Convergence of Random SP Iterative Scheme

Transcription

1 Applied Mathematical Scieces, Vol. 7, 2013, o. 46, HIKARI Ltd, Covergece of Radom SP Iterative Scheme 1 Reu Chugh, 2 Satish Narwal ad 3 Vivek Kumar 1,2,3 Departmet of Mathematics, M. D. Uiversity, Rohtak, Idia 1 chughreu@yahoo.com, 2 arwalmaths@gmail.com, 3 ratheevivek15@yahoo.com Correspodig author: Vivek Kumar Copyright 2013 Reu Chugh 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. Abstract I this article, we prove the covergece of radom SP iterative scheme to a commo radom fixed poit for a certai class of radom operators i Baach spaces. A example is also provided to prove the validity of our result. Mathematics Subject Classificatio: 47H10, 54H25 Keywords: SP iteratio, Measurable mappigs, Cotractive mappigs, Separable Baach spaces 1 Itroductio ad Prelimiaries Radom oliear aalysis is a importat mathematical disciplie which is maily cocered with the study of radom oliear operators ad their properties ad is much eeded for the study of various classes of radom equatios. Radom approximatios ad radom fixed poit theorems are stochastic geeralizatios of approximatios as well as fixed poit theorems. The iterplay betwee radom approximatio ad radom fixed poit results is of great value [11]. Radom techiques have played a crucial role i pure mathematics as well as applied scieces. No doubt, famously radom methods have revolutioized the fiacial markets.

2 2284 Reu Chugh, Satish Narwal ad Vivek Kumar The developmet of radom fixed poit iteratios was iitiated by Choudhury i [4, 5, 6, 7, 8, 9] where radom Ishikawa iteratio scheme was defied ad its strog covergece to a radom fixed poit i Hilbert spaces was discussed. After that several authors [2, 12, 15, 16, 17, 20] have worked o radom fixed poit iteratios to obtai fixed poits i determiistic operator theory. The followig iteratio schemes are ow well kow: Suppose (, Ω deotes a measurable space cosistig of a set Ω ad sigma algebra of subsets of Ω, X stads for a separable Baach space ad C is a oempty subset of X. The x+ 1 ( w = ( 1 α x( w +αt( w, x( w, for each w Ω, = 1, 2,..., (1 where 0 α 1 ad x : Ω F 0, a arbitrary measurable mappig ad is kow as Radom Ma iterative scheme [7]. Also x+ 1 ( w = (1 α x( w +α T( w, y( w, y( w = ( 1 β x( w +β T( w, x( w, for each w Ω, = 1, 2,..., (2 where 0 α, β 1 ad x 0 : Ω F, a arbitrary measurable mappig, is kow as Radom Ishikawa iterative scheme [12]. Obviously x ad y are sequeces of mappigs from Ω ito F. Remark.1.1 If β = 0, the Ishikawa iterates process reduces to the Ma iterates process. Ishikawa ad Ma iterative schemes have bee successfully applied to fid solutio of determiistic operator equatios [7, 15, 20]. I 2006, Beg ad Abbas [12] studied the behavior of the sequece of measurable mappigs costructed through radom Ishikawa ad radom Ma iterative procedures ivolvig strogly pseudo-cotractive radom operators i Baach spaces. I 2011, Rashwa [15] show that for a pair of mappigs S, T satisfyig some cotractive coditios, if the sequece of Ishikawa iterates associated with S or T is coverget, the its limit poit is a commo fixed poit of S ad T. Very recetly, SP iterative scheme was itroduced by Phuegrattaa ad Suatai [23] as follows: x = 1 ( 1 α y + + αty y = ( 1 β z + βtz z = 1 γ x + γ Tx, 3 ( (

3 Covergece of radom SP iterative scheme 2285 where { α }, { β } ad { γ } are sequeces of positive umbers i [0,1]. Chugh ad Kumar [18, 19] studied the covergece of SP iterative scheme usig differet kid of operators. Remark.1.2 If β = γ = 0, the SP iterative scheme reduces to the Ma iterates scheme. We defie the radom SP iterative scheme i a aalogous maer as SP iterative scheme as follows: Let ST, : Ω C C be two operators o a oempty covex subset C of a separable Baach space X. The the sequece { x } of radom SP iterates associated with S or T is defied as follows: Let x : C 0 Ω by ay give measurable mappig. x+ 1 ( w = ( 1 α y( w +αs( w, y( w, y w = 1 β z w +β S w, z w, or ( ( ( ( ( ( ( ( ( ( z w = 1 γ x w +γ S w, x w for > 0, w Ω where{ },{ },{ } ( ( ( ( ( ( = ( 1 β ( +β (, (, ( ( ( ( ( x+ 1 w = 1 α y w +αt w, y w, y w z w T w z w z w = 1 γ x w +γ T w, x w for > 0, w Ω, α β γ satisfyig the followig coditios (a 0 α, β, γ 1, > 0 (b lim α = 0. (c lim β = h, 0 < h< 1. The aim of this paper is to prove covergece of radom SP iterative scheme to a radom commo fixed poit for a certai class of radom operators i Baach spaces. The obtaied result is stochastic geeralizatio of the results i [16] ad some other kow results [13, 20] i the literature of fixed poit theory. We shall eed the followig defiitios to prove our mai result. 1 Defiitio 1.3 A mappig f : Ω C is said to be measurable if f ( B C for every Boral subset B of X. ( 4 (5

