# On Probabilistic Alpha-Fuzzy Fixed Points and Related Convergence Results in Probabilistic Metric and Menger Spaces under Some Pompeiu-Hausdorff-Like Probabilistic Contractive Conditions

Authors: M. De la Sen
Journal: Journal of Function Spaces
 (2015)

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## Abstract

In the framework of complete probabilistic metric spaces and, in particular, in probabilistic Menger spaces, this paper investigates some relevant properties of convergence of sequences to probabilisticα-fuzzy fixed points under some types of probabilistic contractive conditions.


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## Body

## 1. Introduction

Fixed point theory is an important tool to investigate the convergence of sequences to limits and unique limits in metric spaces and normed spaces. See, for instance, [1–34] and the wide list of references cited in those papers. In particular, fixed point theory is also a relevant tool to investigate iterative schemes and stability theory of continuous-time and discrete-time dynamic systems, boundedness of the trajectory solutions, stability of equilibrium points, convergence to stable equilibrium points, and the existence oscillatory solution trajectories [15–17, 23, 35]. See also references therein. On the other hand, fixed point theory is nowadays receiving important research attention in the framework of probabilistic metric spaces. See, for instance, [25, 26, 28, 33, 34, 36–38] and references therein. It has also to be pointed out that Menger probabilistic metric spaces are a special class of the wide class of probabilistic metric spaces which are endowed with a triangular norm [25, 26, 28, 33, 34, 36, 37] and which are very useful in the context of fixed point theory. Note that the triangular norm plays a close role to that of the norm in normed spaces. In probabilistic metric spaces, the deterministic notion of distance is revisited as being probabilistic in the sense that, given any two points x and y of a certain metric space, a measure of the distance between them is a probabilistic metric Fx,yt, rather than the deterministic distance dx,y, which is interpreted as the probability of the distance between x and y being less than tt>0 [33, 34].Fixed point theorems in complete Menger spaces for probabilistic concepts ofB and C-contractions can be found in [33] together with a new notion of contraction, referred to as Ψ,C-contraction. Such a contraction was proved to be useful for multivalued mappings while it generalizes the previous concept of C-contraction. On the other hand, fuzzy metric spaces have been investigated more recently and some ad hoc versions of fixed point theorems have been obtained in that framework. See, for instance, [4–6, 32] and some references therein.This paper investigates some relevant properties of convergence of sequences to the so-called and defined probabilisticα-fuzzy fixed points under some types of probabilistic contractive conditions. The concept of probabilistic α-fuzzy fixed point is defined as an “ad hoc” conceptual extension of that of α-fuzzy fixed points of [6, 32] and it is oriented to the derivation of convergence properties of fuzzy mappings defined on probabilistic metric spaces and, in particular, in probabilistic Menger spaces.Notation, Preliminaries, and Some Basic Concepts. Denote by R+=z∈R:z>0, R0+=R+∪0, Z+=z∈Z:z>0, Z0+=Z+∪0, n-=1,2,…,n, and denote by ΔF (a common used name for this class being D+) the set of probability distribution functions F:R→0,1, [1], which are nondecreasing and left-continuous such that F0=inft∈R⁡Ft=0 and supt∈R⁡Ft=1. Let X be a nonempty set and let the probabilistic metric (or distance) F:X×X→ΔF be a symmetric mapping from X×X to ΔF, where X is an abstract set, to the set of distance distribution functions ΔF of the form F:R→0,1 which are functions of elements Fx,y for every x,y∈X×X. Then, the ordered pair X,F is a probabilistic metric space (PM) [16, 28, 33, 34, 36–38] if the following constraints hold:(1)
∀x,y∈XFx,yt=1;∀t∈R+⇔x=y, equivalently, (1)Fx,yt=Ft⟺x=y,whereF∈ΔF is defined by (2)Ft=0,ift≤0,1,ift>0;(2)
Fx,yt=Fy,xt; ∀x,y∈X, ∀t∈R(3)
(3)∀x,y,z∈X;∀t1,t2∈R+Fx,yt1=Fy,zt2=1⟹Fx,zt1+t2=1.A particular distance distribution function Fx,y∈ΔF is a probabilistic metric (or distance) which takes values Fx,yt identified with a probability distance density function F:R→0,1 in the set of all the distance distribution functions ΔF.A Menger PM-space is a tripletX,F,Δ, where X,F is a PM-space which satisfies(4)Fx,yt1+t2≥ΔFx,zt1,Fz,yt2;∀x,y,z∈X,∀t1,t2∈R0+under Δ:0,1×0,1→0,1 which is a t-norm (or triangular norm) belonging to the set T of t-norms which satisfy the following properties:(1)
Δa,1=a,(2)
Δa,b=Δb,a,(3)
Δc,d≥Δa,b if c≥a, d≥b,(4)
(5)ΔΔa,b,c=Δa,Δb,c.A property which follows from the above ones is Δa,0=0 for a∈0,1. Typical continuous t-norms are the minimum t-norm defined by ΔMa,b=min⁡a,b, the product t-norm defined by ΔPa,b=a·b, and the Lukasiewicz t-norm defined by ΔLa,b=max⁡a+b-1,0 which are related by the inequalities ΔL≤ΔP≤ΔM.(i)
The tripletX,F,Δ is a Menger space, where X,F is a PM-space and Δ:0,1×0,1→0,1 is a triangular norm, which satisfies the inequality Fx,zt+s≥ΔFx,yt,Fy,zs; ∀x,y,z∈X; ∀t,s∈R+.(ii)
ΔM:0,1×0,1→0,1 is the minimum triangular norm defined by ΔMa,b=min⁡a,b.(iii)
A sequencexn⊆X in a probabilistic space X,F is said to be
(1)
convergent to a pointx∈X, denoted by xn→x (a.s.), if for every ε∈R+ and λ∈0,1, there exists some N=Nε,λ∈Z0+ such that (6)Fxn,xε>1-λ;∀n∈Z0+≥N,(2)
Cauchy if for everyε∈R+ and λ∈0,1, there exists some N=Nε,λ∈Z0+ such that (7)Fxn,xmε>1-λ;∀n,m∈Z0+≥N.A PM-space X,F is complete if every Cauchy sequence is convergent.


