R E S E A R C H
Open Access
Random fixed point theorems in partially
ordered metric spaces
Juan J Nieto
1*, Abdelghani Ouahab
2and Rosana Rodríguez-López
1*Correspondence: [email protected] 1Departamento de Análise Matemática, Estatística e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, 15782, Spain Full list of author information is available at the end of the article
Abstract
We present the random version in partially ordered metric spaces of the classical Banach contraction principle and some of its generalizations to ordered metric spaces.
The results are used to prove the existence of solutions for random differential equations with boundary conditions.
MSC: 47H10; 47H30; 54H25
Keywords: fixed point; partially ordered set; random operator; random differential equation
1 Introduction
In , the Polish mathematician Stefan Banach established a remarkable fixed point the-orem, the famous contraction principle, which is one of the most important results of analysis. It is the most widely applied fixed point result in different areas of mathemat-ics and applications. It requires the structure of a complete metric space with contractive condition on the map which is easy to test in many situations. It has been generalized in many different directions. Moreover, the proof of the Banach contraction principle gives a sequence of approximate solutions and useful information as regards the rate of conver-gence toward the fixed point. This is very important since a fundamental principle both in mathematics and computer science is iteration. Particularly, fixed point iterations and monotone iterative techniques are the core methods when solving a large class of abstract and applied mathematical problems and play an important role in many algorithms.
The existence of fixed points for self-mappings in partially ordered sets has been con-sidered in [, ], where some applications to matrix equations are presented. This result was extended by Nietoet al.[] and Nieto and Rodríguez-López [, ] in partially ordered sets and applied to study ordinary differential equations.
Probabilistic functional analysis is an important mathematical area of research due to its applications to probabilistic models in applied problems. Random operator theory is needed for the study of various classes of random equations. Indeed, in many cases, the mathematical models or equations used to describe phenomena in biology, physics, en-gineering, and systems sciences contain certain parameters or coefficients which have specific interpretations, but whose values are unknown. Therefore, it is more realistic to consider such equations as random operator equations. These equations are much more
difficult to handle mathematically than deterministic equations. Important contributions to the study of the mathematical aspects of such random equations have been presented in [, ] among others. The problem of fixed points for random mappings was initiated by the Prague school of probability research. The first results were studied in - by ˘Spa˘cek and Han˘s in the context of Fredholm integral equations with random kernel. In a separable metric space, random fixed point theorems for contraction mappings were proved by Han˘s [, ], Han˘s and ˘Spa˘cek [] and Mukherjee [, ]. Then random fixed point theorems of Schauder or Krasnosel’skii type were given by Mukherjea (cf. Bharucha-Reid [], p. ), Bharucha-Bharucha-Reid [] and Itoh []. Now it has become a full fledged re-search area and a vast amount of mathematical activities have been carried out in this direction (see, for examples, [–]). The existence of a random fixed point for mappings in partially ordered metric spaces and partially ordered probabilistic metric spaces was studied, for example, in [, ].
The goal of this paper is to establish a random version of some fixed point theorems in partially ordered and orderedL-spaces. Finally, we apply our results to random differential equations.
2 Some basic concepts on order andL-spaces
LetX be a nonempty set and letF:X→Xbe an operator. We denote the successive iterations ofFasF= X,F=F,Fn=F◦Fn–, forn∈N,n≥.
Definition . If (X,) is a partially ordered set andf :X→X, we say thatf is monotone nondecreasing if
xy, x,y∈X ⇒ F(x)F(y).
Remark . The above definition coincides with the notion of a nondecreasing function in the case whereX=Randrepresents the usual total order inR.
LetXbe a metric space andF:X→Xbe an operator.
Definition . The set of all nonempty invariant subsets ofFis
I(F) =Y⊂X:F(Y)⊆Y.
We denote bys(X) the set of sequences inX, that is,
s(X) =(xn)n∈N:xn∈X,n∈N
.
Fréchet introduced in [] the notion ofL-space as follows.
Definition . AnL-space is a triple (X,c(X),lim), whereXis a set,c(X)⊆s(X) is a family of sequences of elements ofX, andlim:c(X)→Xis a mapping having the following two properties:
(i) Ifxn=x, for alln∈N, then(xn)n∈N∈c(X)andlim(xn)n∈N=x.
(ii) If(xn)n∈N∈c(X)andlim(xn)n∈N=x, then for all subsequences(xni)i∈Nof(xn)n∈Nwe
The elements ofc(X) are called convergent sequences andlim(xn)n∈N=xis the limit of
the sequence, also writtenxn→xasn→ ∞. AnL-space is denoted by (X,→).
Definition . A mapping F:X→X is said to be orbitally continuous if x∈X, and Fni(x)→aasi→ ∞implies thatFni+(x)→F(a) asi→ ∞.
Definition . Let (X,) be a partially ordered set and
X:=(x,y)∈X×X:xyoryx.
For eachx,y∈Xwithxy, we denote
[x,y]:={z∈X:xzy}.
Definition . ForXa nonempty set, (X,→,) is an orderedL-space if:
(i) (X,→)is anL-space.
(ii) (X,)is a partially ordered set.
(iii) If(xn)n∈N→x,(yn)n∈N→y, andxnyn, for eachn∈N, thenxy.
ForL-spaces equipped with a partial ordering, we obtain the following result, which requires us to define first the concept of orbitally-continuous function.
