Journal of Applied Mathematics and Stochastic Analysis, 15:1 (2002), 29-37.
ON THE PROBABILISTIC DOMAIN
INVARIANCE
ISMAT BEG
Lahore University of Management Sciences
Department of Mathematics
54792-Lahore, Pakistan
SORIN GAL
University of Oradea
Department of Mathematics
3700-Oradea, Romania
(Received October, 1999; Revised August, 2001)
Probabilistic version of the invariance of domain for contractive field and
Schauder invertibility theorem are proved. As an application, the stability of
probabilistic open embedding is established.
Key words: Probabilistic Banach Space, Domain Invariance, Linear Operator, Invertibility, Open Mapping, Contraction Mapping, Fixed Point.
AMS subject classifications: 47S50, 46S40, 47A99.
1. Introduction
Menger [7] was the first who introduced the notion of probabilistic metric space which is a
generalization of the metric space. The study of this space was expanded rapidly with the
pioneering work of Scherwood [9], Schweizer and Sklar [10] and many others. Recently
Sehgal and Bharucha-Reid [8], Hadzic [5], Cho [3] and Beg, Rehman and Shahzad [1] have
studied fixed point theorems for probabilistic metric spaces. This paper deals with the
problem of showing that certain mappings on probabilistic normed spaces are
homeomorphisms and gives the sufficient conditions under which addition of open mappings
results in open mapping. We also study domain invariance and its applications in
probabilistic normed spaces.
2. Preliminaries
Let W be the set of all left continuous functions defined on e such that for all 0 − W,
0 ÐBÑ œ ", for all B Ÿ !;
0 Ð∞Ñ œ !;
Ð"Ñ
Ð#Ñ
and 0 is a nonincreasing function. On W, we consider the natural ordering, that is, 0 Ÿ 1
Ð0 ß 1 − WÑ if and only if 0 ÐBÑ Ÿ 1ÐBÑ for all B − e and 0  1 if 0 Ÿ 1 and there exists an B!
Printed in the U.S.A. ©2002 by North Atlantic Science Publishing Company
29
30
ISMAT BEG and SORIN GAL
such that 0 ÐB! Ñ  1ÐB! Ñ. We denote by M the function in W with the property that MÐBÑ œ !
for B  !.
Let X be a subset of W containing M . A triangular function . on X with values in X is any
associative and commutative composition law having the following properties: for +ß +" ß +# ß ,
in X ,
.Ð+ß MÑ œ +;
Ð$Ñ
.Ð+" ß +Ñ Ÿ .Ð+# ß +Ñ whenever +" Ÿ +# ;
Ð%Ñ
.Ð+ß ,Ñ Ÿ ." Ð+ß ,Ñ,
Ð&Ñ
where ." Ð+ß ,ÑÐBÑ œ inf minÖ+Ð>BÑ  ,ÒÐ"  >ÑBÓß "× Öhere the infimum is taken over all
>
> − Ò!ß "Ó×.
Note that among a number of possible universal choices for ., .Ð+ß ,Ñ œ maxÐ+ß ,Ñ is the
strongest possible universal ..
A linear space P over e (real field) is called a probabilistic normed space if there exists a
mapping ² † à † ² À P Ä X such that the following properties hold:
² 9à † ² œ M if and only if 9 œ !à
Ð'Ñ
² +9à B ² œ ² 9à ± + ± " B ² for any B − e and + − e, + Á !à
Ð(Ñ
² 9  <à B ² Ÿ .Ð ² 9à B ² ß ² <à B ² ÑÞ
Ð)Ñ
The mapping 9 Ä ² 9à † ² is called a probabilistic norm or a random norm on P. A
sequence Ð98 Ñ of points in a probabilistic normed space P is said convergent to a point 9 if
for any !  % Ÿ ", !  $  ∞ there exists a positive integer R%ß$ such that
² 98  9à $ ²  % for all 8  R%ß$ . The sequence Ð98 Ñ is said Cauchy sequence if for any
!  % Ÿ ", !  B   ∞, there exists a positive integer R%ßB such that
² 98:  98 à B ²  %, for all 8  R%ßB ß : − . A complete probabilistic normed space is
called probabilistic Banach space. Denote Y%ß$ Ð9Ñ œ Ö< − Pà ² 9  <à $ ²  %×, where
!  % Ÿ " and ! Ÿ $  ∞. Whenever there is no confusion, we write Y Ð9Ñ for Y%ß$ Ð9Ñ.
