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  • 1. International Refereed Journal of Engineering and Science (IRJES)ISSN (Online) 2319-183X, (Print) 2319-1821Volume 2, Issue 4(April 2013), PP.22-28www.irjes.comwww.irjes.com 22 | PageUnique Common Fixed Point Theorem for Three Pairs of WeaklyCompatible Mappings Satisfying Generalized ContractiveCondition of Integral Type1Kavita B. Bajpai, 2Manjusha P. GandhiKarmavir Dadasaheb Kannamwar College of Engineering, Nagpur, IndiaYeshwantrao Chavan College of Engineering, Wanadongri, Nagpur, IndiaAbstract: We prove some unique common fixed point result for three pairs of weakly compatible mappingssatisfying a generalized contractive condition of Integral type in complete G-metric space.The present theoremis the improvement and extension of Vishal Gupta and Naveen Mani [5] and many other results existing inliterature.Keywords: Fixed point , Complete G- metric space , G-Cauchy sequence , Weakly compatible mapping ,Integral Type contractive condition.I. IntroductionGeneralization of Banach contraction principle in various ways has been studied by many authors.One may refer Beg I. & Abbas M.[2] , Dutta P.N. & Choudhury B.S.[3] ,Khan M.S., Swaleh M. & Sessa, S.[9] ,Rhoades B.E.[12] , Sastry K.P.R. & Babu G.V.R.[13] , Suzuki T.[15] . Alber Ya.I. & Guerre-Delabriere S. [1]had proved results for weakly contractive mapping in Hilbert space , the same was proved by Rhoades B.E.[12]in complete metric space.Jungck G.[6] proved a common fixed point theorem for commuting mappings which is the extension of Banachcontraction principle. Sessa S.[14] introduced the term “Weakly commuting mappings” which was generalizedby Jungck G.[6] as “Compatible mappings”. Pant R.P.[11] coined the notion of “R-weakly commutingmappings”, whereas Jungck G.& Rhoades B.E. [8] defined a term called “weakly compatible mappings” inmetric space.Fisher B. [4] proved an important Common Fixed Point theorem for weakly compatible mapping in completemetric space.Mustafa in collaboration with Sims [10] introduced a new notation of generalized metric space called G- metricspace in 2006. He proved many fixed point results for a self mapping in G- metric space under certainconditions.Now we give some preliminaries and basic definitions which are used through-out the paper.Definition 1.1: Let X be a non empty set, and let RXXXG : be a function satisfying thefollowing properties:)( 1G 0),,( zyxG if zyx )( 2G ),,(0 yxxG for all Xyx , ,with yx )( 3G ),,(),,( zyxGyxxG  for all Xzyx ,, , with zy )( 4G ),,(),,(),,( xzyGyzxGzyxG  (Symmetry in all three variables))( 5G ),,(),,(),,( zyaGaaxGzyxG  , for all Xazyx ,,, (rectangle inequality)Then the function G is called a generalized metric space, or more specially a G- metric on X, and the pair (X, G)is called a G−metric space.Definition 1.2: Let ),( GX be a G - metric space and let }{ nx be a sequence of points of X , a point Xxis said to be the limit of the sequence }{ nx , if 0),,(lim,mnmnxxxG , and we say that the sequence }{ nxis G - convergent to x or }{ nx G -converges to x .
