ON RANDOM COINCIDENCE AND FIXED POINTS FOR A PAIR OF MULTIVALUED AND SINGLE-VALUED MAPPINGS LJUBOMIR B. ´ CIRI ´ C, JEONG S. UME, AND SINI ˇ SA N. JE ˇ SI ´ 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 of X,and(Ω,Σ) a measurable space. A pair of a hybrid measurable m appings f : Ω × X → X and T : Ω × X → CB(X), satisfying the inequality (1.2), are introduced and investigated. It is proved that if X is complete, T(ω, ·), f (ω,·) are continuous for all ω ∈ Ω, T(·,x), f (·,x) are measurable for all x ∈ X,and f (ω × X) = X for each ω ∈ Ω, then there is a measurable mapping ξ : Ω → X such that f (ω, ξ(w)) ∈ T(ω,ξ(w)) for all ω ∈ 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 dist ributed 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], Han ˇ s[6, 7], Itoh [8], Lin [12], Papageorgiou [13, 14], Shahzad and Hussian [19, 20], ˇ Spa ˇ cek [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 and T : X → X a mapping. The class of mappings T satis- fying the following contractive condition: d(Tx,Ty) ≤ αmax  d(x, y),d(x,Tx),d(y,Ty), d(x,Ty)+d(y,Tx) 2  + βmax  d(x,Tx),d(y,Ty)  + γ  d(x,Ty)+d(y,Tx)  (1.1) for all x, y ∈ X,whereα, β, γ are nonnegative real numbers such that β>0, γ>0, and α + 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, Pages 1–12 DOI 10.1155/JIA/2006/81045 2 On random coincidence and fixed points 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 with Σ a sigma algebra of subsets of Ω and let (X,d) beametricspace.Wedenoteby2 X the family of all subsets of X,byCB(X) the family of all nonempty closed and bounded subsets of X,andbyH the Hausdorff metric on CB(X), induced by the metric d.Foranyx ∈ X and A ⊆ X,byd(x,A) we denote the distance between x and A, that is, d(x,A) = inf{d(x,a):a ∈ A}. AmappingT : Ω → 2 X is cal led 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 wil l mean Σ-measurability. A mapping f : Ω × X → X is called a random operator if for any x ∈ X, f ( ·,x) is measurable. A mapping T : Ω ×X → CB(X)iscalledamultivalued random oper- ator if for every x ∈ X, T(·,x) is measurable. A mapping s : Ω → X is called a measurable selector of a measurable multifunction T : Ω → 2 X if s is measurable and s(ω) ∈ T(ω) for all ω ∈ Ω. A measurable mapping ξ : Ω → X is called a random fixed point ofaran- dom multifunction T : Ω × X → CB(X)ifξ(w) ∈ T(w,ξ(w)) for every w ∈ Ω.Amea- surable mapping ξ : Ω → X is called a random coincidence of T : Ω × X → CB(X)and f : Ω × X → X if f (ω,ξ(w)) ∈ T(w,ξ(w)) for every w ∈ 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 operator T : Ω × X → CB(X), satisfying the following nonexpansive-type condition: for each ω ∈ Ω, H  T(ω,x),T(ω, y)  ≤ α(ω)max  d  f (ω, x), f (ω, y)  ,d  f (ω, x),T(ω, x)  ,d  f (ω, y),T(ω, y)  ,  1 2   d  f (ω, x),T(ω, y)  + d  f (ω, y),T(ω,x)   + β(ω)max  d  f (ω, x),T(ω, x)  ,d  f (ω, y),T(ω, y)  + γ(ω)  d  f (ω, x),T(ω, y)  + d  f (ω, y),T(ω,x)  (1.2) for every x, y ∈ X,whereα,β,γ : Ω → [0, 1) are measurable mapping s 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. Ljubomir B. ´ Ciri ´ cetal. 