a generalization of a contraction principle in probabilistic metric spaces part ii

A GENERALIZATION OF A CONTRACTION PRINCIPLE IN PROBABILISTIC METRIC SPACES PART II DOREL MIHEŢ Received 7 June 2004 and in revised form 3 December 2004 A fixed point theorem concerning probabilistic[.]

PROBABILISTIC METRIC SPACES PART II

DOREL MIHET¸

Received 7 June 2004 and in revised form 3 December 2004

A fixed point theorem concerning probabilistic contractions satisfying an implicit rela-tion, which generalizes a well-known result of Hadˇzi´c, is proved

1 Preliminaries

In this section we recall some useful facts from the probabilistic metric spaces theory For more details concerning this problematic we refer the reader to the books [1,3,9]

1.1.t-norms A triangular norm (shortly t-norm) is a binary operation T : [0,1]×[0, 1]→

[0, 1] := I which is commutative, associative, monotone in each place, and has 1 as the

unit element

Basic examples areT L:I × I → I, T L(a,b) =Max(a + b − 1, 0) (Łukasiewicz t-norm),

T P(a,b) = ab, and T M(a,b) =Min{a,b} We also mention the following families of

t-norms:

(i) Sugeno-Weber family ( T SW

λ )λ ∈(−1,∞), defined byT SW

λ =max(0, (x + y −1 +λxy)/

(1 +λ)),

(ii) Domby family ( T λ D)λ ∈(0,∞), defined by T λ D =(1 + (((1− x)/x) λ+ ((1− y)/ y) λ)1/λ)−1,

(iii) Aczel-Alsina family ( T λ AA)λ ∈(0,∞), defined byT λ AA = e −(|logx | λ+|logy | λ) 1/λ

Definition 1.1 [2,3] It is said that thet-norm T is of Hadˇzi´c-type (H-type for short) and

T ∈Ᏼ if the family{T n } n ∈ Nof its iterates defined, for eachx in [0,1], by

T0(x) =1, T n+1(x) = T

T n(x),x

is equicontinuous atx =1, that is,

∀ε ∈(0, 1)∃δ ∈(0, 1) such thatx > 1 − δ =⇒ T n(x) > 1 − ε, ∀n ≥1. (1.2) There is a nice characterization of continuoust-norms T of the class Ᏼ [8]

Copyright©2005 Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences 2005:5 (2005) 729–736

DOI: 10.1155/IJMMS.2005.729

(i) If there exists a strictly increasing sequence (b n)n ∈ Nin [0, 1] such that limn →∞ b n =

1 andT(b n,b n)= b n ∀n ∈ N, then T is of Hadˇzi´c-type.

(ii) IfT is continuous and T ∈ Ᏼ, then there exists a sequence (b n)n ∈ Nas in (i) Thet-norm T M is an trivial example of at-norm of H-type, but there are t-norms T

of Hadˇzi´c-type withT = T M(see, e.g., [3])

Definition 1.2 [3] IfT is a t-norm and (x1,x2, ,x n)∈[0, 1]n (n ∈ N), then T n

i =1x i is defined recurrently by 1, ifn =0 andT i n =1x i = T(T i n = −11x i,x n) for alln ≥1 If (x i)i ∈ Nis a sequence of numbers from [0, 1], thenT i ∞ =1x iis defined as limn →∞ T i n =1x i(this limit always exists) andT i ∞ = n x iasT i ∞ =1x n+i In fixed point theory in probabilistic metric spaces there are

of particular interest thet-norms T and sequences (x n)⊂[0, 1] such that limn →∞ x n =1 and limn →∞ T i ∞ =1x n+i =1 Some examples oft-norms with the above property are given in

the following proposition

Proposition 1.3 [3] (i) For T ≥ T L the following implication holds:

lim

n →∞ T i ∞ =1x n+i =1⇐⇒

∞



n =1

1− x n

(ii) ( 1.3 ) also holds for T = T λ SW

(iii) If T ∈ Ᏼ, then for every sequence (x n)n ∈ N in I such that lim n →∞ x n = 1, one has

limn →∞ T i ∞ =1x n+i = 1.

