R E S E A R C H
Open Access
Clark’s fixed point theorem on a complete
random normed module
Yujie Yang
**Correspondence:
[email protected] Department of Basic Courses, Beijing Union University, Beijing, 100101, P.R. China
Abstract
Motivated by the recent results in random metric theory, we establish Clark’s fixed point theorem on complete random normed modules under two kinds of topologies. When the base space of the random normed module is trivial, our results
automatically degenerate to the classical case.
MSC: 58E30; 47H10; 46H25; 46A20
Keywords: random normed module; (
ε
,λ
)-topology; locallyL0-convex topology; Clark’s fixed point theorem1 Introduction
Random metric theory is based on the idea of randomizing the classical space theory of functional analysis. All the basic notions such as random normed modules, random inner product modules and random locally convex modules, together with their random conju-gate spaces, were naturally presented by Guo in the course of the development of random functional analysis [–]. In the last ten years, random metric theory and its applications in the theory of conditional risk measures have undergone a systematic and deep develop-ment [–]. Especially after , in [] Guo gives the relations between the basic results currently available derived from two kinds of topologies, namely the (ε,λ)-topology and the locallyL-convex topology. In [], Guo gives some basic results onL-convex
analy-sis together with some applications to conditional risk measures and studies the relations among three kinds of conditional convex risk measures. These results pave the way for further research of the random metric theory and conditional convex risk measures.
In , Clark presented Clark’s fixed pointed theorem []. It has been applied in many fields such as optimization theory, different equations, and fixed point theory. Based on the recent work of random metric theory, in this paper, we establish Clark’s fixed pointed theorem on complete random normed modules under two kinds of topologies. A ran-dom normed module is a ranran-dom generalization of an ordinary normed space. Different from ordinary normed spaces, random normed modules possess the rich stratification structure, which is introduced in this paper. It is this kind of rich stratification structure that makes the theory of random normed modules deeply developed and also becomes the most useful part of random metric theory. When the probability (,F,P) is trivial, namelyF={∅,}, our results reduce to the classical Clark’s fixed pointed theorem. So the extension of our results is nontrivial.
The reminder of this article is organized as follows. In Section we briefly recall some necessary notions and facts. In Section we present and prove our main results.
2 Preliminary
Throughout this paper, (,F,P) denotes a probability space,Kthe real number fieldRor the complex number fieldC,Nthe set of positive integers,L¯(F) the set of equivalence
classes of extended real-valued random variables onandL(F,K) the algebra of
equiva-lence classes ofK-valuedF-measurable random variables onunder the ordinary scalar multiplication, addition and multiplication operations on equivalence classes, denoted by L(F) whenK=R.
The pleasant properties of the complete latticeL¯(F) (see the introduction for the
no-tationL¯(F)) are summarized as follows.
Proposition .([]) For every subset A ofL¯(F),there exist countable subsets{an|n∈
N}and{bn|n∈N}of A such thatn≥an=∨A andn≥bn=∧A.Further,if A is directed (dually directed)with respect to≤,then the above{an|n∈N}(accordingly,{bn|n∈N}) can be chosen as nondecreasing(correspondingly,nonincreasing)with respect to≤.
Specially,L
+(F) ={ξ∈L(F)|ξ≥},L++(F) ={ξ∈L(F)|ξ> on}.
As usual,ξ>ηmeansξ≥ηandξ=η, whereasξ>ηonAmeansξ(ω) >η(ω) a.s. onA for anyA∈FandξandηinL¯(F), whereξandηare arbitrarily chosen representatives
ofξandη, respectively.
For anyA∈F,Acdenotes the complement ofA,A˜ ={B∈F|P(AB) = }denotes the equivalence class ofA, whereis the symmetric difference operation,IAthe characteristic function ofA, andIA˜ is used to denote the equivalence class ofIA; given twoξ andηin
¯
L(F), andA={ω∈:ξ=η}, whereξandηare arbitrarily chosen representatives of
ξ andηrespectively, then we always write [ξ=η] for the equivalence class ofAandI[ξ=η]
forIA. One can also understand the implication of such notations as˜ I[ξ≤η],I[ξ<η]andI[ξ=η].
For an arbitrarily chosen representativeξofξ ∈L(F,K), define twoF-measurable
random variables (ξ)–and|ξ|by (ξ)–(ω) =
ξ(ω) ifξ(ω)= , and (ξ)–(ω) = other-wise, and by|ξ|(ω) =|ξ(ω)|,∀ω∈. Then the equivalence classξ–of (ξ)–is called
the generalized inverse ofξand the equivalence class|ξ|of|ξ|is called the absolute value
ofξ. It is clear thatξ·ξ–=I [ξ=].
