t/c\right)\right),{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(t/c\right)\right)\right\}\end{array}$

for all $t>0$.

We should prove that
$min\left\{{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(t/c\right)\right),{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(t/c\right)\right)\right\}={F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(t/c\right)\right)$
(3.2)

for all $t>0$.

If it is not, there exists $p>0$ such that
${F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/c\right)\right)>{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/c\right)\right).$
(3.3)
Then, using (3.1), we have
$\begin{array}{rl}{F}_{{x}_{n+1}{x}_{n}}\left(\varphi \left(p\right)\right)& \ge {F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/c\right)\right)\\ \ge min\left\{{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/{c}^{2}\right)\right),{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{2}\right)\right)\right\}.\end{array}$
If
$min\left\{{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/{c}^{2}\right)\right),{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{2}\right)\right)\right\}={F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/{c}^{2}\right)\right),$
(3.4)
then by (3.3) and (3.4)
${F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/c\right)\right)>{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/c\right)\right)\ge {F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/{c}^{2}\right)\right).$

Accordingly, $min\left\{{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/{c}^{2}\right)\right),{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{2}\right)\right)\right\}={F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{2}\right)\right)$.

Now, we have
$\begin{array}{rl}{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p\right)\right)& \ge {F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/c\right)\right)\\ \ge {F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{2}\right)\right)\\ \ge min\left\{{F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/{c}^{3}\right)\right),{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{3}\right)\right)\right\}.\end{array}$
Repeating the same procedure, we conclude that
${F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(p/c\right)\right)>{F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/c\right)\right)\ge {F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{2}\right)\right)\ge \dots \ge {F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{k}\right)\right).$
Since ${F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(p/{c}^{k}\right)\right)\to 1$, $k\to \mathrm{\infty }$, we get a contradiction. Accordingly, (3.2) is true. Therefore,
${F}_{{x}_{n}{x}_{n+1}}\left(\varphi \left(t\right)\right)\ge {F}_{{x}_{n-1}{x}_{n}}\left(\varphi \left(t/c\right)\right)\ge {F}_{{x}_{1}{x}_{0}}\left(\varphi \left(t/{c}^{n}\right)\right)\to 1$

as $n\to \mathrm{\infty }$.

By the property of ϕ, given $s>0$, there exists $t>0$ such that $s>\varphi \left(t\right)$. Thus,
(3.5)

for all $s>0$.

Now, we claim that $\left\{{x}_{n}\right\}$ is a Cauchy sequence. If not, then $\mathrm{\exists }ϵ>0$ and $\lambda >0$, and subsequences $\left\{{x}_{m\left(k\right)}\right\}$ and $\left\{{x}_{n\left(k\right)}\right\}$ such that $m\left(k\right) and
${F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)}}\left(ϵ\right)<1-\lambda ,$
(3.6)
${F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)-1}}\left(ϵ\right)\ge 1-\lambda .$
(3.7)
Since F is non-decreasing, we have
$\left\{x:{F}_{xp}\left({ϵ}^{″}\right)\ge 1-\lambda \right\}\subseteq \left\{x:{F}_{xp}\left(ϵ\right)\ge 1-\lambda \right\}$

for all $p\in X$, $\lambda >0$, and $0<{ϵ}^{″}<ϵ$. It follows that whenever the above construction is possible for $ϵ>0$, $\lambda >0$, it is also possible to construct $\left\{{x}_{m\left(k\right)}\right\}$ and $\left\{{x}_{n\left(k\right)}\right\}$ satisfying (3.6) and (3.7) corresponding to ${ϵ}^{″}>0$, $\lambda >0$ whenever ${ϵ}^{″}<ϵ$.

