# $(\psi ,\phi ,\u03f5,\lambda )$-Contraction theorems in probabilistic metric spaces for single valued case

- Arman Beitollahi
^{1}Email author and - Parvin Azhdari
^{2}

**2013**:109

https://doi.org/10.1186/1687-1812-2013-109

© Beitollahi and Azhdari; licensee Springer 2013

**Received: **20 April 2012

**Accepted: **9 April 2013

**Published: **23 April 2013

## Abstract

In this article, we prove some fixed-point theorems for $(\psi ,\phi ,\u03f5,\lambda )$-contraction in probabilistic metric spaces for single valued case. We will generalize the definition of $(\psi ,\phi ,\u03f5,\lambda )$-contraction and present fixed-point theorem in the generalized $(\psi ,\phi ,\u03f5,\lambda )$-contraction.

## Keywords

## 1 Introduction

The probabilistic metric space was introduced by Menger [1]. Mihet presented the class of $(\psi ,\phi ,\u03f5,\lambda )$-contraction for a single valued case in fuzzy metric spaces [2, 3]. This class is a generalization of the $(\u03f5,\lambda )$-contraction which was introduced in [4]. We defined the class of $(\psi ,\phi ,\u03f5,\lambda )$-contraction for the multi-valued case in a probabilistic metric space before [5]. Now, we obtain two fixed-point theorems of $(\psi ,\phi ,\u03f5,\lambda )$-contraction for single valued case. Also, we extend the concept of $(\psi ,\phi ,\u03f5,\lambda )$-contraction to the generalized $(\psi ,\phi ,\u03f5,\lambda )$-contraction.

The structure of this paper is as follows: Section 2 is a review of some concepts in probabilistic metric spaces and probabilistic contractions. In Section 3, we will show two theorems for $(\psi ,\phi ,\u03f5,\lambda )$-contraction in the single-valued case and explain the generalized $(\psi ,\phi ,\u03f5,\lambda )$-contraction.

## 2 Preliminary notes

We recall some concepts from probabilistic metric space, convergence and contraction. For more details, we refer the reader to [6–8].

Let ${D}_{+}$ be the set of all distribution of functions *F* such that $F(0)=0$ (*F* is a non-decreasing, left continuous mapping from ℝ into $[0,1]$ such that ${lim}_{x\to \mathrm{\infty}}F(x)=1$).

*S*is a nonempty set and $F:S\times S\to {D}_{+}$ ($F(p,q)$ written by ${F}_{pq}$ for every $(p,q)\in S\times S$) satisfies the following conditions:

- (1)
${F}_{uv}(x)=1$ for every $x>0\iff u=v$ ($u,v\in S$),

- (2)
${F}_{uv}={F}_{vu}$ for every $u,v\in S$,

- (3)
${F}_{uv}(x)=1$ and ${F}_{vw}(y)=1\Rightarrow {F}_{u,w}(x+y)=1$ for every $u,v,w\in S$, and every $x,y\in {\mathbb{R}}^{+}$.

*T*is a triangular norm (abbreviated

*t*-norm) and the following inequality holds:

Recall the mapping $T:[0,1]\times [0,1]\to [0,1]$ is called a triangular norm (a *t*-norm) if the following conditions are satisfied: $T(a,1)=a$ for every $a\in [0,1]$; $T(a,b)=T(b,a)$ for every $a,b\in [0,1]$; $a\ge b,c\ge d\Rightarrow T(a,c)\ge T(b,d)$, $a,b,c,d\in [0,1]$; $T(T(a,b),c)=T(a,T(b,c))$, $a,b,c\in [0,1]$. Basic examples of *t*-norms are ${T}_{L}$ (Lukasiewicz *t*-norm), ${T}_{P}$ and ${T}_{M}$, defined by ${T}_{L}(a,b)=max\{a+b-1,0\}$, ${T}_{P}(a,b)=ab$ and ${T}_{M}(a,b)=min\{a,b\}$. If *T* is a *t*-norm and $({x}_{1},{x}_{2},\dots ,{x}_{n})\in {[0,1]}^{n}$ ($n\in {\mathbb{N}}^{\ast}$), one can define recurrently ${\mathrm{\top}}_{i=1}^{n}{x}_{i}=T({\mathrm{\top}}_{i=1}^{n-1}{x}_{i},{x}_{n})$ for all $n\ge 2$. One can also extend *T* to a countable infinitary operation by defining ${\mathrm{\top}}_{i=1}^{\mathrm{\infty}}{x}_{i}$ for any sequence ${({x}_{i})}_{i\in {\mathbb{N}}^{\ast}}$ as ${lim}_{n\to \mathrm{\infty}}{\mathrm{\top}}_{i=1}^{n}{x}_{i}$.

