Research Article On Common Random Fixed Points of a New Iteration with Errors for Nonself Asymptotically Quasi-Nonexpansive Type Random Mappings

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1 Iteratioal Joural of Aalysis Volume 2013, Article ID , 10 pages Research Article O Commo Radom Fixed Poits of a New Iteratio with Errors for Noself Asymptotically Quasi-Noexpasive Type Radom Mappigs R. A. Rashwa, 1 P. K. Jhade, 2 ad Dhekra Mohammed Al-Baqeri 1 1 Departmet of Mathematics, Uiversity of Assiut, Assiut 71516, Egypt 2 Departmet of Mathematics, NRI Istitute of Iformatio Sciece & Techology, Bhopal, Madhya Pradesh , Idia Correspodece should be addressed to P. K. Jhade; pmathsjhade@gmail.com Received 1 November 2012; Accepted 11 March 2013 Academic Editor: Stefa Kuis Copyright 2013 R. A. Rashwa et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We prove some strog covergece of a ew radom iterative scheme with errors to commo radom fixed poits for three ad the N oself asymptotically quasi-oexpasive-type radom mappigs i a real separable Baach space. Our results exted ad improve the recet results i Kiziltuc, 2011, Thiawa, 2008, Deg et al., 2012, ad Zhou ad Wag, 2007 as well as may others. 1. Itroductio ad Prelimiaries The theory of radom operators is a importat brach of probabilistic aalysis which plays a key role i may applied areas. The study of radom fixed poits forms a cetral topic i this area. Research of this directio was iitiated by Prague School of Probabilistic i coectio with radom operator theory [1 3]. Radom fixed poit theory has attracted much attetio i recet times sice the publicatio of the survey article by Bharucha-Reid [4] i 1976, i which the stochastic versios of some well-kow fixed poit theorems were proved. A lot of efforts have bee devoted to radom fixed poit theory ad applicatios (e.g. see [5 10]admayothers). Let (Ω, Σ) be a measurable space, C aoemptysubsetof a separable Baach space E.Amappigξ:Ω Cis called measurable if ξ 1 (B C) Σ for every Borel subset B of E. AmappigT:Ω C Cis said to be radom mappig if for each fixed x C,themappigT(, x) : Ω C is measurable. Ameasurablemappigξ : Ω C is called a radom fixed poit of the radom mappig T : Ω C C if T(w, ξ(w)) = ξ(w) for each w Ω. Throughout this paper, we deote the set of all radom fixed poits of radom mappig T by RF(T) ad by T (w, x) for the th iterate T(w,T(,...T(w,x)))of T. The class of asymptotically oexpasive mappigs is a atural geeralizatio of the importat class of oexpasive mappigs. Goebel ad Kirk [11]proved that if C is a oempty closed ad bouded subset of a uiformly covex Baach space, the every asymptotically oexpasive self-mappig has a fixed poit. Iterative techiques for asymptotically oexpasive selfmappigs i Baach spaces icludig Ma type ad Ishikawa type iteratio processes have bee studied extesively by various authors (e.g. see [12 15]). The strog ad weak covergeces of the sequece of Ma iterates to a fixed poit of quasi-oexpasive mappigs were studied by Petryshy ad Williamso [16]. Subsequetly, the covergece of Ishikawa iterates of quasioexpasive mappigs i Baach spaces were discussed by Ghosh ad Debath [17]. The previous results ad some obtaied ecessary ad sufficiet coditios for Ishikawa iterativesequecetocovergeafixedpoitforasymptotically quasi-oexpasive mappigs were exteded by Liu [18, 19]. I 2000, Noor [20] itroduced a three-step iterative scheme ad studied the approximate solutios of variatioal iclusio i Hilbert spaces. Xu ad Noor [21] itroduced ad studied a three-step iterative scheme for asymptotically oexpasive mappigs, ad they proved weak ad strog covergeces theorems for asymptotically oexpasive mappigs i Baach spaces. I 2005, Suatai [22] defied