4 2286 Reu Chugh, Satish Narwal ad Vivek Kumar Defiitio 1.4 A fuctio F : Ω C C is said to be a radom operator if ( F., x : Ω C is measurable for every x C. Defiitio 1.5 A measurable mappig g : Ω C is said to be radom fixed poit of the radom operator F : C C F w, g w = g w for all w Ω. Ω, if ( ( ( Defiitio 1.6 A radom operator F : Ω C C is said to be cotiuous if, for fixed w Ω, F( w,. : C C is cotiuous. Now, we prove our mai result. 2. Mai Result Theorem 2.1 Let X be a separable Baach space ad C be a oempty, closed ad covex subset of X. Let ST, : Ω C C be two cotiuous radom operators defied o C such that at least oe of the followig coditios hold for all x, y C ad w Ω : (i S w, x T w, y ( ( ( ( ( ( a x y + b x S w, x + y T w, y + c x T w, y + y S w, x, a > 0, b 0, c 0,1 b c> 0. ( ii S( w, x T( w, y q { x y x S( w x + y T( w y x T( w y + y S( w x } max,,,,,,, 0< q < 1. iii S w, x T w, y ( ( ( α { β x y x S( wx y T( xy x T( wy y S( wx } max,,,,,,,,, αβ, 0, 0 α< 1. ( iv S( w, x T( w, y q { x y x S( w x y T( w y x T( w y y S( w x } max,,,,,,,,, 0< q < 1. If the radom SP iterative scheme associated with S (4 or T (5, satisfyig (i-(iii coverges, the it coverges to a commo radom fixed poit of S ad T. Further, if (iv holds, the this commo fixed poit will be uique.

5 Covergece of radom SP iterative scheme 2287 Proof. Let us assume that the sequece{ x } defied by (4 has a poitwise limit, lim x w = u w, for all w Ω, As X is a separable Baach space, the that is, ( ( mappig x ( w A( w f ( w =,, is measurable mappig for ay radom operator A: Ω C X ad ay measurable mappig f : Ω C [10]. Now, the sequece { x } costructed by the radom SP iterative scheme (4, (5 is a sequece of measurable mappigs as x ( w is measurable ad C is covex. Therefore, x : Ω C is also measurable beig limit of measurable mappig sequece. S w, x w x w x w C. The after First of all we assume that ( ( = ( for ( puttig x ( w = y( w = u( w ito ay of the iequalities( i ( iv, it is easy to see that S( w, u( w = u( w. I a similar maer T( w, u( w u( w S( w, u( w = u( w. Suppose the sequece { x } geerated by SP iterative scheme associated with S coverges to u, that is, lim x ( w = u( w. The, from (4 we have ( ( α ( α ( ( x + 1 w = 1 y w + S w, y w Hece ( ( ( ( ( ( ( ( ( ( ( ( ( ( = implies z w S( wz w S( w y ( w { x w S( wx w S( wz ( w } S( w y ( w { x w x w Swx ( w Swz ( ( w } Swy ( ( w = 1 α 1 β + β, + α, = 1 α 1 β 1 γ + γ, + β, + α, = 1 α 1 β γ + γ, + β, + α, ( ( αswy (, ( w ( ( (, ( (, (, ( ( (, ( ( Swx ( ( w Swz ( ( w Swy ( ( w x( w γx( w + γs wx, ( w βx w + βγ x ( w βγ S wx, ( w + β Swz, ( w = ( 1 α + = x w x w + S wx w x w ( γ ( γ ( β ( ( ( ( + βγ x w βγ S wx w + βs wz w αx w + αγ x w αγ S wx w + αβx w αβγ x w + αβγ, αβ, + α, (6

6 2288 Reu Chugh, Satish Narwal ad Vivek Kumar ( β ( ( ( ( + γ ( x( w + S( w, x( w + αβ ( x( w S( w, z( w + βγ ( x( w S( w, x( w + γα ( x( w S( w, x( w + αβγ ( x( w + S( w, x( w ( x ( w S( w, y ( w β ( x ( w S( w, z ( w + αβ ( x( w S( w, z( w ( x ( w S( w, y ( w ( 1 ( x ( w S( w, z ( w ( ( α ( ( ( x+ 1 w x w = x w + S w, y w + x w + S w, z w = α = α + + β α = α( S( w y( w x( w + β( α S( w z( w x( w But x ( w = u( w implies x ( w x ( w lim ( (, 1, Therefore usig coditio (b ad coditio (c, (7 yields S w, z w x w 0. ( ( ( If S, T satisfy (i, the S w, z w T w, u w a z w u w ( ( ( ( ( ( + b z( w S( w z( w + u( w T w u( w, (, ( (, ( ( (, (.( 8 + c z w T w u w + u w S w z w From (4 we have the followig estimates: z w u w 1 γ x w u w +γ S w, x w u w, 9 ( ( ( ( ( ( ( ( ( (, ( 1 γ, ( ( ( ( ( +γ S( w, x( w S( w, z( w ( 10 z w S w z w x w S w z w ad z w T w, u w 1 γ x w T w, u w ( ( ( ( ( ( ( +γ S( w, x( w T( w, u( w ( 11 Substitutig (9, (10 ad (11 i (8, we obtai