## 2. Concepts and Results on Probabilisticα-Fuzzy Fixed Points

LetA,B be nonempty subsets of an abstract nonempty set X. Then the probabilistic point-to-set distance mapping F:X×A→ΔF from X to A, denoted by Fx,At and the probabilistic set-to-set distance mapping F:A×B→ΔF from A to B are, respectively, defined by(8)Fx,At=sup⁡Fx,yt:y∈A,x∈X,t∈R,FA,Bt=sup⁡Fx,yt:x∈A,y∈B,t∈R.The Pompeiu-Hausdorff-like probabilistic set-to-set distance is defined by (9)HA,Bt=mininfx∈A⁡Fx,Bt,infy∈B⁡Fy,At;∀t∈R.Note that Fx,S0=0 and FS,W0=0 since H0=0. If A,B∈CBX, where CBX is the set of all nonempty closed bounded subsets of X, then supt∈R⁡HA,Bt=supx∈A⁡Fx,Bt=supy∈B⁡Fy,At=1.A fuzzy setA in X is a function from X to 0,1 whose grade of membership of x in A is the function-value Ax. The α-level set of A is denoted by Aα defined by(10)Aα=x∈X:Ax≥α⊆X,ifα∈0,1,A0=x∈X:Ax>0¯⊆X,where B- denotes the closure of B. Let FX be the collection of all fuzzy sets in a PM-space X,F. Let T:X→FX be a fuzzy mapping from an arbitrary set X to FX, which is a fuzzy subset in X×X, and the grade of membership of y in Tx is Txy.ForA,B∈FX, A⊂B means Ax≤Bx, ∀x∈X. Note also that if α∈β,1 and β∈0,1 then Aα⊆Aβ. If there exists α∈0,1 such that Aα, Bα∈CBX, then define(11)HTxαx,Tyαyt=min⁡infa∈Tx⁡Fa,Tyαyt,infb∈Ty⁡Fb,Txαxt;∀t∈R.The collection of all the approximate quantities in a metric linear space X is denoted by WX. T:X→FY is a fuzzy mapping from an arbitrary set X to FY, which is a fuzzy subset in X×Y, and the grade of membership of y in Tx is Txy.The notationf:X∣Y→Z means that the domain of the function f from X to Z is restricted to the subset Y of X.The next definition characterizes probabilistic fuzzy fixed points in an appropriate way to establish some results of this paper.Definition 1.
IfFX is the collection of all fuzzy sets in the PM-space X,F, where X  is a nonempty abstract set and T:X→FX is a fuzzy mapping, then x∈X is a probabilistic α-fuzzy fixed point of T:X→FX if, for some α∈0,1, Txα∈CBX and x∈Txα; that is, Txx≥α.
Note that ifX  is a nonempty abstract set, X,F is a PM-space, A∈FX, and for some α∈0,1, Aα∈CBX, T:X→FX then (1)
FAα,Aαt=1; ∀t∈R+,(2)
ifx∈X is a probabilistic α-fuzzy fixed point of T:X→FX then Fx,Txαt=1; ∀t∈R+,(3)
ifTxαx∈CBX and x∈X is not a probabilistic α-fuzzy fixed point ofT:X→FX then x∉Txαx, equivalently, Txx<α, and Fx,Txαt<1; t∈0,t1 for some t1=t1x∈R+.The following result holds.Theorem 2.
LetFX be the collection of all fuzzy sets in a PM-space X,F, where X  is a nonempty abstract set and let T:X→FX be a fuzzy mapping. Assume that the following conditions hold:(1)
for eachx∈X, there exists αx∈0,1 such that Txαx is a nonempty closed bounded subset of X, and each sequence xn⊂X of the form x1∈X, xn+1∈Txnαxn with Txnαxn∈CBX; ∀n∈Z0+ which satisfies the following contractive constraint for some real constant k∈0,1: (12)HTxαx,Tyαykt≥a1x,y,ktFx,Txαxt+a2x,y,ktFy,Tyαyt+a3x,y,ktFx,yt;∀x∈X,∀y∈Txαx, where ai:X×X×R0+→0,1 for i=1,3; a2:X×X×R0+→0,1,(2)
(13)0<ax,y,t=∑i=13aix,y,t≤1;∀t∈R0+,∀x∈X,∀y∈Txαx⊂CBX,(3)
(14)limN→∞⁡∏i=0Na1xi+1,xi+2,k-i+1t+a3xi+1,xi+2,k-i+1t1-a2xi+1,xi+2,k-i+1t=1;xn+1∈Txnαxn∈CBX,∀t∈R+,∀n∈Z0+.Then, a sequence xn may be built for any given arbitrary x1=x∈X satisfying xn+1∈Txnαxn∈CBX, with αxn⊆0,1 satisfying limn→∞⁡Fxn,xn+1t=1; ∀t∈R+.