Definition .([]) An ordered pair (E, · ) is called a random normed space (briefly, anRNspace) overKwith base (,F,P) ifEis a linear space and · is a mapping from EtoL
+(F) such that the following three axioms are satisfied:
() x= if and only ifx=θ(the null vector ofE); () αx=|α|x,∀α∈Kandx∈E;
() x+y ≤ x+y,∀x,y∈E,
where the mapping · is called the random norm onEandxis called the random norm of a vectorx∈E.
In addition, ifEis left module over the algebraL(F,K) such that the following is also satisfied:
() ξx=|ξ|x,∀ξ∈L(F,K)andx∈E,
There are two important topologies in random metric theory as follows.
Proposition .([]) Let(E, · )be an RN module over K with base(,F,P).For any real numbersε> , <λ< ,let Nθ(ε,λ) ={x∈E|P{ω∈| x(ω) <ε}> –λ}andUθ= {Nθ(ε,λ)|ε> , <λ< },thenUθis a local base atθ of some Hausdorff linear topology,
called the(ε,λ)-topology induced by · .Further,we have the following statements: () L(F,K)is a topological algebra overKendowed with its(ε,λ)-topology,which is
exactly the topology of convergence in probabilityP;
() Eis a topological module over the topological algebraL(F,K)whenEandL(F,K)
are endowed with their respective(ε,λ)-topologies;
() A net{xα,α∈ ∧}inEconverges in the(ε,λ)-topology tox∈Eiff{xα–x,α∈ ∧}
converges in probabilityPto.
From now on, for allRNmodules, (ε,λ)-topology is denoted byTε,λ.
Proposition .([]) Let(E, · )be an RN module over K with base(,F,P).For any
ε∈L++(F),let B(ε) ={x∈E| x ≤ε}andUθ={B(ε)|ε∈L++(F)}.A set G⊂E is called
Tc-open if for every x∈G there exists some B(ε)∈Uθsuch that x+B(ε)⊂G.LetTcbe the family ofTc-open subsets,thenTcis a Hausdorff topology on E,called the locally L-convex topology induced by · .Further,the following statements are true:
() L(F,K)is a topological ring endowed with its locallyL-convex topology;
() Eis a topological module over the topological ringL(F,K)whenEandL(F,K)are
endowed with their respective locallyL-convex topologies;
() A net{xα,α∈ ∧}inEconverges in the locallyL-convex topology tox∈Eiff {xα–x,α∈ ∧}converges in the locallyL-convex topology ofL(F,K)toθ.
From now on, for allRNmodules, locallyL-convex topology is denoted byTc. SinceTc is not necessarily a linear topology as proved in [], but (E,Tc) is always a topological group with respect to the addition operation for anyRNmodule (E, · ), and henceTc-Cauchy nets andTc-completeness are still well defined.
Let (E, · ) be anRNmodule overK with base (,F,P),pA=IA˜ ·pis called the A-stratification ofpfor each givenA∈F andpinE. The so-called stratification structure ofEmeans thatEincludes every stratification of an element inE. Clearly,pA=θ when P(A) = andpA=pwhenP(\A) = , which are both called trivial stratifications ofp. Further, when (,F,P) is a trivial probability space, every element inEhas merely two trivial stratifications sinceF={,∅}; when (,F,P) is arbitrary, every element inEcan possess arbitrarily many nontrivial intermediate stratifications. It is this kind of rich strat-ification structure ofRNmodules that makes the theory ofRNmodules deeply developed and also becomes the most useful part of random metric theory.
To introduce the main results of this paper, let us first recall the definition of the count-able concatenation property as follows.
Definition . ([]) Let E be a left module over the algebra L(F,K). A formal sum
n∈NIAnxnis called acountable concatenationof a sequence{xn|n∈N}inE with re-spect to a countable partition{An|n∈N}oftoF. Moreover, a countable concatena-tionn∈NIAnxnis well defined or
n∈NIAnxn∈Eif there isx∈Esuch thatIAnx=IAnxn,
countable concatenationn∈N˜IAnxnwithxn∈Gfor eachn∈Nstill belongs toG, namely
n∈N˜IAnxnis well defined and there existsx∈Gsuch thatx=
n∈N˜IAnxn.
