11.a common fixed point theorem for compatible mapping
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Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.1, 2011 A Common Fixed Point Theorem for Compatible Mapping Vishal Gupta* Department Of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana, (India) * E-mail of the corresponding author: vishal.gmn@gmail.com, vkgupta09@rediffmail.comAbstractThe main aim of this paper is to prove a common fixed point theorem involving pairs ofcompatible mappings of type (A) using five maps and a contractive condition. This articlerepresents a useful generalization of several results announced in the literature.Key words: Complete metric space, Compatible mapping of type (A), Commutingmapping, Cauchy Sequence, Fixed points.1. IntroductionBy using a compatibility condition due to Jungck many researchers establish lot ofcommon fixed point theorems for mappings on complete and compact metric spaces. Thestudy of common fixed point of mappings satisfying contractive type conditions are alsostudied by many mathematicians. Jungck.G.(1986) prove a theorem on Compatiblemappings and common fixed points. Also Sessa.S(1986) prove a weak commutativitycondition in a fixed point consideration. After that Jungck, Muthy and Cho(1993) definethe concept of compatible map of type (A) and prove a theorem on Compatible mappingof type (A) and common fixed points. Khan, M.S., H.K. Path and Reny George(2007)find a result of Compatible mapping of type (A-1) and type (A-2) and common fixedpoints in fuzzy metric spaces. Recently Vishal Gupta(2011) prove a common fixed pointtheorem for compatible mapping of type (A).2.PreliminariesDefinition 2.1. Self maps S and T of metric space (X,d) are said to be weakly commutingpair iff d(STx,TSx)≤d(Sx,Tx) for all x in X.Definition 2.2. Self maps S and T of a metric space (X,d) are said to be compatible oftype (A) if lim d(TSxn,SSxn) =0 and lim d(STxn, TTxn)=0 as n→∞ whenever {xn} is a Sequence in X such that lim Sxn=lim Txn =t as n→∞ for some t in X.1|P a gewww.iiste.org
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Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.1, 2011Definition 2.3. A function Φ: [0, ∞) → [0, ∞) is said to be a contractive modulus if Φ (0)=0 and Φ (t) <t for t > 0.3.Main ResultTheorem 3.1. Let S,R,T,U and I are five self maps of a complete metric space (X,d) intoitself satisfying the following conditions: (i) SR ∪ TU ⊂ I ( x) (ii) d ( SRx, TUy ) ≤ α d ( Ix, Iy ) + β [d ( SRx, Ix) + d (TUy, Iy )] + γ [d ( Ix, TUy ) + d ( Iy, SRx) for all x, y ∈ X and α , β and γ are non negative real’s such that α + 2β + 2γ p 1 (iii) One of S,R,T,U and I is continuous (iv) (SR,I) and (TU,I) are compatible of type (A). then SR,TU and I have a unique common fixed point. Further if the pairs (S,R), (S,I), (R,I),(T,U), (T,I) ,(U,I) are commuting pairs then S,R,T,U,I have a unique common fixed point.Proof: Let x0 in X be arbitrary. Construct a sequence {Ixn} as follows:Ix2n+1 = SRx2n , Ix2n+2 =TUx2n+1 n=0,1,2,3…From condition (ii),we haved ( Ix2 n +1 , Ix2 n + 2 ) = d ( SRx2 n , TUx2 n +1 ) ≤ α d ( Ix2 n , Ix2 n +1 ) + β [ d ( SRx2 n , Ix2 n ) + d (TUx2 n +1 , Ix2 n +1 )] + γ [ d ( Ix2 n , TUx2 n +d ( Ix2 n +1 , SRx2 n )]= α d ( Ix2 n , Ix2 n +1 ) + β [d ( Ix2 n+1 , Ix2 n ) + d ( Ix2 n + 2 , Ix2 n+1 )] + γ [d ( Ix2 n , I 2 n+ 2 ) + d ( Ix2 n +1 , Ix2 n +1 )]≤ (α + β + γ )d ( Ix2 n , Ix2 n +1 ) + ( β + γ )d ( Ix2 n +1 , Ix2 n + 2 )Therefore, we have α + β +γ d ( Ix2 n+1 , Ix2 n + 2 ) ≤ d ( Ix2 n , Ix2 n +1 ) 1 − (β + γ ) α + β +γi.e d ( Ix2 n +1 , Ix2 n+ 2 ) ≤ hd ( Ix2 n , Ix2 n +1 ) where h = p1 1 − (β + γ )Similarly, we can show thatd ( Ix2 n +1 , Ix2 n + 2 ) ≤ h 2 n+1d ( Ix0 , Ix1 )For k f n , we haved ( Ixn , Ixn + k ) ≤ ∑ d ( I n +i −1 , I n +i )2|P a gewww.iiste.org