Definition . Let (X,→) be anL-space equipped with a partial ordering. We say that F: (X,→)→(X,→) is orbitally-continuous ifx∈Xand
Fni(x)→a asi→ ∞,
Fni(x),a∈X
, ∀i∈N,
imply that
Fni+(x)→F(a) asi→ ∞.
3 Random variable
IfXis a metric space, we shall useB(X) to denote the Borelσ-algebra onX. Let (,F) be a measurable space. The expressionF⊗B(X) denotes the smallestσ-algebra on×X which contains all the setsQ×S, whereQ∈F andS∈B(X). Given a topological space E, we denoteP(E) ={Y⊂E: Y=∅},Pcl(E) ={Y∈P(E) :Yclosed}. LetF:X→P(Y) be a multivalued map. A single-valued mapf :X→Y is said to be a selection ofF(and we writef ⊂F) wheneverf(x)∈F(x) for everyx∈X. Let (,) be a measurable space and F:→P(X) a multivalued mapping,Fis called measurable ifF+–(Q) ={ω∈:F(ω)⊂ Q}is measurable for everyQ∈Pcl(X); equivalently, for everyUopen subset ofX, the set F––(U) ={ω∈:F(ω)∩U=∅}is measurable.
Proposition .([]) Let(,U)be a measurable space and F:→Pcl(X)be a multi-valued map.If,for every C∈P(X),we have F–
Definition . Recall that a mappingf :×X→Xis said to be a random operator if, for anyx∈X,f(·,x) is measurable.
Definition . A random fixed point off is a measurable functiony:→Xsuch that
y(ω) =fω,y(ω) for allω∈.
Equivalently, it is a measurable selection for the multivalued mapFixF:→P(X)
de-fined by
FixF(ω) =x∈X:x=f(ω,x), ω∈.
Theorem .([], Theorem .) Let X be a separable metric space,Y a metric space and f :×X→Y a continuous random operator.Then f is a measurable random operator.
Theorem . Let(,)be a measurable space,Y be a separable metric space,andφ:
→Pcl(Y)be a measurable multivalued function.Thenφhas a measurable selection.
LetX,Y be two locally compact metric spaces andf :×X→Y. ByC(X,Y), we de-note the space of continuous functions fromXintoY endowed with the compact-open topology.
Lemma .([]) f is a Carathéodory function if and only ifω→r(ω)(·) =f(ω,·)is a measurable function fromto C(X,Y).
4 Random fixed point theorems
Theorem . Let(,F)be a measurable space, (X,d,)be a separable complete partially ordered metric space,and F:×X→X be a continuous random operator such that,for eachω∈,the function F(ω,·)is a monotone(either order-preserving or order-reversing) operator.Suppose that the following assertions hold:
(H) For eachω∈,there existsk(ω)∈[, )such that
dF(ω,x),F(ω,x)
≤k(ω)d(x,x) for eachx,x∈X,xx.
(H) There exists a random variablex:→Xwith
x(ω)F
ω,x(ω)
, ∀ω∈
or
x(ω)F
ω,x(ω)
, ∀ω∈.
Then there exists a random variable x:→X which is a random fixed point of F.
Proof If, for eachω∈,F(ω,x(ω)) =x(ω), thenxis a random fixed point ofF. Suppose
F(ω,yn–(ω)),ω∈,n∈N. From the condition (H), we have, for eachω∈andn∈N,
one of the following relations:
yn(ω)yn+(ω) or yn(ω)yn+(ω).
Using (H), we get
dyn(ω),yn+(ω)
≤k(ω)ndx(ω),F
ω,x(ω)
, ω∈,
and, as a consequence,
dyn(ω),yn+m(ω)
≤k(ω)n+k(ω)n++· · ·+k(ω)n+m–dy(ω),x(ω)
.
So (yn(ω))n∈N is a Cauchy sequence, for everyω∈. Lety∗(ω) =limn→∞yn(ω),ω∈. Since y(·) is measurable, then y(·) is measurable. Hence, by induction, we can easily
prove that, for eachn∈N, the functionω→yn(ω) is measurable. This implies thaty∗(·) is measurable. Now, we show that y∗(ω) =F(ω,y∗(ω)). It is clear that, for eachω∈, d(y∗(ω),F(ω,y∗(ω))) =limn→∞d(yn(ω),F(ω,yn(ω))). Then
dyn(ω),Fω,yn(ω)=dFω,yn–(ω)
,Fω,yn(ω)≤k(ω)dyn–(ω),yn(ω)
,
for eachω∈. Hence, for eachω∈,
dyn(ω),Fω,yn(ω)≤k(ω)dyn–(ω),y∗(ω)+k(ω)dy∗(ω),yn(ω)→ asn→ ∞.
Thus,
y∗(ω) =Fω,y∗(ω) for eachω∈.
Next, we consider the following additional condition proposed in []:
(H) Every pair of elements ofXhas a lower bound or an upper bound.
Remark .([]) The above condition (H) is equivalent to:
(H) For everyx,y∈X, there existsz∈Xthat is comparable toxandy.
Remark . In the statement of Theorem ., if we impose, additionally, the condition (H) (equivalently, (H)), then we can conclude the existence of a random variablex:
→Xwhich is the unique random fixed point ofF.
