On random coincidence and fixed points for a pair of multivalued and single valued mappings

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A PAIR OF MULTIVALUED AND SINGLE-VALUED MAPPINGS

LJUBOMIR B. ĆIRI Ć, JEONG S. UME, AND SINIˇSA N. JEˇSI Ć

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 mappingsT satis-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 Ćirić [1]. Ćirić proved that in a complete

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 81045, Pages1–12

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metric space such mappings have a unique fixed point. This class of mappings was further studied by many authors ( Ćirić [2,3], Singh and Mishra [21], and Rhoades et al. [16]). Singh and Mishra [21] have generalized Ćirić’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 of}_{X, 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→2X_{is called}_Σ_-_measurable_{if for any open subset}_U_of_X,_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 arandom operatorif for anyx∈X, f(·,x) is measurable. A mappingT:Ω×X→CB(X) is called amultivalued random oper-atorif for everyx∈X,T(·,x) is measurable. A mappings:Ω→Xis called ameasurable selector of a measurable multifunction T:Ω→2X _if_s _{is measurable and}_s(ω)_∈_T_(ω) for allω∈Ω. A measurable mappingξ:Ω→X is called arandom 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 arandom coincidenceofT: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 theotheo-rem 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

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Theorem2.1. 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) ∈T

ω,ξn(ω) , (2.5)

dfω,ξn(ω) ,f

ω,ξn+1(ω)

≤k(ω)HTω,ξn−1(ω) ,T

ω,ξn(ω)

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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),

≤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)

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Thus, from (1.4) and (2.10), we have

< 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) ,T

ω,ξn+1(ω)

≤dfω,ξn(ω) ,f

ω,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),

≤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) .

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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)),

≤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)

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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−β

2_(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) ,f

ω,ξn+1(ω)

≤

1−β2(ω)γ2(ω) 4

[n/2]

×maxdfω,ξ0(ω) ,fω,ξ1(ω) ,dfω,ξ1(ω) ,fω,ξ2(ω) ,

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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

lim n→∞f

ω,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) ,T

ω,ξ(ω)

+HTω,ξn(ω) ,T

ω,ξ(ω)

+α(ω) max

dfω,ξn(ω) ,fω,ξ(ω) ,·,·, 1 2 [·+·]

+β(ω) maxdfω,ξn(ω) ,T

ω,ξn(ω)

,dfω,ξ(ω) ,Tω,ξ(ω) +γ(ω)dfω,ξn(ω) ,T

ω,ξ(ω) +dfω,ξ(ω) ,Tω,ξn(ω)

.

(2.34)

Thus

dfω,ξ(ω) ,Tω,ξ(ω)

+α(ω) max

dfω,ξn(ω) ,f

ω,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)

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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.

Corollary2.3. 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 thatT(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.

Corollary2.4. 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

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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)<3

α(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 Ćirić [2], Ćirić 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 Ćirić [1] and Rhoades [15].

Acknowledgment

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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]