A PAIR OF MULTIVALUED AND SINGLE-VALUED MAPPINGS
LJUBOMIR B. ´CIRI ´C, JEONG S. UME, AND SINIˇSA N. JEˇSI ´C Received 2 February 2006; Revised 21 June 2006; Accepted 22 July 2006
Let (X,d) be a Polish space, CB(X) the family of all nonempty closed and bounded subsets ofX, and (Ω,Σ) a measurable space. A pair of a hybrid measurable mappings f :Ω×X→X andT:Ω×X→CB(X), satisfying the inequality (1.2), are introduced and investigated. It is proved that ifXis complete,T(ω,·), f(ω,·) are continuous for all ω∈Ω,T(·,x), f(·,x) are measurable for allx∈X, and f(ω×X)=X for eachω∈Ω, then there is a measurable mappingξ:Ω→X such that f(ω,ξ(w))∈T(ω,ξ(w)) for allω∈Ω. This result generalizes and extends the fixed point theorem of Papageorgiou (1984) and many classical fixed point theorems.
Copyright © 2006 Ljubomir B. ´Ciri´c et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
Random fixed point theorems are stochastic generalizations of classical fixed point the- orems. Random fixed point theorems for contraction mappings on separable complete metric spaces have been proved by several authors (Zhang and Huang [25], Hanˇs [6,7], Itoh [8], Lin [12], Papageorgiou [13,14], Shahzad and Hussian [19,20], ˇSpaˇcek [22], and Tan and Yuan [23]). The stochastic version of the well known Schauder’s fixed point theorem was proved by Sehgal and Singh [18].
Let (X,d) be a metric space andT:X→Xa mapping. The class of mappingsTsatis- fying the following contractive condition:
d(Tx,T y)≤αmax
d(x,y),d(x,Tx),d(y,T y),d(x,T y) +d(y,Tx) 2
+βmaxd(x,Tx),d(y,T y)+γd(x,T y) +d(y,Tx)
(1.1)
for allx,y∈X, whereα,β,γare nonnegative real numbers such thatβ >0,γ >0, andα+ β+ 2γ=1, was introduced and investigated by ´Ciri´c [1]. ´Ciri´c proved that in a complete
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 81045, Pages1–12 DOI 10.1155/JIA/2006/81045
metric space such mappings have a unique fixed point. This class of mappings was further studied by many authors ( ´Ciri´c [2,3], Singh and Mishra [21], and Rhoades et al. [16]).
Singh and Mishra [21] have generalized ´Ciri´c’s [2] fixed point theorem to a common fixed point theorem of a pair of mappings and presented some application of such theorems to dynamic programming.
Let (Ω,Σ) be a measurable space withΣa sigma algebra of subsets ofΩand let (X,d) be a metric space. We denote by 2X the family of all subsets ofX, by CB(X) the family of all nonempty closed and bounded subsets ofX, and by H the Hausdorffmetric on CB(X), induced by the metricd. For anyx∈X and A⊆X, by d(x,A) we denote the distance betweenxandA, that is,d(x,A)=inf{d(x,a) :a∈A}.
A mappingT:Ω→2Xis calledΣ-measurable if for any open subsetUofX,T−1(U)= {ω:T(w)∩U= ∅} ∈Σ. In what follows, when we speak of measurability we will mean Σ-measurability. A mapping f :Ω×X→Xis called a random operator if for anyx∈X, f(·,x) is measurable. A mappingT:Ω×X→CB(X) is called a multivalued random oper- ator if for everyx∈X,T(·,x) is measurable. A mappings:Ω→Xis called a measurable selector of a measurable multifunction T:Ω→2X ifs is measurable ands(ω)∈T(ω) for allω∈Ω. A measurable mappingξ:Ω→X is called a random fixed point of a ran- dom multifunctionT:Ω×X→CB(X) ifξ(w)∈T(w,ξ(w)) for everyw∈Ω. A mea- surable mappingξ:Ω→X is called a random coincidence ofT:Ω× X→CB(X) and
f :Ω×X→Xif f(ω,ξ(w))∈T(w,ξ(w)) for everyw∈Ω.