(iv) If T ∈ {T D

λ,T AA

λ }, then lim n →∞ T i ∞ =1x n+i =1⇔∞ n =1(1− x n)λ < ∞.

Note [4, Remark 13] that ifT is a t-norm for which there exists a sequence (x n)⊂[0, 1] such that limn →∞ x n =1 and limn →∞ T i ∞ =1x n+i =1, then supt<1 T(t,t) =1

1.2 Menger spaces and generalized Menger spaces Probabilistic contractions of Sehgal type Let ∆+be the class of distance distribution functions [9], that is, the class of all functionsF : [0,∞)→[0, 1] with the properties

(a)F(0) =0;

(b)F is nondecreasing;

(c)F is left continuous on (0, ∞)

D+ is the subset of∆+ containing the functions F which also satisfy the condition

limx →∞ F(x) =1

A special element ofD+is the functionε0, defined by

ε0(t) =







0, ift =0,

A sequence (F n) in∆+is said to be weakly convergent to F ∈∆+(shortlyF n −−→F) if w

limn →∞ F n(x) = F(x) for every continuity point x of F.

IfX is a nonempty set, a mapping F : X × X →∆+is called a probabilistic distance on X

andF(x, y) is denoted by F xy

The triple (X,F,T), where X is a nonempty set, F is a probabilistic distance on X,

andT is a t-norm, is called a generalized Menger space (or a Menger space in the sense of

Schweizer and Sklar) if the following conditions hold:

F xy = F yx, ∀x, y ∈ X, (1.6)

F xy(t + s) ≥ T

F xz(t),F zy(s)

, ∀x, y,z ∈ X, ∀t,s > 0. (1.7)

A Menger space is a generalized Menger space with the property Range ( F) ⊂ D+.

If (X,F,T) is a generalized Menger space with sup t<1 T(t,t) =1, then the family



U ε,λ

ε>0,λ ∈(0,1), U ε,λ =(x, y) ∈ X × X : F xy(ε) > 1 − λ

(1.8)

is a base for a metrizable uniformity onX, named the F-uniformity and denoted by ᐁ F

ᐁFnaturally determines a topology onX, called the F-topology:

O ∈᐀F ⇐⇒ ∀x ∈ O ∃ε > 0, ∃λ ∈(0, 1) such thatU ε,λ(x) ⊂ O. (1.9)

ᐁFis also generated by the family{V δ } δ>0 whereV δ:= U δ,δ In what follows the topo-logical notions refer to theF-topology Thus, a sequence (x n)n ∈ NisF-convergent to x ∈ X

if for allε > 0, λ ∈(0, 1) there existsk ∈ N such that F xx n(ε) > 1 − λ for all n ≥ k.

Definition 1.4 A sequence (x n)n ∈ N in X is called F-Cauchy if for each ε > 0, λ ∈(0, 1) there existsk ∈ N such that F x r x s(ε) > 1 − λ for all s ≥ r ≥ k.

Probabilistic contractions were first defined and studied by V M Sehgal in his doctoral

dissertation at Wayne State University

Definition 1.5 [10] LetS be a nonempty set and let F be a probabilistic distance on S.

A mapping f : S → S is called a probabilistic contraction (or B-contraction) if there exists

k ∈(0, 1) such that

F f (p) f (q)(kt) ≥ F pq(t), ∀p,q ∈ S, ∀t > 0. (1.10)

In [10] it is showed that any contraction map on a complete Menger space in which the triangle inequality is formulated under the strongest triangular normT M has a unique fixed point In [11] Sherwood showed that one can construct a complete Menger space

underT L and a fixed-point-free contraction map on that space Hadˇzi´c [2] introduced the classᏴ which have the property that Sehgal’s result can be extended to any continuous

triangular norm in that class Completing the result of Hadˇzi´c, Radu solved the problem

of the existence of fixed points for probabilistic contractions in complete Menger spaces (S,F,T) with T continuous Namely, the following theorem holds.