Indeed, following the ideas in [], we prove that, if we take any random variablex¯:→
Xand we define the sequence
¯
y(ω) =x¯(ω), yn(¯ ω) =F
ω,yn¯ –(ω)
, ω∈,n∈N,
is obvious, sinceF(w,x¯(w)) is comparable toF(w,x(w)) for everyω∈, so thatyn(¯ ω) is
comparable toyn(ω) for everyω∈. Hence
dyn(ω),yn(¯ ω)=dFω,yn–(ω)
,Fω,¯yn–(ω)
≤k(ω)dyn–(ω),yn¯ –(ω)
≤k(ω)ndy(ω),y¯(ω)
, ω∈,
and (yn(¯ ω))n∈Ntends toy∗(ω), asn→ ∞, for everyw∈.
On the other hand, for an arbitrary random variablex¯:→Xthen, for eachω∈,
there existsz(ω)∈Xthat is comparable tox(ω) andx¯(ω) simultaneously, then if we
define
z(ω) =z(ω), zn(ω) =F
ω,zn–(ω)
, ω∈,n∈N,
thenyn(ω) is comparable tozn(ω), for everyω∈, andyn(¯ ω) is comparable tozn(ω), for everyω∈. Therefore,
dyn(ω),yn(¯ ω)≤dyn(ω),zn(ω)+dzn(ω),yn(¯ ω) ≤k(ω)ndy(ω),z(ω)
+k(ω)ndz(ω),x¯(ω)
, ω∈,
which proves that (yn(¯ ω))n∈N→y∗(ω), asn→ ∞, for everyω∈.
In the following result, we replace the monotonicity ofF(ω,·) by a more general condi-tion.
Theorem . Let(X,)be a partially ordered set, (X,d)be a complete separable metric space,and(,U)be a measurable space.Let F:×X→X be a joint measurable operator such that,for everyω∈,F(ω,·)maps comparable elements into comparable elements, that is,
xy ⇒ for everyω∈, F(ω,x)F(ω,y)or F(ω,y)F(ω,x).
(H) If(xn)n∈Nis a sequence inXwhose consecutive terms are comparable and(xn)n∈N→x,
then there exists a subsequence(xnp)p∈Nof(xn)n∈Nsuch that every term is comparable
to the limitx.
Suppose that(H)and(H)hold.If there exists x:→X,a random variable with x(ω)
comparable to F(ω,x(ω))for everyω∈,then F has a unique random fixed point.
Proof If x(ω) =F(ω,x(ω)), for every ω∈, then the proof is finished. Suppose that
F(ω,x(ω))=x(ω), for some ω∈. We define the sequence y(·) =x(·) and yn(·) =
F(·,yn–(·)), forn∈N.
Since, for eachω∈, we can comparey(ω) =x(ω) toy(ω) =F(ω,x(ω)), by the
hy-pothesis onF, we can comparey(ω) =F(w,y(ω)) toy(ω) =F(w,y(ω)), so that, for each
n∈N. Then from (H), we get
dyn(ω),yn+(ω)
≤k(ω)ndx(ω),F
ω,x(ω)
, ω∈.
This implies that (yn(ω))n∈Nis a Cauchy sequence inX, for eachω∈. SinceXis a com-plete metric space, then, for eachω∈, there existsy∗(ω)∈Xsuch thatlimn→∞yn(ω) = y∗(ω). From the definition ofynand sinceFis joint measurable,y∗:→Xis a measurable function. Finally, we prove thaty∗is a random fixed point ofF, that is,F(ω,y∗(ω)) =y∗(ω), for everyω∈. From (H), for eachω∈, there exists a subsequence (ynp(ω))p∈N of
(yn(ω))n∈Nsuch that every term is comparable toy∗(ω). Therefore,
dy∗(ω),Fω,y∗(ω)≤dy∗(ω),ynp(ω)
+k(ω)dynp–(ω),y∗(ω)
→ asp→ ∞.
Hence
y∗(ω) =Fω,y∗(ω) for eachω∈.
The uniqueness follows similarly to the proof in Remark ..
Very recently, the random fixed point theory in vector metric spaces was studied by Sinaceret al.in []. Next, we present the random version of the Perov fixed point theorem in partially ordered generalized metric spaces in the following sense.
Definition . LetXbe a nonempty set. By a vector-valued metric onX, we mean a map d:X×X→Rkwith the following properties:
(i) d(u,v)≥for allu,v∈X; ifd(u,v) = thenu=v; (ii) d(u,v) =d(v,u)for allu,v∈X;
(iii) d(u,v)≤d(u,w) +d(w,v)for allu,v,w∈X.
We call the pair (X,d) a generalized metric space with
d(x,y) :=
⎛ ⎜ ⎜ ⎝
d(x,y)
.. . dk(x,y)
⎞ ⎟ ⎟ ⎠.
Notice thatdis a generalized metric space onXif and only ifdi,i= , . . . ,k, are metrics onX. The inequalities in Definition . are understood componentwise.
By the same method as used in Theorem ., we can easily prove the following result.
Theorem . Let(,U)be a measurable space,X be a real separable complete generalized metric space with partial orderingand F:×X→X be a continuous random operator such that,for eachω∈,the function F(ω,·)is a monotone(either order-preserving or order-reversing)operator.Suppose that the following assertions hold:
(H) for everyω∈,there existsM(ω)∈Mk×k(R+),a random variable matrix such that,
for everyω∈,M(ω)< and
dF(ω,x),F(ω,x)
and(H),then there exists a random variable x:→X that is a random fixed point of F
(the unique random fixed point of F if the condition(H)holds).
