Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES Dr Neetu Vishwakarma a Dr M S Chauha Sagar Istitute of Research a Techology - Excellece, Bhopal (MP Iia, Email: eetuvishu@gmailcom Govt Nehru PG College Agar Malwa (MP Iia, Email: rmsc@reiffmailcom Abstract I this paper we obtai commo raom fixe poit theorems for weakly compatible raom operators uer geeralize cotractive coitio i symmetric space I this paper we geeralize the result of Beg a Abbas [4] Keywors : Symmetric space, weakly compatible, raom operators Mathematical subject classificatio (000 : 47H0, 54H5 Itrouctio : I recet years, the stuy of raom fixe poits have attracte much attetio, some of the recet literatures i raom fixe poit may be ote i [,, 3, 4, 5, 7, 9] I metric space some theorems ca be prove without usig some of the efiig properties of metric Hicks [6] establishe some commo fixe poit theorems i symmetric space Recetly Beg a Abbas [4] prove some raom fixe poit theorems for weakly compatible raom operator uer geeralize cotractive coitio i symmetric space Prelimiaries : Throughout this paper, ( Ω, Σ eotes a measurable space (Σ - sigma algebra A symmetric o a set X is a o - egative real value fuctio o X X such that for all x, y X we have (a (x, y = 0 if a oly if x = y a (b (x, y = (y, x Let be a symmetric o a set X For ε > 0 a x X, B(x, ε eotes the spherical ball cetre at x with raius ε, efie as the set { y X:(x,y < ε } A topology t ( o X is give by U t ( if a oly if for each x U, B (x, ε U for some ε > 0 Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 Note that lim (x, x = 0 iff x x i the topology t ( Let F be a subset of X A mappig ξ : Ω X is measurable if ξ (U Σ for each ope subset U of X The mappig T : Ω F F is a raom map if a oly if for each fixe x F, the mappig T (, x : F Ω is measurable The mappig T is cotiuous if for each the mappig T(ω, : F X is cotiuous A measurable mappig ξ: Ω X is a raom fixe poit of raom operator T: Ω F X if a oly if T(ω, ω = ω for each ω Ω We eote the set of raom fixe poits of a raom map T by RF (T a the set of all measurable mappig from Ω ito a symmetric space by M( Ω,X Let φ + + : R R be a fuctio satisfyig the coitio 0 ( t < t, < φ for each t > 0 Defiitio Raom operators S, T : Ω X X are sai to be commutative if S(ω, a T(ω, are commutative for each Defiitio [5] Let X be a Polish space, that is separable complete metric space Mappig f,g: X X are compatible if lim ( fg x, gfx 0, = provie that lim f (x a lim g(x exists i X a lim f (x = lim g (x Raom operators S, T : Ω X X are compatible if S (ω, a T (ω, are compatible for each Defiitio 3 Let X be a Polish space Raom operators S,T : Ω X X are weakly compatible