If, in addition,X,F is endowed with the minimum triangular norm ΔM:0,1×0,1→0,1 and X,F,ΔM is a complete Menger space then each of such sequences xn is a Cauchy sequence which is convergent to a probabilistic α∗-fuzzy fixed point x∗ of T:X→FX.Proof.
Take arbitrary pointsx1=x∈X, x2=y∈Tx1αx1 for some given existing αx1∈0,1 such that Tx1αx1 is nonempty and take also some existing αx2∈0,1 such that Tx2αx2 is nonempty. Note that since Fx,At=sup⁡Fx,yt:y∈A, for any x∈X and t∈R; then Fx1,Tx1αx1t≥Fx1,x2t. Thus, one gets from the contractive condition (12) that (15)
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Case (a). One has(16)HTx1αx1,Tx2αx2kt=infa∈Tx1αx1⁡Fa,Tx2αx2kt=min⁡infb∈Tx2αx2Fb,Tx1αx1kt,Fx2,Tx2αx2kt≤Fx2,Tx2αx2ktfor some given t∈R+. Thus, from (15) and (16), one gets(17)Fx2,Tx2αx2kt≥a1x1,x2,kt+a3x1,x2,ktFx1,x2t+a2x1,x2,ktFx2,Tx2αx2kt;∀t∈R+and one gets for the given t∈R+that since Ht is nondecreasing and left-continuous, then (18)Fx,Txαxt≥Fx,Txαxkt;∀t∈R+,∀x∈Xand since k∈0,1, one gets from (17) that(19)Fx2,Tx2αx2t≥Fx2,Tx2αx2kt≥a1x1,x2,kt+a3x1,x2,ktFx1,x2t+a2x1,x2,ktFx2,Tx2αx2ktand then since Tx2αx2 is closed and nonempty, there exists x3∈Tx2αx2 such that from (19) and the fact that Fx2,Tx2αx2t≥Fx2,Tx2αx2kt; ∀t∈R+,(20)Fx2,x3t≥Fx2,x3kt=Fx2,Tx2αx2kt≥a1x1,x2,kt+a3x1,x2,kt1-a2x1,x2,ktFx1,x2t;∀t∈R+and, equivalently, (21)Fx2,x3t≥gx1,x2,tFx1,x2k-1t;∀t∈R+,where gx1,x2,t=a1x1,x2,t+a3x1,x2,t/1-a2x1,x2,t; ∀t∈R+.
Case (b). One has(22)HTx1αx1,Tx2αx2kt=infb∈Tx2αx2⁡Fb,Tx1αx1kt=min⁡infa∈Tx1αx1⁡Fa,Tx2αx2kt,Fx1,Tx1αx1kt≤Fx1,Tx1αx1ktand some t∈R+ and x3∈Tx2αx2 can be chosen for the previously taken x2∈Tx1αx1 so that Fx2,x3kt=infa∈Tx1αx1⁡Fa,Tx2αx2kt. Thus,(23)HTx1αx1,Tx2αx2kt=infb∈Tx2αx2⁡Fb,Tx1αx1kt=Fx2,x3ktand since Fx2,Tx2αx2kt≥Fx2,x3kt, one gets for the given t∈R+(24)Fx2,x3t≥Fx2,x3kt≥a1x1,x2,tFx1,Tx1αx1t+a3x1,x2,tFx1,x2t+a2x1,x2,tFx2,Tx2αx2kt≥a1x1,x2,t+a3x1,x2,tFx1,x2t+a2x1,x2,tFx2,x3kt.Then, one gets from (24) that (25)1-a2x1,x2,tFx2,x3kt≥a1x1,x2,t+a3x1,x2,tFx1,x2twhich implies (21). So, from Cases (a)-(b), for each given t∈R+, and x1∈X, there exist αx1,αx2∈0,1 and points x2∈Tx1αx1 and x2∈Tx2αx2 in nonempty level sets Tx1αx1 and Tx2αx2 such that (22) holds. Proceeding recursively, one gets that a sequence xn may be built for any arbitrary x1=x∈X and xn+1∈Txnαxn∈CBX with αxn⊆0,1; ∀n∈Z+ which satisfies the recursion(26)Fx2,x3t≥gx1,x2,tFx1,x2k-1t;∀t∈R+,Fx3,x4t≥gx2,x3,tFx2,x3k-1t≥gx2,x3,tgx1,x2,k-1tFx1,x2k-2t;∀t∈R+,⋮Fxn+1,xn+2t≥∏i=1ngxi,xi+1,k-i+1tFx1,x2k-nt;∀t∈R+,∀n∈Z+,where (27)0<gxi,xi+1,t=a1xi,xi+1,t+a3xi,xi+1,t1-a2xi,xi+1,t≤1;∀t∈R+,∀i∈Z+.Note that since(28)0<axi,xi+1,t≤1,0≤a2xi,xi+1,t<1;∀t∈R+,∀i∈Z+,(29)∏i=0na1xi,xi+1,k-i+1t+a3xi,xi+1,k-i+1t1-a2xi,xi+1,k-i+1t⟶1,asn⟶∞;∀t∈R+then δΓ1t/δΓ0t=+∞; ∀t∈R+, ∀i∈Z+, where δΓ0t=∏j=0∞δjt0=∏j=0∞1-δjt1 and δΓ1t=∏j=0∞δjt1=∏j=0∞1-δjt0; ∀t∈R+ are discrete measures of the subsequent sets (30)Γ0t=Γ0txj=j=jt∈Z0+:gxj,xj+1,t∈0,1;∀t∈R+,Γ1t=Γ1txj=j=jt∈Z0+:gxj,xj+1,t=1;∀t∈R+,where δjt0 and δjt1 are Dirac measures defined