7 Covergece of radom SP iterative scheme 2289 (, ( (, ( ( 1 γ ( ( +γ (, ( ( S w z w T w u w a x w u w S w x w u w ( 1 γ x( w S( w, z( w +γ S( w, x S( w, z( w + u( w T( w, u( w + b ( 1 γ x( w T( w, u( w (, ( (, ( + u( w S( w, z ( w + c +γ S w x w T w u w If S, T satisfy (ii, the z w u w z w S w z w S( w, z ( w T( w, u( w qmax + u w T w, u w, z w T w u w u w S w z w ( 12 ( (, ( (, ( ( ( ( ( (, ( + ( (, ( ( 13 Usig (9, (10 ad (11, (13 yields ( 1 γ x( w u( w +γ S( w, x( w u( w, ( 1 γ x( w S( w, z( w +γ S( w, x( w S( w, z( w + u( w T( w, u( w, ( 1 γ x( w T( w, u( w +γ S( w, x( w T( w, u( w + u( w S( w, z ( w S( w, z ( w T( w, u( w qmax 14 (

8 2290 Reu Chugh, Satish Narwal ad Vivek Kumar Also, if S,T satisfy (iii, the γ S( w, z ( w T( w, u( w αmax u w T w, u w, z w T w u w u w S w z w O puttig (9, (10 ad (11 i (15, we get z ( w u( w, z ( w S( w, z ( w, ( ( ( ( (, (, ( (, ( ( 15 γ ( 1 γ x ( w u( w +γ S ( w, x ( w u( w, ( 1 γ x ( w S ( w, z ( w +γ S ( w x ( w S ( w z ( w u( w T ( w, u( w, ( 1 γ x ( w T ( w, u( w +γ S ( w x ( w T ( w u( w u( w S ( w, z ( w,,, S ( w, z ( w T ( w, u( w α max (16,,, Further, if T, S satisfy (iv, the obviously satisfy (ii as well. O applyig the limit, i (7, (9, (11, we get u( w T( w, u( w λ u( w T( w, u( w, ( 17 where λ= max { b+ c, q, α } < 1. The T( w, u( w = u( w. Similarly, we ca show that S( w, u( w = u( w. To prove the uiqueess of u( w i the case (iv, let us assume that v( w commo fixed poit of S ad T other tha u( w, the usig (iv we have ( ( = (, ( (, ( u( w Su( w, u( w S( w, v( w, q max v( w Tv( w, u( w Tv( w, v( w Sv( w u w v w S w u w T w v w qu( w v( w, which further yields q u( w v( w (1 0.

9 Covergece of radom SP iterative scheme 2291 But 0< q < 1, hece u( w v( w 0 oegative, therefore u( w v( w <, which is a cotradictio as orm is always = always. Remark 2.2 By Puttig β = γ = 0, i radom SP iterative scheme, results of [13, 16] ca be obtaied from Theorem 2.1. Remark 2.3 By takig S = T = Id (idetity mappig, it is easy to see that uiqueess of commo fixed poit does ot hold i the case of the coditios (i- (iii. The followig example prove the validity of our result. Example 2.4 Let Ω= [ 0,1] ad be the sigma algebra of Lebesgue s measurable subsets of Ω. Take X = R with d( x, y = x y for x, y R. Defie radom operator T from X T w, x = w x ad radom operator S from Ω X to X as S( w, x Ω to X as ( 1 w = x The the measurable mappig : X w ξ Ω defied by ξ ( w =, for every w Ω, 2 serve as a radom fixed poit of S ad T. It is easy to see that the give operators S ad T satisfies all the coditios (i-(iv give i Theorem 2.1. Refereces [1] A. K. Gaguly, O commo fixed poit of two mappigs, Math. Semiar Notes. Kobe Uiversity, 8(2(1980, [2] A. R. Kha, A. B. Thaheem ad N. Hussio, Radom fixed poits ad radom approximatios i ocovex domais, J. Appl. Math. Stoch. Aal.,15(2002, o. 3, [3] B. E. Rhoades, Iteratio to obtai radom solutios ad fixed poits of operators i uiformly covex Baach spaces, Soochow J. Math., 27(2001, o. 4, [4] B. S. Choudhury, Covergece of a radom iteratio scheme to a radom fixed poit, J. Appl. Math. Stoc. Aal., 8 (1995,

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