Definition .([]) LetEbe a left module over the algebraL(F) andf be a function
fromEtoL¯(F), then
() f isL(F)-convex iff(ξx+ ( –ξ)y)≤ξf(x) + ( –ξ)f(y)for allxandyinEand
ξ∈L
+(F)such that≤ξ≤. (Here, we make the convention that·(±∞) = and
∞–∞=∞!)
() f is said to have the local property ifIAf˜ (x) =˜IAf(˜IAx)for allx∈EandA∈F.
Now, we introduce a kind of lower semicontinuity forL¯-valued functions, which is very
suitable for the study of conditional risk measures [].
Definition .([]) Let (E, · ) be anRNmodule overRwith base (,F,P). A function f :E→ ¯L(F) is calledT
ε,λ-lower semicontinuous ifepi(f) :={(x,r)∈E×L(F)|f(x)≤r}
is closed in (E,Tε,λ)×(L(F),Tε,λ). A functionf :E→ ¯L(F) is calledTc-lower
semicon-tinuous ifepi(f) is closed in (E,Tc)×(L(F),Tc).
LetEbe a left module over the algebraL(F,K), a nonempty subsetMofEis called L(F)-convex ifξx+ηy∈Mfor anyxandy∈Mandξandη∈L+(F) such thatξ+η= . It is well known from [] thatf :E→ ¯L(F) isL(F)-convex ifff has the local property
andepi(f) isL(F)-convex.
3 Main results
The main results in this section are Theorem . and . below. To introduce them, we first give some necessary notions and terminology.
Definition . LetEbe a left module over the algebraL(F). A nonempty subsetKofE is called a random cone ofEifξ·x∈K,∀ξ∈L
+(F),x∈K.
Definition . LetKbe a random cone of anL(F)-moduleE.Kis called () L(F)-convex ifx
+x∈K,∀x,x∈K;
() pointed ifx∈K,–x∈K⇒x= .
Lemma . Let(E, · )be an RN module over R with base(,F,P)andα∈L
++(F)with
<α< .Define Kα:={(x,r)∈E×L(F) :αx ≤–r}.Then Kα is aTε,λ-closed L(F
)-convex random cone.
Further,if E has the countable concatenation property,then Kαhas the countable
con-catenation property.
Proof It is easy to see thatKαisL(F)-convex andTε,λ-closed.
Let{(xn,rn),n∈N}be inKα, namelyαxn ≤–rn,∀n≥. For any countable partition {An,n≥}oftoF, it follows thatα
k
n=I˜An·xn ≤α·
k
n=˜IAn·xn ≤
k
n=(–I˜An·rn). By the countable concatenation properties ofEandL(F), one can haveα∞
n=IA˜ n·xn ≤
∞
n=(–˜IAn·rn), namely
∞
n=IA˜ n(xn,rn)∈Kα.
Remark . IfK is a pointed andL(F)- convex random cone of anL(F)-moduleE,
() x≤Ky⇔λx≤Kλy,∀λ∈L+(F);
() x≤Ky⇒x+z≤Ky+z,∀x,y,z∈E.
In classical metric spaces, the following result Lemma . is clear. But when we come to RNmodules, it is not easy to be proved.
Lemma .([]) Let(E,·)be an RN module over R with base(,F,P),G⊂E be a subset with the countable concatenation property and f :E→ ¯L(F)have the local property.If f|
G is proper and bounded from below on G(resp.,bounded from above on G),then,for each
ε∈L
++(F),there exists xε∈G such that f(xε)≤ ∧f(G) +ε(accordingly,f(xε)≥ ∨f(G) –ε). Lemma . Let(E, · )be aTε,λ-complete RN module over R with base(,F,P),A⊂
E×L(F) be a nonemptyT
ε,λ-closed subset such that A has the countable
concatena-tion property.Ifα∈L
++(F)with <α< and
{r∈L(F) : (x,r)∈A}= ,then for each
(x,r)∈A,there exists(x,¯ r)¯ ∈A such that:
() (x,¯ ¯r)∈A∩[Kα+ (x,r)];
() (x,¯ ¯r) =A∩[Kα+ (x,¯ ¯r)].
Proof Define a functionf :E×L(F)→L(F) byf(x,r) =r,∀(x,r)∈E×L(F). And define a mapping · ∗:E×L(F)→L
+(F) byx∗=ξ ∨ |r|,∀x= (ξ,r)∈E×L(F).
It is easy to check that (E×L(F), ·
∗) is an RN space.