*t*-norm

*T*is

*q*-convergent if ${lim}_{n\to \mathrm{\infty}}{\mathrm{\top}}_{i=n}^{\mathrm{\infty}}(1-{q}^{i})=1$. We remark that if

*T*is

*q*-convergent, then

Also, note that if the *t*-norm *T* is *q*-convergent, then ${sup}_{0\le t<1}T(t,t)=1$.

**Proposition 2.1**

*Let*$(S,F,T)$

*be a Menger space*.

*If*${sup}_{0\le t<1}T(t,t)=1$,

*then the family*${\{{U}_{\u03f5}\}}_{\u03f5>0}$,

*where*

*is a base for a metrizable uniformity on* *S*, *called the* *F*-*uniformity* [6–8]. *The* *F*-*uniformity naturally determines a metrizable topology on* *S*, *called the strong topology or* *F*-*topology* [9], *a subset* *O* *of* *S* *is* *F*-*open if for every* $p\in O$ *there exists* $t>0$ *such that* ${N}_{p}=\{q\in S|{F}_{pq}(t)>1-t\}\subset O$.

**Definition 2.1** [6]

A sequence ${({x}_{n})}_{n\in \mathrm{N}}$ is called an F-convergent sequence to $x\in S$ if for every $\u03f5>0$ and $\lambda \in (0,1)$ there exists $N=N(\u03f5,\lambda )\in \mathrm{N}$ such that ${F}_{{x}_{n}x}(\u03f5)>1-\lambda $, $\mathrm{\forall}n\ge N$.

**Definition 2.2** [6]

*t*-norm

*T*is

*φ*-convergent if

**Definition 2.3** [6]

A sequence ${({x}_{n})}_{n\in \mathrm{N}}$ is called a convergent sequence to $x\in S$ if for every $\u03f5>0$ and $\lambda \in (0,1)$ there exists $N=N(\u03f5,\lambda )\in \mathrm{N}$ such that ${F}_{{x}_{n}x}(\u03f5)>1-\lambda $, $\mathrm{\forall}n\ge N$.

**Definition 2.4** [6]

A sequence ${({x}_{n})}_{n\in \mathrm{N}}$ is called a Cauchy sequence if for every $\u03f5>0$ and $\lambda \in (0,1)$ there exists $N=N(\u03f5,\lambda )\in \mathrm{N}$ such that ${F}_{{x}_{n}{x}_{n+m}}(\u03f5)>1-\lambda $, $\mathrm{\forall}n\ge N$, $\mathrm{\forall}m\in \mathbb{N}$.

A probabilistic metric space $(S,F,T)$ is called sequentially complete if every Cauchy sequence is convergent.

The concept of $(\psi ,\phi ,\u03f5,\lambda )$-contraction has been introduced by Mihet [3].

*ϕ*of all mapping $\phi :(0,1)\to (0,1)$ with the properties:

- (1)
*φ*is an increasing bijection; - (2)
$\phi (\lambda )<\lambda $ $\mathrm{\forall}\lambda \in (0,1)$.

Since every such a comparison mapping is continuous, if $\phi \in \varphi $, then ${lim}_{n\to \mathrm{\infty}}{\phi}^{n}(\lambda )=0$ $\mathrm{\forall}\lambda \in (0,1)$.

**Definition 2.5** [3]

*ψ*be a map from $(0,\mathrm{\infty})$ to $(0,\mathrm{\infty})$. A mapping $f:S\to S$ is called a $(\psi ,\phi ,\u03f5,\lambda )$-contraction on

*S*if it satisfies in the following condition:

In the rest of paper we suppose that *ψ* is increasing bijection.

**Example 2.1**Let $S=\{0,1,2,\dots \}$ and (for $x\ne y$)

Suppose that $f:S\to S$, $f(r)=r+1$.

Then $(S,F,{T}_{L})$ is a probabilistic metric space [10].

*x*,

*y*,

*ϵ*,

*λ*be such that ${F}_{x,y}(\u03f5)>1-\lambda $.