The aim of this paper is to prove a stochastic analog of the ´Ciri´c [1] fixed point theo- rem for single-valued mappings, extended to a coincidence theorem for a pair of a ran- dom operator f :Ω×X→X and a multivalued random operatorT:Ω×X→CB(X), satisfying the following nonexpansive-type condition: for eachω∈Ω,
HT(ω,x),T(ω,y)
≤α(ω) max
df(ω,x),f(ω,y) ,df(ω,x),T(ω,x) ,df(ω,y),T(ω,y) , 1
2
df(ω,x),T(ω,y) +df(ω,y),T(ω,x)
+β(ω) maxdf(ω,x),T(ω,x) ,df(ω,y),T(ω,y) +γ(ω)df(ω,x),T(ω,y) +df(ω,y),T(ω,x)
(1.2)
for everyx,y∈X, whereα,β,γ:Ω→[0, 1) are measurable mappings such that for all ω∈Ω,
β(ω)>0, γ(ω)>0, (1.3)
α(ω) +β(ω) + 2γ(ω)=1. (1.4) 2. Main results
Now we are proving our main result.
Theorem 2.1. Let (X,d) be a complete separable metric space, let (Ω,Σ) be a measurable space, and letT:Ω×X→CB(X) and f :Ω×X→Xbe mappings such that
(i)T(ω,·), f(ω,·) are continuous for allω∈Ω, (ii)T(·,x), f(·,x) are measurable for allx∈X,
(iii) they satisfy (1.2), whereα(ω),β(ω),γ(ω) :Ω→Xsatisfy (1.3) and (1.4).
If f(ω×X)=Xfor eachω∈Ω, then there is a measurable mappingξ:Ω→Xsuch that f(ω,ξ(w))∈T(w,ξ(w)) for allω∈Ω(i.e.,Tandf have a random coincidence point).
Proof. LetΨ= {ξ:Ω→X}be a family of measurable mappings. Define a functiong: Ω×X→R+as follows:
g(ω,x)=dx,T(ω,x) . (2.1)
Sincex→T(ω,x) is continuous for allω∈Ω, we conclude thatg(ω,·) is continuous for allω∈Ω. Also, sinceω→T(ω,x) is measurable for allx∈X, we conclude thatg(·,x) is measurable (see Wagner [24, page 868]) for allω∈Ω. Thusg(ω,x) is the Caratheodory function. Therefore, ifξ:Ω→X is a measurable mapping, thenω→g(ω,ξ(w)) is also measurable (see [17]).
Now we will construct a sequence of measurable mappings{ξn}inΨand a sequence {f(ω,ξn(ω))}inXas follows. Letξ0∈Ψbe arbitrary. Then the multifunctionG:Ω→ CB(X) defined byG(ω)=T(w,ξ0(w)) is measurable.
From the Kuratowski and Ryll-Nardzewski [11] selector theorem, there is a measurable selectorμ1:Ω→Xsuch thatμ1(ω)∈T(w,ξ0(w)) for allω∈Ω. Sinceμ1(ω)∈T(w,ξ0(w))
⊆X= f(ω×X), letξ1∈Ψbe such that f(ω,ξ1(ω))=μ1(ω). Thus f(ω,ξ1(ω))∈T(ω, ξ0(ω)) for allω∈Ω.