3 Theorem 2.1. Let (X,d) be a complete separable metric space, let (Ω,Σ) be a measurable space, and let T : Ω × X → CB(X) and f : Ω ×X → X be mappings such that (i) T(ω, ·), f (ω,·) are continuous for all ω ∈ Ω, (ii) T( ·,x), f (·,x) are measurable for all x ∈ X, (iii) they satisfy (1.2), where α(ω),β(ω),γ(ω):Ω → X satisfy (1.3)and(1.4). If f (ω × X) = X for each ω ∈ Ω, then there is a measurable mapping ξ : Ω → X such that f (ω, ξ(w)) ∈ T(w, ξ(w)) for all ω ∈ Ω (i.e., T and f have a random coincidence point). Proof. Let Ψ ={ξ : Ω → X} be a family of measurable mappings. Define a function g : Ω × X → R + as follows: g(ω,x) = d  x, T(ω,x)  . (2.1) Since x → T(ω,x) is continuous for all ω ∈ Ω,weconcludethatg(ω,·)iscontinuousfor all ω ∈ Ω. Also, since ω → T(ω,x)ismeasurableforallx ∈ X,weconcludethatg(·,x)is measurable (see Wagner [24, page 868]) for all ω ∈ Ω.Thusg(ω,x)istheCaratheodory function. Therefore, if ξ : Ω → X is a measurable mapping, then ω → g(ω,ξ(w)) is also measurable (see [17]). Now we will construct a sequence of measurable mappings {ξ n } in Ψ and a sequence { f (ω,ξ n (ω))} in X as follows. Let ξ 0 ∈ Ψ be arbitrary. Then the multifunction G : Ω → CB(X)definedbyG(ω) = T(w,ξ 0 (w)) is measurable. From the Kuratowski and Ryll-Nardzewski [11] selector theorem, there is a measurable selector μ 1 : Ω → X such that μ 1 (ω) ∈ T(w,ξ 0 (w)) for all ω ∈ Ω.Sinceμ 1 (ω)∈T(w,ξ 0 (w)) ⊆ X = f (ω × X), let ξ 1 ∈ Ψ be such that f (ω, ξ 1 (ω)) = μ 1 (ω). Thus f (ω,ξ 1 (ω)) ∈ T(ω, ξ 0 (ω)) for all ω ∈ Ω. Let k : Ω → (1,∞)bedefinedby k(ω) = 1+ β(ω)γ(ω) 2 (2.2) for all ω ∈ Ω.Thenk(ω) is measurable. Since k(ω) > 1and f (ω,ξ 1 (ω)) is a selector of T(w,ξ 0 (w)), from Papageorgiou [13, Lemma 2.1] there is a measurable selector μ 2 (ω) = f (ω, ξ 2 (ω)); ξ 2 ∈ Ψ, such that for all ω ∈ Ω, f  ω,ξ 2 (ω)  ∈ T  ω,ξ 1 (ω)  , d  f  ω,ξ 1 (ω)  , f  ω,ξ 2 (ω)  ≤ k(ω)H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  . (2.3) Similarly, as f (ω,ξ 2 (ω)) is a selector of T(w, ξ 1 (w)), there is a measurable selector μ 3 (ω) = f (ω, ξ 3 (ω)) of T(ω,ξ 2 (ω)) ⊆ f (ω × X)suchthat d  f  ω,ξ 2 (ω)  , f  ω,ξ 3 (ω)  ≤ k(ω)H  T  ω,ξ 1 (ω)  ,T  ω,ξ 2 (ω)  . (2.4) Continuing this process we can construct a sequence of measurable mappings μ n : Ω → X, defined by μ n (ω) = f (ω,ξ n (ω)); ξ n ∈ Ψ,suchthat f  ω,ξ n+1 (ω)  ∈ T  ω,ξ n (ω)  , (2.5) d  f  ω,ξ n (ω)  , f  ω,ξ n+1 (ω)  ≤ k(ω)H  T  ω,ξ n−1 (ω)  ,T  ω,ξ n (ω)  . (2.6) 4 On random coincidence and fixed points 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: H  T(ω,x),T(ω, y)  ≤ α(ω)max  d  f (ω, x), f (ω, y)  ,·,·,  1 2  [· + ·]  + β(ω)max  d  f (ω, x),T(ω, x)  ,d  f (ω, y),T(ω, y)  + γ(ω)  d  f (ω, x),T(ω, y)  + d  f (ω, y),T(ω,x)  . (2.7) From (2.7), H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  ≤ α(ω)max  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  ,·,·,  1 2  [· + ·]  + β(ω)max  d  f  ω,ξ 0 (ω)  ,T  ω,ξ 0 (ω)  ,d  f  ω,ξ 1 (ω)  ,T  ω,ξ 1 (ω)  + γ(ω)  d  f  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  + d  f  ω,ξ 1 (ω)  ,T  ω,ξ 0 (ω)  . (2.8) Since f (ω,ξ 1 (ω)) ∈ T(ω,ξ 0 (ω)), then d  f  ω,ξ 1 (ω)  ,T  ω,ξ 0 (ω)  = 0, d  f  ω,ξ 0 (ω)  ,T  ω,ξ 0 (ω)  ≤ d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  , d  f  ω,ξ 1 (ω)  ,T  ω,ξ 1 (ω)  ≤ H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  . (2.9) Thus from (2.8), H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  ≤ α(ω)max  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  ,·,·,  1 2  [· + ·]  + β(ω)max  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  ,H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  + γ(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  . (2.10) If we assume that H(T(ω,ξ 0 (ω)),T(ω,ξ 1 (ω)))>d( f (ω,ξ 0 (ω)), f (ω,ξ 1 (ω))), then we have, as γ(ω) > 0, γ(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  < 2γ(ω)H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  . (2.11) Ljubomir B. ´ Ciri ´ cetal. 5 Thus, from (1.4)and(2.10), we have H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  <α(ω)H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  + β(ω)H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  +2γ(ω)H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  =  α(ω)+β(ω)+2γ(ω)  H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  = H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  , (2.12) a contradiction. Therefore, H  T  ω,ξ 0 (ω)  ,T  ω,ξ 1 (ω)  ≤ d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.13) Since d( f (ω,ξ 1 (ω)),T(ω,ξ 1 (ω))) ≤ H(T(ω,ξ 0 (ω)),T(ω,ξ 1 (ω))), we have d  f  ω,ξ 1 (ω)  ,T  ω,ξ 1 (ω)  ≤ d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.14) By induction, we can show that H  T  ω,ξ n (ω)  ,T  ω,ξ n+1 (ω)  ≤ d  f  ω,ξ n (ω)  , f  ω,ξ n+1 (ω)  , (2.15) d  f  ω,ξ n (ω)  ,T  ω,ξ n (ω)  ≤ d  f  ω,ξ n−1 (ω)  , f  ω,ξ n (ω)  (2.16) for each n ≥ 1andallω ∈ Ω.From(2.6)and(2.15), d  f  ω,ξ n (ω)  , f  ω,ξ n+1 (ω)  ≤ k(ω)d  f  ω,ξ n−1 (ω)  , f  ω,ξ n (ω)  . (2.17) By (2.17), we get d  f  ω,ξ 0 (ω)  , f  ω,ξ 2 (ω)  ≤ d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + d  f  ω,ξ 1 (ω)  , f  ω,ξ 2 (ω)  ≤  1+k(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.18) From (2.7), H  T  ω,ξ 0 (ω)  ,T  ω,ξ 2 (ω)  ≤ α(ω)max  d  f  ω,ξ 0 (ω)  , f  ω,ξ 2 (ω)  ,·,·,  1 2  [· + ·]  + β(ω)max  d  f  ω,ξ 0 (ω)  ,T  ω,ξ 0 (ω)  ,d  f  ω,ξ 2 (ω)  ,T  ω,ξ 2 (ω)  + γ(ω)  d  f  ω,ξ 0 (ω)  ,T  ω,ξ 2 (ω)  + d  f  ω,ξ 2 (ω)  ,T  ω,ξ 0 (ω)  . (2.19) 6 On random coincidence and fixed points Using (2.15), (2.16), (2.17), and (2.18) and the triangle inequality, we get d  f  ω,ξ 2 (ω)  ,T  ω,ξ 0 (ω)  ≤ H  T  ω,ξ 1 (ω)  ,T  ω,ξ 0 (ω)  ≤ d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  , (2.20) d  f  ω,ξ 0 (ω)  ,T  ω,ξ 2 (ω)  ≤ d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + d  f  ω,ξ 1 (ω)  , f  ω,ξ 2 (ω)  + d  f  ω,ξ 2 (ω)  ,T  ω,ξ 2 (ω)  ≤  1+k(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + d  f  ω,ξ 1 (ω)  , f  ω,ξ 2 (ω)  ≤  1+2k(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.21) Now from (1.4), (2.17), (2.18), and (2.19), we have H  T  ω,ξ 0 (ω)  ,T  ω,ξ 2 (ω)  ≤ α(ω)  1+k(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + β(ω)k(ω)d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  +2γ(ω)  1+k(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  =  1+k(ω)  α(ω)+β(ω)+2γ(ω)  − β(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  =  1+k(ω) − β(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.22) Hence we get, as 1 + k(ω) < 2k( ω), H  T  ω,ξ 0 (ω)  ,T  ω,ξ 2 (ω)  ≤  