  • 2. Unique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfyingwww.irjes.com 23 | PageThus, xxn  in a G - metric space ),( GX if for any 0 there exists Nk  such that),,( mn xxxG , for all knm ,Proposition 1.3: Let ),( GX be a G - metric space. Then the following are equivalent:i ) }{ nx is G - convergent to xii ) 0),,( xxxG nn as niii ) 0),,( xxxG n as niv ) 0),,( xxxG mn as mn,Proposition 1.4 : Let ),( GX be a G - metric space. Then for any x , y , z , a in X it follows thati) If 0),,( zyxG then zyx ii) ),,(),,(),,( zxxGyxxGzyxG iii) ),,(2),,( xxyGyyxG iv) ),,(),,(),,( zyaGzaxGzyxG v)  ),,(),,(),,(32),,( zyaGzaxGayxGzyxG vi)  ),,(),,(),,(),,( aazGaayGaaxGzyxG Definition 1.5: Let ),( GX be a G - metric space. A sequence }{ nx is called a G - Cauchy sequence if forany 0 there exists Nk  such that ),,( lmn xxxG for all klnm ,, , that is 0),,( lmn xxxGas .,, lmnProposition 1.6: Let ),( GX be a G - metric space .Then the following are equivalent:i ) The sequence }{ nx is G - Cauchy;ii ) For any 0 there exists Nk  such that ),,( mmn xxxG for all knm ,Proposition 1.7: A G - metric space ),( GX is called G -complete if every G -Cauchy sequence is G -convergent in ),( GX .Proposition 1.8: Let (X, G) be a G- metric space. Then the function G(x, y, z) is jointly continuous in all three ofits variables.Definition 1.9 : Let f and g be two self – maps on a set X . Maps f and g are said to be commuting ifgfxfgx  , for all XxDefinition 1.10 : Let f and g be two self – maps on a set X . If gxfx  , for some Xx then x is calledcoincidence point of f and g.Definition 1.11: Let f and g be two self – maps defined on a set X , then f and g are said to be weaklycompatible if they commute at coincidence points. That is if gufu  for some Xu  , then gfufgu  .The main aim of this paper is to prove a unique common fixed point theorem for three pairs of weaklycompatible mappings satisfying Integral type contractive condition in a complete G – metric space.The result is the extension of the following theorem of Vishal Gupta and Naveen Mani [5].II. TheoremLet S and T be self compatible maps of a complete metric space (X, d)satisfying the following conditionsi) )()( XTXS ii)   ),(0),(0),(0)()()(TyTxdSySxd TyTxddttdttdtt for each Xyx , where     ,0,0: is a continuous and non decreasing function and    ,0,0: is a lower semi continuous and non decreasing function such that 0)()(  tt  ifand only if 0t also     ,0,0: is a “Lebesgue-integrable function” which is summable on each
  • 3. Unique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfyingwww.irjes.com 24 | Pagecompact subset ofR , nonnegative, and such that for each 0 , 00)( dtt . Then S and T have aunique common fixed point.III. MAIN RESULTTheorem 2.1 : Let ),( GX be a complete G-metric space and XXRQPNML :,,,,,be mappings such thati) )()( XPXL  , )()( XQXM  , )()( XRXN ii)),,(0),,(0),,(0)()()(RzQyPxGRzQyPxGNzMyLxGdttfdttfdttf  ---------------------(2.1.1)for all Xzyx ,, where     ,0,0: is a continuous and non-decreasing function ,    ,0,0: is a lower semi continuous and non-decreasing function such that0)()(  tt  if and only if 0t , also     ,0,0:f is a Lebesgue integrable functionwhich is summable on each compact subset ofR , non negative and such that for each 0 ,00)( dttfiii) The pairs ),( PL , ),( QM , ),( RN are weakly compatible.Then RQPNML ,,,,, have a unique common fixed point in X.Proof : Let 0x be an arbitrary point of X and define the sequence  nx in X such that1 nnn PxLxy , 211   nnn QxMxy , 322   nnn RxNxyConsider , ),,(0),,(02121)()(nnnnnn NxMxLxGyyyGdttfdttf   ),,(0),,(02121)()(nnnnnn RxQxPxGRxQxPxGdttfdttf   ),,(0),,(01111)()(nnnnnn yyyGyyyGdttfdttf  --------------(2.1.2)  ),,(011)(nnn yyyGdttfSince  is continuous and has a monotone property ,   ),,(0),,(021 11)()(nnn nnnyyyG yyyGdttfdttf ------------------(2.1.3)Let us take ),,(021)(nnn yyyGn dttf , then it follows that n is monotone decreasing and lower bounded sequenceof numbers.Therefore there exists 0k such that kn  as n . Suppose that 0kTaking limit as n on both sides of (2.1.2) and using that  is lower semi continuous , we get ,)()()( kkk   )(k , which is a contradiction. Hence 0k .This implies that 0n as n i.e. ),,(0210)(nnn yyyGdttf as n .----------------(2.1.4)