Letk:Ω→(1,∞) be defined by
k(ω)=1 +β(ω)γ(ω)
2 (2.2)
for allω∈Ω. Thenk(ω) is measurable. Sincek(ω)>1 and f(ω,ξ1(ω)) is a selector of T(w,ξ0(w)), from Papageorgiou [13, Lemma 2.1] there is a measurable selectorμ2(ω)=
f(ω,ξ2(ω));ξ2∈Ψ, such that for allω∈Ω,
fω,ξ2(ω) ∈Tω,ξ1(ω) ,
dfω,ξ1(ω) ,fω,ξ2(ω) ≤k(ω)HTω,ξ0(ω) ,Tω,ξ1(ω) . (2.3) Similarly, asf(ω,ξ2(ω)) is a selector ofT(w,ξ1(w)), there is a measurable selectorμ3(ω)=
f(ω,ξ3(ω)) ofT(ω,ξ2(ω))⊆ f(ω×X) such that
dfω,ξ2(ω) ,fω,ξ3(ω) ≤k(ω)HTω,ξ1(ω) ,Tω,ξ2(ω) . (2.4) Continuing this process we can construct a sequence of measurable mappingsμn:Ω→X, defined byμn(ω)=f(ω,ξn(ω));ξn∈Ψ, such that
fω,ξn+1(ω) ∈Tω,ξn(ω) , (2.5)
dfω,ξn(ω) ,fω,ξn+1(ω) ≤k(ω)HTω,ξn−1(ω) ,Tω,ξn(ω) . (2.6)
Observe that condition (1.2) is clumsy. So, for simplicity, in the rest of the paper we will use this condition in the following form:
HT(ω,x),T(ω,y) ≤α(ω) max
df(ω,x),f(ω,y) ,·,·, 1
2
[·+·]
+β(ω) maxdf(ω,x),T(ω,x) ,df(ω,y),T(ω,y) +γ(ω)df(ω,x),T(ω,y) +df(ω,y),T(ω,x) .
(2.7)
From (2.7),
HTω,ξ0(ω) ,Tω,ξ1(ω)
≤α(ω) max
dfω,ξ0(ω) ,fω,ξ1(ω) ,·,·, 1
2
[·+·]
+β(ω) maxdfω,ξ0(ω) ,Tω,ξ0(ω) ,dfω,ξ1(ω) ,Tω,ξ1(ω) +γ(ω)dfω,ξ0(ω) ,Tω,ξ1(ω) +dfω,ξ1(ω) ,Tω,ξ0(ω) .
(2.8)
Since f(ω,ξ1(ω))∈T(ω,ξ0(ω)), then
dfω,ξ1(ω) ,Tω,ξ0(ω) =0,
dfω,ξ0(ω) ,Tω,ξ0(ω) ≤dfω,ξ0(ω) ,fω,ξ1(ω) , dfω,ξ1(ω) ,Tω,ξ1(ω) ≤HTω,ξ0(ω) ,Tω,ξ1(ω) .
(2.9)
Thus from (2.8),
HTω,ξ0(ω) ,Tω,ξ1(ω)
≤α(ω) max
dfω,ξ0(ω) ,fω,ξ1(ω) ,·,·, 1
2
[·+·]
+β(ω) maxdfω,ξ0(ω) ,fω,ξ1(ω) ,HTω,ξ0(ω) ,Tω,ξ1(ω) +γ(ω)dfω,ξ0(ω) ,fω,ξ1(ω) +HTω,ξ0(ω) ,Tω,ξ1(ω) .