Since ϕ is continuous at 0 and strictly monotonic increasing with $\varphi \left(0\right)=0$, it is possible to obtain ${ϵ}_{1}>0$ such that $\varphi \left({ϵ}_{1}\right)<ϵ$. Then, by the above argument, it is possible to obtain increasing sequences of integers $m\left(k\right)$ and $n\left(k\right)$ with $m\left(k\right) such that
${F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)}}\left(\varphi \left({ϵ}_{1}\right)\right)<1-\lambda ,$
(3.8)
${F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}\right)\right)\ge 1-\lambda .$
(3.9)
Since $0 and $\varphi \in \mathrm{\Phi }$, we can choose $\eta >0$ such that $0<\eta <\varphi \left({ϵ}_{1}/c\right)-\varphi \left({ϵ}_{1}\right)$. Since ϕ is strictly increasing, therefore
$\varphi \left({ϵ}_{1}/c\right)-\eta >\varphi \left({ϵ}_{1}\right).$
From (3.9), we get
${F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}/c\right)-\eta \right)>{F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)-1}}\varphi \left({ϵ}_{1}\right)\ge 1-\lambda .$
(3.10)
By (3.5), for ${\lambda }_{1}<\lambda <1$, it is possible to find a positive integer ${N}_{1}$ such that for all $k>{N}_{1}$,
$\begin{array}{l}{F}_{{x}_{m\left(k\right)}{x}_{m\left(k\right)-1}}\varphi \left(\eta \right)\ge 1-{\lambda }_{1},\\ {F}_{{x}_{n\left(k\right)}{x}_{n\left(k\right)-1}}\varphi \left(\eta \right)\ge 1-{\lambda }_{1}.\end{array}\right\}$
(3.11)
By (PM3), we have
${F}_{{x}_{m\left(k\right)-1}{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}/c\right)\right)\ge T\left({F}_{{x}_{m\left(k\right)-1}{x}_{m\left(k\right)}}\left(\eta \right),{F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}/c\right)-\eta \right)\right).$
(3.12)
Let $0<{\lambda }_{2}<{\lambda }_{1}<\lambda <1$ be arbitrary. Then by (3.5) there exists a positive integer ${N}_{2}$ such that for all $k>{N}_{2}$,
${F}_{{x}_{m\left(k\right)-1}{x}_{m\left(k\right)}}\left(\eta \right)\ge 1-{\lambda }_{2}.$
(3.13)
Now, using (3.10), (3.12), and (3.13), we have, for all $k>max\left\{{N}_{1},{N}_{2}\right\}$,
${F}_{{x}_{m\left(k\right)-1}{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}/c\right)\right)\ge T\left(1-{\lambda }_{2},1-\lambda \right).$
As ${\lambda }_{2}$ is arbitrary and T is continuous, we have
${F}_{{x}_{m\left(k\right)-1}{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}/c\right)\right)\ge T\left(1,1-\lambda \right)=1-\lambda .$
(3.14)
Now, using (3.1), (3.9), (3.11), and (3.14), we have
$\begin{array}{rl}1-\lambda >& {F}_{{x}_{m\left(k\right)}{x}_{n\left(k\right)}}\left(\varphi \left({ϵ}_{1}\right)\right)\\ =& {F}_{f{x}_{m\left(k\right)-1}f{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}\right)\right)\\ \ge & min\left\{{F}_{{x}_{m\left(k\right)-1}{x}_{n\left(k\right)-1}}\left(\varphi \left({ϵ}_{1}/c\right)\right),{F}_{{x}_{m\left(k\right)-1}{x}_{m\left(k\right)}}\left(\varphi \left({ϵ}_{1}/c\right)\right),\\ {F}_{{x}_{n\left(k\right)-1}{x}_{n\left(k\right)}}\left(\varphi \left({ϵ}_{1}/c\right)\right),{F}_{{x}_{n\left(k\right)-1}{x}_{m\left(k\right)}}\left(\varphi \left({ϵ}_{1}/c\right)\right)\right\}\\ \ge & min\left\{1-\lambda ,1-\lambda ,1-\lambda ,1-\lambda \right\}\\ =& 1-\lambda ,\end{array}$

which is a contradiction. Therefore $\left\{{x}_{n}\right\}$ is a Cauchy sequence in a complete Menger PM-space X, thus there exists $z\in X$ such that $z={lim}_{n\to \mathrm{\infty }}{x}_{n}$.

Now, we will show that z is a fixed point of f. Since $\varphi \in \mathrm{\Phi }$, we have that for every $x,y\in X$ and all $s>0$, there exists $r>0$ such that $s>\varphi \left(r\right)$ and ${n}_{0}\in N$ such that for all $n\ge {n}_{0}$,
${F}_{fzz}\left(s\right)\ge T\left({F}_{fz{x}_{n}}\left(\varphi \left(r\right)\right),{F}_{{x}_{n}z}\left(s-\varphi \left(r\right)\right)\right).$
(3.15)
Since $s>\varphi \left(r\right)$, thus $\left(s-\varphi \left(r\right)\right)>0$. Also, since $z={lim}_{n\to \mathrm{\infty }}{x}_{n}$, for arbitrary $\delta \in \left(0,1\right)$, we have
${F}_{{x}_{n}z}\left(s-\varphi \left(r\right)\right)>1-\delta .$
(3.16)
Hence, from (3.15) and (3.16), we get
${F}_{fzz}\left(s\right)\ge T\left({F}_{fz{x}_{n}}\left(\varphi \left(r\right)\right),1-\delta \right).$
Since $\delta >0$ is arbitrary and the t-norm T is continuous, we get
$\begin{array}{rl}{F}_{fzz}\left(s\right)& \ge {F}_{fz{x}_{n}}\left(\varphi \left(r\right)\right)\\ \ge {F}_{fzf{x}_{n-1}}\left(\varphi \left(r\right)\right)\\ \ge min\left\{{F}_{z{x}_{n-1}}\left(\varphi \left(r/c\right)\right),{F}_{zfz}\left(\varphi \left(r/c\right)\right),{F}_{{x}_{n-1}f{x}_{n-1}}\left(\varphi \left(r/c\right)\right),{F}_{{x}_{n-1}fz}\left(\varphi \left(r/c\right)\right)\right\}.\end{array}$
Letting $n\to \mathrm{\infty }$ in the above inequality and using the fact that the t-norm T is continuous, we obtain
${F}_{fzz}\left(\varphi \left(r\right)\right)\ge {F}_{zfz}\left(\varphi \left(r/c\right)\right)$

and applying Lemma 2.1, we get $z=fz$.