2 2 Iteratioal Joural of Aalysis a ew three-step iteratio, which is a extesio of Noor iteratios, ad gave some weak ad strog covergeces theorems of such iteratios for asymptotically oexpasive mappigs i uiformly covex Baach spaces. For oself oexpasive mappigs, some authors (e.g., see [23 27]) have studied the strog ad weak covergeces theorems i Hilbert space or uiformly covex Baach spaces. AsubsetC of E is said to be a retract of E if there exists a cotiuous map P:E Csuch that Px = x for all x C. Every closed covex subset of uiformly covex Baach space is a retract. A map P:E Eis a retractio if P 2 =P.It follows that if a map P is a retractio, the Py = y for all y i the rage of P. The cocept of oself asymptotically oexpasive mappigs was itroduced by Chidume et al. [28] i 2003 as the geeralizatio of asymptotically oexpasive selfmappigs. They studied the followig iteratio process: x 1 C, x +1 =P((1 α )x +α T(PT) 1 x ), (1) where T : C E is a asymptotically oexpasive oself mappig, {α } is a real sequece i (0, 1), adpis a oexpasive retractio from E to C. Wag [29] geeralized the result of Chidume et al. [28] ad got some ew results. He defied ad studied the followig iteratio process: x +1 =P((1 α )x +α T 1 (PT 1 ) 1 y ), y =P((1 β )x +β T 2 (PT 2 ) 1 x ), x 1 C, 1, (2) where T 1,T 2 : C E are asymptotically oexpasive oself mappigs ad {α }, {β } are real sequeces i [0, 1). Now, we itroduce the followig cocepts for oself mappigs Defiitio 1 (see [28, 30, 31]). Let C be a oempty subset of arealseparablebaachspaceadt:ω C Eaoself radom mappig. The, T is said to be (1) oexpasive radom operator if for arbitrary x, y C, T(w, x) T(w, y) x y,forallw Ω; (2) oself asymptotically oexpasive radom mappig if there exists a sequece of measurable fuctios r (w) : Ω [1, ) with lim r (w) = 1 for each w Ωsuchthatfor arbitrary x, y C, T(PT) 1 (w, x) T(PT) 1 (w, y) r (w) x y, w Ω, 1; (3) (3) oself asymptotically quasi-oexpasive radom mappig if RF(T) =φad there exists a sequece of measurable fuctios r (w) : Ω [1, ) with lim r (w) = 1 for each w Ωsuch that T(PT) 1 (w, η (w)) ξ(w) r (w) η (w) ξ(w), w Ω, 1, (4) where ξ(w) : Ω C is a radom fixed poit of T ad η(w) : Ω C is ay measurable mappig; (4) oself asymptotically oexpasive-type radom mappig if lim sup { sup { x,y C T(PT) 1 (w, x) T(PT) 1 2 (w, y) x y 2 }} 0, w Ω, 1; (5) Noself asymptotically quasi-oexpasive-type radom mappig if RF(T) =φ,ad lim sup { sup { ξ(w) F T(PT) 1 2 (w, η(w)) ξ(w) η (w) ξ(w) 2 }} 0, w Ω, 1, (5) (6) where ξ(w) : Ω C is a radom fixed poit of T ad η(w) : Ω C is ay measurable mappig. Remark 2. (1) If T:Ω C Eis a oself asymptotically oexpasive radom mappig, the T is a oself asymptotically oexpasive-type radom mappig. (2) If RF(T) =φ ad T : Ω C E is a oself asymptotically quasi-oexpasive radom mappig, the T is a oself asymptotically quasi-oexpasive-type radom mappig. (3) If RF(T) =φ ad T : Ω C E is a oself asymptotically oexpasive-type radom mappig, the T is a oself asymptotically quasi-oexpasive-type radom mappig. Remark 3. Observe that for ay measurable mappig η(w) : Ω Cad ξ(w) F,wehave which implies lim sup { sup { ξ(w) F T(PT) 1 2 (w, η(w)) ξ(w) η (w) ξ(w) 2 }} 0, lim sup { sup {( ξ(w) F T(PT) 1 (w, η (w)) ξ(w) η (w) ξ(w) ) ( T(PT) 1 (w, η (w)) ξ(w) + η (w) ξ(w) )}} 0. (7) (8)