(2.10)
If we assume thatH(T(ω,ξ0(ω)),T(ω,ξ1(ω)))> d(f(ω,ξ0(ω)),f(ω,ξ1(ω))), then we have, asγ(ω)>0,
γ(ω)dfω,ξ0(ω) ,fω,ξ1(ω) +HTω,ξ0(ω) ,Tω,ξ1(ω)
<2γ(ω)HTω,ξ0(ω) ,Tω,ξ1(ω) . (2.11)
Thus, from (1.4) and (2.10), we have HTω,ξ0(ω) ,Tω,ξ1(ω)
< α(ω)HTω,ξ0(ω) ,Tω,ξ1(ω) +β(ω)HTω,ξ0(ω) ,Tω,ξ1(ω) + 2γ(ω)HTω,ξ0(ω) ,Tω,ξ1(ω)
=
α(ω) +β(ω) + 2γ(ω) HTω,ξ0(ω) ,Tω,ξ1(ω)
=HTω,ξ0(ω) ,Tω,ξ1(ω) ,
(2.12)
a contradiction. Therefore,
HTω,ξ0(ω) ,Tω,ξ1(ω) ≤dfω,ξ0(ω) ,fω,ξ1(ω) . (2.13) Sinced(f(ω,ξ1(ω)),T(ω,ξ1(ω)))≤H(T(ω,ξ0(ω)),T(ω,ξ1(ω))), we have
dfω,ξ1(ω) ,Tω,ξ1(ω) ≤dfω,ξ0(ω) ,fω,ξ1(ω) . (2.14) By induction, we can show that
HTω,ξn(ω) ,Tω,ξn+1(ω) ≤dfω,ξn(ω) ,fω,ξn+1(ω) , (2.15) dfω,ξn(ω) ,Tω,ξn(ω) ≤dfω,ξn−1(ω) ,fω,ξn(ω) (2.16) for eachn≥1 and allω∈Ω. From (2.6) and (2.15),
dfω,ξn(ω) ,fω,ξn+1(ω) ≤k(ω)dfω,ξn−1(ω) ,fω,ξn(ω) . (2.17) By (2.17), we get
dfω,ξ0(ω) ,fω,ξ2(ω) ≤dfω,ξ0(ω) ,fω,ξ1(ω) +dfω,ξ1(ω) ,fω,ξ2(ω)
≤
1 +k(ω) dfω,ξ0(ω) ,fω,ξ1(ω) .
(2.18)
From (2.7),
HTω,ξ0(ω) ,Tω,ξ2(ω)
≤α(ω) max
dfω,ξ0(ω) ,fω,ξ2(ω) ,·,·, 1
2
[·+·]
+β(ω) maxdfω,ξ0(ω) ,Tω,ξ0(ω) ,dfω,ξ2(ω) ,Tω,ξ2(ω) +γ(ω)dfω,ξ0(ω) ,Tω,ξ2(ω) +dfω,ξ2(ω) ,Tω,ξ0(ω) .
(2.19)
Using (2.15), (2.16), (2.17), and (2.18) and the triangle inequality, we get dfω,ξ2(ω) ,Tω,ξ0(ω) ≤HTω,ξ1(ω) ,Tω,ξ0(ω)
≤dfω,ξ0(ω) ,fω,ξ1(ω) , (2.20) dfω,ξ0(ω) ,Tω,ξ2(ω) ≤dfω,ξ0(ω) ,fω,ξ1(ω) +dfω,ξ1(ω) ,fω,ξ2(ω)
+dfω,ξ2(ω) ,Tω,ξ2(ω)
≤
1 +k(ω) dfω,ξ0(ω) ,fω,ξ1(ω) +dfω,ξ1(ω) ,fω,ξ2(ω)
≤
1 + 2k(ω) dfω,ξ0(ω) ,fω,ξ1(ω) .
(2.21) Now from (1.4), (2.17), (2.18), and (2.19), we have
HTω,ξ0(ω) ,Tω,ξ2(ω)
≤α(ω)1 +k(ω) dfω,ξ0(ω) ,fω,ξ1(ω) +β(ω)k(ω)dfω,ξ0(ω) ,fω,ξ1(ω) + 2γ(ω)1 +k(ω) dfω,ξ0(ω) ,fω,ξ1(ω)
=
1 +k(ω) α(ω) +β(ω) + 2γ(ω) −β(ω)dfω,ξ0(ω) ,fω,ξ1(ω)
=
1 +k(ω)−β(ω) dfω,ξ0(ω) ,fω,ξ1(ω) .