- (i)
If ${2}^{-min(x,y)<\u03f5\le 1}$, then $1-{2}^{-min(x,y)}>1-\lambda $.

This implies $1-{2}^{-min(x+1,y+1)}>1-\frac{1}{2}\lambda $, that is,${F}_{fx,fy}(\u03f5)>1-\frac{1}{2}\lambda .$ - (ii)
If $\u03f5>1$ then ${F}_{fx,fy}(\u03f5)=1$, hence again ${F}_{fx,fy}(\u03f5)>1-\frac{1}{2}\lambda $. Thus, the mapping

*f*is a $(\psi ,\phi ,\u03f5,\lambda )$-contraction on*S*with $\psi (\u03f5)=\u03f5$ and $\phi (\lambda )=\frac{1}{2}\lambda $.

## 3 Main results

In this section, we will show $(\psi ,\phi ,\u03f5,\lambda )$-contraction is continuous. By using this assumption, we will also prove two theorems.

**Definition 3.1**Let

*F*be a probabilistic distance on

*S*. A mapping $f:S\to S$ is called continuous if for every $\u03f5>0$ there exists $\delta >0$ such that

Before we start to present the theorems, we will explain the following lemma.

**Lemma 3.1** *Every* $(\psi ,\phi ,\u03f5,\lambda )$-*contraction is continuous*.

*Proof* Suppose that $\u03f5>0$ be given and $\delta \in (0,1)$ be such that $\delta <min\{\u03f5,{\psi}^{-1}(\u03f5)\}$ and since *ψ* is increasing bijection then $\psi (\delta )<\u03f5$. If ${F}_{x,y}(\delta )>1-\delta $ then, by $(\psi ,\phi ,\u03f5,\lambda )$-contraction we have ${F}_{fx,fy}(\psi (\delta ))>1-\phi (\delta )$, from where we obtain that ${F}_{fx,fy}(\u03f5)>{F}_{fx,fy}(\psi (\delta ))>1-\phi (\delta )>1-\delta >1-\u03f5$. So *f* is continuous. □

**Theorem 3.1** *Let* $(S,F,T)$ *be a complete Menger space and* *T* *a* *t*-*norm satisfies in* ${sup}_{0\le a<1}T(a,a)=1$. *Also*, $f:S\to S$ *a* $(\psi ,\phi ,\u03f5,\lambda )$-*contraction where* ${lim}_{n\to \mathrm{\infty}}{\psi}^{n}(\delta )=0$ *for every* $\delta \in (0,\mathrm{\infty})$. *If* ${lim}_{t\to \mathrm{\infty}}{F}_{{x}_{0},{f}^{m}{x}_{0}}(t)=1$ *for some* ${x}_{0}\in S$ *and all* $m\in N$, *then there exists a unique fixed point* *x* *of the mapping* *f* *and* $x={lim}_{n\to \mathrm{\infty}}{f}^{n}({x}_{0})$.

*Proof* Let ${x}_{n}={f}^{n}{x}_{0}$, $n\in N$. We shall prove that ${({x}_{n})}_{n\in \mathbb{N}}$ is a Cauchy sequence.

*n*such that ${\psi}^{n}(\eta )<\u03f5$ and ${\phi}^{n}(\xi )<\lambda $, we obtain

Hence, ${({x}_{n})}_{n\in \mathbb{N}}$ is a Cauchy sequence and since *S* is complete, it follows the existence of $x\in S$ such that $x={lim}_{n\to \mathrm{\infty}}{x}_{n}$. By continuity of *f* and ${x}_{n+1}=f{x}_{n}$ for every $n\in \mathbb{N}$, when $n\to \mathrm{\infty}$, we obtain that $x=fx$. □

**Example 3.1**Let $(S,F,T)$ be a complete Menger space where $S=\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\}$, $T(a,b)=min\{a,b\}$ and ${F}_{xy}(t)$ is defined as

$f:S\to S$ is given by $f({x}_{1})=f({x}_{2})={x}_{2}$ and $f({x}_{3})=f({x}_{4})={x}_{1}$. If we take $\phi (\lambda )=\frac{\lambda}{2}$, $\psi (\u03f5)=\frac{\u03f5}{2}$, then *f* is a $(\psi ,\phi ,\u03f5,\lambda )$-contraction where ${lim}_{n\to \mathrm{\infty}}{\psi}^{n}(\delta )={lim}_{n\to \mathrm{\infty}}\frac{\delta}{{2}^{n}}=0$ for every $\delta \in (0,\mathrm{\infty})$ and if we set ${x}_{0}={x}_{2}$, then for all $m\in N$, we have ${f}^{m}{x}_{0}={f}^{m}{x}_{2}={x}_{2}$ and ${lim}_{t\to \mathrm{\infty}}{F}_{{x}_{2}{x}_{2}}(t)=1$, so ${x}_{2}$ is the unique fixed point for *f*.

