A Common Fixed Point Theorem for Compatible Mapping

Transcription

1 ISSN (Paper) ISSN (Online) Vol., No., 20 A Common Fixed Point Theorem for Compatible Mapping Vishal Gupta * Department Of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana, (India) * of the corresponding author: vishal.gmn@gmail.com, vkgupta09@rediffmail.com Abstract The main aim of this paper is to prove a common fixed point theorem involving pairs of compatible mappings of type (A) using five maps and a contractive condition. This article represents a useful generalization of several results announced in the literature. Key words: Complete metric space, Compatible mapping of type (A), Commuting mapping, Cauchy Sequence, Fixed points.. Introduction By using a compatibility condition due to Jungck many researchers establish lot of common fixed point theorems for mappings on complete and compact metric spaces. The study of common fixed point of mappings satisfying contractive type conditions are also studied by many mathematicians. Jungck.G.(986) prove a theorem on Compatible mappings and common fixed points. Also Sessa.S(986) prove a weak commutativity condition in a fixed point consideration. After that Jungck, Muthy and Cho(993) define the concept of compatible map of type (A) and prove a theorem on Compatible mapping of 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-) and type (A-2) and common fixed points in fuzzy metric spaces. Recently Vishal Gupta(20) prove a common fixed point theorem for compatible mapping of type (A). 2.Preliminaries Definition 2.. Self maps S and T of metric space (X,d) are said to be weakly commuting pair 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 of type (A) if a P a g e lim d(tsx n,ssx n ) =0 and lim d(stx n, TTx n )=0 as n whenever {x n } is Sequence in X such that lim Sx n =lim Tx n =t as n for some t in X.

2 ISSN (Paper) ISSN (Online) Vol., No., 20 Definition 2.3. A function Φ: [0, ) [0, ) is said to be a contractive modulus if Φ (0) =0 and 3.Main Result Φ (t) <t for t > 0. Theorem 3.. Let S,R,T,U and I are five self maps of a complete metric space (X,d) into itself satisfying the following conditions: (i) SR TU I( x) (ii) d( SRx, TUy) αd( Ix, Iy) + β[ d( SRx, Ix) + d( TUy, Iy)] + γ[ d( Ix, TUy) + d( Iy, SRx) for all x, y X and α, β andγ are non negative real s such that α + 2β + 2γ p (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 x 0 in X be arbitrary. Construct a sequence {Ix n } as follows: Ix 2n+ = SRx 2n, Ix 2n+2 =TUx 2n+ From condition (ii),we have n=0,,2,3 d( Ix, Ix ) = d( SRx, TUx ) αd( Ix, Ix ) + β[ d( SRx, Ix ) + d( TUx, Ix )] + γ [ d( Ix, TUx d( Ix, SRx )] 2n+ 2n+ 2 2n 2n+ 2n 2n+ 2n 2n 2n+ 2n+ 2n 2n+ 2n+ 2n = αd( Ix, Ix ) + β[ d( Ix, Ix ) + d( Ix, Ix )] + γ[ d( Ix, I ) + d( Ix, Ix )] 2n 2n+ 2n+ 2n 2n+ 2 2n+ 2n 2n+ 2 2n+ 2n+ ( α + β + γ ) d( Ix, Ix ) + ( β + γ ) d( Ix, Ix ) Therefore, we have 2n 2n+ 2n+ 2n+ 2 α + β + γ d( Ix2n, Ix2n 2) d( Ix2n, Ix2n ) + + ( β + γ ) + α + β + γ d( Ix, Ix ) hd( Ix, Ix ) where h = p ( β + γ ) i.e 2n+ 2n+ 2 2n 2n+ Similarly, we can show that d( Ix, Ix ) h d( Ix, Ix ) For k 2n+ 2n+ 2n+ 2 0 f n, we have d( Ix, Ix ) d( I, I ) n n+ k n+ i n+ i 2 P a g e

