# Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

Authors: Adil El-Ghabi; Abdelmjid Khchine; Mohamed Aziz Taoudi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we establish several random fixed point theorems for random operators satisfying some iterative condition w.r.t. a measure of noncompactness. We also discuss the case of monotone random operators in ordered Banach spaces. Our results extend several earlier works, including Itoh’s random fixed point theorem. As an application, we discuss the existence of random solutions to a class of random first-order vector-valued ordinary differential equations with lack of compactness.


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## Body

## 1. Introduction

Random fixed point theorems are stochastic generalizations of deterministic fixed point theorems. In recent years, random fixed point theorems have assumed importance due to their applications to random differential and integral equations (see for instance [1–9] and the references therein). Since Bharucha-Reid published his survey paper [10], numerous works of great importance have appeared and contributed strongly to the enrichment of the random fixed point theory. We quote, for instance, the contributions by Itoh [11], Reich [12], Rybinski [13], Sehgal and Singh [14], Sehgal and Waters [15], Papageorgiou [16], Tan and Yuan [17], Hussain et al. [18], and many others.In 1979, Itoh [11] proved the random version of Sadovskii’s well-known fixed point theorem [19]. Later on, Tan and Yuan [17] established an interesting result which serves as a bridge that links the random fixed point theory with the deterministic fixed point theory. Specifically, their result ensures the existence of a random fixed point for a continuous random operator under some compactness conditions provided that the corresponding deterministic fixed point problem is solvable. Recently, El-Ghabi and Taoudi [20] developed a new fixed point approach that combines the advantages of the strong topology with the advantages of the weak topology and which enables them to draw new and meaningful conclusions about random fixed points for a given random operator and to handle nonlinear random problems with lack of compactness.In the present paper, we prove some new random fixed point theorems for (countably) convex-power condensing random operators in Banach spaces. Our results extend, generalize, and unify several known random fixed point results including the famous Itoh random fixed point theorem [11], Theorem 2.1. We should point out here that the concept of convex-power condensing operator was introduced in [21] as a generalization of the concept of condensing operator. We also prove some new random fixed point results for monotone (countably) convex-power condensing random operators in ordered Banach spaces. To illustrate our theoretical results, we investigate the solvability of a broad class of random first-order vector-valued ordinary differential equations. At this point, it is of significance to note that this class of random differential equations was already examined in [11] under some restriction on the constants used in the condition of measure of noncompactness and we have been successful in using our random fixed point results to remove this restriction (see Remark 34).The paper is arranged as follows. In Section2, we fix the notation and present some key tools that will be used to prove our main results. Section 3 is devoted to random fixed point theorems for (countably) convex-power condensing random operators and their corollaries. In Section 4, we prove some new random fixed point theorems for a class of monotone random operators in an ordered Banach space. Finally, in Section 5, we use some material from the previous sections to solve the random differential equation
(1)ddtut,w=ft,ut,w,w,where t belongs to a bounded and closed interval in the real line ℝ, ω belongs to a set Ω endowed with a σ-algebra, and the functions f,u have values in a Banach space E.


## 2. Preliminaries

Throughout this paper,Ω,Σ denotes a measurable space, where Ω is a nonempty set and Σ is a σ-algebra of subsets of Ω. Let (E,∥.∥) be a Banach space, X be a nonempty subset of E and B1 be the closed unit ball of E. Let 2X, BX, and CDX denote, respectively, the collections of all nonempty subsets, bounded subsets, and closed subsets of X. In addition, let KX and WKX denote, respectively, the classes of compact and weakly compact subsets of X. We say that a mapping T:X⟶E is hemicompact if each sequence xnn∈ℕ in X has a convergent subsequence whenever xn−Txn⟶0 as n⟶∞. A mapping F:Ω⟶2E is said to be measurable if for every open subset U of E, F−1U=ω∈Ω:Fω∩U≠∅∈Σ. It is important to stress here that, in the case where Fω∈KE for each ω∈Ω, F is measurable if and only if F−1C∈Σ for each C∈CDE (cf. [22], Theorem 3]). A mapping T:Ω×X⟶E is called a random operator if T.,x is measurable for each x∈X. A mapping ξ:Ω⟶X is said to be:
(i)
a deterministic fixed point of a random operatorT:Ω×X⟶E if Tω,ξω=ξω for every ω∈Ω,(ii)
a random fixed point of a random operatorT:Ω×X⟶E if it is measurable and Tω,ξω=ξω for every ω ∈ Ω,Now, we present some basic facts regarding measures of (weak) noncompactness in Banach spaces, which we will be needed in the sequel. It is noteworthy that measures of noncompactness have played a substantial role in nonlinear functional analysis. They are very often used in the theory of functional equations, including ordinary differential equations, equations with partial derivatives, integral and integrodifferential equations, and optimal control theory. We also highlight that the interplay between fixed point theory and measures of noncompactness is very powerful and fruitful (see for instance [19, 23–29] and the references therein).Definition 1 (see [30–32]).
LetE be a Banach space. A function ψ:BE⟶0,+∞ is said to be a measure of (weak) noncompactness if it satisfies the following conditions:
(1)
The familykerψ=M∈BE:ψM=0 is nonempty and kerψ is contained in the set of relatively (weakly) compact sets of E.(2)
M1⊂M2⟹ψM1≤ψM2 for all M1,M2 in BE.(3)
ψco¯M=ψM for each M∈BE, where co¯M denotes the closed convex hull of M(4)
ψλM1+1−λM2≤λψM1+1−λψM2 for all λ∈0;1 and M1,M2∈BE.(5)
IfMnn≥1 is a sequence of nonempty (weakly) closed subsets of E with M1 bounded and M1⊇M2⊇⋯⊇Mn⊇⋯ such that limn→∞ψMn=0, then M∞≔∩n=1∞Mn is nonempty.
The familykerψ is said to be the kernel of the measure of (weak) noncompactness ψ. Note that the intersection set M∞ belongs to kerψ (see [33]). A measure of (weak) noncompactness ψ is said to be
(6)
Full ifψM=0 if and only if M is a relatively (weakly) compact set.(7)
Nonsingular if(2)ψM∪x=ψM,for all M∈BE and x∈E(8)
Set additive or has the maximum property if(3)ψM1∪M2=maxψM1,ψM2forallM1,M2∈BE(9)
Homogeneous if(4)ψλM=λψMforallλ∈ℝandM∈BE(10)
Subadditive if(5)ψM1+M2≤ψM1+ψM2forallM1,M2∈BE(11)
Regular ifψ is full, homogeneous, nonsingular, subadditive, and set additive.The first important measure of weak noncompactness was defined by De Blasi [34] as follows:
(6)βM=infr>0:thereexistsW∈WKEwithM⊂W+rB1,for each M in BE. Notice that β is regular (see [34]).Also, the most important regular measures of noncompactness include the Kuratowski measure of noncompactness defined by(7)αM=infr>0:M⊆∪i=1nMi,diamMi≤r,∀i,here diamMi=supx−y:x,y∈Mi. For more details, we refer the reader to the interesting book by Banaś and Goebel [30].Now, we give a short note about (countably) convex-power condensing mappings with respect to a measure of (weak) noncompactness. LetX be a nonempty closed convex subset of E and x0∈X. Let T:X⟶X be a bounded mapping (i.e., T takes bounded sets into bounded ones). Following [33, 35], for any M⊂X, we set
(8)T1,x0M=TM,Tn,x0M=Tco¯Tn−1,x0M∪x0for,n=2,3,.Definition 2.