by(31)δjt0=1-δjt1=1,ifj∈Γ0t,0,ifj∉Γ0t;∀t∈R+,δjt1=1-δjt0=1,ifj∈Γ1t,0,ifj∉Γ1t;∀t∈R+.Then, one gets from (26), (29), and limn→∞⁡Fx1,x2k-nt=limτ→+∞-⁡Hτ=H+∞-=1; ∀t∈R+, since k∈0,1, that limn→∞⁡Fxn+1,xnt=1; ∀t∈R+ with xn+1∈Txnαxn; ∀n∈Z+ since (32)limn→∞⁡Fxn+1,xn+2t≥limn→∞⁡∏i=1ngxi,xi+1,k-i+1t·limn→∞⁡Fx1,x2k-nt;∀t∈R+.Since limn→∞⁡Fxn+1,xnt=limn→∞Fx1,x2k-nt=1; ∀t∈R+ and any given x1∈X then for any given ε∈R+ and λ∈0,1, there is N=Nε,λ∈Z0+ such that Fxn+1,xnε>1-λ; ∀n≥N∈Z0+. Assume on the contrary that there exist nk≥N,nk+2>nk+1>nk such that Fxnk+i+1,xnk+iε≥Fxnk+i+1,xnk+iε/2>1-λ for i=0,1 and 1-λ≥Fxnk+2,xnkε. Then, one has the following contradiction for the subsequence xnk of xn:(33)1-λ≥Fxnk+2,xnkε≥ΔMFxnk+2,xnk+1ε2,Fxnk+1,xnkε2>1-λ.Then, Fxn+1,xnε>1-λ; ∀n≥N∈Z0+ so that xn is a Cauchy sequence. Since X,F,ΔM is complete, one gets xn→x∗ and αxn→αx∗.
It is now proved thatx∗∈Tx∗αx∗. Assume on the contrary that x∗∉Tx∗αx∗; that is, Tx∗x∗<αx∗ and, for some given y∈Tx∗αx∗, there is t1=t1y∈R+ such that Fx∗,Tx∗αx∗t<1 for t∈0,t1. Then, since xn+1∈Txnαxn; ∀n∈Z+ and since y≠x∗, (34)
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,a contradiction. Then, x∗∈Tx∗αx∗ with α∗=αx∗; that is, it is a probabilistic α∗-fuzzy fixed point of T:X→FX.Corollary 3.
LetFX be the collection of all fuzzy sets in a PM-space X,F, where X  is a nonempty abstract set, and let T:X→FX be a fuzzy mapping. Assume that the contractive condition (12) holds for some real constant k∈0,1 subject to condition (2) of Theorem 2 and the specific particular form of condition (3):(3′)
there exists and strictly increasing sequence of nonnegative integersNn which satisfies (36)∏j=Nn+1Nn+2-1a2x,y,k-i+1t+a3x,y,k-i+1t1-a2x,y,k-i+1t=1∏j=NnNn+1-1a2x,y,k-i+1t+a3x,y,k-i+1t/1-a2x,y,k-i+1t;∀t∈R+,∀x∈X,y∈Txαx⊂CBX,∀n∈Z0+.Proof.
It is direct from that of Theorem2 since condition (3′) guarantees condition (3) of Theorem 2 for any finite first element N0∈Z0+ of a strictly increasing sequence Nn subject to 0<M1≤Nn+1-Nn≤M2<+∞.Corollary 4.
Theorem2 and Corollary 3 also hold if the contractive condition (1) is modified as follows:(37)HTx1αx1,Tx2αx2kt≥a1x1,x2,ktFx1,Tx1αx1t+a2x1,x2,ktFx2,Tx2αx2kt+a3x1,x2,ktFx1,x2t=a1x1,x2,kt+a3x1,x2,ktFx1,x2t+a2x1,x2,ktFx2,Tx2αx2kt;∀t∈R+,∀x1=x∈X,xn+1∈Txnαxn⊂CBX,∀n∈Z0+.Proof.
It is direct from that of Theorem2 since the above contraction condition follows from (12) as a lower-bound which has been used in the proof of Theorem 2.Corollary 5.
LetFX be the collection of all fuzzy sets in a PM-space X,F, where X  is a nonempty abstract set, and let T:X→FX be a fuzzy mapping. Assume that the following conditions are fulfilled:(1)
for eachx∈X, there exists αx∈0,1 such that Txαx is a nonempty closed bounded subset of X, and each sequence xn⊂X of the form x1∈X, xn+1∈Txnαxn∈CBX; ∀n∈Z0+ which satisfies for some real constant k∈0,1 the contractive constraint: (38)
H
T
x
α
x
,
T
y
α
y
k
t
≥
min
⁡
1
,
a
1
x
,
y
,
k
t
F
x
,
T
x
α
x
t
+
a
2
x
,
y
,
k
t
F
y
,
T
y
α
y
t
+
a
3
x
,
y
,
k
t
F
x
,
y
t
;∀x∈X,∀y∈Txαx, where  ai:X×X×R0+→0,1 for i=1,3;a2:X×X×R0+→0,1.