(2.22)
Hence we get, as 1 +k(ω)<2k(ω), HTω,ξ0(ω) ,Tω,ξ2(ω) ≤
2k(ω)−β(ω) dfω,ξ0(ω) ,fω,ξ1(ω) . (2.23) From (1.4) and (2.7) we have, asf(ω,ξ2(ω))∈T(ω,ξ1(ω)),
HTω,ξ1(ω) ,Tω,ξ2(ω)
≤α(ω) max
dfω,ξ1(ω) ,fω,ξ2(ω) ,·,·, 1
2
[·+·]
+β(ω) maxdfω,ξ1(ω) ,Tω,ξ1(ω) ,dfω,ξ2(ω) ,Tω,ξ2(ω) +γ(ω)dfω,ξ1(ω) ,Tω,ξ2(ω) .
(2.24)
Since f(ω,ξ1(ω))∈T(ω,ξ0(ω)), by (2.23) we have
dfω,ξ1(ω) ,Tω,ξ2(ω) ≤HTω,ξ0(ω) ,Tω,ξ2(ω)
≤
2k(ω)−β(ω) dfω,ξ0(ω) ,fω,ξ1(ω) . (2.25)
Thus from (2.17) and (2.24), we get HTω,ξ1(ω) ,Tω,ξ2(ω)
≤α(ω)k(ω)dfω,ξ0(ω) ,fω,ξ1(ω) +β(ω)k(ω)dfω,ξ0(ω) ,fω,ξ1(ω) +γ(ω)2k(ω)−β(ω) dfω,ξ0(ω) ,fω,ξ1(ω)
=
k(ω)α(ω) +β(ω) + 2γ(ω) −β(ω)γ(ω)dfω,ξ0(ω) ,fω,ξ1(ω) .
(2.26) Hence, asα(ω) +β(ω) + 2γ(ω)=1,
HTω,ξ1(ω) ,Tω,ξ2(ω) ≤
k(ω)−β(ω)γ(ω) dfω,ξ0(ω) ,fω,ξ1(ω) . (2.27) From (2.6) and (2.27),
dfω,ξ2(ω) ,fω,ξ3(ω) ≤k(ω)HTω,ξ1(ω) ,Tω,ξ2(ω)
≤k(ω)k(ω)−β(ω)γ(ω) dfω,ξ0(ω) ,fω,ξ1(ω) . (2.28) Sincek(ω)=1 +β(ω)γ(ω)/2, we have
k(ω)k(ω)−β(ω)γ(ω) =
1 +β(ω)γ(ω) 2
1 +β(ω)γ(ω)
2 −β(ω)γ(ω)
=
1 +β(ω)γ(ω) 2
1−β(ω)γ(ω) 2
=1−β2(ω)γ2(ω)
4 .
(2.29)
Thus from (2.28),
dfω,ξ2(ω) ,fω,ξ3(ω) ≤
1−β2(ω)γ2(ω) 4
dfω,ξ0(ω) ,fω,ξ1(ω) . (2.30) Analogously,
dfω,ξ3(ω) ,fω,ξ4(ω) ≤
1−β2(ω)γ2(ω)/4 dfω,ξ1(ω) ,fω,ξ2(ω) . (2.31) By induction,
dfω,ξn(ω) ,fω,ξn+1(ω)
≤
1−β2(ω)γ2(ω) 4
[n/2]
×maxdfω,ξ0(ω) ,fω,ξ1(ω) ,dfω,ξ1(ω) ,fω,ξ2(ω) ,
(2.32)
where [n/2] stands for the greatest integer not exceedingn/2. Sinceβ(ω)γ(ω)>0 for all ω∈Ω, from (2.32), we conclude that{f(ω,ξn(ω))}is a Cauchy sequence in f(ω×X).