if ω = ω for some ξ Μ(Ω, X the T(ω, S(ω, ω = S(ω, Τ(ω,ω for every Defiitio 4[8] Let { x } a { } y X The space X is sai to satisfy the followig axioms: (W lim ( x, x = lim ( x, y = 0 implies that x = y y be two sequeces i a symmetric space (X, a x, (W lim ( x, x = lim ( x, y = 0 implies that ( y, x = 0 Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 Defiitio 5 Let { x } a { } y be two sequeces i a symmetric space (X, a x X The space X is sai to satisfy axioms (H E ; if lim ( ( x,x = lim y,x = 0 implies that lim ( x, y 0 = Defiitio 6 Let be a symmetric fuctio o X Two raom mappigs S a T from Ω X X are sai to satisfy property (I if there exists a sequece { } ξ, some M ( Ω, X lim ( ( ξ i M (,X T( ω, ξ ( ω, ω = lim S( ω, ξ ( ω, ω = 0 for every Ω such that for Theorem 3 Let (X, be a separable symmetric space that satisfies (W a (H E Let T,S: Ω X CB(X be two weakly compatible raom multivalue operators satisfyig the property (I Moreover, for all x,y X we have ( T( ω,x,t( ω, y φ( max { (s( ω,x, S( ω, y, ( s( ω, x, T( ω, y, ( S( ω, y, T( ω, y, ( S( ω,x,t( ω,y + ( S( ω,y,t( ω,x } for every Ω for every ω If T ( ω,x S ( ω,x a oe of T( ω,x or (,X, the T a S have uique commo raom fixe poit S ω is a complete subspace of X Proof Sice raom multivalue operators T a S satisfy the property (I, so there exists a sequece { ξ } i M ( Ω, X such that : lim ( ξ ( ω = lim ( S( ω, ξ ( ω, ω = 0 ξ M ( Ω,X Therefore by property ( H E, we have lim Suppose (,X ( T( ω ξ ( ω, ξ ( ω = 0 for every S ω is a complete subspace of X for every ω Ω Let for every, for some ξ : Ω X be the limit of the sequece of measurable mappigs { ( } ( { } ( S ω, ξ ( ω a S ω, ξ ( ω S ω,x for every a N Now sice X is Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 separable, therefore ξ M ( Ω, X Moreover ξ ( ω S( ω,x for every This allows obtaiig the measurable mappigs Now we show that T (, ξ ( ω = ξ ( ω If ot the for some Cosier ξ : Ω X such that ξ ( ω = ξ ( ω ω for every ( { ( ω ξ ω ( ω ξ ω ϕ ( ω ξ ω ω ξ ω T(, (,T, ( max S(, (,S(, (, ( S( ω, ξ ( ω, ξ ( ω, ( S( ω, ξ ( ω, T( ωξ ( ω, ( ( S(, (,T (, ( S(, (,T (, ( } ω ξ ω ω ξ ω + ω ξ ω ω ξ ω < max{ ( ω,s( ω, ξ ( ω, ( ω, T( ω, ξ ( ω, Takig ( ω ξ ω ω ξ ω ( ξ ω ω ξ ω + ( ω ξ ω ( ω ξ ω S(, (,T(, (, (,T(, ( S(, (,T, ( we have ( T( ω, ξ ( ω, T( ω, ξ ( ω < max ( 0,0,0, 0 ( (,T(, ( + ξ ω ω ξ ω or ( T ( ω, ω, ω < ( ξ ( ω, ω which is a cotraictio, so T (, ξ ( ω = ξ ( ω ω for every The weak compatibility of raom mappigs T a S implies that S( ω, ω = S( ω,t( ω, ω, The T( ω,t( ω, ω = T( ω,s( ω, ω = S( ω,t( ω, ω = S( ω,s( ω, ω for every Let us show that T( ω,t( ω, ω = ω for each If ot, the for some, cosier ( { (T( ω, ω, T( ω, ω ϕ Max (S( ω, ω,s( ω, ω ( S( ω, ξ ( ω, ξ ( ω, (s( ω, ω, ω, Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 [ (S(, (,T, ( S(, (,T, ( ] ( ω ξ ω ( ω ξ ω + ( ω ξ ω ( ω ξ ω } ( max { (T(, (, T(, (,(T(, (,T (, ( ϕ ω ξ ω ω ξ ω ω ξ ω ω ξ ω T(, (,T, (, [ (T(, (,T, ( ( ω ξ ω ( ω ξ ω ( ω ξ ω ( ω ξ ω + ( T( ω, ω, ω ]} ( { ( ϕ max T( ω, ξ ( ω,t( ω, ξ ( ω,0,(t( ω, ξ ( ω,t( ω, ξ ( ω } ϕ ( T( ω, ω, ω < ( T ( ω, ω, ω ie ( T( ω, ω,t( ω, ω < (T( ω, ω, ω which is a cotraictio, so T (, ξ ( ω T ( ω, ξ ( ω = T( ω, ξ ( ω ξ ( ω Therefore T (, ξ ( ω whe (,X ω is a raom fixe poit of T Now = for every ω is a commo raom fixe poit of T a S The proof is similar T ω is suppose to be a complete subspace of X for every, as T( ω,x S( ω,x for each ω Ω To prove the uiqueess of commo raom fixe poit, let η, η : Ω X be two commo raom fixe poits of raom operators T a S such that η ( ω η ( ω for some, cosier ( η ( ω, ω = ( T( ω, ω, T( ω, ω ( { ϕ max (S( ω, ω,s( ω, ω,(s( ω, ω,t( ω, ω, (S( ω, ω,t( ω, ω, [ (S( ω, η ( ω,t( ω, η ( ω + (S( ω, η ( ω,t( ω, η ( ω } ] Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 { ( ϕ(max ( ω, ω,( ω, ω, ω, ω, φ < ( η ( ω, η ( ω + ( η ( ω, η ( ω } ( η ( ω, ω ( ω, ω ie ( η ( ω, η ( ω < ( η ( ω, ω This cotraictio shows η ( ω = η ( ω for every Theorem 3 Let (X, be a separable symmetric space that satisfies (W, (W a (H E Let (A, S a (B, T be two pairs of weakly compatible raom operators from Ω X X such that oe of the pairs (A, S or (B, T satisfies the property (I Moreover ( A( ω,x,b( ω,y ϕ(max { (A( ω,x,s( ω,x,(b( ω,yt( ω,y, for every Ω ( S (,x, y, ( ω (A( ω,x,t( ω,y + (B( ω,y,s( ω,x} ω If A (,X T (,X ab(,x S (,X T (,X,S (,X,B (,X or A (,X ω ω ω ω a oe of ω ω ω ω is a complete subspace of X for every, the A, B, T, a S have uique commo raom fixe poit Proof : Suppose the pair (B,T of raom mappigs satisfies the property (I So there exists a sequece { ξ } i (, X M Ω such that lim (B(, (, ( ω ξ ω ξ ω = lim ( T( ω, ξ ( ω, ω = 0 for every for some ξ M( Ω,X As { B ( ω, ξ ( ω } is a sequece of measurable mappigs a (, ξ ( ω B(,X B ω ω for every a N obtaiig the sequece of measurable mappigs for every Ω ω Hece lim ( S( ω, η ( ω, ω = 0, ow the fact B (,X S (,X ω ω allows η : Ω X such that B( ω, ξ ( ω = S( ω, η ( ω for every Now we show that (A(ω,η (ω, ω = 0 for every ω Ω Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 For this cosier ( A( ω, η ( ω, B ( ω, ξ ( ω every Let ( { ϕ max (A( ω, η ( ω,s( ω, η ( ω,(b( ω, ξ ( ω,t( ω, ξ ( ω, ( ω η ω ω ξ ω S(, (,T(, (, ( (A( ω, η ( ω,t( ω, ξ ( ω + ( B( ω, ξ ( ω, B( ω, ω } ( { ϕ max (A( ω, η ( ω,b( ω, ξ ( ω,(b( ω, ξ ( ω,t( ω, ξ ( ω, φ ( (A( ω, η ( ω, ξ ( ω 0 } + ( (B( ω, ξ ( ω, ξ ( ω for every ( A ( ω, η ( ω, B ( ω, ξ ( ω < ( B( ω, ξ ( ω, ξ ( ω Therefore by property (H E, we have Hece: ( ( lim B( ω, ξ ( ω, T ω, ξ ( ω = 0 for every lim ( B ( ω, ξ ( ω, A ( ω, η ( ω = 0 for every By (W, we euce that lim ( A ( ω, η ( ω, ω = 0, (, X for every S ω is a complete subspace of X Now S(, η ( ω S( ω, X suppose for ω for every ξ : Ω X be the limit of the sequece of measurable mappigs { S ( ω, η ( ω } Sice X is separable, therefore ξ M(,X Moreover ( ω S (, X Ω obtaiig the measurable mappig lim ξ ω for every This allows ξ : Ω X such that ξ ( ω = ξ ( ω Now cosier ( A ( ω, η ( ω, ω = lim ( B ( ω, ξ ( ω, S( ω, ξ ( ω = lim = lim ( ξ ( ω, S( ω, ω ( η ( ω, ω Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 Thus, = 0 for every ( ω ξ ω ω ξ ω ϕ { ω ξ ω ω ξ ω A(, (,B(, ( (max (A(, (,S(, (, ( B ( ω, ξ ( ω, ξ ( ω, ( S( ω, ω, T( ω, ξ ( ω (A(, (,T(, ( B(, (,S(, ( ( ω ξ ω ω ξ ω + ( ω ξ ω ω ξ ω } for each This immeiately gives : ( lim A( ω, ω,b( ω, ξ ( ω = 0 for every By (W, we have A( ω, ω = S( ω, ω for every S The weak compatibility of raom operators A a S implies that ( ω, A ( ω, ω = A( ω, S( ω, ξ ( ω for every Now A (, A ( ω, ω = A( ω, ξ ( ω = A( ω, ξ ( ω = ω A ω for every As A (, ξ ( ω A ( ω, X (,X X ω for every Ω ω for every Ω ( ( A ω, ω = T ω, ξ ( ω for every some We ow show that for every cosier ( ω ξ ω ( ω ξ ω A(, (, B, ( ω where ξ M( Ω, X ω allows obtaiig (, the assumptio ξ M Ω,X such that, B ( ω, ξ ( ω = ξ ( ω If ot, the for ϕ(max{(a( ω, ω,s( ω, ω,(b( ω, ξ( ω,t( ω, ξ( ω, (S( ω, ω,t( ω, ξ( ω, ((A(, (,T(, ( ω ξ ω ω ξ ω + ω ξ ω ω ξ ω } (B(, (,S(, ( ϕ(max{(a( ω, ω,a( ω, ω,(b( ω, ξ( ω,a( ω, ω, (A( ω, ω,t( ω, ξ( ω, Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 ( { ((A(, (,T(, ( ϕ max 0,(B( ω, ξ ( ω,a( ω, ω,0, ω ξ ω ω ξ ω + ω ξ ω ω ξ ω } 0 (B(, (,A(, ( ( + ω ξ ω ω ξ ω } ( { ϕ max (A( ω, ω,b( ω, ξ ( ω, A(, (,B(, ( ( ω ξ ω ω ξ ω } ( ( ϕ A( ω, ω,b ω, ω ( ( < A( ω, ω, B ω, ξ ( ω ie ( A( ω, ω, B ( ω, ξ( ω < ( A( ω, ω,b( ω, ξ( ω which is a cotraictio Hece ( (B(, (,A(, ( B ω, ξ ( ω = T( ω, ξ ( ω = A( ω, ω = S( ω, ω for every The weak compatibility of raom operators B a T implies that ( B ω,t( ω, ξ ( ω = T( ω,b( ω, ξ ( ω for every, ( ω ω ξ ω = ( ω ω ξ ω = ( ω ω ξ ω T,T(, ( T,B(, ( B,T(, ( B (,B(, ( Let us show that A (, A( ω, ω = A ( ω, ξ ( ω ω If ot, the for some = ω ω ξ ω for each ω for each Ω, cosier ( A( ω, ω,a( ω,a( ω, ω = ( A( ω,a( ω, ω,b( ω, ξ( ω ϕ(max { (A( ω,a( ω, ω,s( ω,a( ω, ω, (B( ω, ξ( ω,t( ω, ξ( ω,(s( ω,a( ω, ω,t( ω, ξ( ω ( ( (A( ω,a( ω, ξ ( ω,t( ω, ξ ( ω + (B( ω, ξ ( ω, S ω,a( ω, ω } Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 ϕ(max { (A( ω,a( ω, ω,a( ω,a( ω, ω, ( A( ω, ω,a( ω, ω, ( A( ω,a( ω, ω,a( ω, ω ( (A( ω,a( ω, ξ ( ω,a( ω, ξ ( ω + A ( ω, ω, A ( ω, A( ω, ξ ( ω } φ(max { 0, 0, (A ( ω, A ( ω, ξ ( ω, A ( ω, ξ ( ω, ( ω ω ξ ω ω ξ ω } A(,A(, (,A(, ( < ϕ ( A( ω,a( ω, ω,a( ω, ω < ( A( ω,a( ω, ω,a( ω, ω which is a cotraictio Therefore ( A ( ω,a ( ω, ω = A ( ω, ξ ( ω = A( ω, ω for every So A (, ξ ( ω a commo raom fixe poit of raom operators A a S Similarly, B(, ξ ( ω raom fixe poit of raom operators B a T Sice A (, ξ ( ω = B ( ω, ξ ( ω, thus A (, ω proof is similar whe for every A ( ω,x or B (,X ω is ω is commo ω for every ω is commo raom fixe poit of raom operators A, B, S a T The, T( ω, X is complete subspace of X The cases i which ω is a complete subspace of X for every are similar to the cases i which ( ω X or S( ω X, respectively, is a complete subspace of X, sice A (,X T T ( ω,x a B ( ω,x B (,X ω for every ω UNIQUENESS: To establish the uiqueess of commo raom fixe poit of raom operators, let ξ a η be two commo raom fixe poits of the raom operators such that ( ω ω cosier ( ξ ( ω, ω = ( A ( ω, ω, B ( ω, ω φ (max{ (A ( ω, ω, ω, (B( ω, ω, ω, ( S( ω, ω, T( ω, ω, ( (A( ω, ω, ω + ( B ( ω, ω, ω } ξ for some Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 φ (max{ ( ω, ω, ( ω, ω, ( ω, ω, φ φ ( ( ξ ( ω, ω + ( ω, ω } ( max{ 0, 0, ( ω, ω } ( ω, ω ( ξ ( ω, ω < ( ω, ω which is a cotraictio So the result follows REFERENCES Bashah, VH a Sayye, F Raom fixe poit of raom multivalue o expasive o-self raom operators, Joural of Applie Mathematics a Stochastic Aalysis, Vol 006, Article ID 43796, Pages -9 Bashah, VH a Gagrai, S, Commo raom fixe poit of raom multivalue operators o Polish spaces, Jour Chugcheog Math Soc, Vol 8 (, (005, 33-39 3 Bashah, VH a Shrivastava, N, Semi compatibility a raom fixe poits o Polish spaces, Varahmihir Joural of Math Sci Vol 6, No (006, 56-568 4 Beg, I a Abbas, M Commo raom fixe poit of compatible raom operators, Iter, Joural of Mathematics a Mathematical Sciece Volume 006, Article ID 3486, pages -5 5 Beg, I a Shahza, N Raom fixe poit of raom multivalue operator o Polish spaces, o liear Aalysis 0 (993, o 7, 835-847 6 Hicks, TL a Rhoaes, BE Fixe poit theory i symmetric spaces with applicatios to probabilistic spaces, Noliear Aalysis 36 (999 No 3, 33-344 7 Plubtig, S a Kumar, P Copyright 03 SciResPub
Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 Raom fixe poit theorems for multivalue o expasive o-self-raom operators joural of applie mathematics a stochastic aalysis, Vol 006, Article ID 43796, pages -9 8 Wilso, WA O semi - metric spaces, America Joural of Mathematics, 53 (93, o, 36-373 9 Xu, HK, Multivalue o-expasive mappigs i Baach spaces, oliear aalysis 43 (00, o 6, 693-706 Copyright 03 SciResPub