2k(ω) − β(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.23) From (1.4)and(2.7)wehave,as f (ω,ξ 2 (ω)) ∈ T(ω,ξ 1 (ω)), H  T  ω,ξ 1 (ω)  ,T  ω,ξ 2 (ω)  ≤ α(ω)max  d  f  ω,ξ 1 (ω)  , f  ω,ξ 2 (ω)  ,·,·,  1 2  [· + ·]  + β(ω)max  d  f  ω,ξ 1 (ω)  ,T  ω,ξ 1 (ω)  ,d  f  ω,ξ 2 (ω)  ,T  ω,ξ 2 (ω)  + γ(ω)d  f  ω,ξ 1 (ω)  ,T  ω,ξ 2 (ω)  . (2.24) Since f (ω,ξ 1 (ω)) ∈ T(ω,ξ 0 (ω)), by (2.23)wehave d  f  ω,ξ 1 (ω)  ,T  ω,ξ 2 (ω)  ≤ H  T  ω,ξ 0 (ω)  ,T  ω,ξ 2 (ω)  ≤  2k(ω) − β(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.25) Ljubomir B. ´ Ciri ´ cetal. 7 Thus from (2.17)and(2.24), we get H  T  ω,ξ 1 (ω)  ,T  ω,ξ 2 (ω)  ≤ α(ω)k(ω)d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + β(ω)k(ω)d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  + γ(ω)  2k(ω) − β(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  =  k(ω)  α(ω)+β(ω)+2γ(ω)  − β(ω)γ(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.26) Hence, as α(ω)+β(ω)+2γ(ω) = 1, H  T  ω,ξ 1 (ω)  ,T  ω,ξ 2 (ω)  ≤  k(ω) − β(ω)γ(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.27) From (2.6)and(2.27), d  f  ω,ξ 2 (ω)  , f  ω,ξ 3 (ω)  ≤ k(ω)H  T  ω,ξ 1 (ω)  ,T  ω,ξ 2 (ω)  ≤ k(ω)  k(ω) − β(ω)γ(ω)  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.28) Since k(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)γ 2 (ω) 4 . (2.29) Thus from (2.28), d  f  ω,ξ 2 (ω)  , f  ω,ξ 3 (ω)  ≤  1 − β 2 (ω)γ 2 (ω) 4  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  . (2.30) Analogously, d  f  ω,ξ 3 (ω)  , f  ω,ξ 4 (ω)  ≤  1 − β 2 (ω)γ 2 (ω)/4  d  f  ω,ξ 1 (ω)  , f  ω,ξ 2 (ω)  . (2.31) By induction, d  f  ω,ξ n (ω)  , f  ω,ξ n+1 (ω)  ≤  1 − β 2 (ω)γ 2 (ω) 4  [n/2] ×max  d  f  ω,ξ 0 (ω)  , f  ω,ξ 1 (ω)  ,d  f  ω,ξ 1 (ω)  , f  ω,ξ 2 (ω)  , (2.32) 8 On random coincidence and fixed points where [n/2] stands for the greatest integer not exceeding n/2. Since β(ω)γ(ω) > 0forall ω ∈ Ω,from(2.32), we conclude that { f (ω,ξ n (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,ξ n (ω)  = f  ω,ξ(ω)  . (2.33) Now by the triangle inequality and (1.2), we have d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  ≤ d  f  ω,ξ(ω)  , f  ω,ξ n+1 (ω)  + d  f  ω,ξ n+1 (ω)  ,T  ω,ξ(ω)  ≤ d  f  ω,ξ(ω)  , f  ω,ξ n+1 (ω)  + H  T  ω,ξ n (ω)  ,T  ω,ξ(ω)  ≤ d  f  ω,ξ(ω)  , f  ω,ξ n+1 (ω)  + α(ω)max  d  f  ω,ξ n (ω)  , f  ω,ξ(ω)  ,·,·,  1 2  [· + ·]  + β(ω)max  d  f  ω,ξ n (ω)  ,T  ω,ξ n (ω)  ,d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  + γ(ω)  d  f  ω,ξ n (ω)  ,T  ω,ξ(ω)  + d  f  ω,ξ(ω)  ,T  ω,ξ n (ω)  . (2.34) Thus d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  ≤ d  f  ω,ξ(ω)  , f  ω,ξ n+1 (ω)  + α(ω)max  d  f  ω,ξ n (ω)  , f  ω,ξ(ω)  ,·,·,  1 2  [· + ·]  + β(ω)max  d  f  ω,ξ n (ω)  , f  ω,ξ n+1 (ω)  ,d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  + γ(ω)  d  f  ω,ξ n (ω)  ,T  ω,ξ(ω)  + d  f  ω,ξ(ω)  , f  ω,ξ n+1 (ω)  . (2.35) Taking the limit as n →∞,weget d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  ≤ α(ω)d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  + β(ω)d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  + γ(ω)d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  =  1 − γ(ω)  d  f  ω,ξ(ω)  ,T  ω,ξ(ω)  . (2.36) Hence d( f (ω,ξ(ω)),T(ω,ξ(ω))) = 0, as 1 − γ(ω) < 1forallω ∈ Ω.Hence,asT(ω,ξ(ω)) is closed, f  ω,ξ(ω)  ∈ T  ω,ξ(ω)  ∀ ω ∈ Ω. (2.37)  Ljubomir B. ´ Ciri ´ cetal. 