Proof Similarly to the proof of Theorem . and with an analogous construction, using (H), we get
dyn(ω),yn+(ω)
≤M(ω)ndx(ω),F
ω,x(ω)
for everyω∈,
and, therefore, for eachω∈,
dyn(ω),yn+m(ω)
≤M(ω)n+M(ω)n++· · ·+M(ω)n+m–dy(ω),x(ω)
=M(ω)nI+M(ω) +· · ·+M(ω)m–dy(ω),x(ω)
.
SinceM(ω)< , the seriesn∞=M(ω)nis convergent and, in consequence, the series
∞
n=[M(ω)]nis also convergent and, therefore,
∞
n=[M(ω)]nis convergent. Since all
the coefficients ofM(ω) are nonnegative, we get
dyn(ω),yn+m(ω)
≤M(ω)n+M(ω)n++· · ·+M(ω)n+m–dy(ω),x(ω)
=M(ω)n ∞
l=
M(ω)ldy(ω),x(ω)
.
Taking norms in the above inequality and usingM(ω)< or, in other words, that the matrix [M(ω)]nconverges to the null matrix asn→ ∞, it is easy to deduce that (y
n(ω))n∈N is a Cauchy sequence, for every ω∈. We take y∗(ω) =limn→∞yn(ω), ω∈. Simi-larly to the proof of Theorem ., y∗ is measurable. It is obvious, for eachω∈, that d(y∗(ω),F(ω,y∗(ω))) =limn→∞d(yn(ω),F(ω,yn(ω))). It comes from
diy∗(ω),Fω,y∗(ω)–diyn(ω),Fω,yn(ω) ≤diy∗(ω),yn(ω)+diFω,yn(ω),Fω,y∗(ω),
forω∈andi= , . . . ,k. The proof is finished analogously to the proof of Theorem . by
replacingk(ω) byM(ω).
Now, we present the random fixed point theorem in partially ordered spaces for expan-sive operators.
Lemma . Let(X,d,)be a partially ordered metric space,and F:X→X be an operator. Suppose that the following assertions hold:
(C) There existsk> such that
dF(x),F(x)
≥kd(x,x) for eachx,x∈X,xx
and
Then F has inverse operator F–:X→X.If F–preserves(or reverses)the order of elements,
then
dF–(x),F–(x)
≤
kd(x,x) for each x,x∈X,xx.
Proof Letx,y∈Xbe such thatF(x) =F(y), thenF(x)F(y) andF(y)F(x), hence
=dF(x),F(y)≥kd(x,y) ⇒ x=y.
This implies thatFhas inverse operatorF–. In the case whenF–preserves the order, we
show that
dF–(x),F–(x)
≤
kd(x,x) for eachx,x∈X,xx.
Indeed, letx,x∈Xwithxx, thenF–(x)F–(x). By (C), we have
dFF–(x)
,FF–(x)
≥kdF–(x),F–(x)
.
This implies that
dF–(x),F–(x)
≤
kd(x,x).
The case whenF–reserves the order is similar.
We can easily prove the following result.
Corollary . Let(X,)be a partially ordered set and F:X→X be a bijective mapping. Suppose that,for every x,y∈X,we can compare x to y(that is,the ordering is total).Then F is monotone if and only if F–is monotone(of the same type).
Proof Suppose thatFis order-preserving. Letx,y∈Xsuch thatxy. ThenF–xandF–y are comparable. We prove thatF–xF–y. In other case, ifF–xF–yis not true, then
F–xF–y, hencex=F(F–x)F(F–y) =y. This contradicts the assumption.
We state the following auxiliary result.
Theorem .([]) Let(X,)be a partially ordered set and suppose that there exists a met-ric d in X such that(X,d)is a complete metric space.Let N:X→X be a monotone func-tion(nondecreasing or nonincreasing)such that there exists k∈[, )with d(N(x),N(y))≤ kd(x,y), xy.Suppose that N is continuous. If there exists x∈X with xN(x)or
xN(x),then N has a unique fixed point.
(C) for eachω∈,there existsk(ω) > such that
dF(ω,x),F(ω,x)
≥k(ω)d(x,x) for eachx,x∈X,xx.
Suppose also that,for eachω∈,we have X⊆F(ω,X). If,for everyω∈,the operator F–
ω (·) = [F(ω,·)]–is monotone(either order-preserving
or order-reversing)and continuous,then there exists a random variable x:→X that is a random fixed point of F(the unique random fixed point of F in the case of validity of the condition(H)).
Proof Letω∈, then from Lemma .,F–
ω (·) = [F(ω,·)]–exists and
dFω–(x),Fω–(y)
≤
kd(x,y), xy.
Using (H), we can conclude from Theorem . that, for everyω∈, there exists a unique
xω∈Xfixed point of [F(ω,·)]–. Hence,
Fω–(xω) =xω, ω∈ ⇒ F(ω,xω) =xω, ω∈.
Define a multivalued mappingG:→P(X) by
G(ω) =x∈X:x=F(ω,x).
SinceF(ω,·) is a continuous mapping, for everyω∈, we haveG(ω)∈Pcl(X). LetC∈ Pcl(X), then
G––(C) =ω∈:G(ω)∩C=∅=ω∈: there existsx∈Cwithx=F(ω,x).
SinceXis a separable metric space, there exists{xi:i∈N} ⊆Csuch that
{xi:i∈N}=C,
G––(C) = ∞
i=
∞
n=
ω∈:dxi,F(ω,xi)< n
.
Therefore,G–
–(C) is measurable. From Theorem ., there exists a measurable selection
x:→XofG, that is,x(ω)∈G(ω), for everyω∈, orx(ω) =F(ω,x(ω)), for everyω∈,
so thatxis a random fixed point ofF.