3 ISSN (Paper) ISSN (Online) Vol., No., 20 k n+ i h d I0 I i= (, ) n h d( Ixn, Ixn+ k ) d( Ix0, Ix ) 0 h 3 P a g e as n Hence { n } that Ix z. Then the subsequence of { Ix },{ SRx } and { } n i.e { SRx2n} Ix is a Cauchy sequence. Since X is complete metric space z X such TUx + also converges to z. z & { TUx2n+ } z. n 2n 2n Suppose that I is continuous and the pair {SR,I} is compatible of type (A), then from condition (ii), we have d( SR( Ix ), TUx ) αd( I x, Ix ) + β[ d( SR( Ix ), I x ) + d( TUx, Ix )] + γ 2 2 2n 2n+ 2n 2n+ 2n 2n 2n+ 2n+ 2 [ d( I x2n, TUx2n+ ) + d( Ix2n+, SR( Ix2n ))] Since I is continuous, 2 I x2n Iz as n. The pair (SR,I) is compatible of type (A), then Letting n, we have SRIx2n d( Iz, z) αd( Iz, z) + β[ d( Iz, Iz) + d( z, z)] + γ[ d( Iz, z) + d( z, Iz)] = ( α + 2 γ ) d( Iz, z) Hence d( Iz, z ) = 0 and Iz = z since α + 2γ < Again, we have Iz as n d( SRz, TUx ) αd( Iz, Ix ) + β[ d( SRz, Iz) + d( TUx, Ix )] + γ[ d( Iz, TUx ) + d( Ix, SRz)] 2n+ 2n+ 2n+ 2n+ 2n+ 2n+ Letting n and using Iz = z, we have d( SRz, z) αd( 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 Since SR I, z X such that z = SRz = Iz = Iz Now d( z, TUz ) = d( SRz, TUz ) = z since β + γ < αd( Iz, Iz ) + β[ d( SRz, Iz) + d( TUz, Iz )] + γ[ d( Iz, TUz ) + d( Iz, SRz)]

4 ISSN (Paper) ISSN (Online) Vol., No., 20 = d( z, z) + [ d( z, z) + d( TUz, z)] + [ d( z, TUz ) + d( SRz, SRz)] α β γ Implies d( TUz, z) = 0 as α + γ < Hence TUz = z = Iz Take yn = z for n, we have Now, again d( z, TUz) = d( SRz, TUz) Hence TUz = z TUyn Tz αd( Iz, Iz) + β[ d( SRz, Iz) + d( TUz, Iz)] + γ[ d( Iz, TUz) + d( Iz, SRz)] = αd( z, z) + β[ d( Iz, Iz) + d( TUz, z)] + γ[ d( z, TUz) + d( z, z)] = ( β + γ ) d( TUz, z) For uniqueness, Let z and w be the common fixed point of SR,TU and I, then by condition (ii) d( z, w) = d( SRz, TUw) αd( Iz, Iw) + β[ d( SRz, Iz) + d( TUw, Iw)] + γ[ d( Iz, TUw) + d( Iw, SRz)] = αd( z, w) + β[ d( z, z) + d( w, w)] + γ[ d( z, w) + d( w, z)] = ( α + 2 γ ) d( z, w) Since α + 2γ < we have z = w. Further assume that (S, R), (S, I), (T, U) and (T, I) are commuting pairs and z be the unique common fixed point of pairs SR, TU and I then Sz = 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 thereby Sz = z = Rz = Iz = SRz In view of uniqueness of the common fixed point of SR and I. therefore z is the unique common fixed point of S, R and I. Similarly using commutatively of (T, U) and (T, I) it can be shown that Tz = z = Uz = Iz = TUz Thus z is the unique common fixed point of S, R, T, U and I. Hence the proof. References 4 P a g e

5 ISSN (Paper) ISSN (Online) Vol., No., 20 Aage.C.T., Salunke.J.N. (2009), On Common fixed point theorem in complete metric space, International Mathematical Forum,Vol.3,pp Jungck.G. (986), Compatible mappings and common fixed points Internat. J. Math. Math. Sci., Vol. 9, pp Jungck.G., P.P.Murthy and Y.J.Cho (993), Compatible mapping of type (A) and common fixed points Math.Japonica, Vol.38, issue 2, pp Gupta Vishal (20) Common Fixed Point Theorem for Compatible Mapping of Type (A) Computer Engineering and Intelligent System(U.S.A), Volume 2, No.8, pp Khan, M.S., Pathak.H.K and George Reny (2007) Compatible mapping of type (A-) and type (A-2) and common fixed points in fuzzy metric spaces. International Mathematical Forum, 2 (): Sessa.S (986), On a weak commutativity condition in a fixed point consideration. Publ. Inst. Math., 32(46), pp P a g e

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