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(\psi ,\phi ,\u03f5,\lambda )Contraction theorems in probabilistic metric spaces for single valued case
Fixed Point Theory and Applications volume 2013, Article number: 109 (2013)
Abstract
In this article, we prove some fixedpoint 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 fixedpoint theorem in the generalized (\psi ,\phi ,\u03f5,\lambda )contraction.
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 multivalued case in a probabilistic metric space before [5]. Now, we obtain two fixedpoint 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 singlevalued 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 nondecreasing, left continuous mapping from ℝ into [0,1] such that {lim}_{x\to \mathrm{\infty}}F(x)=1).
The ordered pair (S,F) is said to be a probabilistic metric space if 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}}^{+}.
A Menger space is a triple (S,F,T) where (S,F) is a probabilistic metric space, T is a triangular norm (abbreviated tnorm) and the following inequality holds:
Recall the mapping T:[0,1]\times [0,1]\to [0,1] is called a triangular norm (a tnorm) 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 tnorms are {T}_{L} (Lukasiewicz tnorm), {T}_{P} and {T}_{M}, defined by {T}_{L}(a,b)=max\{a+b1,0\}, {T}_{P}(a,b)=ab and {T}_{M}(a,b)=min\{a,b\}. If T is a tnorm 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}^{n1}{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}.
If q\in (0,1) is given, we say that the tnorm T is qconvergent if {lim}_{n\to \mathrm{\infty}}{\mathrm{\top}}_{i=n}^{\mathrm{\infty}}(1{q}^{i})=1. We remark that if T is qconvergent, then
Also, note that if the tnorm T is qconvergent, 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 Funiformity [6–8]. The Funiformity naturally determines a metrizable topology on S, called the strong topology or Ftopology [9], a subset O of S is Fopen if for every p\in O there exists t>0 such that {N}_{p}=\{q\in S{F}_{pq}(t)>1t\}\subset O.
Definition 2.1 [6]
A sequence {({x}_{n})}_{n\in \mathrm{N}} is called an Fconvergent 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]
Let \phi :(0,1)\to (0,1) be a mapping, we say that the tnorm T is φconvergent 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}.
We also have
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].
We will consider comparison functions from the class 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]
Let (S,F) be a probabilistic space, \phi \in \varphi and 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].
Let 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 tnorm 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.
Let n,m\in N, \u03f5>0, \lambda \in (0,1). Since {lim}_{t\to \mathrm{\infty}}{F}_{{x}_{0},{f}^{m}{x}_{0}(t)}(t)=1, it follows that for every \xi \in (0,1) there exists \eta >0 such that {F}_{{x}_{0},{f}^{m}({x}_{0})}(\eta )>1\xi and by induction {F}_{{f}^{m}{x}_{0},{f}^{n+m}{x}_{0}}({\psi}^{n}(\eta ))>1{\phi}^{n}(\xi ) for all n\in \mathbb{N}. By choosing 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
and
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 tnorm 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 tnorm T is φconvergent, 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
Suppose that \mu \in (0,1), M>0 are such that
Let {n}_{1} be such that
From (1), it follows 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}(1M{({\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}\}.
Now
Since T is φconvergent, 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.
Let ℵ be the family of all the mappings m:\overline{R}\to \overline{R} such that the following conditions are satisfied:

(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)=1a for every a\in [0,1], we obtain the Mihet definition.
Theorem 3.3 Let (S,F,T) be a complete Menger space with tnorm 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
Let ϵ be such that {m}_{2}(\psi (\u03f5))<\zeta and \lambda \in (0,1) such that
Since {m}_{1} and {m}_{2} are continuous at zero, and {m}_{1}(0)={m}_{2}(0)=0 such numbers 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
Since h is decreasing, it follows that ho{F}_{p,q}({m}_{2}(\u03f5))<{m}_{1}(\lambda ). Hence,
Using (2), we conclude that
and since h is decreasing we have
Therefore, (f(p),f(q))\in N(\zeta ,\eta ) if (p,q)\in N(\overline{\zeta},\overline{\eta}). We prove that for every \zeta >0 and \eta \in (0,1) there exists {n}_{0}(\zeta ,\eta )\in N such that for every p,q\in S
By assumption, there is a \lambda \in (0,1) such that h(0)<{m}_{1}(\lambda ). From {F}_{p,q}({m}_{2}(\u03f5))\ge 0, it follows that
which implies that ho{F}_{f(p),f(q)}({m}_{2}(\psi (\u03f5)))<{m}_{1}(\phi (\lambda )), and continuing in this way we obtain that for every n\in N
Let {n}_{0}(\zeta ,\eta ) be a natural number such that {m}_{2}({\psi}^{n}(\u03f5))<\zeta and {m}_{1}({\phi}^{n}(\lambda ))<h(1\eta ), for every n\ge {n}_{0}(\zeta ,\eta ). Then n>{n}_{0}(\zeta ,\eta ) implies that
If q={f}^{m}(p), from (3) we obtain that
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.
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PA defined the definitions and wrote the Introduction, preliminaries and abstract. AB proved the theorems. AB has approved the final manuscript. Also, PA has verified the final manuscript.
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Beitollahi, A., Azhdari, P. (\psi ,\phi ,\u03f5,\lambda )Contraction theorems in probabilistic metric spaces for single valued case. Fixed Point Theory Appl 2013, 109 (2013). https://doi.org/10.1186/168718122013109
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DOI: https://doi.org/10.1186/168718122013109