5 Random fixed point theorems inL-spaces
In the following results, we consider the notationFω=F(ω,·), forω∈.
Theorem . Let(X,→)be an L-space with a partial ordering,and(,U)be a mea-surable space.Let F:×X→X be a joint measurable operator.Suppose that:
(H) For everyx,y:→X,random variables,there exists a random variablez:→X
(H) There exist random variablesx,x∗:→Xsuch that the sequence defined as
y(ω) :=x(ω), ω∈, yn(ω) :=Fωn
x(ω)
=Fω,yn–(ω)
, ω∈,n∈N,
satisfies
yn(ω)n∈N→x∗(ω) asn→ ∞, for everyω∈,
and,for eachω∈fixed,there exists(ynk(ω))k∈N,a subsequence of(yn(ω))n∈Nsuch
that
ynk(ω),x∗(ω)
∈X for eachk∈N.
(H) F:×X→Xis such thatFω=F(ω,·)is orbitally-continuous for everyω∈.
(H) Ifx,y:→Xare two random variables such that,for everyω∈,we have
x(ω),y(ω)∈X and Fωn
x(ω)n∈N→x∗(ω) asn→ ∞,
then
Fωny(ω)n∈N→x∗(ω) asn→ ∞, for everyω∈.
Then F has a unique random fixed point.In fact,x∗is the unique random fixed point of F.
Proof If the random variablex∗:→Xis such thatF(ω,x∗(ω)) =x∗(ω) for everyω∈ , then it is clear thatx∗ is a random fixed point ofF. Consider that, for someω∈, F(ω,x∗(ω))=x∗(ω). From (H), for everyω∈, we have
yn(ω)n∈N=Fωn
x(ω)
n∈N→x∗(ω) asn→ ∞.
Besides, if we fixω∈, there exists (ynk(ω))k∈N, a subsequence of (yn(ω))n∈Nsuch that
ynk(ω),x∗(ω)∈X for eachk∈N. (.)
Therefore, by (H), due to
Fnk
ω
x(ω)
k∈N→x∗(ω) ask→ ∞,
and (.), we obtain
ynk+(ω)
k∈N=
Fnk+
ω
x(ω)
k∈N→Fω
x∗(ω)=Fω,x∗(ω) ask→ ∞.
Since (ynk+(ω))k∈N= (Fωnk+(x(ω)))k∈Nis a subsequence of (yn(ω))n∈N= (Fωn(x(ω)))n∈N,
ynk+(ω)
and, hence,
Fω,x∗(ω)=x∗(ω).
Since this identity is valid for everyω∈, we have proved thatx∗is a random fixed point ofF.
Letx:→Xbe an arbitrary random variable, and we distinguish two cases:
• Consider that, for everyω∈, we have
x(ω),x(ω)
∈X.
By(H), we obtain
yn(ω)n∈N=Fωn
x(ω)
n∈N→x∗(ω) asn→ ∞, for everyω∈,
and, by(H), we get
Fωn
x(ω)n∈N→x∗(ω) asn→ ∞, for everyω∈.
• Consider that, for someω∈,(x(ω),x(ω)) /∈X. From(H), there exists a random variablez:→Xsuch that, for everyω∈,(x(ω),z(ω)),(x(ω),z(ω))∈X. Since
Fωn
x(ω)
n∈N→x∗(ω) asn→ ∞, for everyω∈,
by(H), we obtain
Fωn
z(ω)n∈N→x∗(ω) asn→ ∞, for everyω∈
and
Fωn
x(ω)n∈N→x∗(ω) asn→ ∞, for everyω∈.
This justifies thatx∗is the unique random fixed point ofF.
Theorem . Let(X,→)be an L-space with a partial ordering,and(,U)be a mea-surable space.Let F:×X→X be a joint measurable operator.Suppose that conditions (H)and(H)hold and that the following conditions hold:
(H) For everyx,y∈X,there existsz∈Xsuch that(x,z),(y,z)∈X(that is,the condition
(H)).
(H) Ifx,y∈Xandω∈fixed are such that(x,y)∈Xand(Fωn(x))n∈N→x∗(ω),asn→
∞,then
Fωn(y)
n∈N→x∗(ω) asn→ ∞.
Proof To prove thatx∗is a random fixed point, we proceed similarly to the proof of The-orem .. To prove the uniqueness, we considerx:→Xan arbitrary random variable and we takeω∈fixed. Again, we distinguish two cases:
• Suppose that(x(ω),x(ω))∈X. By(H),
yn(ω)n∈N=Fωn
x(ω)
n∈N→x∗(ω) asn→ ∞,
and, by(H), we get
Fn
ω
x(ω)n∈N→x∗(ω) asn→ ∞.
• On the other hand, consider that(x(ω),x(ω)) /∈X. Then, from(H), there exists
z(ω)∈Xsuch that(x(ω),z(ω)),(x(ω),z(ω))∈X. Since
Fωnx(ω)
n∈N→x∗(ω) asn→ ∞,
by(H), we obtain
Fωn
z(ω)n∈N→x∗(ω) asn→ ∞
and
Fωn
x(ω)n∈N→x∗(ω) asn→ ∞.
Sinceω∈is arbitrary, this justifies thatx∗is the unique random fixed point ofF.
6 Boundary value problem
Nonlocal boundary value problems arise in many applied sciences. For example, the vibra-tions of a guy wire of uniform cross section and composed ofNparts of different densities, and some problems in the theory of elastic stability, can be modeled by multi-point bound-ary value problems (see [, ]). The existence of solutions for systems of local and nonlo-cal boundary value problems (BVPs) has received increased attention by researchers; see, for example, the papers of Agarwal, O’Regan, and Wong [–], Henderson, Ntouyas, and Purnaras [], Precup [, ] and the references therein. Very recently, the coupled systems of BVPs with local and nonlocal conditions were studied by Bolojan-Nicaet al. [] and Precup [] among others.
In this section, we are interested in getting a random solution to the following coupled system:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x(t,ω) =f(t,x(t,ω),y(t,ω),ω), t∈(, ),ω∈,
y(t,ω) =g(t,x(t,ω),y(t,ω),ω), t∈(, ),ω∈,
x(,ω) = , x(,ω) =L(
x(t,ω)dα(t)), ω∈,
y(,ω) = y(,ω) =L(
y(t,ω)dβ(t)), ω∈,
(.)
where L,L ∈ C(R,R) and
x(t,ω)dα(t),
y(t,ω)dβ(t) denote, respectively,
Lemma . Let f,g: [, ]×R×R→Rbe two continuous functions and consider the problem
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x(t) =f(t,x(t),y(t)), t∈(, ),
y(t) =g(t,x(t),y(t)), t∈(, ),
x() = , x() =L(
x(t)dα(t)),
y() = , y() =L(
y(t)dβ(t)),
(.)
where L,L∈C(R,R)and
x(t)dα(t),
y(t)dβ(t)denote Riemann-Stieltjes integrals.
The pair(x,y)∈C([, ],R)×C([, ],R)is a solution to problem(.)if and only if
⎧ ⎨ ⎩
x(t) =K(t,¯ s)f(s,x(s),y(s))ds+L(
x(s)dα(s))t, t∈(, ),
y(t) =K(t,¯ s)g(s,x(s),y(s))ds+L(
y(s)dβ(s))t, t∈(, ),
where
¯ K(t,s) =
⎧ ⎨ ⎩
–t( –s), ≤t≤s≤,
–s( –t), ≤s≤t≤.
In particular, we can take the set C([a,b],R) of continuous functions u: [a,b]→R equipped with the partial ordering
x,y∈C[a,b],R, x≤y ⇐⇒ x(t)≤y(t) for everyt∈[a,b],
and, inC([a,b],R)×C([a,b],R), we define the following partial ordering:
(x,y), (x∗,y∗)∈C[a,b],R×C[a,b],R, (x,y)≤(x∗,y∗) ⇐⇒ x≤x∗,y≤y∗.
Theorem . Let f,g: [, ]×R×R×→Rbe two Carathéodory functions.Assume that the following conditions hold:
(L) There existp,p,p,p:→R,random variables such that
f(t,x,y,ω) –f(t,x,y,ω)≤p(ω)|x–x|+p(ω)|y–y|
and
g(t,x,y,ω) –g(t,x,y,ω)≤p(ω)|x–x|+p(ω)|y–y|,
for eacht∈[, ],x,y,x,y∈Rwith(x,y)≤(x,y)andω∈.
(L) There exist <K< , <K< two positive real constants such that
L(x) –L(y)≤K|x–y|, L(x) –L(y)≤K|x–y|,
(L) For eachω∈fixed,the functionsf(t,·,·,ω),g(t,·,·,ω)(for everyt∈[, ]),L and
Lare monotonic of the same type(that is,all of them are nondecreasing or all of them
are nonincreasing).
(L) One of the following conditions holds:
≤f(t, , ,ω), ≤g(t, , ,ω), ∀t∈[, ],∀ω∈,
≤L(), ≤L()
or
≥f(t, , ,ω), ≥g(t, , ,ω), ∀t∈[, ],∀ω∈,
≥L(), ≥L().
Suppose that,for everyω∈,the matrix
M(ω) =
p(ω) +K p(ω)
p(ω) p(ω) +K
has norm less than.Then problem(.)has a unique random solution.
Proof Consider the operatorN:C([, ],R)×C([, ],R)×→C([, ],R)×C([, ],R),
(x,y,ω)→N(x,y,ω) :=N(·,x,y,ω),N(·,x,y,ω)
,
where
N(t,x,y,ω) =
¯
K(t,s)fs,x(s),y(s),ωds+L
x(s)dα(s)
t, t∈[, ]
and
N(t,x,y,ω) =
¯
K(t,s)gs,x(s),y(s),ωds+L
y(s)dβ(s)
t, t∈[, ].
First, we show thatN is a random operator onC([, ],R)×C([, ],R). Given (x,y)∈ C([, ],R)×C([, ],R), sincef andg are Carathéodory functions, thenω→f(t,x(t), y(t),ω) andω→g(t,x(t),y(t),ω) are measurable maps in view of Lemma .. Further, the integral is a limit of a finite sum of measurable functions, therefore, the maps
ω→N(t,x,y,ω), ω→N(t,x,y,ω)
are measurable. As a result,N is a random operator fromC([, ],R)×C([, ],R)×
intoC([, ],R)×C([, ],R).
Clearly, the random fixed points ofNare solutions to (.) and conversely. Indeed, given a random variable (x,y) :→C([, ],R)×C([, ],R), we get
is equivalent to
x(ω)(t) =
¯
K(t,s)fs,x(ω)(s),y(ω)(s),ωds+L
x(ω)(s)dα(s)
t, t∈[, ],
y(ω)(t) =
¯
K(t,s)gs,x(ω)(s),y(ω)(s),ωds+L
y(ω)(s)dβ(s)
t, t∈[, ],
so that the corresponding solution to (.) is defined asx(t,ω) =x(ω)(t),y(t,ω) =y(ω)(t), fort∈[, ] andω∈.
