A common fixed point theorem for two random operators using random mann iteration scheme
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This academic article presents a common fixed point theorem for two random operators using a random Mann iteration scheme. It proves that if a sequence defined by the random Mann iteration of two random operators converges, then the limit point is a common fixed point of the two operators. The paper defines relevant concepts such as random operators and random fixed points. It then presents the main theorem and proof that under a contractive condition, the limit of the random Mann iteration is a common fixed point. The proof uses properties of measurable mappings and the convergence of the iterative sequence.
![Mathematical Theory and Modeling www.iiste.org
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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](https://image.slidesharecdn.com/acommonfixedpointtheoremfortworandomoperatorsusingrandommanniterationscheme-130612021907-phpapp02/85/A-common-fixed-point-theorem-for-two-random-operators-using-random-mann-iteration-scheme-1-320.jpg)
![Mathematical Theory and Modeling 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](https://image.slidesharecdn.com/acommonfixedpointtheoremfortworandomoperatorsusingrandommanniterationscheme-130612021907-phpapp02/85/A-common-fixed-point-theorem-for-two-random-operators-using-random-mann-iteration-scheme-2-320.jpg)
![Mathematical Theory and Modeling 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.](https://image.slidesharecdn.com/acommonfixedpointtheoremfortworandomoperatorsusingrandommanniterationscheme-130612021907-phpapp02/85/A-common-fixed-point-theorem-for-two-random-operators-using-random-mann-iteration-scheme-3-320.jpg)
![Mathematical Theory and Modeling 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.](https://image.slidesharecdn.com/acommonfixedpointtheoremfortworandomoperatorsusingrandommanniterationscheme-130612021907-phpapp02/85/A-common-fixed-point-theorem-for-two-random-operators-using-random-mann-iteration-scheme-4-320.jpg)
A common fixed point theorem for two random operators using random mann iteration scheme
- 1. Mathematical Theory and Modeling 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:[email protected] 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
- 2. Mathematical Theory and Modeling 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
- 3. Mathematical Theory and Modeling 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.
- 4. Mathematical Theory and Modeling 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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