Then, a sequencexn may be built for any given arbitrary x1=x∈X satisfying xn+1∈Txnαxn∈CBX, with αxn⊆0,1 and limn→∞⁡Fxn,xn+1t=1; ∀t∈R+.
If, in addition,X,F is endowed with the minimum triangular norm ΔM:0,1×0,1→0,1 and X,F,ΔM is a complete Menger space then each above sequence xn is a Cauchy sequence which is convergent to a probabilistic α∗-fuzzy fixed point x∗ ofT:X→FX.Proof.
Define the indicator function sequenceσi:X×X×R0+→0,1; i∈Z+ as(39)σixi,xi+1,k-i+1t=1,ifgxi,xi+1,k-i+1tFxi,xi+1k-it≤1,0,otherwise,∀t∈R+with gxi,xi+1,t defined as in Theorem 2. Then, even if for some j∈Z+ and all i≥j∈Z+, σixi,xi+1,t=0 because gxi,xi+1,k-i+1tFxi,xi+1k-it>1; ∀i≥j∈Z+, it follows from (38)-(39) and (26) that limn→∞⁡Fxn+1,xnt=1; ∀t∈R+ with xn+1∈Txnαxn; ∀n∈Z+. The rest of the proof is close to that of Theorem 2.Example 6.
It is claimed through this example to revisit the idea of fuzzy fixed point addressed in [4–6, 32] to that of probabilistic fuzzy fixed point according to the definition and the formalism given above. Consider X=0.6,0.7,0.8,0.6,0.7,0.8 and let T:X→FX be a probabilistic fuzzy mapping defined as follows:(40)
T
0.6
t
=
T
0.8
t
=
3
4
,
i
f
t
=
0.8
,
1
2
,
i
f
t
=
0.7
,
0
,
i
f
t
=
0.6
;
T
0.7
t
=
0
,
i
f
t
=
0.8
,
1
3
,
i
f
t
=
0.7
,
3
4
,
i
f
t
=
0.6
.The α-level sets are(41)T0.61/2=T0.81/2=0.7,0.8,T0.63/4=T0.83/4=0.8,T0.73/4=0.6;T0.71/3=0.7.Note that Fx,Txαxt=sup⁡Fx,yt:y∈Txαx, for any x∈X and t∈R and some αx∈0,1. The probability density functions are as follows with Fx,yt=t/t+x-y; ∀x,y∈X, ∀t∈R+: (42)F0.6,T0.63/4t=F0.6,0.8t=tt+0.2;∀t∈R+,F0.6,T0.61/2t=F0.6,0.7t=tt+0.1;∀t∈R+,F0.8,T0.81/2t=F0.8,0.7t=tt+0.1;∀t∈R+,F0.8,T0.83/4t=F0.8,0.8t=1;∀t∈R+,F0.7,T0.73/4t=F0.7,0.6t=tt+0.1;∀t∈R+,F0.7,T0.71/3t=F0.7,0.7t=1;∀t∈R+,HTx1/2,Tx1/2t=tt+0.1,forx=0.6,0.8;∀t∈R+,HTx3/4,Tx3/4t=1,forx=0.6,0.8;∀t∈R+,HTx1/2,Ty3/4t=tt+0.1,forx,y=0.6,0.8;∀t∈R+,HTx1/2,T0.71/3t=tt+0.1,forx,y=0.6,0.8;∀t∈R+,HTx1/2,T0.73/4t=tt+0.2,forx,y=0.6,0.8;∀t∈R+,HT0.71/3,T0.71/3t=1;∀t∈R+,HT0.73/4,T0.73/4t=tt+0.1;∀t∈R+,HT0.71/3,T0.73/4t=tt+0.1;∀t∈R+.Assume that the contractive condition of Theorem 2 holds under the form (43)
H
T
x
α
x
,
T
y
α
y
t
≥
α
x
n
,
x
n
+
1
,
t
,
n
·
F
x
n
,
T
x
n
α
x
n
k
-
1
t
+
F
x
n
+
1
,
T
x
n
+
1
α
x
n
+
1
k
-
1
t
+
F
x
n
,
x
n
+
1
k
-
1
t
;
n
∈
Z
0
+for sequences xn⊂X with initial points x0=x∈X, x1∈Tx0α, where αx,y,t,n=αix,y,t,n; i=1,2,3, ∀t∈R+ satisfies (44)αx,y,t,n=αat≤kt3kt+0.2;i=1,2,3,∀t∈R+,∀n≤l∈Z0+,αx,y,t,n=αbt≤131-e-λt;i=1,2,3,∀t∈R+,∀n≤l∈Z0+,for some l≥2∈Z0+, some k∈0,1, and some λ∈R+. Note that T0.71/3=0.7 with F0.7,T0.71/3t=F0.7,0.7t=1; ∀t∈R+ is a probabilistic 1/3-fuzzy fixed point of T:X→FX to which the sequences 0.6,0.8,0.7,0.7,…, 0.8,0.6,0.7,0.7,… converge.


## 3. Further Results

Note that the uniqueness of probabilistic fuzzy fixed points is not an interesting property to study in the context of probabilistic fuzzy fixed point since distinct level sets associated with mappings of the formT:X→FX can easily have intersections of cardinal greater than one in many problems in the fuzzy context. The subsequent result gives conditions for the case when several distinct probabilistic fuzzy fixed points, if they exist, are in the intersections of their respective level sets under slightly extended contractive conditions of those given in Section 2. The extended conditions are of direct applicability.Theorem 7.