3 Iteratioal Joural of Aalysis 3 Therefore, lim sup { sup {( ξ(w) F T(PT) 1 (w, η (w)) ξ(w) (9) η (w) ξ(w) )}} 0. I [25],Shahzadstudiedthefollowigiterativesequeces: x +1 = P ((1 α )x +α TP [(1 β )x +β Tx ]), x 1 C, 1, (10) where T : C E is a oexpasive oself mappig, C is a oempty closed covex oexpasive retract of a real uiformly covex Baach space E with P beig a oexpasive retractio from E to C,ad{α }, {β } are real sequeces i [0, 1). Recetly, Thiawa [32] geeralized the iteratio process (10) as follows: x +1 =P((1 α γ )x +α TP [(1 β )y +β Ty ]+γ u ), y =P((1 α γ )x +α TP [(1 β )x +β Tx ]+γ V ), x 1 C, 1, (11) where {α }, {β }, {γ }, {α }, {β }, {γ } are appropriate sequeces i [0, 1) ad {u }, {V } are bouded sequeces i C. He proved weak ad strog covergeces theorems for oexpasive oself mappigs i uiformly covex Baach spaces. I 2011, Kiziltuc [33] studied the strog covergece to a commo fixed poit of a ew iterative scheme for two oself asymptotically quasi-oexpasive-type mappigs i Baach spaces defied as follows: The iterative scheme is defied as follows: x +1 =α 1 x +β 1 (PT 1 ) y +γ 1 (PT 2 ) y, y =α 2 x +β 2 (PT 1 ) z +γ 2 (PT 2 ) z, z =α 3 x +β 3 (PT 1 ) x +γ 3 (PT 2 ) x, (13) where {α i }, {β i }, {γ i } (i = 1, 2, 3) are appropriate sequeces i [a, 1 a] for some a (0, 1) satisfyig α i +β i +γ i = 1 (i = 1, 2, 3). For radom operators, Beg ad Abbas [30] studied the differet radom iterative algorithms for weakly cotractive ad asymptotically oexpasive radom operators o arbitrary Baach space. They also established covergece of a implicit radom iterative process to a commo fixed poit for a fiite family of asymptotically quasi-oexpasive operators. Plubtieg et al. [35, 36] studied weak ad strog covergeces theorems for a modified radom Noor iterative scheme with errors for three asymptotically oexpasive self-mappigs i Baach space defied as follows: ξ +1 (w) =α T 1 (w, η (w))+β ξ (w) +γ f (w), η (w) =α T 2 (w, ζ (w))+β ξ (w) +γ f (w), ζ (w) =α T 3 (w, ξ (w))+β ξ (w) +γ f (w), 1, w Ω, (14) where T 1,T 2,T 3 : Ω C C are three asymptotically oexpasive radom self-mappigs, ξ 1 : Ω C is a arbitrary give measurable mappig from Ω to C, {f (w)}, {f (w)}, {f (w)} are bouded sequece of measurable fuctios from Ω to C, ad{α }, {α }, {α }, {β }, {β }, {β }, {γ }, {γ }, {γ } are sequeces of real umbers i [0, 1] with α +β +γ =α +β +γ =α +β +γ =1. Remark 4. If T 1 =T 2 =T 3 =Tad γ =γ =γ =0,the (14) becomes as follows: ξ +1 (w) =α T (w, η (w))+β ξ (w), η (w) =α T (w, ζ (w))+β ξ (w), x +1 =P((1 a )x +a S(PS) 1 ((1 α )y +α S(PS) 1 y )), y =P((1 b )x +b T(PT) 1 ((1 β )x +β T(PT) 1 x )), x 1 C, 1, (12) ζ (w) =α T 3 (w, ξ (w))+β ξ (w), 1, w Ω, which was studied by Beg ad Abbas i [30]. (15) For oself radom mappigs, Zhou ad Wag [37] studied the approximatio of the followig iteratio process: ξ +1 (w) =P((1 α )ξ (w) where {a }, {b }, {α }, {β } are appropriate sequeces i [0, 1). More recetly, Deg et al. [34] obtaied the strog ad weak covergeces theorems for commo fixed poits of two oself asymptotically oexpasive mappigs i Baach spaces. +α T(PT) 1 (w, η (w))), η (w) =P((1 β )ξ (w) +β T(PT) 1 (w, ξ (w))), 1, w Ω, (16)