**Theorem 3.2**

*Let*$(S,F,T)$

*be a complete Menger space*,

*T*

*be a*

*t*-

*norm such that*${sup}_{0\le a<1}T(a,a)=1$

*and*$f:S\to S$

*a*$(\psi ,\phi ,\u03f5,\lambda )$-

*contraction where the series*${\sum}_{i}{\psi}^{i}(\delta )$

*is convergent for all*$\delta >0$

*and suppose that for some*$p\in S$

*and*$j>0$

*If* *t*-*norm* *T* *is* *φ*-*convergent*, *then there exist a unique fixed point* *z* *of mapping* *f* *and* $z={lim}_{l\to \mathrm{\infty}}{f}^{l}p$.

*Proof*Choose $\u03f5>0$ and $\lambda \in (0,1)$. Let ${z}_{l}={f}^{l}p$, $l\in N$. We shall prove that ${({z}_{l})}_{l\in N}$ is a Cauchy sequence. It means we prove that there exists ${n}_{0}(\u03f5,\lambda )\in N$ such that

Since *f* is $(\psi ,\phi ,\u03f5,\lambda )$-contraction, we derived by induction ${F}_{{f}^{l}p,{f}^{l+1}p}({\psi}^{l}(\frac{1}{{\mu}^{{n}_{1}}}))>1-{\phi}^{l}(1-M{({\mu}^{j})}^{{n}_{1}})$ $\mathrm{\forall}l>1$. Since the series ${\sum}_{i=1}^{\mathrm{\infty}}{\psi}^{i}(\delta )$ is convergent, there exists ${n}_{2}={n}_{2}(\u03f5)\in N$ such that ${\sum}_{i=l}^{\mathrm{\infty}}{\psi}^{i}(\delta )\le \u03f5$ $\mathrm{\forall}l\ge {n}_{2}$. We know ${\sum}_{i=l}^{\mathrm{\infty}}{\psi}^{i}(\frac{1}{{\mu}^{{n}_{1}}})\le \u03f5$ for every $l>max\{{n}_{1},{n}_{2}\}$.

Since *T* is *φ*-convergent, we conclude that ${({f}^{l}p)}_{l\in \mathbb{N}}$ is a Cauchy sequence. On the other hand, *S* is complete, therefore, there is a $z\in S$ such that $z={lim}_{l\to \mathrm{\infty}}{f}^{l}p$. By the continuity of the mapping *f* and ${z}_{l+1}=f{z}_{l}$ when $l\to +\mathrm{\infty}$, it follows that $fz=z$. □

**Example 3.2** Let $(S,F,T)$ and the mappings *f*, *ψ* and *φ* be the same as in Example 3.1. Since ${\sum}_{i}{\psi}^{i}(\delta )={\sum}_{i}\frac{\delta}{{2}^{i}}=\delta $ for all $\delta >0$ and if we set $p={x}_{2}\in S$, $j>0$ then ${t}^{j}(1-{F}_{{x}_{2}{x}_{2}}(t))=0$ for every $t>0$ or ${sup}_{t>0}{t}^{j}(1-{F}_{{x}_{2}{x}_{2}}(t))<\mathrm{\infty}$, so ${x}_{2}$ is the unique fixed point for *f*.

Mihet in [3] showed, if $f:S\to S$ is a $(\psi ,\phi ,\u03f5,\lambda )$-contraction and
is a complete fuzzy metric space, then *f* has an unique fixed point. Now we present a generalization of the $(\psi ,\phi ,\u03f5,\lambda )$-contraction. First, we define the class of functions ℵ as follows.

- (1)
$\mathrm{\forall}t,s\ge 0:m(t+s)\ge m(t)+m(s)$;

- (2)
$m(t)=0\iff t=0$;

- (3)
*m*is continuous.