Consider a complete probabilistic metric spaceX,F,ΔM under all the assumptions of Theorem 2 with an extended contractive condition (12) such that it also holds for any x,y∈X being probabilistic αx∗-fuzzy fixed points x=x∗∈Tx∗αx∗ and y=y∗∈Ty∗αy∗ of T:X→FX. Then, the set Ty∗αy∗∩Tx∗αx∗ is nonempty and x∗,y∗∈Ty∗αy∗∩Tx∗αx∗.Proof.
Ifx∗=y∗ the result is obvious. Assume that there exist a probabilistic αx∗-fuzzy fixed point x∗∈Tx∗αx∗ and a probabilistic αy∗-fuzzy fixed point y∗≠x∗∈Ty∗αy∗. Consider two convergent sequences xn→x∗ and yn→y∗ in X. Then,(45)
F
x
n
,
y
n
t
≥
Δ
M
F
x
∗
,
x
n
t
2
,
Δ
M
F
x
∗
,
y
∗
t
4
,
F
y
n
,
y
∗
t
4
;
∀
t
∈
R
+
,
liminf
n
→
∞
⁡
F
x
n
,
y
n
t
≥
liminf
n
→
∞
⁡
Δ
M
F
x
∗
,
x
n
t
2
,
Δ
M
F
x
∗
,
y
∗
t
4
,
F
y
n
,
y
∗
t
4
≥
Δ
M
lim
n
→
∞
⁡
F
x
∗
,
x
n
t
2
,
Δ
M
F
x
∗
,
y
∗
t
4
,
lim
n
→
∞
⁡
F
y
n
,
y
∗
t
4
=
Δ
M
1
,
Δ
M
F
x
∗
,
y
∗
t
4
,
1
≥
F
x
∗
,
y
∗
t
4
;
∀
t
∈
R
+
.Assume that Fy∗,Tx∗αx∗kt≥Fx∗,Ty∗αy∗kt; ∀t∈R+. Then, from the extended contractive condition (12) and since 1=Fx∗,Tx∗αx∗t=Fy∗,Ty∗αy∗t≥Fx∗,y∗t; ∀t∈R+, one gets for some zx∈Tx∗αx∗ and zy∈Ty∗αy∗ since Tx∗αx∗ and Ty∗αy∗ are members of CBX, that is, nonempty, closed and bounded sets,(46)
F
y
∗
,
T
x
∗
α
x
∗
k
t
≥
F
x
∗
,
T
y
∗
α
y
∗
k
t
=
min
⁡
F
x
∗
,
T
y
∗
α
y
∗
k
t
,
F
y
∗
,
T
x
∗
α
x
∗
k
t
=
min
⁡
F
x
∗
,
z
y
k
t
,
F
y
∗
,
z
x
k
t
≥
a
1
x
∗
,
y
∗
,
k
t
·
F
x
∗
,
T
x
∗
α
x
∗
t
+
a
2
x
∗
,
y
∗
,
k
t
F
y
∗
,
T
y
∗
α
y
∗
t
+
a
3
x
∗
,
y
∗
,
k
t
F
x
∗
,
y
∗
t
≥
a
1
x
∗
,
y
∗
,
k
t
+
a
2
x
∗
,
y
∗
,
k
t
+
a
3
x
∗
,
y
∗
,
k
t
F
x
∗
,
y
∗
tso that(47)min⁡Fx∗,zykt,Fy∗,zxkt≥a1x∗,y∗,t+a2x∗,y∗,t+a3x∗,y∗,tFx∗,y∗k-1t≥a1x∗,y∗,t+a2x∗,y∗,t+a3x∗,y∗,tΔMFx∗,zyk-1t2,Fzy,y∗k-1t2≥∏i=1ma1x∗,y∗,k-i+1t+a2x∗,y∗,k-i+1t+a3x∗,y∗,k-i+1tΔMFx∗,zy2k-mt,Fzy,y∗2kmt≥limn→∞⁡∏i=1na1x∗,y∗,k-i+1t+a2x∗,y∗,k-i+1t+a3x∗,y∗,k-i+1t·liminfn→∞⁡ΔMFx∗,zy2k-nt,Fzy,y∗2k-nt≥limn→∞⁡∏i=1na1x∗,y∗,k-i+1t+a2x∗,y∗,k-i+1t+a3x∗,y∗,k-i+1t·ΔMFx∗,zylimn→∞⁡2k-nt,Fzy,y∗limn→∞⁡2k-nt=limn→∞⁡∏i=1na1x∗,y∗,k-i+1t+a2x∗,y∗,k-i+1t+a3x∗,y∗,k-i+1tH+∞-=1;∀t∈R+,∀m∈Z+and then y∗=zx, x∗=zyTy∗αy∗∩Tx∗αx∗ so that Ty∗αy∗∩Tx∗αx∗ is nonempty. We can now assume that Fy∗,Tx∗αx∗kt≤Fx∗,Ty∗αy∗kt; ∀t∈R+. It is direct to prove in a close way to the above proof that y∗∈Ty∗αy∗∩Tx∗αx∗.Remark 8.
Note that a direct consequence of Theorem7, from the definition of the level sets, is that (48)x∗,y∗∈⋂β∈0,αx∗Tx∗β∩⋂β∈0,αy∗Ty∗βif x∗∈Tx∗αx∗ and y∗∈Ty∗αy∗ are any probabilistic αx∗ and αy∗-fuzzy fixed points.Remark 9.