4 4 Iteratioal Joural of Aalysis where T:Ω C Eis a asymptotically oexpasive oself radom mappig, ξ 1 : Ω C is a arbitrary give measurable mappig from Ω to C, {α }, {β } are sequeces i [0, 1],adP is a oexpasive retractio from E to C. Saluja [38] ad may other authors exteded the results of Zhou ad Wag [37] by studyig multistep radom iteratio scheme with errors for commo radom fixed poit of a fiite family of oself asymptotically oexpasive radom mappigirealuiformlyseparablebaachspaces. Ispired ad motivated by [32 34, 37] adothers, we itroduced a ew three-step ad N-step radom iterative scheme with errors for asymptotically quasi-oexpasivetype oself radom mappigs i a separable Baach space. Some strog covergeces theorems are established for these ew radom iterative schemes with errors i separable Baach space. The iterative scheme for three oself radom mappigs is defied as follows. Defiitio 5. Let T 1,T 2,T 3 :Ω C Cbe three oself radom mappigs, where C is a oempty closed covex subset of a separable Baach space E, adp:e Cis a oexpasive retractio of E oto C. Letξ 1 (w) : Ω C be a measurable mappig. Suppose that {ξ (w)} is geerated iteratively by ξ 1 (w) C,havig ξ +1 (w) =P[(1 a σ )ξ (w) +a T 1 (PT 1 ) 1 (w,(1 α )η (w) +α T 1 (PT 1 ) 1 (w, η (w))) +σ f (w)], η (w) = P [(1 b δ )ξ (w) +b T 2 (PT 2 ) 1 (w,(1 β )ζ (w) +β T 2 (PT 2 ) 1 (w, ζ (w))) + δ g (w)], ζ (w) =P[(1 c λ )ξ (w) +c T 3 (PT 3 ) 1 (w,(1 γ )ξ (w) +γ T 3 (PT 3 ) 1 (w, ξ (w))) + λ h (w) ], (17) for all 1, w Ω,where{a }, {b }, {c }, {α }, {β }, {γ }, {σ }, {δ },ad{λ } are sequeces i [0, 1] such that a +σ 1, b +δ 1, c +λ 1,ad{f (w)}, {g (w)}, {h (w)} areboudedsequecesofmeasurablefuctiosfromω to C for all w Ω. Defiitio 5 ca be exteded to N oself radom mappigs as follows. Defiitio 6. Let T 1,T 2,...,T N :Ω C Cbe N oself radom mappigs, where C is a oempty closed covex subset of a separable Baach space E, adp:e Cis a oexpasive retractio of E oto C.Letξ 1 (w) : Ω C be a measurable mappig. Defie sequeces fuctio {ξ (N) (w)}, {ξ (N 1) (w)},...,{ξ (1) (w)} i C as follows: ξ +1 (w) =ξ (N) (w) = P [(1 a (N) σ (N) (w,(1 α (N) ξ (N 1) (w) = P [(1 a (N 1) = = )ξ (w) +a (N) T N (PT N ) 1 )ξ (N 1) (PT N ) 1 (w, ξ (N 1) σ (N 1) )ξ (w) +a (N 1) T N 1 (PT N 1 ) 1 (w,(1 α (N 1) +α (N 1) +σ (N 1) f (N 1) ], (w) +α (N) T N )ξ (N 2) (w) (w)))+σ (N) f (N) ], T N 1 (PT N 1 ) 1 (w, ξ (N 2) (w))) ξ (1) (w) = P [(1 a (1) σ (1) )ξ (w) +a (1) T 1(PT 1 ) 1 (w,(1 α (1) )ξ (w) +α (1) T 1(PT 1 ) 1 (w,ξ (w)))+σ (1) f(1) ], 1, w Ω, (18) where {a (i) }, {α(i) },ad{σ(i) } (i = 1,2,...,N)are sequeces i [0, 1] such that a (i) +σ(i) 1,forall(i = 1,2,...,N),ad {f (i) } (i=1,2,...,n)are bouded sequeces of measurable fuctios from Ω to C for all w Ω. The followig lemma is useful for provig our results. Lemma 7 (see [39]). Let {a }, {b } ad {m } be oegative real sequeces satisfyig a +1 (1+m )a +b, 1. (19) If =1 m < ad =1 b <,the (1) lim a exists; (2) lim a =0wheever lim if a =0. 2. Mai Results I this sectio, we will first prove the strog covergece of the iterative scheme (17) to a commo radom fixed poit for three asymptotically quasi-oexpasive-type oself radom mappigs i a separable Baach space. The, we exted theobtaiedresultsto N asymptotically quasi-oexpasivetype oself radom mappigs by usig the iterative scheme (18). Fially, we use Theorem 8 ad Coditio (A) [40] to obtai a covergeces theorem for scheme (17).

5 Iteratioal Joural of Aalysis 5 Theorem 8. Let E be a real separable Baach space ad C a oempty closed covex subset of E with P beig a oexpasive retractio. Let T i : Ω C E, i = 1,2,3,be three asymptotically quasi-oexpasive-type oself radom mappigs with F= 3 i=1 RF(T i) =φ,forallw Ω.Suppose that {ξ (w)}, {η (w)} ad {ζ (w)} are the sequeces defied as i (17) where {a }, {b }, {c }, {α }, {β }, {γ }, {σ }, {δ }, ad {λ } are sequeces i [0, 1] such that a +σ 1,b + δ 1, c +λ 1 ad {f (w)}, {g (w)}, {h (w)} are bouded sequeces of measurable fuctios from Ω to C with the followig restrictios: =1 σ <, =1 δ <,ad =1 λ <.The,{ξ (w)} coverge to a commo radom fixed poit of T 1, T 2,adT 3 if ad oly if lim if (w),f)=0, w Ω. (20) Proof. The ecessity of (20)isobvious. Next, we prove the sufficiecy of (20). Let ξ(w) F = 3 i=1 RF(T i); by the boudedess of the sequeces of measurable fuctios {f (w)}, {g (w)}, {h (w)}, weputfor each w Ω, M (w) = max { sup f (w) ξ(w) 1,ξ F sup g (w) ξ(w) 1,ξ F sup h (w) ξ(w) }. 