**Definition 3.2**Let $(S,F)$ be a probabilistic metric space and $f:S\to S$. The mapping

*f*is a generalized $(\psi ,\phi ,\u03f5,\lambda )$-contraction if there exist a continuous, decreasing function $h:[0,1]\to [0,\mathrm{\infty}]$ such that $h(1)=0$, ${m}_{1},{m}_{2}\in \mathrm{\aleph}$, and $\lambda \in (0,1)$ such that the following implication holds for every $p,q\in S$ and for every $\u03f5>0$:

If ${m}_{1}(a)={m}_{2}(a)=a$, and $h(a)=1-a$ for every $a\in [0,1]$, we obtain the Mihet definition.

**Theorem 3.3** *Let* $(S,F,T)$ *be a complete Menger space with* *t*-*norm* *T* *such that* ${sup}_{0\le a<1}T(a,a)=1$ *and* $f:S\to S$ *be a generalized* $(\psi ,\phi ,\u03f5,\lambda )$-*contraction such that* *ψ* *is continuous on* $(0,\mathrm{\infty})$ *and* ${lim}_{n\to \mathrm{\infty}}{\psi}^{n}(\delta )=0$ *for every* $\delta \in (0,\mathrm{\infty})$. *Suppose that there exists* $\lambda \in (0,1)$ *such that* $h(0)<{m}_{1}(\lambda )$ *and* *φ*, *ψ* *satisfy* $\phi (0)=\psi (0)=0$. *Then* $x={lim}_{n\to \mathrm{\infty}}{f}^{n}(p)$ *is the unique fixed point of the mapping* *f* *for an arbitrary* $p\in S$.

*Proof*First we shall prove that

*f*is uniformly continuous. Let $\zeta >0$ and $\eta \in (0,1)$. We have to prove that there exists $N(\overline{\zeta},\overline{\eta})=\{(p,q)|(p,q)\in S\times S,{F}_{p,q}(\overline{\zeta})>1-\overline{\eta}\}$ such that

*ϵ*be such that ${m}_{2}(\psi (\u03f5))<\zeta $ and $\lambda \in (0,1)$ such that

*ϵ*and

*λ*exist. We prove that $\overline{\zeta}={m}_{2}(\u03f5)$, $\overline{\eta}=1-{h}^{-1}({m}_{1}(\lambda ))$. If $(p,q)\in N(\overline{\zeta},\overline{\eta})$, we have

*h*is decreasing, it follows that $ho{F}_{p,q}({m}_{2}(\u03f5))<{m}_{1}(\lambda )$. Hence,

*h*is decreasing we have

Relation (4) means that ${({f}^{n}(p))}_{n\in N}$ is a Cauchy sequence, and since *S* is complete there exists $x={lim}_{n\to \mathrm{\infty}}{f}^{n}(p)$, which is obviously a fixed point of *f* since *f* is continuous.

For every $p\in S$ and $q\in S$ such that $f(p)=p$ and $f(q)=q$ we have for every $n\in N$ that ${f}^{n}(p)=p$, ${f}^{n}(q)=q$ and, therefore, from (3) we have ${F}_{p,q}(\zeta )>1-\eta $ for every $\eta \in (0,1)$ and $\zeta >0$. This implies that ${F}_{p,q}(\zeta )=1$ for every $\zeta >0$ and, therefore, $p=q$. □

**Example 3.3** Let $(S,F,T)$ and the mappings *f*, *ψ* and *φ* be the same as in Example 3.1. Set $h(a)={e}^{-a}-{e}^{-1}$ for every $a\in [0,1]$ and ${m}_{1}(a)={m}_{2}(a)=a$. The mapping *f* is generalized $(\psi ,\phi ,\u03f5,\lambda )$-contraction and ${lim}_{n\to \mathrm{\infty}}{\psi}^{n}(\delta )={lim}_{n\to \mathrm{\infty}}\frac{\delta}{{2}^{n}}=0$ for every $\delta \in (0,\mathrm{\infty})$. On the other hand, there exists $\lambda \in (0,1)$ such that $h(0)=1-\frac{1}{e}<\lambda $ and $\psi (0)=\phi (0)=0$. So ${x}_{2}$ is the unique fixed point for *f*.

## Declarations

### Acknowledgements

The authors thank the reviewers for their useful comments.

## Authors’ Affiliations

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