LetE be a Banach space and ψ be a measure of (weak) noncompactness on E. Let X be a nonempty closed convex subset of E and T:X⟶X be a bounded mapping. We say that T is (countably) ψ-convex-power condensing about x0∈X and n0∈ℕ if
(9)ψTn0,x0M<ψM,for any (countable) bounded subset M of X with ψM>0.It is customary to say “(countably)ψ-condensing” in place of “ψ-convex-power condensing about x0 and 1.” We also say that T is (countably) ψ-convex-power condensing if there exist x0∈X and n0∈ℕ such that T is (countably) ψ-convex-power condensing about x0 and n0.


## 3. Random Fixed Point Theorems for (Countably) Convex-Power Condensing Random Operators in Banach Spaces

Throughout this section,Ω,Σ is a measurable space, E is a Banach space and X is a nonempty closed convex subset of E. Let T:Ω×X⟶E be a random operator. We say that T is continuous (resp., uniformly continuous, hemicompact) if for each ω∈Ω, Tω,. is continuous (resp., uniformly continuous, hemicompact). If T has values in X, we say that T is (countably) ψ-convex-power condensing if Tω,. is (countably) ψ-convex-power condensing for every ω∈Ω. We say also that T is k.-lipschitzian (resp. a k.-contraction) if there exists a mapping k:Ω⟶0,+∞(resp.:Ω⟶0,1) satisfying Tω,x−Tω,y≤kωx−y for all x,y∈X and all ω∈Ω.The following result guarantees the existence of a random fixed point for a continuous random operator provided that the corresponding deterministic fixed point problem is solvable.Theorem 3 (see [17], Theorem 2.3).
LetΩ,Σ be a measurable space and X be a nonempty separable complete subset of a Banach space E. Suppose that T:Ω×X⟶CDE is a continuous hemicompact random operator. Then, T has a deterministic fixed point if and only if T has a random fixed point.Our main purpose in the immediate sequel is to prove some new random fixed point theorems for countably convex-power condensing mappings. Before proceeding further with the first main theorem, we obtain some auxiliary results.Lemma 4.
LetE be a Banach space, X be a closed convex subset of E and ψ be a regular measure of noncompactness on E. Let T:X⟶X be a uniformly continuous countably ψ-convex-power condensing operator. If TX is bounded, then T is hemicompact.Proof.
Let∆=xkk≥0 be a sequence of elements of X such that
(10)Txk−xk⟶0ask⟶∞.We show that for eachn∈ℕ,ψ∆≤ψTn∆, where Tn is the nth iterate of T. To see this, notice first that if T is uniformly continuous, then so is Tn for each integer n. Let ε>0 be fixed and take any integer n. Then, there exists δ>0 such that for all x,y∈X with x−y<δ, we have Tnx−Tny<ε. By virtue of (10), there exists an integer N such that for each k≥N, we have Txk−xk<δ, so that,
(11)Tn+1xk−Tnxk=TnTxk−Tnxk<ε,whenever k≥N. With this in mind, we can easily see that
(12)Tnxkk≥N⊆Tn+1xkk≥N+εB1⊆Tn+1Δ+εB1,where B1 is the closed unit ball of E. From the properties of the measure of noncompactness, it follows that
(13)ψTnxkk≥N≤ψTn+1Δ+εψB1,which amounts to
(14)ψTnΔ=ψTnxkk≥N≤ψTn+1Δ+εψB1.The arbitrariness ofε yields ψTnΔ≤ψTn+1Δ. Thus, ψTn∆n∈ℕ is an increasing sequence and therefore
(15)ψ∆=ψT0∆≤ψTn∆,for any integer n≥0. Let a∈X and n0∈ℕ such that T is countably ψ-convex-power condensing about a and n0. By mathematical induction, we can see that
(16)Tn∆⊆Tn,a∆,for any integer n≥1. Going back to (15), we get
(17)ψ∆≤Tn0,a∆.From our hypotheses, we know thatψ∆=0 and so ∆ is relatively compact. Thus, the sequence xkk≥0 has a convergent subsequence and consequently T is hemicompact.Now, we are ready to state and prove the following deterministic fixed point theorem.Theorem 5.
LetE be a Banach space and X be a closed convex subset of E. Let T:X⟶X be a continuous mapping such that TX is bounded. Assume further that there are an integer n0≥1 and a vector x0∈X such that for any countable subset C⊂X, we have
(18)C⊆co¯Tn0,x0C∪x0⇒Cisrelativelycompact.
Then,T has a fixed point.Proof.
Consider the iterative sequenceDn of sets given by
(19)D0=x0,Dn=co¯TDn−1∪x0foreveryn≥1.By mathematical induction, it is easy to show thatDn⊆Dn+1 and Dn is convex compact for each n≥0. Let D=⋃n≥0Dn. Plainly, D is convex. Furthermore, observe that for each n∈ℕ, Dn is separable (since it is compact), and hence, there exists a countable set Cn⊆Dn such that C¯n=Dn. Let us consider the countable subset C=⋃n≥0Cn of D. It is readily apparent that D=⋃n≥0C¯n⊆⋃n≥0Cn¯=C, and so D¯=C¯.We claim that for eachk≥1,
(20)C⊆co¯Tk,x0C∪x0.We proceed by induction. For the base casek=1, we can easily verify that
(21)C⊂D¯⊆co¯TD∪x0=co¯TD¯∪x0=co¯TC¯∪x0=co¯TC∪x0=co¯T1,x0C∪x0.Letk≥1 be fixed and suppose that
(22)C⊆co¯Tk,x0C∪x0.Then,(23)TC⊆Tco¯Tk,x0C∪x0=Tk+1,x0C,so that
(24)C⊂co¯TC∪x0⊆co¯Tk+1,x0C∪x0.This proves our claim. From our hypotheses, we deduce thatC¯ (and so D¯) is compact. Furthermore, it is not difficult to see that
(25)TD¯⊆⋃n≥0TDn¯⊆⋃n≥0Tco¯TDn∪x0=¯⋃n≥0Dn+1¯⊆D¯.An appeal to Schauder’s fixed point theorem yields a fixed point forT.Remark 6.