1,ξ F (21) The, M(w) < for each w Ω. Sice ξ(w) F ad η(w) : Ω C is ay measurable mappig, we have lim sup { sup { ξ(w) F T i(pt i ) 1 (w, η (w)) ξ(w) { T 2(PT 2 ) 1 (w, ζ (w)) ξ(w) ζ (w) ξ(w) } ε, 0, ξ F, (25) { T 3(PT 3 ) 1 (w, ξ (w)) ξ(w) ξ (w) ξ(w) } ε, Settig for w Ω, 0, ξ F. (26) μ (w) =(1 α )η (w) +α T 1 (PT 1 ) 1 (w, η (w)), V (w) =(1 β )ζ (w) +β T 2 (PT 2 ) 1 (w, ζ (w)), τ (w) =(1 γ )ξ (w) +γ T 3 (PT 3 ) 1 (w, ξ (w)). (27) Thus, for ξ(w) F ad w Ω,usig(17)ad(24), we have ξ +1 (w) ξ(w) = P[(1 a σ )ξ (w) +a T 1 (PT 1 ) 1 (w,μ (w))+σ f (w) ] ξ(w) (1 a σ )ξ (w) +a T 1 (PT 1 ) 1 (w, (1 α )η (w) +α T 1 (PT 1 ) 1 (w, η (w))) +σ f (w) ξ(w) = (1 a σ )ξ (w) +a ξ (w) +σ ξ (w) ξ(w) +a (T 1 (PT 1 ) 1 (w, μ (w)) ξ(w)) +σ (f (w) ξ(w)) η (w) ξ(w) }} 0, i=1,2,3. (22) (1 a σ ) ξ (w) ξ(w) +a T 1(PT 1 ) 1 (w, μ (w)) ξ(w) It follows that for ay give ε > 0, there exists a positive iteger 0 such that for 0 ad ξ(w) F,wehave sup { ξ(w) F T i(pt i ) 1 (w, η (w)) ξ(w) (23) η (w) ξ(w) } ε, i=1,2,3. Sice {ξ (w)}, {η (w)}, ad{ζ (w)} E, thewehavefor w Ω, { T 1(PT 1 ) 1 (w, η (w)) ξ(w) η (w) ξ(w) } ε, 0, ξ F, (24) +σ f (w) ξ(w) (1 a σ ) ξ (w) ξ(w) +a [ T 1(PT 1 ) 1 (w, μ (w)) ξ(w) μ (w) ξ(w) ] +a μ (w) ξ(w) +σ M (w) (1 a σ ) ξ (w) ξ(w) +a ε +a μ (w) ξ(w) +σ M (w). (28)

6 6 Iteratioal Joural of Aalysis I additio, by (24), we obtai μ (w) ξ(w) = (1 α )η (w) +α T 1 (PT 1 ) 1 (w, η (w)) ξ(w) (1 α ) η (w) ξ(w) +α T 1(PT 1 ) 1 (w, η (w)) ξ(w) (1 α ) η (w) ξ(w) +α ε+α η (w) ξ(w) = η (w) ξ(w) +α ε. (29) Agai usig (17)ad(25), we have η (w) ξ(w) = P[(1 b δ )ξ (w) +b T 2 (PT 2 ) 1 (w, V (w)) +δ g (w) ] ξ(w) (1 b δ )ξ (w) +b T 2 (PT 2 ) 1 (w,(1 β )ζ (w) +β T 2 (PT 2 ) 1 (w, ζ (w))) +δ g (w) ξ(w) = (1 b δ )ξ (w) +b ξ (w) +δ ξ (w) ξ(w) +b (T 2 (PT 2 ) 1 (w, V (w)) ξ(w)) +δ (g (w) ξ(w)) (1 b δ ) ξ (w) ξ(w) +b T 2(PT 2 ) 1 (w, V (w)) ξ(w) +δ g (w) ξ(w) (1 b δ ) ξ (w) ξ(w) +b ε +b V (w) ξ(w) +δ M (w). (30) I additio, by (25), we have V (w) ξ(w) = (1 β )ζ (w) +β T 2 (PT 2 ) 1 (w, ζ (w)) ξ(w) (1 β ) ζ (w) ξ(w) +β T 2(PT 2 ) 1 (w, ζ (w)) ξ(w) (1 β ) ζ (w) ξ(w) +β ε+β ζ (w) ξ(w) = ζ (w) ξ(w) +β ε. (31) Also, by (17)ad(26), we have ζ (w) ξ(w) = P[(1 c λ )ξ (w) +c T 3 (PT 3 ) 1 (w,τ (w))+λ h (w) ] ξ(w) (1 c λ )ξ (w) +c T 3 (PT 3 ) 1 (w,(1 γ )ξ (w) +γ T 3 (PT 3 ) 1 (w, ξ (w))) +λ h (w) ξ(w) = (1 c λ )ξ (w) +c ξ (w) +λ ξ (w) ξ(w) +c (T 3 (PT 3 ) 1 (w, τ (w)) ξ(w)) +λ (h (w) ξ(w)) (1 c λ ) ξ (w) ξ(w) +c T 3(PT 3 ) 1 (w, τ (w)) ξ(w) +λ h (w) ξ(w) (1 c λ ) ξ (w) ξ(w) +c ε +c τ (w) ξ(w) +λ M (w). I additio, by (26), we have τ (w) ξ(w) (32) = (1 γ )ξ (w) +γ T 3 (PT 3 ) 1 (w, ξ (w)) ξ(w) (1 γ ) ξ (w) ξ(w) +γ T 3(PT 3 ) 1 (w, ξ (w)) ξ(w) (1 γ ) ξ (w) ξ(w) +γ ε+γ ξ (w) ξ(w) = ξ (w) ξ(w) +γ ε. (33) Substitutig (29), (30), (31), (32), ad (33) ito(28) ad simplifyig, we obtai ξ +1 (w) ξ(w) (1 a σ ) ξ (w) ξ(w) +a ε +a μ (w) ξ(w) +σ M (w) (1 a σ ) ξ (w) ξ(w) +a ε +a [ η (w) ξ(w) +α ε] + σ M (w) =(1 a σ ) ξ (w) ξ(w) +a ε +a η (w) ξ(w) +a α ε+σ M (w)