Since f(ω×X)=X is complete, there is a measurable mapping f(ω,ξ(ω))∈f(ω×X) such that
nlim→∞fω,ξn(ω) = fω,ξ(ω) . (2.33) Now by the triangle inequality and (1.2), we have
dfω,ξ(ω) ,Tω,ξ(ω)
≤dfω,ξ(ω) ,fω,ξn+1(ω) +dfω,ξn+1(ω) ,Tω,ξ(ω)
≤dfω,ξ(ω) ,fω,ξn+1(ω) +HTω,ξn(ω) ,Tω,ξ(ω)
≤dfω,ξ(ω) ,fω,ξn+1(ω) +α(ω) max
dfω,ξn(ω) ,fω,ξ(ω) ,·,·, 1
2
[·+·]
+β(ω) maxdfω,ξn(ω) ,Tω,ξn(ω) ,dfω,ξ(ω) ,Tω,ξ(ω) +γ(ω)dfω,ξn(ω) ,Tω,ξ(ω) +dfω,ξ(ω) ,Tω,ξn(ω) .
(2.34)
Thus
dfω,ξ(ω) ,Tω,ξ(ω)
≤dfω,ξ(ω) ,fω,ξn+1(ω) +α(ω) max
dfω,ξn(ω) ,fω,ξ(ω) ,·,·, 1
2
[·+·]
+β(ω) maxdfω,ξn(ω) ,fω,ξn+1(ω) ,dfω,ξ(ω) ,Tω,ξ(ω) +γ(ω)dfω,ξn(ω) ,Tω,ξ(ω) +dfω,ξ(ω) ,fω,ξn+1(ω) .
(2.35)
Taking the limit asn→ ∞, we get
dfω,ξ(ω) ,Tω,ξ(ω) ≤α(ω)dfω,ξ(ω) ,Tω,ξ(ω) +β(ω)dfω,ξ(ω) ,Tω,ξ(ω) +γ(ω)dfω,ξ(ω) ,Tω,ξ(ω)
=
1−γ(ω) dfω,ξ(ω) ,Tω,ξ(ω) .
(2.36)
Henced(f(ω,ξ(ω)),T(ω,ξ(ω)))=0, as 1−γ(ω)<1 for allω∈Ω. Hence, asT(ω,ξ(ω)) is closed,
fω,ξ(ω) ∈Tω,ξ(ω) ∀ω∈Ω. (2.37)
Remark 2.2. If inTheorem 2.1, f(ω,x)=xfor all (ω,x)∈Ω×X, then we get the follow- ing random fixed point theorem.
Corollary 2.3. Let (X,d) be a separable complete metric space, let (Ω,Σ) be a measurable space, and let a mapping T:Ω×X→CB(X) be such that T(ω,·) is continuous for all ω∈Ω,T(·,x) is measurable for allx∈X, and
HT(ω,x),T(ω,y)
≤α(ω) max
d(x,y),dx,T(ω,x) ,dy,T(ω,y) , 1
2
dx,T(ω,y) +dy,T(ω,x)
+β(ω) maxdx,T(ω,x) ,dy,T(ω,y) +γ(ω)dx,T(ω,y) +dy,T(ω,x) (2.38) for everyx,y∈X, whereα,β,γ:Ω→(0, 1) are measurable mappings satisfying (1.2). Then there is a measurable mappingξ:Ω→Xsuch thatξ(w)∈T(w,ξ(w)) for allω∈Ω.
Corollary 2.4. Let (X,d) be a complete separable metric space, let (Ω,Σ) be a measurable space, and let f :Ω×X→XandT:Ω×X→CB(X) be two mappings satisfying the con- ditions (i) and (ii) inTheorem 2.1. If f(ω×X)=Xfor eachω∈Ωand f andTsatisfy the following condition:
HT(ω,x),T(ω,y)
≤λ(ω) max
df(ω,x),f(ω,y) ,df(ω,x),T(ω,x) ,df(ω,y),T(ω,y) , df(ω,x),T(ω,y) +df(ω,y ,T(ω,x)
2
,
(2.39) whereλ:Ω→(0, 1) is a measurable function, then there is a measurable mappingξ:Ω→X such that f(ω,ξ(w))∈T(w,ξ(w)) for allω∈Ω.