Next, we show that N satisfies all the conditions of Theorem . on C([, ],R)× C([, ],R). The proof will be given in several steps:
Step . Forω∈fixed, we check thatN(·,·,ω) = (N(·,·,ω),N(·,·,ω)) is continuous.
Take ω∈ fixed. Let (xn,yn) be a sequence in C([, ],R)×C([, ],R) such that (xn,yn)→(x,y)∈C([, ],R)×C([, ],R), asn→ ∞. Then, fort∈[, ],
N(t,xn,yn,ω) –N(t,x,y,ω)
≤
fs,xn(s),yn(s),ω–fs,x(s),y(s),ωds
+L
xn(s)dα(s)
t–L
x(s)dα(s)
t. Then, by (L),
N(·,xn,yn,ω) –N(·,x,y,ω) ∞
≤
fs,xn(s),yn(s),ω
–fs,x(s),y(s),ωds+K
xn(s) –x(s)dα(s)
≤
f
s,xn(s),yn(s),ω–fs,x(s),y(s),ωds+K
xn–x∞dα(s).
Sincef is a Carathéodory function, by the Lebesgue dominated convergence theorem, we get
N(·,xn,yn,ω) –N(·,x,y,ω) ∞→ asn→ ∞.
Similarly,
N(·,xn,yn,ω) –N(·,x,y,ω) ∞→ asn→ ∞.
Thus,Nis a continuous random operator.
Step . For eachω∈, the functionN(·,·,ω) is a monotone operator.
Letω∈be fixed. Let (x,y), (x,¯ y)¯ ∈C([, ],R)×C([, ],R) be such that (x,y)≤(x,¯ y),¯ that is,
x(t)≤ ¯x(t), y(t)≤ ¯y(t), ∀t∈[, ].
If, for everyt∈[, ],f(t,·,·,ω),g(t,·,·,ω),LandLare nondecreasing operators, then
gt,x(t),y(t),ω≤gt,x(t),¯ y(t),¯ ω, ∀t∈[, ], L
x(t)dα(t)
≤L
¯
x(t)dα(t)
,
and
L
y(t)dβ(t)
≤L
¯
y(t)dβ(t)
.
This implies that
N(t,x,y,ω)≤N(t,x,¯ y,¯ ω), ∀t∈[, ]
and
N(t,x,y,ω)≤N(t,x,¯ y,¯ ω), ∀t∈[, ].
Hence
N(x,y,ω)(t)≤N(x,¯ y,¯ ω)(t), ∀t∈[, ].
On the other hand, if, for everyt∈[, ],f(t,·,·,ω),g(t,·,·,ω),LandLare nonincreasing
operators, then
N(x,y,ω)(t)≥N(x,¯ y,¯ ω)(t), ∀t∈[, ].
Step . We show that N satisfies the following property: for everyω∈ and every (x,y), (x,y)∈C([, ],R)×C([, ],R) such that (x,y)(x,y), we have
dN(x,y,ω),N(x,y,ω)≤M( ω)d(x,y), (x,y).
Considerω∈ fixed. Let (x,y), (x,y)∈C([, ],R)×C([, ],R) be such that (x,y) (x,y), then
N(t,x,y,ω) –N(t,x,y,ω)
≤
¯
K(t,s)fs,x(s),y(s),ω–fs,x(s),y(s),ωds
+L
x(t)dα(t)
–L
x(t)dα(t) ≤p(ω)x–x∞+p(ω)y–y∞+Kx–x∞. Then
N(·,x,y,ω) –N(·,x,y,ω) ∞≤
p(ω) +K
x–x∞+p(ω)y–y∞.
Similarly, we obtain
N(·,x,y,ω) –N(·,x,y,ω) ∞≤p(ω)x–x∞+p(ω) +K
Hence
N(·,x,y,ω) –N(·,x,y,ω)∞
N(·,x,y,ω) –N(·,x,y,ω)∞
≤
p(ω) +K p(ω)
p(ω) p(ω) +K
x–x∞ y–y∞
,
that is,
dN(x,y,ω),N(x,y,ω)≤M( ω)d(x,y), (x,y),
where
M(ω) =
p(ω) +K p(ω)
p(ω) p(ω) +K
and
d(x,x), (y,y)
=
x–y∞
x–y∞
,
for (x,x), (y,y)∈C([, ],R)×C([, ],R).
Step . By (L), we can easily show that
≤N(·, , ,ω) and ≤N(·, , ,ω), ∀ω∈
or
≥N(·, , ,ω) and ≥N(·, , ,ω), ∀ω∈,
that is,N(, ,ω)≥(, )∀ω∈, orN(, ,ω)≤(, )∀ω∈.
This means that, for the null random variable defined as (, ) :→C([, ],R)× C([, ],R), (, )(ω) = (, ),∀ω∈, one of the following conditions holds:N((, )(ω),
ω)≥(, )(ω)∀ω∈, orN((, )(ω),ω)≤(, )(ω)∀ω∈. Note that the uniqueness condition (H) also holds.
Then, from Theorem ., there exists a unique random solution to problem (.).
Competing interests
None of the authors have any competing interests in the manuscript.
Authors’ contributions
The three authors contributed equally in this paper.