7 Iteratioal Joural of Aalysis 7 (1 a σ ) ξ (w) ξ(w) +a ε +a [(1 b δ ) ξ (w) ξ(w) +b ε +b V (w) ξ(w) +δ M (w)] +a α ε+σ M (w) =(1 a σ ) ξ (w) ξ(w) +a ε +a (1 b δ ) ξ (w) ξ(w) +a b ε +a b V (w) ξ(w) +a δ M (w) +a α ε+σ M (w) =(1 σ a b a δ ) ξ (w) ξ(w) +a ε+a b ε+a b V (w) ξ(w) +a δ M (w) +a α ε+σ M (w) (1 σ a b a δ ) ξ (w) ξ(w) +a ε+a b ε+a b [ ζ (w) ξ(w) +β ε] +a δ M (w) +a α ε+σ M (w) =(1 σ a b a δ ) ξ (w) ξ(w) +a ε+a b ε+a b ζ (w) ξ(w) +a b β ε+a δ M (w) +a α ε+σ M (w) (1 σ a b a δ ) ξ (w) ξ(w) +a ε +a b ε+a b (1 c λ ) ξ (w) ξ(w) +a b c ε+a b c τ (w) ξ(w) +a b λ M (w) +a b β ε+a δ M (w) +a α ε+σ M (w) =(1 σ a δ a b c a b λ ) ξ (w) ξ(w) +a ε+a b ε =(1 σ a δ a b λ ) ξ (w) ξ(w) +a ε+a b ε+a b c ε+a b c γ ε +a b λ M (w) +a b β ε+a δ M (w) +a α ε+σ M (w) =(1 σ a δ a b λ ) ξ (w) ξ(w) +[a +a b +a b c +a b c γ +a b β +a α ]ε +[a b λ +a δ +σ ]M(w) ξ (w) ξ(w) +6ε+(λ +δ +σ )M(w). (34) Let R (w) = 6ε + (λ +δ +σ )M(w);the, =1 R (w) < for all w Ω. It follows by (34)that if ξ(w) F ξ +1 (w) ξ(w) if ξ(w) F ξ (w) ξ(w) +R (w), 0, w Ω. (35) From (35)ad =1 R (w) < for all w Ω,wehave d (ξ +1 (w),f) d(ξ (w),f) +R (w), w Ω. (36) By Lemma 7 ad (36), it follows that lim d(ξ (w), F) exists for all ξ(w) F = 3 i=1 RF(T i) ad w Ω. Sice lim if d(ξ (w), F) = 0,thewehave lim d(ξ (w),f)=0, w Ω. (37) Next, we prove that ξ (w) is a Cauchy sequece i E for each w Ω. For 0, m 1,adξ(w) F,wehaveby(35)that ξ +m (w) ξ(w) ξ +m 1 (w) ξ(w) +R +m 1 (w) ξ +m 2 (w) ξ(w) +R +m 1 (w) +R +m 2 (w). +a b c ε+a b c τ (w) ξ(w) +a b λ M (w) +a b β ε+a δ M (w) +a α ε+σ M (w) (1 σ a δ a b c a b λ ) ξ (w) ξ(w) +a ε+a b ε+a b c ε+a b c ξ (w) ξ(w) +a b c γ ε+a b λ M (w) +a b β ε+a δ M (w) +a α ε+σ M (w) +m 1 ξ (w) ξ(w) + R k (w). k= Therefore, by usig (38), we have for each w Ω, ξ +m (w) ξ (w) ξ +m (w) ξ(w) + ξ (w) ξ(w) 2 ξ (w) ξ(w) + R k (w). k= (38) (39)

8 8 Iteratioal Joural of Aalysis Sice ξ(w) F ad by (39), we have for each w Ω, ξ +m (w) ξ (w) 2d(ξ (w),f)+ k= R k (w), 0. (40) Sice lim d(ξ (w), F) = 0 ad =1 R (w) <, for give ε>0, there exists a positive iteger 1 0 such that d(ξ (w), F) < ε/4 ad =1 R (w) < ε/2.wehave or ξ +m (w) ξ (w) <ε, w Ω, (41) lim ξ +m (w) ξ (w) =0, w Ω; (42) this shows that ξ (w) is a Cauchy sequece i C for each w Ω. Sice E is complete ad C is a closed subset of E ad so it is complete, the there exists a p(w) C such that ξ (w) p(w) as,for all w Ω. Now, we show that p(w) F. By cotradictio, we assume that p(w) does ot belog to F.SiceF is closed set, d(p(w), F) > 0. By usig the fact that lim d(ξ (w), F) = 0,itfollowsthatforallξ(w) F, This implies that p (w) ξ(w) (43) p (w) ξ (w) + ξ (w) ξ(w). d(p(w),f) p (w) ξ (w) +d(ξ (w),f) 0 (as ), which is a cotradictio. Hece, p(w) F. (44) Corollary 9. Suppose that the coditios i Theorem 8 are satisfied. The the radom iterative sequece ξ (w) geerated by (17) coverges to a commo radom fixed poit ξ(w) if ad oly if for all w Ω, there exists a subsequece ξ j (w) of ξ (w) which coverges to ξ(w). Theorem 10. Let E be a real separable Baach space ad C a oempty closed covex subset of E with P as a oexpasive retractio. Let T i : Ω C E, i = 1,2,3,bethree asymptotically quasi-oexpasive oself radom mappigs with F = 3 i=1 RF(T i) =φ,forallw Ω.Supposethat {ξ (w)}, {η (w)}, ad{ζ (w)} are the sequeces defied as i (17) where {a }, {b }, {c }, {α }, {β }, {γ }, {σ }, {δ },ad {λ } are sequeces i [0, 1] such that a +σ 1, b + δ 1, c +λ 1,ad{f (w)}, {g (w)}, {h (w)} are bouded sequeces of measurable fuctios from Ω to C with the followig restrictios: =1 σ <, =1 δ < ad =1 λ <.The,{ξ (w)} coverge to a commo radom fixed poit of T 1, T 2,adT 3 if ad oly if lim if d (ξ (w),f) =0, w Ω. (45) Proof. Sice T i :Ω C E,i=1,2,3, are three asymptotically quasi-oexpasive oself radom mappigs, by Remark 2, they are asymptotically quasi-oexpasive-type oself radom mappigs the