Let 0 be a mapping from a probabilistic normed space P" into a probabilistic normed
space P# , then 0 is said to be continuous at 9! − P" if 9 − Y Ð9! Ñ implies 0 Ð9Ñ − Y Ð0 Ð9! ÑÑ.
Let 0 be a linear operator from a probabilistic normed space P" into a probabilistic normed
space P# , then 0 is called bounded if there exists some 7 ! such that
² 0 Ð9Ñà † ² Ÿ ² 79à † ² for all 9 − P" , and the probabilistic norm
of 0 is defined as ² 0 à B ² œ inf
² ² 9à B ² 0 Ð9Ñà B ² . If 0 is a bounded linear
! Á 9 − P"
operator from a probabilistic normed space P" into a probabilistic normed space P# , then
² 0 Ð9Ñà B ² Ÿ ² ² 0 à B ² 9à B ² Þ Let P" and P# be two probabilistic normed spaces and
J À P" Ä P# be a mapping satisfying ² J Ð9Ñ  J Ð<Ñà † ² Ÿ ² αÐ9  <Ñà † ² then the
constant α, is called Lipschitzian. The smallest such α is called Lipschitz constant of J and
denote it by _ÐJ Ñ. If _ÐJ Ñ  " then the map J is called contraction. If _ÐJ Ñ œ " then the
map J is said to be nonexpansive.
Let LÐBÑ denote the distribution function defined as follows:
LÐBÑ œ œ
!
"
BŸ!
B  !.
On the Probabilistic Domain Invariance
31
A probabilistic metric space (abbreviated as PM-space) is an ordered pair ÐWß IÑ where W
is a nonempty set and I is a function defined on W ‚ W into the set
J À e Ä Ò!ß "Óà J nondecreasing and
Y œœ
,
left continuous on eß J Ð  ∞Ñ œ !ß J Ð  ∞Ñ œ "
such that if :ß ; are points of W , then
I:; Ð!Ñ œ ! (I:; denotes IÐT ß ;ÑÑà
Ð*Ñ
I:; ÐBÑ œ LÐBÑ for all B − e if and only if : œ ;à
Ð"!Ñ
I:; œ I;: à
Ð""Ñ
if I:; ÐBÑ œ " and I;< ÐCÑ œ ", then I:< ÐB  CÑ œ "Þ
Ð"#Ñ
A mapping OÀ Ò!ß "Ó ‚ Ò!ß "Ó Ä Ò!ß "Ó is called T-norm if it satisfies:
OÐBß "Ñ œ Bß OÐ!ß !Ñ œ !à
Ð"$Ñ
OÐBß CÑ OÐ?ß @Ñ for B ?ß C @à
Ð"%Ñ
OÐBß CÑ œ OÐCß BÑà
Ð"&Ñ
OÐOÐBß CÑß ?Ñ œ OÐBß OÐCß ?ÑÑà
Ð"'Ñ
for all Bß Cß ?ß @ in Ò!ß "Ó.
A Menger PM-space is a triplet ÐWß Iß OÑ where ÐWß IÑ is a PM-space and the T-norm O
is such that the inequality
I:; ÐB  ;Ñ OÐI:< ÐBÑß I<; ÐCÑÑà
Ð"#‡ Ñ
holds for all :ß ;ß < in W and all B !, C !Þ
A sequence Ð98 Ñ in the PM-space ÐWß Iß OÑ is said Cauchy sequence if
lim I98: 98 ÐBÑ œ ", for any B  ! and : − .
8Ä∞
A mapping J À ÐWß Iß OÑ Ä ÐWß Iß OÑ is called a contraction if there exists α − Ð!ß "Ñ
such that
IJ Ð:ÑJ Ð;Ñ ÐαBÑ I:; ÐBÑß
for any B  ! and :ß ; − W .
Lemma 2.1: [4, Theorem 3.4.12]. Let J be a linear operator from a probabilistic normed
space P" into a probabilistic normed space P# . If J is bounded then J is continuous.
Theorem 2.2: [6, Theorem 11.2.2]. Every contraction mapping on a PM-space has at
most one fixed point.