It is of significance to note that condition (18) is fulfilled whenever T is (countably) ψ-convex-power condensing about x0 and n0.After these preparations, we are now ready to state the main result of this section. This result ensures the existence of a random fixed point for a uniformly continuous random operator under general compactness conditions.Theorem 7.
LetE be a Banach space and ψ be a regular measure of noncompactness on E. Let X be a separable closed convex subset of E and let T:Ω×X⟶X be a uniformly continuous and countably ψ-convex-power condensing random operator. Assume that Tω,X is bounded for each ω∈Ω, then, T has a random fixed point.Proof.
Invoking Theorem5, we infer that the random operator T has a deterministic fixed point. In addition, since for each ω∈Ω, Tω,.:X⟶X is a uniformly continuous countably ψ-convex-power condensing mapping and Tω,X is bounded, then by Lemma 4, T is hemicompact. The desired result follows from Theorem 3.Next, we turn our attention to the case when the boundary condition is of Rothe type. Before making a formal statement of our next random fixed point result, we need some auxiliary results. We first recall the following lemma.Lemma 8 (see [15], Lemma 2).
Letf:Ω⟶E and l:Ω⟶0,1 be measurable mappings. Then, for any y∈E, the mapping h:Ω⟶E defined by hω=lωfω+1−lωy is measurable.Recall that, ifU is a convex neighborhood of the origin in E, then the Minkowski functional p of U is defined on E by
(26)px=infε>0:x∈εU.It is well known [36], Lemma 5.12.1 that p is continuous, subadditive, positively homogeneous, and satisfies
(27)x∈E:px<1⊆U⊆x∈E:px≤1⊆U.Note also that ifU has a nonempty interior, then p is uniformly continuous.Now, we state the following Browder-Fan type result [37], which is crucial for our purposes.Theorem 9.
LetX be a separable closed convex subset of a Banach space E with IntX≠∅ and let ψ be a regular measure of noncompactness on E. Let x0∈IntX, n0:Ω⟶ℕ be a mapping and T:Ω×X⟶E be a uniformly continuous random operator. Assume that Tω,X is bounded and Tω,. is (countably) ψ-convex-power condensing about x0 and n0ω for each ω∈Ω. Then, there exists a measurable mapping ξ:Ω⟶X satisfying
(28)pTω,ξω−ξω=minpTω,ξω−x:x∈X,for each ω∈Ω, where p is the Minkowski functional of X−x0. Further, if pTω,ξω−x0≤1 for some ω∈Ω, then Tω,ξω=ξω.Proof.
Letp be the Minkowski functional of X−x0 and define the mapping l:Ω×X⟶0,1 by
(29)lω,x=max1,pTω,x−x0−1,for each ω,x∈Ω×X. Consider the mapping g:Ω×X⟶E defined by
(30)gω,x=lω,xTω,x+1−lω,xx0.The reasoning in [15], Theorem 1 yields that g is a continuous random operator such that for each ω∈Ω, gω,X is bounded. We will show that g satisfies all conditions of Theorem 7,. To this end, let ω∈Ω be fixed. First notice that
(31)pgω,x−x0=lω,xpTω,x−x0≤1,for any x∈X, so that, gΩ×X⊂X. In addition, it is easy to check that
(32)lω,x−lω,y≤pTω,x−x0−pTω,y−x0,for all x,y∈E. Since p and Tω,. are uniformly continuous, so is lω,.. Furthermore, for all x,y∈X, we obtain
(33)gω,x−gω,y=lω,xTω,x−x0−lω,yTω,y−x0=lω,xTω,x−Tω,y+lω,x−lω,yTω,y−x0≤lω,xTω,x−Tω,y+lω,x−lω,yTω,y+x0≤Tω,x−Tω,y+∣lω,x−lω,y∣Tω,.∞+x0.It follows thatgω,. is uniformly continuous. Next, we will illustrate that gω,. is countably ψ-convex-power condensing about x0 and n0ω. To see this, take any bounded countable subset M of X with ψM>0. We claim that gk,x0ω,M⊆co¯Tk,x0ω,M∪x0 for each k≥1. Indeed, for k=1, by (30), we have gω,M⊂co¯Tω,M∪x0, so that
(34)g1,x0ω,M⊂co¯T1,x0ω,M∪x0.Letk≥1 be fixed and assume that gk,x0ω,M⊆co¯Tk,x0ω,M∪x0.Then,(35)gk+1,x0ω,M=gco¯gk,x0ω,M∪x0⊆co¯Tω,co¯Tk,x0ω,M∪x0∪x0=co¯Tk+1,x0ω,M∪x0.This proves our claim. Thus,(36)ψgn0ω,x0ω,M≤ψco¯Tn0ω,x0ω,M∪x0=ψTn0ω,x0ω,M<ψM.Consequently,gω,. is countably ψ-convex-power condensing about x0 and n0ω. Now, by applying Theorem 7, we infer that there is a measurable mapping ξ:Ω⟶X such that for each ω∈Ω, we have gω,ξω=ξω, that is,
(37)lω,ξωTω,ξω+1−lω,ξωx0=ξω.Now, for anyω∈Ω, either (a) pTω,ξω−x0≤1 or (b) pTω,ξω−x0>1. In case of (a), it follows by (29) that lω,ξω=1 and hence by (37), Tω,ξω=ξω. If (b) holds, then by (29), we have
(38)lω,ξωpTω,ξω−x0=1,and since for every x∈X,px−x0≤1, it follows by (37) that for every x∈X, we have
(39)Tω,ξω−ξω=1−lω,ξωTω,ξω−x0.Then,(40)pTω,ξω−ξω≤pTω,ξω−x0−1pξω−x0.Sinceξω∈X, the last inequality implies that
(41)pTω,ξω−ξω≤pTω,ξω−x0−1≤pTω,ξω−x,and
(42)pTω,ξω−ξω=minpTω,ξω−x:x∈X.Thus, (a) and (b) provide the desired conclusion.With these preliminaries, we can proceed to the following interesting random fixed point theorem under Rothe-type boundary conditions.Theorem 5.
LetX be a separable closed convex subset of a Banach space E with IntX≠∅ and let ψ be a regular measure of noncompactness on E. Let n0:Ω⟶ℕ be a mapping and T:Ω×X⟶E be a uniformly continuous random operator. Let x0∈IntX such that for each ω∈Ω,Tω,X is bounded and Tω,. is countably ψ-convex-power condensing about x0 and n0ω. In addition, assume that TΩ×∂X⊆X (Rothe boundary condition). Then, T has a random fixed point.Proof.