Theorem 1.6 [7] Every B-contraction in a complete Menger space (S,F,T) with T contin-uous has a (unique) fixed point if and only if T is of Hadˇzi´c-type.

However, under some additional growth conditions on the probabilistic metricF one

may replace the t-norm of H-type in the above theorem, as in Tardiff ’s paper [13]

Corollary 2.6in our paper gives another result in this respect

2 Main results

The main result of this paper isTheorem 2.4concerning contractive mappings satisfying

an implicit relation similar to that in [6,12] This theorem generalizes the mentioned result of Hadˇzi´c (seeCorollary 2.7) Note that we work in generalized Menger spaces

We begin with an auxiliary result, which is formulated as follows

Lemma 2.1 Let ( X,F,T) be a generalized Menger space and let (x n)n ∈ N be a sequence in X such that, for some k ∈ (0, 1),

F x n x n+1(kt) ≥ F x n −1x n(t), ∀n ≥1,∀t > 0. (2.1)

If there exists γ > 1 such that

lim

n →∞ T i ∞ = n F x0x1

γ i

then (x n)n ∈ N is an F-Cauchy sequence.

Proof First note [4] that if the condition limn →∞ T i ∞ = n F x0x1(γ i)=1 holds for someγ =

γ0> 1, then it is satisfied for all γ > 1 Indeed, if lim n →∞ T i ∞ = n F x0x1(γ i0)=1 andγ ≥ γ0, then limn →∞ T i ∞ = n F x0x1(γ i)≥limn →∞ T i ∞ = n F x0x1(γ0)i =1 and therefore limn →∞ T i ∞ = n F x0x1(γ i)=1, while if γ < γ0, then γ s > γ0, for some s ∈ N, and now lim n →∞ T i ∞ = n+s F x0x1(γ i)≥

limn →∞ T i ∞ = n F x0x1(γ i0)=1

We will prove that

∀ε > 0, ∃n0= n0(ε) : F x n x n+m(ε) > 1 − ε, ∀n ≥ n0,∀m ∈ N. (2.3) Letµ ∈(k,1) and let δ = k/µ From the above remark it follows that

lim

n →∞ T i ∞ = n F x0x1

1

µ i

Letε > 0 be given and y i:= F x0x1(1/µ i) From limn →∞ T i ∞ =1y n+i =1 it follows that there existsn1∈ N such that T i m =1y n+i −1> 1− ε, for all n ≥ n1, for allm ∈ N.

Since the series∞

n =1δ nis convergent, there existsn2∈ N such that∞

n = n2δ n < ε.

Letn0=max{n1,n2} Then, for alln ≥ n0andm ∈ N, we have

F x n x n+m(ε) ≥ F x n x n+m

n+m−1

i = n

δ i

≥ T i m = −01F x n+i x n+i+1

δ n+i

≥ T i m = −01y n+i > 1 − ε,

(2.5)

where the last “≥” inequality follows fromF x s x s+1(δ s)= F x s x s+1(k/µ) s ≥ F x0x1(1/µ s) for all

In the following we deal with the classΦ of all continuous functions ϕ : [0,1]4→ R

with the property:

Next we give some examples of functions inΦ

Example 2.2 If a,b,c,d ∈ Randa + b + c + d =0, thenϕ(t1,t2,t3,t4) := at1+bt2+ct3+

dt4∈ Φ if and only if a + d > 0.

Indeed,a + d ≤0⇒ b + c ≥0 Choosingu =0,v =1 we haveu < v and ϕ(u,v,v,u) =

(a + d)u + (b + c)v = b + c ≥0

Conversely, if a + d > 0 and ϕ(u,v,v,u) ≥0, then (a + d)u ≥ −(b + c)v, that is (a + d)u ≥(a + d)v, which implies that u ≥ v.