Next, we prove the uniqueness of a fixed point. Let $w\in X$ be another fixed point of f, i.e., $fw=w$. Since $\varphi \in \mathrm{\Phi }$, for all $s>0$, there exists $r>0$ such that $s>\varphi \left(r\right)$. Then we have
$\begin{array}{rl}{F}_{zw}\left(s\right)& \ge {F}_{zw}\left(\varphi \left(r\right)\right)\\ ={F}_{fzfw}\left(\varphi \left(r\right)\right)\\ \ge min\left\{{F}_{zw}\left(\varphi \left(r/c\right)\right),{F}_{zfz}\left(\varphi \left(r/c\right)\right),{F}_{wfw}\left(\varphi \left(r/c\right)\right),{F}_{wfz}\left(\varphi \left(r/c\right)\right)\right\}\\ =min\left\{{F}_{zw}\left(\varphi \left(r/c\right)\right),{F}_{zz}\left(\varphi \left(r/c\right)\right),{F}_{ww}\left(\varphi \left(r/c\right)\right),{F}_{wz}\left(\varphi \left(r/c\right)\right)\right\}\\ ={F}_{zw}\left(\varphi \left(r/c\right)\right).\end{array}$

From Lemma 2.1, it follows that $z=w$, i.e., z is the unique fixed point of f. □

## 4 Connection with metric spaces

It is well known that every metric space $\left(X,d\right)$ is also a Menger space $\left(X,F,{T}_{M}\right)$ if F is defined in the following way:
${F}_{xy}\left(t\right)=\left\{\begin{array}{ll}1,& d\left(x,y\right)
If ψ is an altering distance function defined in Definition 2.4 with additional property $\psi \left(t\right)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$, then the function
$\varphi \left(t\right)=\left\{\begin{array}{ll}inf\left\{\alpha :\psi \left(\alpha \right)\ge t\right\},& t>0,\\ 0,& t=0\end{array}$

is a Φ-function (see ).

We will present that (3.1) in this case implies
$\psi \left(d\left(fx,fy\right)\right)\le cmax\left\{\psi \left(d\left(fx,x\right)\right),\psi \left(d\left(fy,y\right)\right),\psi \left(d\left(x,y\right)\right),\psi \left(d\left(fx,y\right)\right)\right\},$

$c\in \left(0,1\right)$ and $x,y\in X$ in a metric space.

Suppose the contrary, i.e., there exists $t>0$ such that ${F}_{fxfy}\left(\varphi \left(t\right)\right)=0$ and all of
${F}_{xfx}\left(\varphi \left(t/c\right)\right)=1,\phantom{\rule{2em}{0ex}}{F}_{yfy}\left(\varphi \left(t/c\right)\right)=1,\phantom{\rule{2em}{0ex}}{F}_{x,y}\left(\varphi \left(t/c\right)\right)=1,\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{F}_{fxy}\left(\varphi \left(t/c\right)\right)=1.$
So, ${F}_{fxfy}\left(\varphi \left(t\right)\right)=0$ implies that $d\left(fx,fy\right)\ge \varphi \left(t\right)$, and since ψ is continuous, we have
$\psi \left(d\left(fx,fy\right)\right)\ge t.$
Similarly, since ${F}_{xfx}\left(\varphi \left(t/c\right)\right)=1$, we have that $d\left(fx,x\right)<\varphi \left(t/c\right)$, which implies
$\psi \left(d\left(fx,x\right)\right)<\frac{t}{c}.$
Also, we have the following:
$\psi \left(d\left(fy,y\right)\right)<\frac{t}{c},\phantom{\rule{2em}{0ex}}\psi \left(d\left(x,y\right)\right)<\frac{t}{c},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\psi \left(d\left(fx,y\right)\right)<\frac{t}{c}.$
Thus, we have
$\psi \left(d\left(fx,fy\right)\right)>cmax\left\{\psi \left(d\left(fx,x\right)\right),\psi \left(d\left(fy,y\right)\right),\psi \left(d\left(x,y\right)\right),\psi \left(d\left(fx,y\right)\right)\right\},$

## Declarations

### Acknowledgements

The authors would like to express their thanks to the referees for their helpful comments and suggestions. The first and fifth author are supported by MNTRRS-174009. The third author gratefully acknowledges the support from the CSIR, Govt. of India, Grant No.-25(0215)/13/EMR-II. The fourth author is thankful to S. V. National Institute of Technology, Surat, India for awarding Senior Research Fellow.

## Authors’ Affiliations

(1) 