Furthermore,E×L(F) is an RN module if we define a module multiplication·:L(F)·
(E×L(F))→E×L(F) byγ ·(ξ,r) = (γ ξ,γr),∀γ ∈L(F),∀(ξ,r)∈E×L(F).
For eachx= (ξ,r)∈E×L(F), it follows thatIB·f(IB·x) =IB·f(IB·(ξ,r)) =IB·f(IB·ξ,
IB·r) =IB·(IB·r) =IB·r=IB·f(x),∀B∈F, which meansf has the local property. SinceEhas the countable concatenation property, by Lemma .Kαhas the countable
concatenation property. Further, sinceAhas the countable concatenation property, it im-plies thatA∩[Kα+ (x,r)],∀(x,r)∈E×L(F) has the countable concatenation property.
Obviously,{r∈L(F) : (x,r)∈A}= implies thatf is bounded from below onA.
Lemma . yields a sequence {(xn,rn) :n≥}such that (xn+,rn+)∈An=A∩[Kα+
(xn,rn)] andrn+=f(xn+,rn+) <∧f(An) +n+,∀n≥.
One can have An+ ⊂An, ∀n≥, which follows from Kα+ (xn+,rn+)⊂Kα + [Kα+
(xn,rn)] =Kα+ (xn,rn).
We now prove that{xn,n≥}and{rn,n≥}areTε,λ-Cauchy sequences ofEandL(F),
respectively.
For each (y,s)∈An, it is easy to sees≥ ∧f(An) and (y,s)∈Kα+ (xn,rn), namelyαy–
xn ≤rn–s. Thus one can have αy–xn ≤rn–s≤ ∧f(An–) + n –s≤ ∧f(An) + n–
s≤ n,∀(xn,rn)∈An–, (y,s)∈An. Then it follows thatαy–y ≤αy–xn+αy–
xn ≤ n and|s–s| ≤ |s–rn|+|rn–s| ≤n,∀(y,s), (y,s)∈An. Hencediam(An) :=
(y,s),(y,s)∈An(y,s) – (y,s)∗=
(y,s),(y,s)∈An(y–y ∨ |s–s|)≤
nα→ in the
(ε,λ)-topology asn→ ∞. Since for eachε∈Rwithε> and eachλ∈Rwith <λ< , there existsNsuch thatP{w:
nα(w)≤ε}> –λ,∀n≥N, then(xn,rn) – (xm,rm)∗=xn–
xm ∨ |rn–rm| ≤diam(AN)≤Nα,∀m,n≥N. ThereforeP{w:xn–xm(w)≤ε} ≥P{w:
Nα(w)≤ε}> –λ,∀m,n≥N, which verifies that{xn,n≥}is aTε,λ-Cauchy sequence
ofE. Similarly,{rn,n≥}is aTε,λ-Cauchy sequence ofL(F).
)-topology tor, which implies¯ {(xn,rn) :n∈N} converges in the (ε,λ)-topology to (x,¯ ¯r). BothAandKαareTε,λ-closed, soAn=A∩[Kα+ (xn,rn)],∀n≥ is alsoTε,λ-closed. From
An+⊂An,∀n≥, one can have (x,¯ r)¯ ∈∞n=An.
For each (x,ˆ ˆr)∈∞n=An, (x,ˆ ˆr)∈Animpliesˆx–xn ≤αn,∀(xn,rn)∈An–. Then it fol-lows that{xn:n∈N}converges in the (ε,λ)-topology tox. Similarly, we can prove thatˆ
{rn:n∈N}converges in the (ε,λ)-topology tor. Sinceˆ Tε,λis Hausdorff onEandL(F),
one can have (x,¯ r) = (¯ x,ˆ ˆr). Thus we have{(x,¯ r)¯}=∞n=An. From (x,¯ r)¯ ∈A, () is proved.
If (y,s)∈A∩[Kα+ (x,¯ ¯r)], then (y,s)∈ ∞
n=An, which follows fromKα+ (x,¯ r)¯ ⊂Kα+
[Kα+ (xn,rn)] =Kα+ (xn,rn),∀n≥. Hence (y,s) = (x,¯ r), namely (¯ x,¯ r) =¯ A∩[Kα+ (x,¯ ¯r)].
Thus () is proved.
To prove Theorem . below, we need Lemma ., which is very easy and thus its proof is omitted.
Lemma . Let(E, · )be an RN module over R with base(,F,P)such that E has the countable concatenation property,and let f:E→L(F)have the local property.Then
epi(f)has the countable concatenation property.