Letp be the Minkowski functional of X−x0. By Theorem 9, there is a measurable mapping ξ:Ω⟶X satisfying (28). To prove that ξ is a random fixed point of T, it suffices to show that pTω,ξω−x0≤1 for each ω∈Ω. Suppose that for some ω∈Ω, we have pTω,ξω−x0>1, then Tω,ξω∉X. This implies that ξω∈IntX. Consequently, there is 0<η<1 such that z=ηξω+1−ηTω,ξω∈∂X. Hence, by (28), we have
(43)pTω,ξω−ξω≤pTω,ξω−z=ηpTω,ξω−ξω.Thus,pTω,ξω−ξω=0. This obviously yields that
(44)pTω,ξω−x0≤pξω−x0≤1,which contradicts the assumption.Our next concern will be the existence of random fixed points for the sum of two random operators. We should mention here that the need for such random fixed point theorems arose out of the study of random differential equations. Specifically, the inversion of a random differential operator may yield the sum of a contraction and an operator satisfying some compactness conditions. Before, to state our next random fixed point result, we need to recall some basic facts.Lemma 3 (see [20], Lemma 2.11).
LetE be a separable Banach space and S:Ω×E⟶E be a k.-contraction random operator. Then,
(i)
for eachx∈E there is a unique measurable function τx:Ω⟶E such that for each ω∈Ω:τxω=Sω,τxω+x(ii)
the mappingτ:Ω×E⟶E defined by τω,x=τxω is a Lipschitzian random operatorLemma 4 (see [20], Lemma 2.12).
LetX be a closed and convex subset of a separable Banach space E, T:Ω×X⟶E be a continuous random operator, and x:Ω⟶X be measurable. Then, the mapping ω⟶Tω,xω is measurable.Now, we are in a position to state the following random fixed point theorem for the sum of two random operators. For convenient purposes, we list some necessary definitions below for completeness. LetX be a nonempty bounded closed convex subset of a Banach space E and T:Ω×X⟶E and S:Ω×E⟶E be two random operators. Following [24, 32, 38], we set for any ω∈Ω and for any subset M of X:
(45)F1,x0ω,T,S,M=Fω,T,S,M=x∈X:x=Sω,x+Tω,y,forsomey∈Mand(46)Fn,x0ω,T,S,M=Fω,T,S,co¯Fn−1,x0ω,T,S,M∪x0,n=2,3,.⋯In the case whenS=0, we have
(47)F1,x0ω,T,0,M=T1,x0ω,M=Tω,Mand
(48)Fn,x0ω,T,0,M=Tω,c¯o¯Fn−1,x0ω,T,0,M∪x0=Tn,x0ω,M.Theorem 6.
LetX be a nonempty bounded closed convex subset of a separable Banach space E and ψ be a regular measure of noncompactness on E. Let T:Ω×X⟶E and S:Ω×E⟶E be random operators satisfying the following conditions:
(i)
T is uniformly continuous(ii)
there are mappingsn0:Ω⟶N and x0:Ω⟶X such that
(49)ψFn0ω,x0ωω,T,S,M<ψM,for every ω∈Ω and for any countable subset M of X with ψM>0,(iii)
S is a k.−contraction(iv)
for eachω∈Ω, [x=Tω,y+Sω,x and y∈X]⟹x∈X
Then,T+S has a random fixed point.Proof.
Consider the mappingf:Ω×X⟶E defined by
(50)fω,x=τω,Tω,x,where τ is as described in Lemma 3. Referring to Lemma 4, we see that every x∈X,f.,x=τ.,T.,x is measurable. Thus, f is a random operator. In addition, the fact that T is uniformly continuous and τ is Lipschitzian imply that f is uniformly continuous. Now, let ω∈Ω be fixed. We claim that fω,X⊆X. Indeed, by virtue of Lemma 3 (i), we infer that for any x∈X we have
(51)τω,Tω,x=Sω,τω,Tω,x+Tω,x,so that,
(52)fω,x=Sω,fω,x+Tω,x.Hence, by (iv), we havefω,x∈X. This proves our claim.Furthermore, for any subsetM of X, by (52), we obtain
(53)f1,x0ωω,M=fω,M=x∈X:x=fω,yforsomey∈M⊆x∈X:x=Sω,x+Tω,yforsomey∈M=F1,x0ωω,T,S,M.Letn≥1, suppose that
(54)fn,x0ωω,M⊆Fn,x0ωω,T,S,M.We have(55)fn+1,x0ωω,M=fω,c¯o¯fn,x0ωω,M∪x0ω⊆fω,c¯o¯Fn,x0ωω,T,S,M∪x0ω⊆Fω,T,S,c¯o¯Fn,x0ωω,T,S,M∪x0ω=Fn+1,x0ωω,T,S,M.Hence, for eachn≥1, we have
(56)fn,x0ωω,M⊆Fn,x0ωω,T,S,M.Thus, by (ii), we have(57)ψfn0ω,x0ωω,M<ψM,for any countable subset M of X with ψM>0. As a result, fω,. is countably ψ-convex-power condensing about x0ω and n0ω. Consequently, f is countably ψ-convex-power condensing. Invoking Theorem 7, we see that f has a random fixed point ξ:Ω⟶X which is in turn a random fixed point of T+S.Corollary 10.
LetX be a nonempty bounded closed convex subset of a separable Banach space E and ψ be a measure of noncompactness on E. Let T:Ω×X⟶E and S:Ω×E⟶E be random operators satisfying the following conditions:
(i)
T is uniformly continuous(ii)
there are mappingsn0:Ω⟶ℕ, x0:Ω⟶X and λ:Ω⟶0,1 such that
(58)ψFn0ω,x0ωω,T,S,M≤λωψM,for every ω∈Ω and for any countable subset M of X(iii)
S is a k.−contraction(iv)
for eachω∈Ω, [x=Tω,y+Sω,x and y∈X]⟹x∈X. Then, T+S has a random fixed point


## 4. Random Fixed Point Theorems for Monotone Random Operators

In this section, we prove some random fixed point theorems for monotone random operators in ordered real Banach spaces. We combine the advantages of the strong topology (continuity of random operators with respect to the strong topology) with the advantages of the weak topology (the random operators will satisfy some compactness conditions relative to the weak topology) to draw new conclusions about random fixed points for a given monotone random operator. Our results are random versions of the results in [39].We start this section by recalling some definitions and auxiliary results which will be used further on. Throughout this section,Ω,Σ is a measurable space, E is a real Banach space, and X is a nonempty subset of E.Definition 11.
A subsetP of E is called an order cone if it satisfies the following conditions:
(i)
P is closed, nonempty and P≠0,(ii)
∀a,b∈ℝ+, ∀x,y∈P:ax+by∈P(iii)
x∈Pand−x∈P⇒x=0.
An order cone permits to define a partial order inE by
(59)x≤y⇔y−x∈P.Conversely, letE be a real Banach space with a partial order compatible with the algebraic operations in E, that is,
(60)x≥0andλ∈ℝ+impliesλx≥0,x1≤y1andx2≤y2impliesx1+x2≤y1+y2.The positive cone ofE is defined by
(61)E+=x∈E:0≤x.Letx,y∈E with x≤y, the order interval x,y is defined by
(62)x,y=z∈E:x≤z≤y=x+P∩y−P.Definition 12.