Author details
1Departamento de Análise Matemática, Estatística e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, 15782, Spain.2Laboratory of Mathematics, Sidi-Bel-Abbès University, PoBox 89, Sidi-Bel-Abbès, 22000, Algeria.
Acknowledgements
The authors would like to thank the anonymous referees for their careful reading of the manuscript and pertinent comments; their constructive suggestions substantially improved the quality of the work.
References
1. Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.132, 1435-1443 (2004)
2. Petrusel, A, Rus, IA: Fixed point theorems in orderedL-spaces. Proc. Am. Math. Soc.134, 411-418 (2006)
3. Nieto, JJ, Pouso, RL, Rodríguez-López, R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc.135, 2505-2517 (2007)
4. Nieto, JJ, Rodríguez-López, R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser.23, 2205-2212 (2007)
5. Nieto, JJ, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order22, 223-239 (2005)
6. Bharucha-Reid, AT: Random Integral Equations. Academic Press, New York (1972) 7. Skorohod, A: Random Linear Operators. Reidel, Boston (1985)
8. Han˘s, O: Random Operator Equations. In: Proc. 4th Berkeley Sympos. Math. Statist. and Prob. vol. II, pp. 185-202. Univ. California Press, Berkeley (1961)
9. Han˘s, O: Random fixed point theorems. In: 1957 Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processes Held at Liblice Near Prague from November 28 to 30 pp. 105-125. Publishing House of the Czechoslovak Academy of Sciences, Prague (1956)
10. Han˘s, O, ˘Spacek, A: Random fixed point approximation by differentiable trajectories. In: Trans. 2nd Prague Conf. Information Theory, pp. 203-213. Publ. House Czechoslovak Acad. Sci, Prague (1960)
11. Mukherjea, A: Transformations aléatoires separables. Théorème du point fixe aléatoire. C. R. Acad. Sei. Paris Sér. A-B 263, 393-395 (1966)
12. Mukherjea, A: Random Transformations of Banach Spaces. Ph. D. Dissertation, Wayne State Univ. Detroit, Michigan (1968)
13. Bharucha-Reid, AT: Fixed point theorems in probabilistic analysis. Bull. Am. Math. Soc.82, 641-657 (1976)
14. Itoh, S: Random fixed point theorems with an application to random differential equations in Banach spaces. J. Math. Anal. Appl.67, 261-273 (1979)
15. Anh, TN: Some general random coincidence point theorems. N.Z. J. Math.41, 17-24 (2011)
16. Thang, HD, Anh, PT: Random fixed points of completely random operators. Random Oper. Stoch. Equ.21(1), 1-20 (2013)
17. Beg, I, Shahzad, N: Some random approximation theorems with applications. Nonlinear Anal.35, 609-616 (1999) 18. Shahzad, N: Random fixed point theorems for various classes of 1-set-contractive maps in Banach spaces. J. Math.
Anal. Appl.203, 712-718 (1996)
19. Ciric, LJB, Mihet, D, Saadati, R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topol. Appl.17, 2838-2844 (2009)
20. Li, G, Duan, H: On random fixed point theorems of random monotone operators. Appl. Math. Lett.18, 1019-1026 (2005)
21. Fréchet, M: Les Espaces Abstraits. Gauthier-Villars, Paris (1928)
22. Hu, S, Papageorgiou, NS: Handbook of Multi-Valued Analysis, Volume I: Theory. Kluwer, Dordrecht (1997) 23. Himmelberg, CJ: Measurable relations. Fundam. Math.87, 53-72 (1975)
24. Papageorgiou, NS: Random fixed point theorems for measurable multifunctions in Banach spaces. Proc. Am. Math. Soc.97, 507-514 (1986)
25. Sinacer, ML, Nieto, JJ, Ouahab, A: Random fixed point theorem in generalized Banach space and applications. Random Oper. Stoch. Equ.24, 93-112 (2016)
26. Moshinsky, M: Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas. Bol. Soc. Mat. Mexicana7, 1-25 (1950)
27. Palamides, PK: Positive and monotone solutions of anm-point boundary value problem. Electron. J. Differ. Equ.2002, 18 (2002)
28. Agarwal, RP, O’Regan, D, Wong, PJY: Constant-sign solutions of a system of integral equations with integrable singularities. J. Integral Equ. Appl.19, 117-142 (2007)
29. Agarwal, RP, O’Regan, D, Wong, PJY: Constant-sign solutions for singular systems of Fredholm integral equations. Math. Methods Appl. Sci.33, 1783-1793 (2010)
30. Agarwal, RP, O’Regan, D, Wong, PJY: Constant-sign solutions for systems of singular integral equations of Hammerstein type. Math. Comput. Model.50, 999-1025 (2009)
31. Henderson, J, Ntouyas, SK, Purnaras, IK: Positive solutions for systems of second order four-point nonlinear boundary value problems. Commun. Appl. Anal.12, 29-40 (2008)
32. Precup, R: Componentwise compression-expansion conditions for systems of nonlinear operator equations and applications. In: Mathematical Models in Engineering, Biology and Medicine. AIP Conf. Proc., vol. 1124, pp. 284-293. Amer. Inst. Phys, Melville (2009)
33. Precup, R: Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems. J. Math. Anal. Appl.352, 48-56 (2009)
34. Bolojan-Nica, O, Infante, G, Precup, R: Existence results for systems with coupled nonlocal initial conditions. Nonlinear Anal.94, 231-242 (2014)