Theorem . Let(E, · )be aTε,λ-complete RN module over R with base(,F,P),G be a
Tε,λ-closed subset of E,ε∈L++(F)andϕ:G→ ¯L(F)be proper,Tε,λ-lower semicontinuous
and bounded from below on G.Then for each point x∈G satisfyingϕ(x)≤ ∧ϕ(G) +εand
eachα∈L
++(F),there exists z∈G such that the following are satisfied:
() ϕ(z)≤ϕ(x) –αz–x;
() z–x ≤α–·ε;
() for eachx∈Gsuch thatx=z,ϕ(x)ϕ(z) –αx–z.
Proof We can, without loss of generality, suppose{ϕ(x),x∈E}= . Then ϕ(x)≤ε,
∀ε∈L ++(F).
SinceϕisTε,λ-l.s.c , it follows thatepi(ϕ) is closed in (E,Tε,λ)×(L(F),Tε,λ).
TakeA=epi(ϕ) and (x,r) = (x,ϕ(x))∈A.
By Lemma .,Ahas the countable concatenation property. According to Lemma ., there exists (z,r)∈Asuch that: (a) (z,r)∈A∩[Kα+ (x,ϕ(x))];
(b) (z,r) =A∩[Kα+ (z,r)].
From (a), one can haveαz–x ≤ϕ(x) –r≤ϕ(x) –ϕ(z)≤ϕ(x)≤ε, which yields ()
and ().
Now, we prove () as follows.
We can deduce ϕ(z) = r. Otherwise, one can have (z,r)= (z,ϕ(z)). By (b), we have (z,ϕ(z))¯∈Kα+ (z,r), namely ≤r–ϕ(z) does not hold. That is in contradiction to (z,r)∈A.
() is obvious, when ϕ(x) =∞. Ifϕ(x) <∞andx=z, then (x,ϕ(x))= (z,ϕ(z)) = (z,r). From (b), one can have (x,ϕ(x))¯∈Kα+ (z,ϕ(z)), namelyαx–zϕ(z) –ϕ(x).
Theorem . Let(E, · )be aTε,λ-complete RN module over R with base(,F,P),φ:
E→ ¯L(F)be a properTε,λ-lower semicontinuous function which is bounded from below,
Proof It follows from Theorem . that there exists some point z∈E, for every x=z, there existsAx∈Fsuch thatP(Ax) > andα· x–z>ψ(z) –ψ(x) onAx. We deduce thatTz=z. IfTz=zholds, it follows from Theorem . that there existsATz∈F such that P(ATz) > andα· Tz–z>ψ(z) –ψ(Tz) onATz, which is in contradiction with
α· u–Tu ≤ψ(u) –ψ(Tu),∀u∈E.
In , Clark presented Clark’s fixed point theorem [], which means that in a com-plete metric space, ‘directional contraction’ admits a fixed point. Now we establish ran-dom version of Clark’s fixed point theorem on aTε,λ-completeRNmodule, namely
The-orem . below.
Theorem . Let(E, · )be aTε,λ-complete RN module over R with base(,F,P),λ∈
L
++(F)with <λ< onand f :E→E be aTε,λ-continuous function with the local
property.If for each v∈E,there exists x∈E satisfying x=v and
() v–x+f(v) –x=v–f(v);
() f(v) –f(x) ≤λv–x.
Then f has a fixed point.
Proof Define a functiong:E→Ebyg(v) =x, whenf(v)=vandg(v) =v, whenf(v) =v.
It is obvious thatf andghave the same fixed points.
Define a functionφ:E→L(F) byφ(v) = ( –λ)–·v–f(v). Sincef isT
ε,λ-continuous, φ isTε,λ-continuous. Thus we haveφisTε,λ-l.s.c. From the local property off, it follows
thatIA·φ(IA·v) =IA·( –λ)–· IAv–f(IAv)= ( –λ)–· IAv–IAf(v)=IA·φ(v),∀v∈E,
A∈F, which meansφhas the local property. Clearly, is the lower bound ofφ.
In order to prove thatghas a fixed point, we only need to provev–g(v) ≤φ(v) –φ(g(v)) by Theorem ..
Ifv=g(v), it is obvious thatv–g(v) ≤φ(v) –φ(g(v)) .
Ifv=g(v), theng(v) =x. By () and (), one can have ≤λv–x–f(v) –f(x) ≤
λv–x–f(x) –x+x–f(v) ≤(λ– )v–x–f(x) –x+v–f(v), which
meansv–g(v) ≤φ(v) –φ(g(v)).