Proof. It is clear that if f andTsatisfy (2.39), thenf andTsatisfy (1.2) with α(ω)=λ(ω), β(ω)=1−λ(ω)
2 , γ(ω)=1−λ(ω)
4 . (2.40)
Remark 2.5. If inCorollary 2.4, f(ω,x)=xfor all (ω,x)∈Ω×X, then we obtain the corresponding theorems of Hadˇzi´c [5] and Papageorgiou [13].
Finally, we give a simple example which shows thatTheorem 2.1and Corollaries2.3 and2.4are actually an improvement of the results of Kubiak [10] and Papageorgiou [13].
Example 2.6. Let (Ω,Σ) be any measurable space and letK= {0, 1, 2, 4, 6}be the subset of the real line. Let the mappings f :Ω×K→KandT:Ω×K→Kbe defined such that
for eachω∈Ω,
f(ω, 0)=2, f(ω, 1)=4, f(ω, 2)=6, f(ω, 4)=0, f(ω, 6)=1, T(ω, 0)=1, T(ω, 1)=2, T(ω, 2)=4, T(ω, 4)=0, T(ω, 6)=0.
(2.41) Then f andTdo not satisfy the contractive-type condition (2.39). Indeed, forx=1 and y=2, we have
dT(ω, 1),T(ω, 2) =2> λ(ω) max
4−6,4−2,6−4,0 +6−2 2
=2λ(ω) (2.42) for anyλ(ω)<1. On the other hand,
dT(ω, 1),T(ω, 2) =4 5·2 + 1
10·2 + 1
20(4 + 0). (2.43)
Thus, forx=1 andy=2,f andTsatisfy (1.2) withα(ω)=4/5,β(ω)=1/10, andγ(ω)= 1/20. It is easy to show thatf andTsatisfy (1.2) for allx,y∈K, with the sameα(ω),β(ω), andγ(ω). Also, the rest of assumptions ofTheorem 2.1is satisfied and forξ(ω)=4 we have
fω,ξ(ω) =0=Tω,ξ(ω) . (2.44) Note thatTdoes not satisfy (2.38) either, as for instance, forx=0 andy=2, we have α(ω) max
0−2,0−1,2−4,0−4+2−1 2
+β(ω) max0−1,2−4
+γ(ω)0−4+2−1
=5
2α(ω) + 2β(ω) + 5γ(ω)<3α(ω) +β(ω) + 2γ(ω)=3=dT(ω, 0),T(ω, 2) . (2.45) Remark 2.7. Corollary 2.4is a stochastic generalization and improvement of the corre- sponding fixed point theorems for contractive-type multivalued mappings of ´Ciri´c [2],
´Ciri´c and Ume [4], Kubiaczyk [9], Kubiak [10], Papageorgiou [14], and several other au- thors. AlsoTheorem 2.1generalizes and extends the corresponding fixed point theorems for nonexpansive-type single-valued mappings of ´Ciri´c [1] and Rhoades [15].
Acknowledgment
This research was financially supported by Changwon National University in 2006.
References
[1] L. B. ´Ciri´c, On some nonexpansive type mappings and fixed points, Indian Journal of Pure and Applied Mathematics 24 (1993), no. 3, 145–149.
[2] , Nonexpansive type mappings and a fixed point theorem in convex metric spaces, Ac- cademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I 19 (1995), 263–271.
[3] , On some mappings in metric spaces and fixed points, Acad´emie Royale de Belgique. Bul- letin de la Classe des Sciences. 6e S´erie 6 (1995), no. 1–6, 81–89.
[4] L. B. ´Ciri´c and J. S. Ume, Some common fixed point theorems for weakly compatible mappings, Journal of Mathematical Analysis and Applications 314 (2006), no. 2, 488–499.
[5] O. Hadˇzi´c, A random fixed point theorem for multivalued mappings of ´Ciri´c’s type, Matematiˇcki Vesnik 3(16)(31) (1979), no. 4, 397–401.