Note that corollaries to Theorem7 can also be stated “mutatis-mutandis” under extended contractive conditions for probabilistic fuzzy fixed points to those given in Corollaries 3–5.Theorem 10.
Consider a complete probabilistic metric spaceX,F,ΔM under all the assumptions of Theorem 2 and conditions (1)–(3) where the contractive condition (49)HTxαx,Tyαykt≥a1x,y,ktFx,Txαxt+a2x,y,ktFy,Tyαyt+a3x,y,ktFx,ytis extended to hold for any x,y∈X with ai:X×X×R0+→0,1 for i=1,3; a2:X×X×R0+→0,1.Proof.
Note, from Theorem2, a sequence xn⊂X can be built being a (convergent) Cauchy sequence such that xn→x∗, xn+1=Txn∈Txnαxn; ∀n∈Z0+ for any given x0∈X. Then, (50)
lim
n
→
∞
⁡
F
x
n
,
T
x
n
t
=
lim
n
→
∞
⁡
F
x
n
,
x
∗
t
=
1
;
∀
t
∈
R
+
,
lim
n
→
∞
⁡
F
T
x
n
,
x
∗
t
=
1
;
∀
t
∈
R
+
,
s
i
n
c
e
F
T
x
n
,
x
∗
t
≥
Δ
M
F
x
∗
,
x
n
t
2
,
Δ
M
F
x
n
,
x
n
+
1
t
4
,
F
T
x
n
,
T
x
n
t
4
>
min
⁡
1
-
λ
,
1
-
λ
,
1
=
1
-
λfor any given t∈R+, λ∈0,1, n≥N0∈Z0+, and N0=N0ε,λ≥max⁡N1,N2 such that(51)Fx∗,xnt2>1-λ,forn≥N1∈Z0+andsomeN1=N1ε,λ∈Z0+,Fxn+1,xnt4>1-λ,forn≥N2∈Z0+andsomeN2=N2ε,λ∈Z0+,since FTxn,Txnt/4=1; ∀t∈R+, ∀n∈Z0+ from property (1) of (3) for PM-spaces.Theorem 11.
LetFX be the collection of all fuzzy sets in a PM-space X,F, where X  is a nonempty abstract set, and let T:X→FX be a fuzzy mapping. Assume that the following conditions are fulfilled:(1)
for eachx∈X, there exists αx∈0,1 such that Txαx is a nonempty closed bounded subset of X, and each sequence xn⊂X of the form x1∈X, xn+1∈Txnαxn with Txnαxn∈CBX; ∀n∈Z0+ which satisfies the following contractive constraint for some real constant k∈0,1: (52)HTxαx,Tyαykt≥a1Fx,Txαxt+a2Fx,Tyαyt+a3Fy,Txαxt+a4Fy,Tyαyt+a5Fx,yt;∀x∈X,∀y∈Txαx, where ai∈0,1 for i=1,3,4,5; a2:X×X×R0+→0,1,(2)
(53)0<a=∑i=15ai≤1;∀t∈R0+,∀x∈X,∀y∈Txαx⊂CBX,(3)
(54)limN→∞⁡∏i=0Na1xi+1,xi+2,k-i+1t+a3xi+1,xi+2,k-i+1t1-a2xi+1,xi+2,k-i+1t=1;xn+1∈Txnαxn∈CBX,∀t∈R+,∀n∈Z0+.Then, a sequence xn may be constructed for any given arbitrary x1=x∈X such that xn+1∈Txnαxn∈CBX, with αxn⊆0,1 satisfying limn→∞⁡Fxn,xn+1t=1; ∀t∈R+.
If, in addition,X,F is endowed with the minimum triangular norm ΔM:0,1×0,1→0,1 and X,F,ΔM is a complete Menger space then each of such sequences xn is a Cauchy sequence which is convergent to a probabilistic α∗-fuzzy fixed point x∗ of T:X→FX.Proof.