Theorem 2.3: [6, Theorem 11.2.4]. Let ÐWß Iß OÑ be a complete Menger PM-space and
let J be a contraction mapping on W . Then J has a unique fixed point where
OÐBß CÑ œ 738ÐBß CÑ.
Remark 2.4: Let ÐPß ² † à † ² ß .Ñ be a probabilistic Banach space with .Ð+ß ,Ñ œ
maxÖ+ß ,×. If Ð98 Ñ is a Cauchy sequence in ÐPß ² † à † ² ß .Ñ and J is a contraction on
ÐPß ² † à † ² ß .Ñ, then ÐPß Iß OÑ, with OÐBß CÑ œ minÖBß C× and I:; ÐBÑ œ "  ² :  ;à
B ² , becomes a complete Menger PM-space (i.e., Ð98 Ñ becomes a Cauchy sequence in
32
ISMAT BEG and SORIN GAL
ÐPß Iß OÑ and the convergence of a sequence in ÐPß ² † à † ² ß .Ñ is equivalent to the
convergence in ÐPß Iß OÑÑ and J becomes a contraction on ÐPß Iß OÑÑ.
For more details, we refer to Serstnev [12], Istratescu [6], Schweizer and Sklar [11] and
Chang et al [2].
We will prove the following fixed point theorem in probabilistic Banach spaces for
subsequent use in the next sections.
Theorem 2.5: Let
Xœœ
0 − Wà 0 is nonconvex on Ò!ß  ∞Ñ i.e. +0 ÐBÑ  Ð"  +Ñ0 ÐCÑ
Ÿ 0 Ð+B  Ð"  +ÑCÑß a+ − Ò!ß "Óß aBß C !
and .À X ‚ X Ä X be a triangular function. Also, let ÐP ² † à † ² ß .Ñ be a probabilistic
Banach space Ðwhere ² Pß † ² § X Ñ. If J À ÐPß ² † à † ² ß .Ñ Ä ÐPß ² † à † ² ß .Ñ is a
contraction mapping, then J has a unique fixed point.
Proof: Let 9! − P be fixed and consider the iterations 98" œ J Ð98 Ñ, 8 − . For any
B  ! we have
² 98"  98 à B ² œ ² J Ð98 Ñ  J Ð98" Ñà B ² Ÿ ² αÐ98  98" Ñà B ²
Ÿ á Ÿ ² α8 Ð9"  9! Ñà B ² œ ² 9"  9! à αB8 ² Þ
Then
² 98#  98 à B ² Ÿ .Ð ² 98#  98" à B ² ß ² 98"  98 à B ² Ñ
B
Ÿ .Ð ² 9"  9! à α8"
² ß ² 9"  9! à αB8 ² Ñ
Ÿ .Ð ² 9"  9! αB8 ² ß ² 9"  9! à αB8 ² Ñ
Ÿ ." Ð ² 9"  9! à αB8 ² ß ² 9"  9! à αB8 ² Ñ
œ inf šminš ² 9"  9! à α>B8 ²  ² 9"  9! à Ð">ÑB
² ß "››
α8
> − Ò!ß "Ó
(because ² 9"  9! à † ² is nonconvex)
Ÿ inf ˜min˜ ² 9"  9! à αB8 ² ß "™™ œ ² 9"  9! à αB8 ² Þ
> − Ò!ß "Ó
By induction, it easily follows that in general,
² 98:  98 à B ² Ÿ ² 9"  9! à αB8 ² ß a8 − ß : − Þ
Passing to limit with 8 Ä  ∞, we obtain that Ð98 Ñ is a Cauchy sequence. Because
ÐPß ² † à † ² ß .Ñ is a probabilistic Banach space, therefore there exists 9‡ − P, such that
aB  !, lim ² 98  9‡ à B ² œ !. Because
8Ä∞
² 9‡  J Ð9‡ Ñà B ² Ÿ .Ð ² 9‡  98" à B ² ß ² 98"  J Ð9‡ Ñà B ² Ñ
œ .Ð ² 9‡  98" à B ² ß ² J Ð98 Ñ  J Ð9‡ Ñà B ² Ñ
Ÿ .Ð ² 9‡  98" à B ² ß ² αÐ98  9‡ Ñà B ² Ñ
On the Probabilistic Domain Invariance
33
Ÿ .Ð ² 9‡  98" à B ² ß ² 98  9‡ à B ² ÑÞ
Let B  ! be fixed and !  %  "# . Then there exists RBß%  ! such that for all 8  RBß% , we
have
² 9‡  98" à B ²  % and ² 98  9‡ à B ²  %.