coclusio of Theorem 10 ca be proved from Theorem 8 immediately. Theorem 11. Let E be a real separable Baach space ad C be a oempty closed covex subset of E with P as a oexpasive retractio. Let T i :Ω C E, i=1,2,3,be three asymptotically oexpasive oself radom mappigs with F = 3 i=1 RF(T i) =φ,forallw Ω.Supposethat {ξ (w)}, {η (w)} ad {ζ (w)} are the sequeces defied as i (17) where {a }, {b }, {c }, {α }, {β }, {γ }, {σ }, {δ },ad {λ } are sequeces i [0, 1] such that a +σ 1, b + δ 1, c +λ 1,ad{f (w)}, {g (w)}, {h (w)} are bouded sequeces of measurable fuctios from Ω to C with the followig restrictios: =1 σ <, =1 δ <,ad =1 λ <.The,{ξ (w)} coverge to a commo radom fixed poit of T 1, T 2,adT 3 if ad oly if lim if d(ξ (w),f)=0, w Ω. (46) Proof. Sice T i : Ω C E, i = 1,2,3, are three asymptotically oexpasive oself radom mappigs, by Remark 2, they are asymptotically oexpasive-type oself radom mappigs, ad therefore they are asymptotically quasi-oexpasive-type oself radom mappigs; the coclusio of Theorem 11 ca be obtaied from Theorem 8 immediately. Now, we ca exted ad geeralize Theorems 8, 10, ad 11 by usig radom iterative scheme (18) as follows. Theorem 12. Let E be a real separable Baach space ad C a oempty closed covex subset of E with P as a oexpasive retractio. Let T i : Ω C E, i = 1,2,...,N,beN asymptotically quasi-oexpasive-type oself radom mappigs with F= N i=1 RF(T i) =φ,forallw Ω.Supposethat {ξ (w)} is the sequece defied as i (18) where {a (i) }, {α(i) }, ad {σ (i) } (i = 1,2,...,N) are sequeces i [0, 1] such that a (i) +σ (i) 1 for all i = 1,2,...,N ad {f (i) (w)} (i = 1,2,...,N) are bouded sequeces of measurable fuctios from Ω to C with the followig restrictios: =1 σ(i) <, for all (i = 1,2,...,N).The{ξ (w)} coverge to a commo radom fixed poit of T 1,T 2,...,T N if ad oly if lim if d(ξ (w),f)=0, w Ω. (47) Theorem 13. Let E be a real separable Baach space ad C be aoemptyclosedcovexsubsetofewith P as a oexpasive retractio. Let T i : Ω C E, i = 1,2,...,N be N asymptotically quasi-oexpasive oself radom mappigs with F = N i=1 RF(T i) =φ,forallw Ω.Supposethat {ξ (w)} be the sequece defied as i (18) where {a (i) }, {α(i) }, ad {σ (i) }, (i = 1,2,...,N) are sequeces i [0, 1] such that a (i) +σ (i) 1 for all i = 1,2,...,N ad {f (i) (w)}, (i = 1,2,...,N) are bouded sequeces of measurable fuctios from Ω to C with the followig restrictios: =1 σ(i) <

9 Iteratioal Joural of Aalysis 9 for all (i = 1,2,...,N).The{ξ (w)} coverge to a commo radom fixed poit of T 1,T 2,...,T N if ad oly if lim if d(ξ (w),f)=0, w Ω. (48) Theorem 14. Let E be a real separable Baach space ad C be a oempty closed covex subset of E with P as a oexpasive retractio. Let T i : Ω C E, i = 1,2,...,N be N asymptotically oexpasive oself radom mappigs with F= N i=1 RF(T i) =φ,forallw Ω.Supposethat{ξ (w)} is the sequece defied as i (18) where {a (i) }, {α(i) },ad{σ(i) } (i = 1,2,...,N) are sequeces i [0, 1] such that a (i) +σ (i) 1 for all i = 1,2,...,N ad {f (i) (w)} (i = 1,2,...,N) are bouded sequeces of measurable fuctios from Ω to C with the followig restrictios =1 σ(i) < for all (i=1,2,...,n). The, {ξ (w)} coverge to a commo radom fixed poit of T 1,T 2,...,T N if ad oly if lim if d(ξ (w),f)=0, w Ω. (49) Seter ad Dotso [40] defied Coditio (A) as follows. Defiitio 15 (see [40]). A mappig T : C C where C is a subset of a Baach space E with F(T) =φ is said to satisfy Coditio (A) if there exists a odecreasig fuctio f : [0, ) [0, ) with f(0) = 0, f(r) > 0, forall r (0, ) such that for all x C, x Tx f(d (x, F (T))), (50) where