9 Remark 2.2. If in Theorem 2.1, f (ω,x) = x for all (ω,x) ∈ Ω × X,thenwegetthefollow- 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 all x ∈ X,and H  T(ω,x),T(ω, y)  ≤ α(ω)max  d(x, y),d  x, T(ω,x)  ,d  y,T(ω, y)  ,  1 2   d  x, T(ω, y)  + d  y,T(ω,x)   + β(ω)max  d  x, T(ω,x)  ,d  y,T(ω, y)  + γ(ω)  d  x, T(ω, y)  + d  y,T(ω,x)  (2.38) for every x, y ∈ X,whereα,β,γ : Ω → (0,1) are measurable mappings satisfying (1.2). Then there is a measurable mapping ξ : Ω → X such that ξ(w) ∈ T(w,ξ(w)) for all ω ∈ 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 → X and T : Ω × X → CB(X) be two mappings satisfying the con- ditions (i) and (ii) in Theorem 2.1.If f (ω × X) = X for each ω ∈ Ω and f and T satisfy the following condition: H  T(ω,x),T(ω, y)  ≤ λ(ω)max  d  f (ω, x), f (ω, y)  ,d  f (ω, x),T(ω, x)  ,d  f (ω, y),T(ω, y)  , d  f (ω, x),T(ω, y)  + d  f (ω, y  ,T(ω,x)  2  , (2.39) where λ : Ω → (0,1) is a measurable function, then there is a measurable mapping ξ : Ω → X such that f (ω,ξ(w)) ∈ T(w, ξ(w)) for all ω ∈ Ω. Proof. It is clear that if f and T satisfy (2.39), then f and T satisfy (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 in Corollary 2.4, f (ω,x) = x for all (ω,x) ∈ Ω × X,thenweobtainthe corresponding theorems of Had ˇ zi ´ c[5] and Papageorgiou [13]. Finally, we give a simple example which shows that Theorem 2.1 and Corollaries 2.3 and 2.4 are actually an improvement of the results of Kubiak [10] and Papageorgiou [13]. Example 2.6. Let (Ω,Σ) be any measurable space and let K ={0,1,2,4,6} be the subset of the real line. Let the mappings f : Ω × K → K and T : Ω × K → K be defined such that 10 On random coincidence and fixed points for each ω ∈ 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 and T do not satisfy the contractive-type condition (2.39). Indeed, for x = 1and y = 2, we have d  T(ω,1),T(ω,2)  = 2 >λ(ω)max   4 − 6,4 − 2, 6 − 4, 0+ 6 − 2 2  = 2λ(ω) (2.42) for any λ(ω) < 1. On the other hand, d  T(ω,1),T(ω,2)  = 4 5 · 2+ 1 10 · 2+ 1 20 (4 + 0). (2.43) Thus, for x = 1andy = 2, f and T satisfy (1.2)withα(ω) = 4/5, β(ω) = 1/10, and γ(ω) = 1/20. It is easy to show that f and T satisfy (1.2)forallx, y ∈ K, with the same α(ω), β(ω), and γ(ω). Also, the rest of assumptions of Theorem 2.1 is satisfied and for ξ(ω) = 4we have f  ω,ξ(ω)  = 0 = T  ω,ξ(ω)  . (2.44) Note that T does not satisfy (2.38) either, as for instance, for x = 0andy = 2, we have α(ω)max   0 − 2,0 − 1, 2 − 4, 0 − 4 + 2 − 1 2  + β(ω)max   0 − 1,2 − 4  + γ(ω)   0 − 4 + 2 − 1  = 5 2 α(ω)+2β(ω)+5γ(ω) < 3  α(ω)+β(ω)+2γ(ω)  = 3 = d  T(ω,0),T(ω,2)  . (2.45) Remark 2.7. Corollary 2.4 is a stochastic generalization and improvement of the corre- sponding fixed point theorems for contractive-ty pe multivalued mappings of ´ Ciri ´ c[2], ´ Ciri ´ candUme[4], Kubiaczyk [9], Kubiak [10], Papageorgiou [14], and several other au- thors. Also Theorem 2.1 generalizes and extends the corresponding fixed point theorems for nonexpansive-type single-valued mappings of ´ Ciri ´ c[1]andRhoades[15]. 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