It follows that
² 9‡  J Ð9‡ Ñà B ² Ÿ .Ð%ß %Ñ Ÿ ." Ð%ß %Ñ  #%ß
and % is arbitrary (sufficiently small), we obtain ² 9‡  J Ð9‡ Ñà B ² œ !, aB  !, i.e.,
9‡ œ J Ð9‡ Ñ.
In order to prove the uniqueness, let us suppose that there exist two fixed points
9 Á < − P, for J . We get
² 9  <à B ² œ ² J Ð9Ñ  J Ð<Ñà B ² Ÿ ² 9  <à αB ² Ÿ ² 9  <à B ² ß
which implies ² 9  <à B ² œ ² 9  <à αB ² , aB  !, and taking into account that
² 9  <à  ∞ ² œ !, we obtain ² 9  <à B ² œ !, aB  !, i.e., 9  < œ !, which proves
the theorem.
Remark 2.6: For reasons of simplicity, throughout the rest of the paper, a probabilistic
Banach space ÐPß ² † à † ² ß .Ñ with .Ð+ß ,Ñ œ maxÐ+ß ,Ñ or satisfying the hypothesis in the
statement of Theorem 2.5, will be called of W-type.
3. Invariance of Domain for Contractive Fields
Lemma 3.1:
Let P be a probabilistic Banach space of S-type and
FÒ9! ß MÐB  <ÑÓ œ Ö9à ² 9  9! à B ²  MÐB  <Ñ×. Let J À F Ä P be a contraction with
_ÐJ Ñ œ α  ". If ² J Ð9! Ñ  9! à B ²  MÐB  Ð"  αÑ<Ñ, then J has a fixed point.
Proof: Choose %  < so that
² J Ð9! Ñ  9! à B ² Ÿ MÐB  Ð"  αÑ%Ñ  MÐB  Ð"  αÑ<ÑÞ

The mapping J maps the closed ball H œ Ö9à ² 9  9! à B ² Ÿ MÐB  %Ñ× into itself.

Indeed, if 9 − H , then
² J Ð9 Ñ  9! à B ² œ ² J Ð 9 Ñ  J Ð 9! Ñ  J Ð 9 ! Ñ  9 ! à B ²
Ÿ .Ð ² J Ð9Ñ  J Ð9! Ñà B ² ß ² J Ð9! Ñ  9! à B ² Ñ
Ÿ .Ð ² 9  9! à αB ² ß MÐB  Ð"  αÑ%Ñ
Ÿ .ÐMÐ αB  %Ñß MÐB  Ð"  αÑ%Ñ
œ .ÐMÐB  α%Ñß MÐB  Ð"  αÑ%Ñ œ MÐB  %w Ñ


where α% œ %w  <, α%  α<  < and !  %  ". Thus if 9 − H then J Ð9Ñ − H .

Because P is complete, it is sufficient to prove that H is closed. In this sense, let Ð98 Ñ be

a sequence of points in H , convergent to a point 9‡ − P. We obtain
34
ISMAT BEG and SORIN GAL
² 9 ‡  9 ! à B ² Ÿ .Ð ² 9 ‡  9 8 à B ² ß ² 9 8  9 ! à B ² Ñ
Ÿ ." Ð ² 9 ‡  98 à B ² ß ² 98  9! à B ² Ñ
œ inf ÖminÖ ² 9‡  98 à >B ²  ² 98  9! à Ð"  >ÑB ² ß "××Þ
> − Ò!ß "Ó
We have two cases:
a)
B − Ð  ∞ß %Óà , ÑB − Ð%ß  ∞Ñ
Case a) obviously MÐB  %Ñ œ " and therefore
² 9‡  9! à B ² Ÿ " œ MÐB  %ÑÞ
b)
For each > − Ò!ß "Ó, fixed, and each !  (  ", there exists R>ß( − , such that for
8  R>ß( we get ² 9‡  98 à >B ²  (.
It follows
² 9‡  9! à B ² Ÿ inf ÖminÖ(  MÐÐ"  >ÑB  %Ñß "××  (ß
> − Ò!ß "Ó
because for >  " 
we get
%
B
we have MÐÐ"  >ÑB  %Ñ œ !. Since ( is arbitrary (independent of BÑ,
² 9‡  9! à B ² œ ! œ MÐB  %Ñ for B  %.