Thus, the functionsϕ1,ϕ2,

ϕ1

t1,t2,t3,t4 

= t1− t2,

ϕ2

t1,t2,t3,t4



are inΦ

Also, the functionϕ defined by ϕ(t1,t2,t3,t4)= t2− t2t3and, more generally,ϕ(t1,t2,

t3,t4)= t2−(at2+bt2)− t2t3witha + b =0 are inΦ

In the proof of Theorem 2.4we need the following lemma, which is the analog of uniform continuity of a metric (note that ([0, 1],T) is rather a semigroup than a group) Lemma 2.3 Let ( S,F,T) be a generalized Menger space with T continuous in (a,1) for all

a ∈ (0, 1), that is,

lim

n →∞ a n = a, lim

n →∞ b n =1=⇒lim

n →∞ T

a n,b n

If p,q ∈ S and (p n ) is a sequence in S such that p n → p, then F p n q −−→F w pq

Proof Let p,q ∈ S, p n → p and t be a continuity point of F pq By (1.7) it follows that for all 0< ε < t,

F p n q(t) ≥ T

F p n p(ε),F pq(t − ε)

,

F pq(t + ε) ≥ T

F p n p(ε),F p n q(t)

Therefore, limninfF p n q(t) ≥ F pq(t − ε) and F pq(t + ε) ≥limnsupF p n q(t) Letting ε →0

we obtain limnsupF p n q(t) ≤ F pq(t) ≤limninfF p n q(t), and thus lim n →∞ F p n q(t) = F pq(t).

Theorem 2.4 Let ( X,F,T) be an F-complete generalized Menger space under a t-norm

T which is continuous in (a,1) for all a ∈ (0, 1), k ∈ (0, 1), and ϕ ∈ Φ If f : X → X is a mapping such that

ϕ f

:ϕ

F f (x) f (y)(kt),F xy(t),F x f (x)(t),F y f (y)(kt)

≥0, ∀x, y ∈ X, ∀t > 0 (2.10)

and there exist x0∈ X and γ > 1 for which lim n →∞ T i ∞ = n F x0f (x0)(γ i)= 1, then f has a fixed point.

Proof Let x0∈ X be such that lim n →∞ T i ∞ = n F x0f (x0)(γ i)=1 and, for alln ≥1,x n = f (x n −1). Note that (ϕ f) implies that

F f (x) f2 (x)(kt) ≥ F x f (x)(t), ∀x ∈ X, ∀t > 0. (2.11)

On taking in this relationx = x nwe obtain

ϕ

F x n+1 x n+2(kt),F x n x n+1(t),F x n x n+1(t),F x n+1 x n+2(kt)

≥0, ∀n ∈ N, ∀t > 0. (2.12)

It follows that F x n+1 x n+2(kt) ≥ F x n x n+1(t), for all n ∈ N, for all t > 0 and therefore, by

Lemma 2.1, (x n) is a Cauchy sequence

By theF-completeness of X it follows that there exists u ∈ X such that lim n →∞ F ux n(t) =

1, for allt > 0.

Notice that from F x n+1 x n+2(kt) ≥ F x n x n+1(t), for all n ∈ N, for all t > 0 it follows that

limn →∞ F x n x n+1(t) = 1, for all t > 0, for lim n →∞ T i ∞ = n F x0f (x0)(γ i) = 1 implies that limn →∞ F x0f (x0)(γ n)=1 (thereforeF x0f (x0)∈ D+) andF x n x n+1(t) ≥ F x0x1(t/k n), for alln ∈ N,

for allt > 0.

Next, on takingx = x n,y = u in (ϕ f) one obtains

ϕ

F x n+1 f (u)(kt),F x n u(t),F x n x n+1(t),F u f (u)(kt)

≥0, ∀n ∈ N, ∀t > 0. (2.13)

Ifkt is a continuity point of F u f (u), then, on takingn → ∞in the above inequality and usingLemma 2.3, we get

ϕ

F u f (u)(kt),1,1,F u f (u)(kt)

ThusF u f (u)(kt) =1 SinceF u f (u)is increasing, the set of its discontinuity points is at most countable HenceF u f (u)(kt) =1 for allt > 0, from which (using (1.5)) we obtainu = f (u).

