A common fixed point theorem for two random operators using random mann iteration scheme

A common fixed point theorem for two random operators using random mann iteration scheme

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  Mathematical Theory andModeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 263 A Common Fixed Point Theorem for Two Random Operators using Random Mann Iteration Scheme S. Saluja Department of Mathematics, Government J. H. P. G. College Betul (MP) India-460001 Devkrishna Magarde Department of Mathematics, Patel College of Science and Technology, Bhopal(MP) India-462044 Email:-devmagarde@gmail.com Alkesh Kumar Dhakde Department of Mathematics, IES College of Technology, Bhopal(MP) India-462044 Abstract: In this paper, we proved that if a random Mann iteration scheme is defined by two random operators is convergent under some contractive inequality the limit point is a common fixed point of each of two random operators in Banach space. Keywords: Mann iteration, fixed point, measurable mappings, Banach space. AMS Subject Classification: 47H10, 47H40. 1.Introduction and Preliminaries: Kasahara [8] had shown that if an iterated sequence defined by using a continuous linear mapping is convergent under certain assumption, then the limit point is a common fixed point of each of two non-linear mappings. Ganguly [6] arrived at same conclusion by taking the same contractive condition and using the sequence of Mann iteration [9]. It this note, it is proved that if a random Mann iteration scheme is defined by two operators is convergent under some contractive inequality the limit point is a common fixed point of each of two random operators in a Banach space. The study of random fixed point has been an active area of contemporary research in mathematics. Random iteration scheme has been elaborately discussed by Choudhury ([1], [2], [3], [4]). Looking to the immense applications of iterative algorithms in signal processing and image reconstruction, it is essential to venture upon random iteration. We first review the following concepts, which are essential for our study. Throughout this paper ( Ω, ∑ ) denotes a measurable space and X stands for a separable Banach space. C is a nonempty subset of X . A mapping :f Ω C→ is said to be measurable if ( ) ∑∈∩− CBf 1 for every Borel subset B of X . A mapping :F Ω ,CC →× is said to be a random operator, if ( ):., xF Ω C→ is measurable for every Cx ∈ . A measurable mapping :g Ω C→ is said to be a random fixed point of the random operator :F Ω ,CC →× if ( ) )()(, tgtgtF = for all ∈t Ω. A random operator :F Ω CC →× is said to be continuous if, for fixed ∈t Ω, ( ) CCtF →:,. is continuous. Definition 1 (Random Mann Iteration scheme): Let :,TS Ω CC →× be two random operators on a nonempty convex subset C of a separable Banach space X . Then the sequence { }nx of random Mann iterates associates with TS or is defined as follows: (1) Let :0x Ω C→ be any given measurable mapping. (2) ( ) ( ))(,1)(1 txtScxctx nnnnn +−=+ for 0>n , ∈t Ω, or (3) ( ) ( ))(,1)(1 txtTcxctx nnnnn +−=+ for 0>n , ∈t Ω, where nc satisfies: (4) 10 =c for 0=n (5) 10 ≤< nc for 0>n
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  Mathematical Theory andModeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 264 (6) 0lim >= ∞→ hcn n 2.Main Result: Theorem 1: Let :,TS Ω ,CC →× where C is a nonempty closed convex subset of a separable Banach space X , be two continuous random operators which satisfy the following inequality: for all Cyx ∈, and ∈t Ω, ( ) ( ) ( ) ( ){ } ( ) ( ){ } ( ){ ( ) })()()(,)( ,)()()(,)(max )(,)(,)(,)(max )(,)(,)(,)(,)()(max,,(7) tytxtytTty tytxtxtStx tytTtytxtStx txtStytytTtxtytxytTxtS −+− −+−+ −−+ −−−≤− γ β α where 1and0,, <++≥ γβαγβα . If the sequence { })(txn of random Mann iterates associated with TS or satisfying (1)-(6) converges, then it converges to a common random fixed point of both TS and . Proof: We may assume that the sequence { })(txn defined by (2) is pointwise convergent, that is, for all ∈t Ω, (8) )(lim)( txtx n n ∞→ = Since X is a separable Banach space, for any continuous random operator :A Ω ,CC →× and any measurable mapping :f Ω C→ , the mapping ( ))(,)( tftAtx = is measurable mapping [7]. Since )(tx is measurable and C is convex, it follows that{ })(txn constructed in the random iteration from (2)-(6) is a sequence of measurable mappings. Hence being limit of measurable mapping sequence is also measurable. Now for ∈t Ω, from (2), (6) and (7) we obtain ( ) ( ))(,)()()()(,)( 11 txtTtxtxtxtxtTtx nn −+−≤− ++ ( ) ( ) ( ))(,)(,)(1)()( 1 txtTtxtSctxctxtx nnnnn −+−+−≤ + ( ) ( ) ( ) ( ))(,)(, )(,)(1)()( 1 txtTtxtSc txtTtxctxtx nn nnn −+ −−+−≤ + ( ) ( ) ( ) [ ( ){ ( )} ( ){ ( )} ( ){ ( ) }])()()(,)(,)()( )(,)(max)(,)( ,)(,)(max)(,)( ,)(,)(,)()(max )(,)(1)()()(,)()9( 1 txtxtxtTtxtxtx txtStxtxtTtx txtStxtxtStx txtTtxtxtxc txtTtxhtxtxtxtTtx nn nn nnn nnn nn −+−− +−+− −+− −−+ −−+−≤− + γ β α Now ( )( ) ( ) )()()()(,)()(, 1 txtxtxctxtSctxtxtSc nnnnnnnnn −=−=− + Implies that ( ) )()( 1 )()(, 1 txtx c txtxtS nn n nn −≤− + This shows that for ∈t Ω, ( ) ∞→→− ntxtxtS nn as0)()(, and so ( ) ∞→→− ntxtxtS n as0)()(, as S is continuous random operator and x is a measurable mapping. Consequently from (9) on taking limit as ∞→n we obtain ( ) ( ) ( ) ( ){ }[ ( ){ } ( ){ }])(,)(,0max)(,)(,0max 0,)(,)(,0max)(,)(10)(,)( txtTtxtxtTtx txtTtxctxtTtxhtxtTtx n −+−+ −+−−+≤− γβ α
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  Mathematical Theory andModeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 265 ( ) ( ))(,)(1 txtTtxhhhh −+++−≤ γβα implies that ( ) )()(, txtxtT = for all ∈t Ω ( )1sin <++ γβαce as T is continuous random operator and x is measurable. Therefore, ( ) ( ) ( ))(,)(,)()(, txtTtxtStxtxtS −=− ( ) ( ){ } ( ) ( ){ } ( ){ ( ) })()()(,)( ,)()()(,)(max )(,)(,)(,)(max )(,)(,)(,)(,)()(max txtxtxtTtx txtxtxtStx txtTtxtxtStx txtStxtxtTtxtxtx −+− −+−+ −−+ −−−≤ γ β α ( ) ( ){ } ( ){ } ( ){ })()()()( ,)()()(,)(max )()(,)(,)(max )(,)(,)()(,)()(max)()(, txtxtxtx txtxtxtStx txtxtxtStx txtStxtxtxtxtxtxtxtS −+− −+−+ −−+ −−−≤− γ β α ( ){ } ( ){ } ( ){ }0,)(,)(max 0,)(,)(max)(,)(,0,0max txtStx txtStxtxtStx −+ −+−≤ γ βα ( ) ( ))(,)( txtStx −++≤ γβα Since 1<++ γβα implies that ( ) )()(, txtxtS = . Uniquness:-Let )()(),( txtvtv ≠ is another common fixed point of S and T ,then, using (7), we have ( ) ( ){ } ( ) ( ){ } ( ){ ( ) })()()(,)( ,)()()(,)(max )(,)(,)(,)(max )(,)(,)(,)(,)()(max)()( tvtxtvtTtv tvtxtxtStx tvtTtvtxtStx txtStvtvtTtxtvtxtvtx −+− −+−+ −−+ −−−≤− γ β α { } { } { })()()()( ,)()()()(max )()(,)()(max )()(,)()(,)()(max tvtxtvtv tvtxtxtx tvtvtxtx txtvtvtxtvtx −+− −+−+ −−+ −−−≤ γ β α )()()( tvtx −+≤ γα 1as)()( <+=⇒ γαtvtx This complete the proof. References: [1] Choudhury B. S., Convergence of a random iteration scheme to a random fixed point, J. Appl. Math. Stoc. Anal., 8(1995), 139-142. [2] Choudhury B. S., Random Mann iteration scheme, Appl. Math. Lett., 16 (2003), 93-96. [3] Choudhury B. S., Ray M., Convergence of an iteration leading to a solution of a random operator equation, J. Appl. Math. Stoc. Anal., 12 (1999), 161-168.i [4] Choudhury B. S., Upadhyay A., An iteration leading to a solution and fixed point of operators, Soochow J. Math., 25 (1999), 394-400. [5] Ciric Lj., Quasi-contractions in Banach space, Publ. Inst. Math., 21(35) (1977), 41-48.
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  Mathematical Theory andModeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications 266 [6] Ganguly A. K., On common fixed point of two mappings, Mathematics Seminar Notes., 8 (1980), 343-345. [7] Himmelberg C.J.,Measurable relations, Fund.Math.,LXXXVII (1975),53-71. [8] Kasahara S., Fixed point iterations using linear mappings, Mathematics Seminar Notes., 6 (1978), 87-90. [9] Rhoades B. E., Extensions of some fixed point theorems of Ciric, Maiti and Pal, Mathematics Seminar Notes., 6 (1978), 41-46.
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