As a conclusion, 9‡ − H . Since H is complete, we get that J has a fixed point by Theorem
2.3 and by Theorem 2.5.
Definition 3.2: Let Q be a subset of a probabilistic Banach space P. Given a map
J À Q Ä P, the map 0 À Q Ä P defined by 0 Ð9Ñ œ 9  J Ð9Ñ is called the field associative
with J . The field 0 determined by a contraction J is called a contractive field.
Theorem 3.3: Let P be a probabilistic Banach space of S-type, Y § P open, and
J À Y Ä P be a contraction. Let 0 À Y Ä P be the field associative with J . Then,
Ð3Ñ
the field 0 is an open mapping; in particular, 0 ÐY Ñ is open in P, and
Ð33Ñ
the mapping 0 À Y Ä 0 ÐY Ñ is a homeomorphism.
Proof: (3Ñ To show that 0 is an open mapping, it is enough to show that for any 9 − Y , if
FÒ9ß <Ó § Y , then FÒ0 Ð9Ñß Ð"  αÑ<Ó § 0 ÐFÒ9ß <ÓÑ.
For this purpose, choose any
<! − FÒ0 Ð9Ñß Ð"  αÑ<Ó and define KÀ FÒ9ß <Ó Ä P by KÐ<Ñ œ <!  J Ð<Ñ. Then K is a
contraction and
² KÐ9Ñ  9à B ² œ ² <!  J Ð9Ñ  9à B ²
œ ² <!  0 Ð9Ñà B ²  MÐB  Ð"  αÑ<ÑÞ
By Lemma 3.1 there exists 9! − FÒ9ß <Ó with 9! œ <!  J Ð9! Ñ, therefore
0 Ð9! Ñ œ <! œ 9!  J Ð9! Ñ. Thus FÒ0 Ð9Ñß Ð"  αÑ<Ó § 0 ÐFÒ9ß <ÓÑ. So 0 is an open mapping
and, in particular, 0 ÐY Ñ is open in P.
Ð33Ñ If 9ß < − Y , then
² 0 Ð9Ñ  0 Ð<Ñà B ² œ ² Ð9  <Ñ  ÐJ Ð9Ñ  J Ð<ÑÑà B ²
² Ð"  αÑÐ9  <Ñà B ² Þ
On the Probabilistic Domain Invariance
35
So that 0 is injective. Since 0 À Y Ä 0 ÐY Ñ is a continuous open bijection, it is a
homeomorphism.
Corollary 3.4: Let P be a probabilistic Banach space of S-type and J À P Ä P be a
contraction. Then the corresponding field 0 œ M.  J is a homeomorphism.
Proof: Let <! − P, define KÀ P Ä P by KÐ9Ñ œ <!  J Ð9Ñ. Then K is a contraction, so
K has a fixed point 9! œ <!  J Ð9! Ñ, therefore 0 Ð9! Ñ œ <! . Thus 0 ÐPÑ œ P.
4. Domain Invariance and Invertibility of Linear Operators
Proposition 4.1: Let J be a linear operator on a probabilistic Banach space P of W -type. If
² M.  J à " ² œ !, then J is invertible.
Proof: The map M.  J À P Ä P is a contraction. Indeed,
² ÐM.  J ÑÐ9  <Ñà B ² Ÿ ² ² ÐM.  J Ñà B ² Ð9  <Ñà B ²
œ ² αÐ9  <Ñà B ² ß α − Ð!ß "ÑÞ
Corollary 3.4 further implies that M.  ÐM.  J Ñ œ J is a homeomorphism. Thus J is
invertible.