SinceTxαx,Tyαy⊂X, (55)max⁡Fx,Tyαykt,Fy,Txαxkt≥min⁡Fx,Tyαykt,Fy,Txαxkt≥minz∈Txαx,ω∈Tyαy⁡Fz,Tyαykt,Fω,Txαxkt=HFTxαxkt,ktTyαy≥a1Fx,Txαxt+a2Fx,Tyαyt+a3Fy,Txαxt+a4Fy,Tyαyt+a5Fx,yt;∀t∈R+,∀x,y∈X.Now for any given x,y∈X. Then, the following cases can occur:(a)
IfFx,Tyαykt≥Fy,Txαxkt, Fx,Txαxt≥Fy,Tyαyt, and Fy,Tyαyt≥Fx,yt; ∀t∈R+ then(56)Fy,Txαxkt≥a1Fx,Txαxt+a2+a3Fy,Txαxt+a4Fy,Tyαyt+a5Fx,yt;∀t∈R+,1-a2-a3Fy,Txαxkt≥a1Fx,Txαxt+a4Fy,Tyαyt+a5Fx,yt≥a1+a4Fy,Tyαyt+a5Fx,yt≥a1+a4+a5Fx,yt;∀t∈R+so that(57)Fx,Tyαykt≥Fy,Txαxkt≥a1+a4+a51-a2-a3Fx,yt;∀t∈R+.(b)
IfFx,Tyαykt≥Fy,Txαxkt, Fx,Txαxt≥Fy,Tyαyt, and Fy,Tyαyt≤Fx,yt; ∀t∈R+ then(58)Fx,Tyαyt≥Fy,Txαxt≥a2+a3Fy,Txαxt+a1+a4+a5Fy,Tyαyt;∀t∈R+so that(59)Fx,Tyαykt≥Fy,Txαxkt≥a1+a4+a51-a2-a3Fy,Tyαyt;∀t∈R+.(c)
IfFx,Tyαykt≥Fy,Txαxkt, Fx,Txαxt≤Fy,Tyαyt, and Fy,Tyαyt≥Fx,yt; ∀t∈R+ then(60)
1
-
a
2
-
a
3
F
x
,
T
y
α
y
t
≥
1
-
a
2
-
a
3
F
y
,
T
x
α
x
t
≥
a
1
F
x
,
T
x
α
x
t
+
a
4
F
y
,
T
y
α
y
t
+
a
5
F
x
,
y
t
≥
max
⁡
a
1
+
a
4
F
x
,
T
x
α
x
t
+
a
5
F
x
,
y
t
,
a
1
F
x
,
T
x
α
x
t
+
a
4
+
a
5
F
x
,
y
t
≥
a
1
+
a
4
+
a
5
min
⁡
F
x
,
T
x
α
x
t
,
F
x
,
y
t
;
∀
t
∈
R
+so that(61)Fx,Tyαykt≥Fy,Txαxkt≥a1+a4+a51-a2-a3min⁡Fx,Txαxt,Fx,yt;∀t∈R+.(d)
IfFx,Tyαykt≥Fy,Txαxkt, Fx,Txαxt≤Fy,Tyαyt, and Fy,Tyαyt≤Fx,yt; ∀t∈R+ then(62)Fx,Tyαykt≥Fy,Txαxkt≥a1+a4Fx,Txαxt+a2+a3Fy,Txαxt+a5Fx,yt;∀t∈R+,Fx,Tyαykt≥Fy,Txαxkt≥a1+a4+a51-a2-a3min⁡Fx,Txαxt,Fy,Tyαyt=a1+a4+a51-a2-a3Fx,Txαxt;∀t∈R+.(e)
IfFx,Tyαykt≤Fy,Txαxkt, Fx,Txαxt≥Fy,Tyαyt, and Fy,Tyαyt≥Fx,yt; ∀t∈R+ then(63)Fy,Txαxkt≥Fx,Tyαykt≥a1+a4+a51-a2-a3Fx,yt;∀t∈R+.(f)
IfFx,Tyαykt≤Fy,Txαxkt, Fx,Txαxt≥Fy,Tyαyt, and Fy,Tyαyt≤Fx,yt; ∀t∈R+ then(64)Fy,Txαxt≥Fx,Tyαyt≥a1+a4+a51-a2-a3Fy,Tyαyt;∀t∈R+.(g)
IfFx,Tyαykt≤Fy,Txαxkt, Fx,Txαxt≤Fy,Tyαyt, and Fy,Tyαyt≥Fx,yt; ∀t∈R+ then(65)Fy,Txαxkt≥Fx,Tyαykt≥a1+a4+a51-a2-a3min⁡Fx,Txαxt,Fx,yt;∀t∈R+.(h)
IfFx,Tyαykt≤Fy,Txαxkt, Fx,Txαxt≤Fy,Tyαyt, and Fy,Tyαyt≥Fx,yt; ∀t∈R+ then(66)Fy,Txαxkt≥Fx,Tyαykt≥a1+a4+a51-a2-a3min⁡Fx,Txαxt,Fy,Tyαyt=a1+a4+a51-a2-a3Fx,Txαxt;∀t∈R+.Now, take x1∈X, x2=y∈Txαx, and any z=x3∈Tyαy such that Fx,yt=Fx,Txt. Then, one gets for Cases (a)–(h)(67)Fx2,x3t≥a1+a4+a51-a2-a3Fx1,x2k-1t=Fx1,x2k-1tand we can construct a sequence xn⊂X with x1=x∈X arbitrary, xn+1∈Txnαxn∈CBX for some sequence αxn⊂0,1, such that since a1+a4+a5/1-a2-a3=1, one gets proceeding recursively:(68)Fxn+1,xnt≥a1+a4+a51-a2-a3Fx1,x2k-1t≥Fx1,x2k-n+1t;∀n≥2∈Z+;∀t∈R+so that limn→∞⁡Fxn+1,xnt=1; ∀t∈R+.Remark 12.
Extensions of Theorem11 are direct for the case when the subsequent contractive condition holds instead of (48):(69)
H
F
T
x
α
x
k
t
,
k
t
T
y
α
y
≥
min
⁡
1
,
a
1
F
x
,
T
x
α
x
t
+
a
2
F
x
,
T
y
α
y
t
+
a
3
F
y
,
T
x
α
x
t
+
a
4
F
y
,
T
y
α
y
t
+
a
5
F
x
,
y
t
;
∀
x
,
y
∈
X
,
∀
t
∈
R
+
.Extensions of Theorem 11 and its variant of Remark 12 to the light of Theorems 2 and 10 are direct concerning the case when the coefficients of the respective contractive condition are functions ai:X×X×R0+→0,1 for i=1,3,4,5; a2:X×X×R0+→0,1. Also, close examples to Example 6 can be given for the more general contractive conditions of this section.



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*Source: 213174-2015-12-31.xml*