Corollary 2.5 [5, Theorem 2.1] Let ( X,F,T) be an F-complete generalized Menger space under a continuous t-norm T ∈ Ᏼ, k ∈ (0, 1), and ϕ ∈ Φ If f : X → X is a mapping such that

ϕ

F f (x) f (y)(kt),F xy(t),F x f (x)(t),F y f (y)(kt)

≥0, ∀x, y ∈ X, ∀t > 0 (2.15)

and there exists x0∈ X for which F x0f (x0)∈ D+, then f has a fixed point.

Proof Choose a µ > 1 Since lim n →∞ µ n = ∞ and F x0x1 ∈ D+, it follows that limn →∞ F x0f (x0)(µ n)=1 Therefore, byProposition 1.3(iii),

lim

n →∞ T i ∞ = n F x0f (x0)

µ i

Corollary 2.6 Let ( X,F,T L ) be an F-complete generalized Menger space and ϕ ∈ Φ If

f : X → X is a mapping such that

ϕ

F f (x) f (y)(kt),F xy(t),F x f (x)(t),F y f (y)(kt)

≥0, ∀x, y ∈ X, ∀t > 0, (2.17)

and∞

n =1(1− F x0f (x0)(γ n))< ∞ for some x0∈ X and γ > 1, then f has a fixed point.

For the proof seeProposition 1.3

Corollary 2.7 Let ( X,F,T) be an F-complete generalized Menger space under T ∈ {T D

λ,T AA

λ }, k ∈ (0, 1), and ϕ ∈ Φ If f : X → X is a mapping such that

ϕ

F f (x) f (y)(kt),F xy(t),F x f (x)(t),F y f (y)(kt)

≥0, ∀x, y ∈ X, ∀t > 0 (2.18)

and∞

n =1(1− F x0f (x0)(γ n))λ < ∞ for some x0∈ X and γ > 1, then f has a fixed point Corollary 2.8 Let ( X,F,T) be an F-complete generalized Menger space under a continu-ous t-norm T ∈ Ᏼ and k ∈ (0, 1) If f : X → X is a mapping satisfying one of the following conditions:

F f (x) f (y)(kt) ≥ F xy(t), ∀x, y ∈ X, ∀t > 0, (2.19)

F2

f (x) f (y)(kt) ≥ F xy(t)F x f (x)(t), ∀x, y ∈ X, ∀t > 0, (2.20)

F f (x) f (y)(kt) ≥2F xy(t) − F x f (x)(t), ∀x, y ∈ X, ∀t > 0 (2.21)

and there exists x0∈ X for which F x0f (x0)∈ D+, then f has a fixed point.

As a final result for this section, we consider an example to see the generality of

Theorem 2.4

Example 2.9 Let X be a set containing at least two elements and the mapping F from

X × X to ∆+, defined by

F xy(t) =







0, ift ≤1 1

2, ift > 1 forx, y ∈ X, x = y, F xx = ε0, ∀x ∈ X. (2.22)

It is easy to show (see [14]) that (X,F,T M) is a complete Menger space

We are going to prove that the mapping f : X → X, f (x) = x satisfies the

contrac-tivity condition (2.21) from the above corollary withb =2,c = −1, however it is not a

B-contraction (here we took advantage of working in ∆+rather than inD+).

First, we show that

F xy(kt) + 1 ≥2F xy(t), ∀x, y ∈ X, ∀t > 0. (2.23) Indeed, the above inequality holds with equality ifx = y, while if x = y then the

right-hand member is at most 1

Next, for everyt ∈(1, 1/k], F xy(kt) =0, whileF xy(t) =1/2, which means that f is not

a Sehgal contraction

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Dorel Mihet¸: Faculty of Mathematics and Computer Science, West University of Timisoara, Bd V Parvan 4, 300223 Timisoara, Romania

E-mail address:mihet@math.uvt.ro

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[5] D Mihet¸, A generalization of a contraction principle in probabilistic metric spaces, The 9th

Inter-national Conference on Applied Mathematics and Computer... 2001.

[4] , New classes of probabilistic contractions and applications... Verw.

Gebiete 20 (1971/72), 117–128.

[12] B Singh and S Jain, A quantitative generalization of Banach contractions, in preparation.