Lemma 4.2: Let P" and P# be two probabilistic normed spaces and J À P# Ä P" and
KÀ P" Ä P# be two bounded linear operators. Then ² J Kà B ² Ÿ ² J ² Kà B ² à B ² Þ
Proof: We have
² J Kà B ² œ inf
² ² 9à B ² ÐJ KÑÐ9Ñà B ²
! Á 9 − P"
œ inf
² J Ð ² 9à B ² KÐ9ÑÑà B ² Ÿ ² J ² Kà B ² à B ² Þ
! Á 9 − P"
Remark 4.3: Let P" and P# be two probabilistic Banach spaces and WÀ P" Ä P# be a
linear operator. If there is some 7  ! such that ² WÐ9Ñà B ² ² 79à B ² for 9 − P" ,
then W is injective; if such an W is also surjective, then it is invertible because
<
² W " Ð<Ñà B ² Ÿ ² 7
à B ² for < − P# . Therefore W " is continuous by Lemma 2.1.
Theorem 4.4: [Schauder Invertibility Theorem] Let P" and P# be two probabilistic
Banach spaces of S-type and Wß X À P" Ä P# be two linear operators, with W invertible.
Assume that there is an 7  ! such that, for each ! Ÿ > Ÿ ", the operator
I> œ Ð"  >ÑW  >X satisfies ² I> Ð9Ñà B ² ² 79à B ² for all 9 − P" . Then I> is
invertible for all ! Ÿ > Ÿ ", and in particular, X is invertible.
Proof: If an operator I= is invertible, then for each > in the open interval
7
N= œ Ö>À ± >  = ±  ²X WàB²
×, the operator I> is invertible or equivalently,
"
I= I> À P" Ä P" is invertible for each > − N= . For this purpose, note that
I> œ W  =ÐX  WÑ  Ð>  =ÑÐX  WÑ
œ I=  Ð>  =ÑÐX  WÑÞ
So that,
I=" I> œ M  Ð>  =ÑI=" ÐX  WÑÞ
Because
36
ISMAT BEG and SORIN GAL
² I= 9à B ² ² 79à B ² ß
so for > − N= , we have
²  Ð>  =ÑI=" ÐX  WÑà B ² œ ² I=" ÐX  WÑà
B
Ð>=Ñ
²
B
Ÿ ² I=" ² X  Wà B ² à Ð>=Ñ
²
7
Ÿ ² I=" ²X WàB²
² X  Wà B ² à B ² Ÿ MÐB  "ÑÞ
Therefore
² M.  I=" I> à B ² Ÿ MÐB  "ÑÞ
Proposition 4.1 further implies that I=" I> is invertible. Thus I> is invertible. Now assume
that Y œ Ö> − Ò!ß "ÓÀ I> is invertible×, then Y is an open set. If >  Y , then the operator I=
with = − N> is not invertible. So that Ò!ß "Ó  Y is also an open set. Because Ò!ß "Ó is
connected and Y is nonempty, we get Y œ Ò!ß "Ó and the proof is complete.
5. Stability of Open Embeddings
Theorem 5.1: Let P be a probabilistic normed space, Q be a probabilistic Banach space of
S-type and J À P Ä Q be an open embedding of P onto an open subset Y § Q . Let
KÀ P Ä Q be a map such that K ‰ J " À Y Ä P is a contraction. Then 9 Ä J Ð9Ñ  KÐ9Ñ
is also an open embedding of P into Q .
Proof:
Consider 2 œ ÐJ  KÑ ‰ J " œ M.  KJ " À Y Ä Q .
By the domain
invariance, this 2 maps Y homeomorphically onto an open 2ÐY Ñ § Q . Since
J  K œ 2 ‰ J œ ÐJ  KÑ ‰ J " ‰ J , it follows that 2 ‰ J is an open embedding of P into
Q.
Theorem 5.2: Let P be a probabilistic metric space, Q be a probabilistic Banach space
of S-type and J À P Ä Q be an open embedding such that J " is Lipschitzian. Let
KÀ P Ä Q be a Lipschitzian map such that _ÐKÑ_ÐJ " Ñ  ". Then 9 Ä J Ð9Ñ  KÐ9Ñ is
also an open embedding of P into Q .
Proof: Since _ÐK ‰ J " Ñ Ÿ _ÐKÑ_ÐJ " Ñ  ", we get that K ‰ J " is a contraction. By
Theorem 5.1, J  K is also an open embedding of P into Q .
Acknowledgement
The authors are thankful to the referee for careful reading and many critical remarks to
improve the presentation of the paper.
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