(i)
A subsetX⊂E is said order bounded if there exist u,v∈E such that X⊂u,v(ii)
The order cone is called normal if and only if there is a numberC>0 such that
(63)∀x,y∈E:0≤x≤y⇒x≤CyThe least positive numberC (if it exists) satisfying (63) is called a normal constant.Remarks 13.
(1)
IfP is normal, then every order interval is norm bounded.(2)
LetK be a Hausdorff space and E be an ordered Banach space with normal cone P. We denote by CK,E the Banach space of all continuous E-valued functions on K equipped with the usual maximum norm. Plainly, CK,E is an ordered Banach space whose positive cone is given by
(64)C+K,E=f∈CK,E:fx∈E+,∀x∈K,and it is normalDefinition 14.
LetX be a subset of E and T:X⟶E be an operator.
(i)
The operatorT is said to be increasing if
(65)∀x,y∈X:x≤y⇒Tx≤Ty(ii)
The operatorT is said to be decreasing if
(66)∀x,y∈X:x≤y⇒Ty≤TxDefinition 15.
LetE be an ordered real Banach space with a normal order cone P. A sequence xn is said to be totally ordered if for all m,n∈ℕ:xm≤xnorxn≤xm.The following lemmas are quite useful below.Lemma 16 (see [27], Lemma 2.1).
LetE be an ordered real Banach space with a normal order cone P. Suppose that xn is a monotone sequence which has a subsequence xnk converging weakly to x∞. Then, xn converges strongly to x∞. Moreover, if xn is an increasing sequence then xn≤x∞ for each n≥1; if xn is a decreasing sequence then x∞≤xn for each n≥1.Lemma 17 (see [39], Lemma 1.8).
LetE be an ordered real Banach space with a normal order cone P. Suppose that xn is a totally ordered sequence which is contained in a relatively weakly compact set, then it converges strongly in E.Letu0,v0:Ω⟶X be two measurable mappings. By u0≤v0 (resp., u0<v0) on Ω, we mean u0ω≤v0ω (resp., u0ω<v0ω) for every ω∈Ω. If u0≤v0, the sector u0,v0 defined by
(67)u0,v0=x:Ω→X:u0≤x≤v0,is called a random interval in X.LetT:Ω×E⟶E be a random operator. For all n∈N, ω∈Ω, and x∈X, we denote by Tnω,x the value at x of the nth iterate of the mapping Tω,.. We say that a random operator T:Ω×X⟶E satisfies the condition Pn0 on u0,v0 if there is a mapping n0:Ω→ℕ such that for each ω∈Ω,
(68)ifV=xnisamonotonesequenceofu0ω,v0ω,suchthatV=F∪Tn0ωω,VwithFisafinitesetofcardinaln0ω,thenVisrelativelyweaklycompact.We say that a random operatorT:Ω×X⟶E satisfies the condition P1 on u0,v0 if T satisfies Pn0 with n0ω=1 for every ω∈Ω.Definition 18.
Letu0,v0 be a random interval in X with u0≤v0. A random operator T:Ω×X⟶E is called increasing (resp., decreasing) on u0,v0 if for each ω∈Ω, Tω,. is increasing (resp., decreasing) on u0ω,v0ω.Theorem 19.
LetE be an ordered separable real Banach space with a normal cone P and T:Ω×E⟶E be a continuous random operator. Let u0,v0 be a random interval in E such that T is increasing on u0,v0 and satisfies the condition Pn0. Assume that
(69)u0.≤T.,u0.,T.,v0.≤v0..Then,T has a random fixed point ξ in u0,v0 which can be obtained by monotone iterative procedure starting from u0 or from v0.Proof.
Consider the iterate sequence of mappingsun:Ω⟶E defined by unω=Tω,un−1ω, for all ω∈Ω and all n∈N. Let ω∈Ω be fixed. We can show by induction that
(70)u0ω≤u1ω≤u2ω≤⋯≤unω⋯≤v0ω.Let us setSω=unω:n∈ℕ. Then,
(71)Sω=Tn0ωω,Sω∪Fω,with Fω=u0ω,u1ω,⋯,un0ω−1ω is a finite set of cardinal n0ω. Therefore, it follows from our hypotheses that Sω is relatively weakly compact. Referring to Lemma 17, we see that the sequence unω converges strongly to some ξω in u0ω,v0ω. The continuity of Tω,. yields that Tω,ξω=ξω. Now, by using Lemma 4, we can show (via induction) that un is measurable for every n∈ℕ. The use of [8], Theorem 1.6 yields that ξ:Ω⟶E is measurable; and therefore, ξ is a random fixed point of T. Similarly, we can show that T has random fixed point obtained by monotone iterative procedure starting from v0.As a convenient specialization of Theorem19, we state the following result.Corollary 20.
LetE be an ordered separable real Banach space with a normal cone P and T:Ω×E⟶E be a continuous random operator. Let u0,v0 be a random interval in E such that T is increasing on u0,v0 and satisfies the condition P1. Assume that
(72)u0.≤T.,u0.,T.,v0.≤v0..Then,T has a random fixed point ξ in u0,v0 which can be obtained by monotone iterative procedure starting from u0 or from v0.Proof.
Apply Theorem19 with n0ω=1 for every ω∈Ω.Another consequence of Theorem19 is the following.Corollary 21.
LetE be an ordered separable real Banach space with a normal cone P and ψ be a nonsingular measure of weak noncompactness on E. Let T:Ω×E⟶E be a continuous random operator and let u0,v0 be a random interval in E such that T is increasing on u0,v0 and
(73)u0.≤T.,u0.,T.,v0.≤v0..
In addition, if there is a mappingn0:Ω⟶N, such that for each ω∈Ω and any monotone sequence Γ=xn in u0ω,v0ω with ψΓ>0, we have
(74)ψTn0ωω,Γ<ψΓ.
Then,T has a random fixed point ξ in u0,v0 which can be obtained by monotone iterative procedure starting from u0 or from v0.Proof.
Letω∈Ω be fixed and let V=xn be a monotone sequence of u0ω,v0ω such that V=F∪Tn0ωω,V with F is a finite set of cardinal n0ω. By (74), V is relatively weakly compact. Thus, Tω,. satisfies the condition (Pn0) on u0ω,v0ω. The arbitrariness of ω yields that T satisfies the condition Pn0 on the random interval u0,v0. The desired result follows from Theorem 19.Corollary 22.
LetE be an ordered separable real Banach space with a normal cone P and ψ be a measure of weak noncompactness on E.