To obtain Clark’s fixed point theorem under the locallyL-convex topology, we need the following key results obtained in [, ].
Proposition .([]) Let(E, · )be an RN module over K with base(,F,P).Then E is Tε,λ-complete if and only if E isTc-complete and has the countable concatenation property.
Proposition .([]) Let(E, · )be an RN module over R with base(,F,P)such that E has the countable concatenation property,let f:E→ ¯L(F)be a function with the local property.Then f isTε,λ-lower semicontinuous iff f isTc-lower semicontinuous.
From both the relations of completeness of a random normed module and lower semi-continuity of a function underTcandTε,λ, we can obtain Clark’s fixed point theorem under
the topologyTcas follows.
Theorem . Let(E, · )be aTc-complete RN module over R with base(,F,P)such that E has the countable concatenation property,λ∈L
E→E be aTc-continuous function with the local property.If for each v∈E,there exists x∈E satisfying x=v and
() v–x+f(v) –x=v–f(v);
() f(v) –f(x) ≤λv–x.
Then f has a fixed point.
Remark . From the results above, it is easy to see that Clark’s fixed point theorem on complete RN modules is essentially independent of a special choice ofTcandTε,λ. It is an
algebra result.
When the base space (,F,P) of theRNmodule is trivial, namelyF={∅,}, our re-sult automatically degenerates to the classical Dane˘s theorem. So our rere-sult is a nontrivial random extension.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work is supported by BeiJing Talents Found (No. 2014000020124G065), Beijing Municipal Education Commission Project (KM201511417002), and the National Natural Science Foundation of China (11501030).
Received: 30 April 2015 Accepted: 27 August 2015
References
1. Guo, TX, Xiao, HX, Chen, XX: A basic strict separation theorem in random locally convex modules. Nonlinear Anal.71, 3794-3804 (2009)
2. Guo, TX, Li, SB: The James theorem in complete random normed modules. J. Math. Anal. Appl.308, 257-265 (2005) 3. Guo, TX: The Radon-Nikodým property of conjugate spaces and thew∗-equivalence theorem forw∗-measurable
functions. Sci. China Ser. A39, 1034-1041 (1996)
4. Guo, TX: Survey of recent developments of random metric theory and its applications in China (II). Acta Anal. Funct. Appl.3, 129-158 (2001)
5. Guo, TX: Relations between some basic results derived from two kinds of topologies for a random locally convex module. J. Funct. Anal.258, 3024-3047 (2010)
6. Guo, TX: Recent progress in random metric theory and its applications to conditional risk measures. Sci. China Ser. A
54, 633-660 (2011)
7. Guo, TX, Shi, G: The algebraic structure of finitely generatedL0(F,K)-modules and the Helly theorem in random
normed modules. J. Math. Anal. Appl.381, 833-842 (2011)
8. Tang, YH: The Wintner theorem in unital complete random normed algebras. Bull. Korean Math. Soc.50(6), 1973-1979 (2013)
9. Guo, TX, Yang, YJ: Ekeland’s variational principle for an¯L0-valued function on a complete random metric space.
J. Math. Anal. Appl.389, 1-14 (2012)
10. Wu, MZ: Farkas’s lemma in random locally convex modules and Minkowski-Weyl type results inL0(F,Rn). J. Math.
Anal. Appl.404, 300-309 (2013)
11. Zhao, SE, Zhao, Y: A minimax theorem for¯L0-valued functions on random normed modules. J. Funct. Spaces Appl.
(2013). doi:10.1155/2013/704251
12. Guo, TX, Zhao, SE, Zeng, XL: The relations among the three kinds of conditional risk measures. Sci. China Ser. A57, 1753-1764 (2014)
13. Zhao, SE, Zhao, Y, Yao, MH: On non-L0-linear perturbations of random isometries in random normed modules.
J. Inequal. Appl.2014, 496 (2014)
14. Clark, FH: Pointwise contraction criteria for the existence of fixed points. Can. Math. Bull.21(1), 7-11 (1978) 15. Dunford, N, Schwartz, JT: Linear Operators (I). Interscience, New York (1957)
16. Guo, TX: Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal. Funct. Appl.1(2), 160-184 (1999)
17. Filipovi´c, D, Kupper, M, Vogelpoth, N: Separation and duality in locallyL0-convex modules. J. Funct. Anal.256,