d(x, F(T)) = if{ x p : p F(T)}. As a applicatio, we ca apply Theorem 8 ad Coditio (A) to obtai a covergeces theorem for scheme (17). Theorem 16. Let E be a real uiformly separable Baach space ad C a oempty closed covex subset of E with P as a oexpasive retractio. Let T i : Ω C E, i = 1,2,3,be three asymptotically quasi-oexpasive-type oself radom mappigs with F= 3 i=1 RF(T i) =φ,forallw Ω.Suppose that {ξ (w)}, {η (w)} ad {ζ (w)} are the sequeces defied as i (17) where {a },{b }, {c }, {α }, {β }, {γ }, {σ }, {δ },ad {λ } are sequeces i [0, 1] such that a +σ 1, b + δ 1, c +λ 1 ad {f (w)}, {g (w)}, {h (w)} are bouded sequeces of measurable fuctios from Ω to C with the followig restrictios: =1 σ <, =1 δ <, ad =1 λ <.SupposeoeofthemappigsT i, i = 1, 2, 3, satisfyig Coditio (A) ad the followig coditio: lim ξ (w) T(w, ξ (w)) = 0, forallw Ω.The, {ξ (w)} coverge to a commo radom fixed poit of T 1,T 2, ad T 3. Proof. From Theorem 8, wehavelim ξ (w) ξ(w), ad lim d(ξ (w), F) exists. Let oe of the mappigs T i,sayt 1 satisfy Coditio (A) ad lim ξ (w) T 1 (w, ξ (w)) = 0;the,wehaveforallw Ω, lim f(d(ξ (w),f)) lim ξ (w) T 1 (w, ξ (w)) =0. (51) By the property of f ad sice lim d(ξ (w), F) exists, we have that lim d (ξ (w),f) =0. (52) By Theorem 8, {ξ (w)} coverge to a commo radom fixed poit of T 1, T 2,adT 3. Ackowledgmet The authors would like to exted their sicerest thaks to the aoymous referees ad editors for the exceptioal review of this work. The suggestios ad recommedatios i the report icreased the quality of their paper. Refereces [1] O. Haš, Reduzierede zufällige Trasformatioe, Czechoslovak Mathematical Joural,vol.7,pp ,1957. [2] O. Haš, Radom operator equatios, i Proceedig of the 4th Barkeley Symposium o Mathematical Statistics ad Probability, vol. 2, pp , Uiversity of Califoria Press, Berkeley, Calif, USA, [3] A. Špaček, Zufällige Gleichuge, Czechoslovak Mathematical Joural,vol.5,pp ,1955. [4] A. T. Bharucha-Reid, Fixed poit theorems i probabilistic aalysis, Bulleti of the America Mathematical Society,vol.82, o.5,pp ,1976. [5]I.Beg, Approximatioofradomfixedpoitsiormed spaces, Noliear Aalysis: Theory, Methods & Applicatios,vol. 51, o. 8, pp , [6] I. Beg, Miimal displacemet of radom variables uder Lipschitz radom maps, Topological Methods i Noliear Aalysis,vol.19,o.2,pp ,2002. [7] I. Beg ad N. Shahzad, Radom fixed poit theorems for oexpasive ad cotractive-type radom operators o Baach spaces, Joural of Applied Mathematics ad Stochastic Aalysis, vol. 7, o. 4, pp , [8] S. Itoh, Radom fixed-poit theorems with a applicatio to radom differetial equatios i Baach spaces, Joural of Mathematical Aalysis ad Applicatios, vol.67,o.2,pp , [9] N. S. Papageorgiou, Radom fixed poit theorems for measurable multifuctios i Baach spaces, Proceedigs of the America Mathematical Society,vol.97,o.3,pp ,1986. [10] H. K. Xu, Some radom fixed poit theorems for codesig ad oexpasive operators, Proceedigs of the America Mathematical Society,vol.110,o.2,pp ,1990. [11] K. Goebel ad W. A. Kirk, A fixed poit theorem for asymptotically oexpasive mappigs, Proceedigs of the America Mathematical Society,vol.35,pp ,1972. [12] J. Goricki, Weak covergece theorems for asymptotically oexpasive mappigs i uiformly covex Baach spaces, Commetatioes Mathematicae Uiversitatis Caroliae,vol.30, o. 2, pp , [13] M.O.OsilikeadS.C.Aiagbosor, Weakadstrogcovergece theorems for fixed poits of asymptotically oexpasive mappigs, Mathematical ad Computer Modellig, vol.32,o. 10, pp , 2000.

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