LetT:Ω×X⟶E be a continuous random operator and let u0,v0 be a random interval in E such that T is increasing on u0,v0 and
(75)u0.≤T.,u0.,T.,v0.≤v0..In addition, assume that there is a mappingn0:Ω⟶N such that for each ω∈Ω,
(76)Tn0ωω,u0ω,v0ω≔Tn0ωω,xω:x∈u0,v0,is relatively weakly compact. Then, T has a random fixed point ξ in u0,v0 which can be obtained by monotone iterative procedure starting from u0 or from v0.


## 5. Applications to Random Differential Equations

Throughout this section,Ω,Σ denotes a measurable space and E is a separable real Banach space. Let r,T>0,x0∈E,B=x∈E:x−x0≤r and I=0,T. Denote by CI;E the Banach space of all continuous mappings u:I⟶E equipped with the supremum norm u∞=suput:t∈I and by C1I;E the Banach space of all continuously differentiable mappings from I to E. Let α (resp., αC) be the Kuratowski measure of noncompactness on Eresp.CI;E.Definition 23.
A mappingu:I×Ω⟶E is said to satisfy the following conditions:
(1)
C,Ω if for each ω∈Ω, u.,ω is continuous and for each t∈I,ut,. is measurable(2)
C1,Ω if u.,ω∈C1I;E for every ω∈Ω and ut,. is measurable for every t∈IIfu satisfies condition C,Ω, then u is considered a mapping of Ω into CI,E. The following result, due to Itoh, discusses the measurability of u.Proposition 24 (see [11], Proposition 4.2).
u satisfies condition C,Ω if and only if u is measurable as a mapping of Ω into CI;E.
Now, we consider the initial random differential equation(77)ddtut,ω=ft,ut,ω,ω,fort,ω∈I×Ω,u0,ω=ηω,where f.,.,ω∈CI×E,E and the function η:Ω⟶E is measurable. Our main purpose in the immediate sequel is to show the existence of a random solution to Eq. (77). Before doing so, it is appropriate to clarify the definition of solution we will consider.Definition 25.
A mappingu:I×Ω⟶E is said to be a random solution of (77) if u satisfies conditions C1,Ω and (77).
For convenience of later reference, we list some necessary results below for completeness.Lemma 26 (see [30]).
LetE be a Banach space and C0,T;E be the space of continuous functions defined on 0,T with values in E. If H⊂C0,T;E is bounded, then αHt≤αCH for any t∈0,T, where Ht=ut:u∈H. Furthermore, if H is equicontinuous, then t⟼αHt is continuous on 0,T,
(78)αCH=supαHt:t∈0,T,α∫0tHsds≤∫0tαHsds,for all t ∈ [0,T], where ∫0tHsds=∫0txsds:x∈H.Lemma 27 (see [21]).
LetE be a Banach space. If H⊆C0,T;E is equicontinuous and x0∈C0,T;E, then c¯o¯H∪x0 is also equicontinuous in C0,T;E.Lemma 28 (see [40]).
LetH be a bounded subset of E. Then, for each ε>0, there exists a sequence un⊆H such that
(79)αH≤2αun+ε.Lemma 29 (see [41], Lemma 2.7).
For eachn∈N, 0<λ<1, and T>0, we put
(80)Sn=λn+Cn1λn−1T+Cn2λn−2T22!+⋯+Tnn!.Then,limn→∞Sn=0.Before we proceed further, we present the following useful lemma.Lemma 30.
LetE be a Banach space and X be a subset of E. Let f:X⟶E be a mapping such that there exists a nonnegative constant γ≥0 satisfying
(81)αfΔ≤γαΔ,for any countable subset ∆ of X. Then, for any subset M of X, we have
(82)αfM≤2γαM.Proof.
LetM be a subset of X and ε>0 be fixed. In view of Lemma 28, there is a sequence yn of fM such that
(83)αfM≤2αyn+ε.Further, for eachn∈N, there is xn∈M such that yn=fxn and so
(84)αyn=αfxn≤γαxn≤γαM.Linking (83) and (84), we obtain
(85)αfM≤2γαM+ε.The arbitrariness ofε yields that
(86)αfM≤2γαM.If we replace the Kuratowski measure of noncompactnessα by the Hausdorff measure of noncompactness χ in Lemma 30, we obtain the following result.Lemma 31.
LetE be a Banach space and X be subset of E. Let f:X⟶E be a mapping such that there exists a nonnegative constant γ≥0 satisfying
(87)χfΔ≤γχΔ,for any countable subset ∆ of X. Then, for any subset M of X, we have
(88)χfM≤γχM.Proof.
By virtue of [42], Lemma 2.9, the reasoning in the proof of Lemma 30 yields the desired result.Now, we are in a position to state the following existence result.Theorem 32.
Letf:I×B×Ω→E and η:Ω⟶E be mappings satisfying the following assumptions:
(i)
For eachω∈Ω,f.,.,ω is uniformly continuous on I×B(ii)
For eacht,x∈I×B, ft,x,. is measurable(iii)
There exists a functionC:I×Ω⟶R+ such that(89)αft,D,ω≤Ct,ωαD,for all ω∈Ω, t∈I, and for any countable subset D of B, where each C.,ω∈L10,T;R+(iv)
m=supft,x,ω;t∈I,x∈B,ω∈Ω<+∞,(v)
η is measurable and(90)supηω−x0:ω∈Ω=r0<rThen, there exists a random solution of (77) on 0,T1×Ω, with T1=minT,r−r0/m.Proof.