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Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.1, 2011 k ≤ ∑ h n +i −1d ( I 0 , I1 ) i =1 hn d ( Ixn , Ixn + k ) ≤ d ( Ix0 , Ix1 ) → 0 as n → ∞ 1− hHence { Ixn } is a Cauchy sequence. Since X is complete metric space ∃z ∈ X suchthat Ixn → z . Then the subsequence of { Ixn } , {SRx2n } and {TUx2 n +1} also converges to z.i.e {SRx2n } → z & {TUx2 n +1} → z .Suppose that I is continuous and the pair {SR,I} is compatible of type (A),then from condition (ii), we haved ( SR ( Ix2 n ), TUx2 n+1 ) ≤ α d ( I 2 x2 n , Ix2 n +1 ) + β [d ( SR ( Ix2 n ), I 2 x2 n ) + d (TUx2 n+1 , Ix2 n +1 )] +γ [d ( I 2 x2 n , TUx2 n+1 ) + d ( Ix2 n +1 , SR ( Ix2 n ))]Since I is continuous, I 2 x2n → Iz as n → ∞ .The pair (SR,I) is compatible of type (A), then SRIx2n → Iz as n→∞Letting n → ∞ , we haved ( Iz, z ) ≤ α d ( Iz, z ) + β [d ( Iz, Iz ) + d ( z, z )] + γ [d ( Iz, z ) + d ( z, Iz )] = (α + 2γ )d ( Iz , z )Henced ( Iz, z ) = 0 and Iz = z since α + 2γ < 1Again, we haved ( SRz , TUx2 n +1 ) ≤ α d ( Iz, Ix2 n +1 ) + β [d (SRz, Iz ) + d (TUx2 n +1 , Ix2 n+1 )] + γ [d ( Iz, TUx2 n +1 ) + d ( Ix2 n+1 , SRz )]Letting n → ∞ and using Iz = z , we haved ( SRz , z ) ≤ α d ( z, z ) + β [d (SRz, Iz ) + d ( z, z )] + γ [d ( z, z ) + d ( z, SRz )] = ( β + γ )d ( SRz, z )Hence d ( SRz, z ) = 0 and ( SR) z = z since β + γ < 1Since SR ⊂ I , ∃z ′ ∈ X such thatz = SRz = Iz′ = IzNowd ( z, TUz′) = d (SRz, TUz′) ≤ α d ( Iz, Iz′) + β [d ( SRz, Iz ) + d (TUz′, Iz′)] + γ [d ( Iz , TUz′) + d ( Iz′, SRz )]3|P a gewww.iiste.org
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Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.1, 2011 = α d ( z, z ) + β [d ( z, z ) + d (TUz′, z )] + γ [d ( z, TUz′) + d ( SRz, SRz )]Implies d (TUz′, z ) = 0 as α + γ < 1Hence TUz′ = z = Iz′Take yn = z′ for n ≥ 1, we have TUyn → Tz′Now, againd ( z, TUz ) = d ( SRz, TUz ) ≤ α d ( Iz, Iz ) + β [d ( SRz, Iz ) + d (TUz, Iz )] + γ [d ( Iz, TUz ) + d ( Iz, SRz )] = α d ( z, z ) + β [d ( Iz, Iz ) + d (TUz, z )] + γ [d ( z, TUz ) + d ( z, z )] = ( β + γ )d (TUz, z )Hence TUz = zFor uniqueness, Let z and w be the common fixed point of SR,TU and I, then bycondition (ii)d ( z, w) = d (SRz, TUw) ≤ α d ( Iz, Iw) + β [d ( SRz, Iz ) + d (TUw, Iw)] + γ [d ( Iz, TUw) + d ( Iw, SRz )] = α d ( z, w) + β [d ( z , z ) + d ( w, w)] + γ [d ( z, w) + d ( w, z )] = (α + 2γ )d ( z, w)Since α + 2γ < 1 we have z = w .Further assume that (S, R), (S, I), (T, U) and (T, I) are commuting pairs and z be theunique common fixed point of pairs SR, TU and I thenSz = S ( SRz ) = S ( RSz ) = SR( Sz ) , Sz = S ( Iz ) = I ( Sz )Rz = R( SRz ) = ( RS )( Rz ) = SR( Rz ) , Rz = R( Iz ) = I ( Rz )Which shows that Sz and Rz is the common fixed point of SR and I, yielding therebySz = z = Rz = Iz = SRzIn view of uniqueness of the common fixed point of SR and I. therefore z is the uniquecommon fixed point of S, R and I. Similarly using commutatively of (T, U) and (T, I) itcan be shown thatTz = z = Uz = Iz = TUzThus z is the unique common fixed point of S, R, T, U and I.Hence the proof.References4|P a gewww.iiste.org
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Journal of Natural Sciences Research www.iiste.orgISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)Vol.1, No.1, 2011Aage.C.T., Salunke.J.N. (2009), “On Common fixed point theorem in complete metricspace,” International Mathematical Forum,Vol.3,pp.151-159.Jungck.G. (1986), “Compatible mappings and common fixed points” Internat. J. Math.Math. Sci., Vol. 9, pp.771-779.Jungck.G., P.P.Murthy and Y.J.Cho (1993), “Compatible mapping of type (A) andcommon fixed points” Math.Japonica, Vol.38, issue 2, pp.381-390.Gupta Vishal (2011)“Common Fixed Point Theorem for Compatible Mapping of Type(A)”Computer Engineering and Intelligent System(U.S.A), Volume 2, No.8, pp.21-24.Khan, M.S., Pathak.H.K and George Reny (2007) “Compatible mapping of type (A-1)and type (A-2) and common fixed points in fuzzy metric spaces”. InternationalMathematical Forum, 2 (11): 515-524.Sessa.S (1986), “On a weak commutativity condition in a fixed point consideration.”Publ. Inst. Math., 32(46), pp. 149-153.5|P a gewww.iiste.org
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