[6] O. Hanˇs, Reduzierende zuf¨allige Transformationen, Czechoslovak Mathematical Journal 7 (1957), no. 82, 154–158.
[7] , Random operator equations, Proceedings of the 4th Berkeley Symposium on Mathe- matical Statistics and Probability, vol. 2, part 1, University of California Press, California, 1961, pp. 185–202.
[8] S. Itoh, A random fixed point theorem for a multivalued contraction mapping, Pacific Journal of Mathematics 68 (1977), no. 1, 85–90.
[9] I. Kubiaczyk, Some fixed point theorems, Demonstratio Mathematica 9 (1976), no. 3, 507–515.
[10] T. Kubiak, Fixed point theorems for contractive type multivalued mappings, Mathematica Japonica 30 (1985), no. 1, 89–101.
[11] K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bulletin de l’Acad´emie Polonaise des Sciences. S´erie des Sciences Math´ematiques, Astronomiques et Physiques 13 (1965), 397–403.
[12] T.-C. Lin, Random approximations and random fixed point theorems for non-self-maps, Proceed- ings of the American Mathematical Society 103 (1988), no. 4, 1129–1135.
[13] N. S. Papageorgiou, Random fixed point theorems for multifunctions, Mathematica Japonica 29 (1984), no. 1, 93–106.
[14] , Random fixed point theorems for measurable multifunctions in Banach spaces, Proceed- ings of the American Mathematical Society 97 (1986), no. 3, 507–514.
[15] B. E. Rhoades, A generalization of a fixed point theorem of Bogin, Mathematics Seminar Notes, Kobe University 6 (1978), no. 1, 1–7.
[16] B. E. Rhoades, S. L. Singh, and C. Kulshrestha, Coincidence theorems for some multivalued map- pings, International Journal of Mathematics and Mathematical Sciences 7 (1984), no. 3, 429–
434.
[17] R. T. Rockafellar, Measurable dependence of convex sets and functions on parameters, Journal of Mathematical Analysis and Applications 28 (1969), no. 1, 4–25.
[18] V. M. Sehgal and S. P. Singh, On random approximations and a random fixed point theorem for set valued mappings, Proceedings of the American Mathematical Society 95 (1985), no. 1, 91–94.
[19] N. Shahzad and N. Hussain, Deterministic and random coincidence point results for f- nonexpansive maps, to appear in Journal of Mathematical Analysis and Applications.
[20] N. Shahzad and A. Latif, A random coincidence point theorem, Journal of Mathematical Analysis and Applications 245 (2000), no. 2, 633–638.
[21] S. L. Singh and S. N. Mishra, On a Ljubomir ´Ciri´c’s fixed point theorem for nonexpansive type maps with applications, Indian Journal of Pure and Applied Mathematics 33 (2002), no. 4, 531–542.
[22] A. ˇSpaˇcek, Zuf¨allige Gleichungen, Czechoslovak Mathematical Journal 5(80) (1955), no. 80, 462–
466.
[23] K.-K. Tan and X.-Z. Yuan, Random fixed-point theorems and approximation in cones, Journal of Mathematical Analysis and Applications 185 (1994), no. 2, 378–390.
[24] D. H. Wagner, Survey of measurable selection theorems, SIAM Journal on Control and Optimiza- tion 15 (1977), no. 5, 859–903.
[25] S. S. Zhang and N.-J. Huang, On the principle of randomization of fixed points for set-valued mappings with applications, Northeastern Mathematical Journal 7 (1991), no. 4, 486–491.
Ljubomir B. ´Ciri´c: Faculty of Mechanical Engineering, University of Belgrade, Aleksinaˇckih Rudara 12-35, Belgrade 11070, Serbia and Montenegro
E-mail address:[email protected]
Jeong S. Ume: Department of Applied Mathematics, Changwon National University, Changwon 641-773, Korea
E-mail address:[email protected]
Siniˇsa N. Jeˇsi´c: Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade 11000, Serbia and Montenegro E-mail address:[email protected]