Let(91)K=u∈CI1,E;us−ut≤ms−t,andut−x0≤r,∀s,t∈I1,where I1=0,T1. Clearly, K is a nonempty bounded closed convex equicontinuous subset of CI1;E and ut∈B for each u,t∈K×I1.Define the mappingW:Ω×K⟶K by
(92)Wω,ut=ηω+∫0tfs,us,ωds,for all ω∈Ω, u∈K, and t∈I1. We show that W is a random operator. To see this, let u∈K be fixed. Since the mapping ω⟶fs,us,ω is measurable for each s∈I1 and the mapping s⟶fs,us,ω is continuous for each ω∈Ω, then the mapping Fu:I1×Ω⟶E, s,ω⟶fs,us,ω satisfies the condition C,Ω. Thus, by Proposition 24, the mapping F^u:Ω→CI1;E defined by F^uω=Fuω,. for every ω∈Ω is measurable. Further, observe that mapping
(93)L:CI1;E→CI1;EdefinedbyLvt=∫0tvsdsfor all v∈CI1;E, and t∈I1 is linear and hence, by [1], Theorem 2.14, L∘F^u is measurable. Therefore, W.,u=η+L∘F^u is measurable. The arbitrariness of u∈K yields that W is a random operator. Now, we will prove that W fulfills all conditions of Theorem 7. To achieve this, let ω∈Ω be fixed. First, by (i) for ε>0, there exists δω>0 such that
(94)ft1,x1,ω−ft2,x2,ω<εT1,for all t1,x1,t2,x2∈I1×B satisfying x1−x2<δω and ∣t1−t2∣<δω. Let u,v∈K such that
(95)v−u∞=supt∈I1vt−ut<δω.We obtain(96)Wω,vt−Wω,ut≤∫0tfs,vs,ω−fs,us,ωds≤εT1t<ε,for each t∈I1. Hence,
(97)Wω,v−Wω,u∞<ε.Thus,Wω,. is uniformly continuous and, therefore, by arbitrariness of ω, W is uniformly continuous. Next, we claim that W is countably αC-convex-power condensing. Let H be a countable subset of K and let u∈H. We see that for ε>0 there exists δω>0 such that
(98)fs,us,ω−ft,ut,ω<ε,for all s,t∈I12 satisfying s−t<δω/max1,m. Then, fs,us,ω−ft,ut,ω→0 as t→s independently to u∈H.Hence,f.,u.,ω:u∈H is equicontinuous. By using Lemma 26 and assumption (iii), we obtain for each t∈I1(99)αWω,Ht=αηω+∫0tfs,us,ωds:u∈H≤∫0tαfs,Hs,ωds≤∫0tCs,ωαHsds≤∫0tCs,ωdsαCH.Let0<λ<1, we know that there is a continuous function φω: I1→ℝ+withMω=1/2maxφωt:t∈I1 such that
(100)∫0TCs,ω−φωsds<λ2.So, for eacht ∈ I1, we have
(101)αWω,Ht≤∫0tCs,ω−φωsds+∫0tφωsdsαCH≤λ+MωtαCH.Thus,(102)αW1,θ0ω,Ht≤λ+MωtαCH,foreacht∈I1,where θ0∈K is the mapping defined by θ0t=x0 for every t∈I1.Now, since forn≥1,c¯o¯Wn,θ0ω,H∪θ0⊂K, then f.,c¯o¯Wn,θ0ω,H∪θ0.,ω is equicontinious. Thus, in view of (101), assumption (iii) and Lemmas 26–30, we obtain
(103)αW2,θ0ω,Ht=αWω,c¯o¯W1,θ0ω,H∪θ0t≤α∫0tf.,c¯o¯W1,θ0ω,H∪θ0.,ωsds≤∫0t2Cs,ωαc¯o¯Wω,H∪θ0sds≤2∫0tCs,ω−φωs+φωsλ+MωsαcHds≤λ+Mωtλ+Mωλt+Mωt22αCH=λ2+2λMωt+Mωt22αCH.Hence, by mathematical induction, for alln≥1 and t∈I1, we obtain
(104)αWn,θ0ω,Ht≤ΓntαCH,where Γnt=λn+Cn1λn−1Mωt+Cn2λn−2Mωt2/2!+⋯+Mωtn/n!.Using the equicontinuity of theWn,θ0ω,H on I1, we get
(105)αCWn,θ0ω,H≤ΓnT1αCH.Since0<λ<1 and MωT1>0, Lemma 29 yields that there exists a positive integer n0ω such that Γn0ωT1<1. Consequently,
(106)αCWn0ω,θ0ω,H<αCH,for any countable subset H of K with αCH>0. Thus, Wω,. is countably αC-convex-power condensing about θ0 and n0ω. Therefore, W is countably αC-convex-power condensing. Now, Theorem 7 guarantees that W has a random fixed point ξ:Ω⟶K⊂CI1;E. By Proposition 24, the mapping ξ satisfies condition C,Ω as a mapping of I1×Ω into E. Further, by (92), ξ satisfies condition C1,Ω and ξ is a random solution of problem (77).As a convenient specialization of Theorem32, we consider the particular case when η=0 and x0=0. We, therefore, obtain the following result.Corollary 33.
Letf:I×B×Ω⟶E be a mapping satisfying the following assumptions:
(i)
For eachω∈Ω,f.,.,ω is uniformly continuous on I×B(ii)
For eacht,x∈I×B, ft,x,. is measurable(iii)
There exists a functionC:I×Ω→ℝ+ such that(107)αft,D,ω≤Ct,ωαD,for all ω∈Ω, t∈I, and for any countable subset D of B, where each C.,ω∈L10,T;ℝ+(vi)
m=supft,x,ω;t∈I,x∈B,ω∈Ω<+∞,
Then, the random differential equation(108)∀t,ω∈I×Ω:ddtut,ω=ft,ut,ω,ω,u0,ω=0,has a random solution on 0,T1×Ω, with T1=minT,r/m.Remark 34.
(1)
Theorem32 improves [11], Theorem 4.3. In our considerations, mapping C is not subject to any restriction(2)
Theorem32 remains true if we replace the Kuratowski measure of noncompactness by the Hausdorff measure of noncompactnessIn the following, we present a new approach to solve (77) based on Theorem 19. To achieve this, let E be an ordered separable real Banach space with a normal cone P and β be the De Blasi measure of weak noncompactness on E and on CI;E.Theorem 35.
Letf:I×E×Ω⟶E be a mapping satisfying the following assumptions:
(i)
For eachω∈Ω, f.,.,ω is continuous on I×E and maps bounded sets into bounded ones(ii)
For allt∈I and x∈E, ft,x,. is measurable(iii)
There exist two mappingsv0,w0:I×Ω⟶E satisfying condition C1,Ω and the following conditions hold(109)∀ω∈Ω,t∈I,dv0dtt,ω≤ft,v0t,ω,ω,ft,w0t,ω,ω≤dw0dtt,ω,(iv)
There is a measurable mappingM:Ω⟶R+\0 satisfying(110)ft,y,ω−ft,x,ω≥−Mωy−x,whenever v0t,ω≤x≤y≤w0t,ω, t,ω∈I×Ω(v)
there is a mappingρ:Ω⟶0,+∞ such that for each ω∈Ω, for any monotone sequence V=un of v0.,ω,w0.,ω and for all a,b∈I with a≤b we have(111)βfa,b×V×ω≤ρωβVa,b,where fa,b×V×ω≔fs,uns,ω;s∈a,b,n≥1Then, the random differential equation (77) has a random solution on I×Ω which can obtained by monotone iterative procedure starting from v0 or from w0.Remark 36.
Setgt,x,ω=ft,x,ω+Mωx, for t,x,ω∈I×E×Ω. Then,
(G1) For each t,ω∈I×Ω,
(112)gt,x,ω≤gt,y,ωwheneverv0t,ω≤x≤y≤w0t,ω
(G2) For each ω∈Ω, for any monotone sequence V=un in v0.,ω,w0.,ω and for all a,b∈I with a≤b, we have
(113)βga,b×V×ω≤γωβVa,b,where γω=ρω+MωProof.
The random problem (77) is equivalent to the random problem
(114)∀t,ω∈I×Ω:ddtut,ω+Mωut,ω=gt,ut,ω,ω,u0,ω=ηω,which is equivalent to the random problem
(115)∀t,ω∈I×Ω:ddteMωtut,ω=eMωtgt,ut,ω,ω,u0,ω=ηω.Let us write (115) as a random integral equation
(116)∀t,ω∈I×Ω:ut,ω=e−Mωtηω+∫0te−Mωt−sgs,us,ω,ωds.Define the mappingΦ:Ω×CI,E⟶CI,E by
(117)Φω,ut=e−Mωtηω+∫0te−Mωt−sgs,us,ωds,for each t,ω∈I×Ω and any u∈CI,EWe will show thatΦ is a random operator. To this end, let u∈CI,E be fixed. Notice first that for each t∈I, the mapping Ω⟶0,1, ω⟶e−Mωt is measurable, since exp:ℝ⟶ℝ+\0, x⟶ex is continuous and M is measurable. Let t∈I and s∈0,t. Using Lemma 8 together with assumptions (ii) and (iv) we infer that ω⟶e−Mωtηω and ω⟶e−Mωt−sgs,us,ω are measurable. Furthermore, since s⟶e−Mωt−sgs,us,ω is continuous for each ω∈Ω, then the mapping Gu,t:0,t×Ω⟶E, s,ω⟶e−Mωt−sgs,us,ω satisfies the condition C,Ω. Thus, by Proposition 24, the mapping G^u,t:Ω→C0,t;E defined by G^u,tω=Gu,tω,. for every ω∈Ω, is measurable. Next, observe that the mapping Lt:C0,t;E⟶C0,t;E defined by Ltvτ=∫0τvsds for all v∈CI;E and τ∈0,t is linear. Hence, by [1], Theorem 2.14, Lt∘G^u,t is measurable. By virtue of Proposition 24, we see that the mapping ω,τ↦Lt∘G^u,tωτ satisfies condition C,Ω.Accordingly, for eachτ∈0,t, the mapping ω↦Lt∘G^u,tωτ=∫0τe−Mωt−sgs,us,ωds is measurable. In particular, the mapping ω↦Lt∘G^u,tωτ=∫0τe−Mωt−sgs,us,ωds is measurable.Therefore,(118)ω↦Φω,uτ=e−Mωtηω+∫0τe−Mωt−sgs,us,ωds,is measurable. This is true for each t∈I. By a simple verification using assumption (i), we can show that t↦Φω,ut=e−Mωtηω+∫0te−Mωt−sgs,us,ωds is continuous for each ω∈Ω. Then, the mapping ω,t↦Φω,ut satisfies condition C,Ω. Finally by invoking Proposition 24, we conclude that Φ.,u:Ω⟶CI,E is measurable. The arbitrariness of u∈CI,E yields that Φ is a random operator.Now, we will prove thatΦ fulfills all conditions of Theorem 19. This will be achieved in three steps:Step 1.
We show thatΦ is continuous. To this end, fixω∈Ω and u∈CI,E. Let un be a sequence in CI,E which converges to u. Then, there is an integer N∈ℕ such that
(119)un−u∞<1whenevern≥N.Hence,(120)un∞≤u∞+1=rforeachn≥N.Now, we put(121)Fω,ns=gs,uns,ω,Fωs=gs,us,ω,and
(122)Kω=supgs,z,ω;s∈0,Tandz≤r<∞.Note that, for alln≥N and s∈0,T we have Fω,ns≤Kω. Furthermore, the continuity of g.,.,ω on I×E implies that
(123)Fω,ns−Fωs→0asn→∞.The dominated convergence theorem yields(124)∫0TFω,ns−Fωsds→0asn→∞.On the other hand, for eacht∈0,T,
(125)Φω,unt−Φω,ut≤∫0tgs,uns,ω−gs,us,ωds≤∫0TFω,ns−Fωsds.Hence,(126)Φω,un−Φω,u∞≤∫0TFω,ns−Fωsds.Therefore,Φω,un−Φω,u∞→0asn→∞ and consequently Φω,. is continuous at u. The arbitrariness of u implies that Φω,. is continuous on CI,E.Step 2.
We illuminate thatΦ is increasing on the random interval
(127)v0,w0=x:Ω→CI,E:v0.,ω≤xω≤w0.,ωforeachω∈Ω.To do this, letω∈Ω be fixed and u,v∈CI,E with v0.,ω≤u≤v≤w0.,ω. Using (G1) we obtain
(128)gt,ut,ω≤gt,vt,ωforeacht∈I.Now, (117) becomes
(129)Φω,ut≤Φω,vtforeacht∈I.Thus,(130)Φω,u≤Φω,v.Therefore,Φ is increasing on v0,w0.Step 3.
We show that the following conditions hold:(131)∀ω∈Ω,v0.,ω≤Φω,v0.,ω,Φω,w0.,ω≤w0.,ω.
To this end, letω∈Ω. We begin by setting
(132)h1t,ω=dv0dtt,w+Mωv0t,ω,h2t,ω=dw0dtt,w+Mωw0t,ω.
Clearly,h1.,ω, h2.,ω∈CI,E and for each t∈I, we have
(133)h1t,ω=dv0dtt,ω+Mωv0t,ω≤ft,v0t,ω,ω+Mωv0t,ω=gt,v0t,ω,ω≤gt,ω0t,ω,ω=ft,w0t,ω,ω+Mωw0t,ω≤dw0dtt,ω+Mωω0t,ω=h2t,ω.
Keeping in mind the fact thatd/dteMωtv0t,ω=eMωth1t,ω and d/dteMωtw0t=eMωth2t,ω, we deduce that for each t∈I have:
(134)eMωtv0t,ω=v00,ω+∫0teMωsh1s,ωds≤ηω+∫0teMωsgt,v0s,ω,ωds≤ηω+∫0teMωsgt,w0s,ω,ωds≤w00,ω+∫0teMωsh2s,ωds=eMωtw0t,ω.
Accordingly,v0.,ω≤Φω,v0.,ω≤Φω,w0.,ω≤w0.,ω.Step 4.
The reasoning in [39], Theorem 3.2 yields that each Φω,. satisfies the condition Pn0. Therefore, the random operator Φ satisfies the condition Pn0.Now, by applying Theorem19, we infer that Φ has a random fixed point ξ which can obtained by monotone iterative procedure starting from v0 (or w0): Ω⟶CI,E. The mapping ξ^:I×Ω⟶E defined by ξ^t,ω=ξωt is measurable as mapping from Ω into CI,E. Hence, by Proposition 24, it satisfies the condition C,Ω. Further, ξb satisfies the random integral equation (116). Then, for every ω∈Ω, we have ξ^.,ω∈C1I,E and so ξ^ satisfies condition C1,Ω. ξ is a random solution of problem (77).



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*Source: 6648938-2021-03-05.xml*