  • 4. Unique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfyingwww.irjes.com 25 | PageNow , we prove that  ny is a G- Cauchy sequence. On the contrary , suppose it is not aG- Cauchy sequence. There exists 0 and subsequences  )(imy and  )(iny such that for each positive integer i , )(in isminimal in the sense that ,  )()()( ,, imimin yyyG and   )()()1( ,, imimin yyyGNow ,      )()()1()1()1()()()()( ,,,,,, imimimimiminimimin yyyGyyyGyyyG   )()()1( ,, imimim yyyG  ----------------(2.1.5)Let 0     0),,(0),,(0)()()( )()()1()()()(imimin imimimyyyG yyyGdttfdttfdttfTaking i , and using (2.1.4) , we get ,  )()()( ,,0)(limimimin yyyGidttf  -------------(2.1.6)Now , using rectangular inequality , we have       )()()1()1()1()1()1()1()()()()( ,,,,,,,, imimimimimininininimimin yyyGyyyGyyyGyyyG   ----(2.1.7)       )1()1()()()()()()()1()1()1()1( ,,,,,,,,   imimimimimininininimimin yyyGyyyGyyyGyyyG ----(2.1.8)        )()()1()1()1()()1()1()()()()( ,,,,,,0,0)()(imimimimimininininimimin yyyGyyyGyyyGyyyGdttfdttfand         )1()1()1( )1()1()()()()()()()1(,,0,,,,,,0)()(imimin imimimimimininininyyyG yyyGyyyGyyyGdttfdttfTaking limit as i and using (2.1.4) , (2.1.6) we get    )1()1()1( )1()1()1(,,0,,0)()(imimin imiminyyyG yyyGdttfdttf This implies that , )1()1()1( ,,0)(limimimin yyyGidttf  ----------------------(2.1.9)Now , from (2.1.1) , we have ,      )1()1()1()1()1()1()()()( ,,0,,0,,0)()()(imiminimiminimimin yyyGyyyGyyyGdttfdttfdttf  Taking limit as i and using (2.1.6) , (2.1.8) we will have , )()()()(  which is a contradiction. Hence we have 0 .Hence  ny is a G- Cauchy sequence. Since ),( GX is a complete G-metric space , there exists a pointXu such that uynnlimi.e. uPxLx nnnn 1limlim , uQxMx nnnn 21 limlim , uRxNx nnnn 32 limlimAs uLxn  and uPxn 1 , therefore we can find some Xh such that uQh  .        111 ,,0,,0,,0,,0)()()()(nnnnnnn RxQhPxGRxQhPxGNxMhLxGMhMhLxGdttfdttfdttfdttf On taking limit as n , we get , )0()0()(,,0 MhMhuGdttf
  • 5. Unique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfyingwww.irjes.com 26 | Page 0)(,,0MhMhuGdttf , which implies that uMh  .Hence uQhMh  i.e. h is the point of coincidence of M and Q.Since the pair of maps M and Q are weakly compatible , we write QMhMQh  i.e. QuMu  .Also , uMxn 1 and uQxn 2 ,  we can find some Xv such that uPv  .        21212111 ,,0,,0,,0,,0)()()()(nnnnnnnn RxQxPvGRxQxPvGNxMxLvGMxMxLvGdttfdttfdttfdttf On taking limit as n , we get , )0()0()(,,0 uuLvGdttf 0)(,,0uuLvGdttf , which implies that uLv  . Hence we have uPvLv  i.e. v is the point ofcoincidence of L and P. Since the pair of maps L and P are weakly compatible , we can write PLvLPv  i.e.PuLu  .Again , uNxn 2 and uRxn 3 , therefore we can find some Xw such that uRw  .      RwQxPxGRwQxPxGNwMxLxG nnnnnndttfdttfdttf,,0,,0,,0111)()()( On taking limit as n , we get , )0()0()(,,0 NwuuGdttfi.e. 0)(,,0NwuuGdttf , which implies that uNw  .Thus we get uRwNw  i.e. w is the coincidence point of N and R.Since the pair of maps N and R are weakly compatible , we have RNwNRw  i.e. RuNu Now , we show that u is the fixed point of L.Consider ,       RwQhPuGRwQhPuGNwMhLuGuuLuGdttfdttfdttfdttf,,0,,0,,0,,0)()()()(      uuLuGuuLuGuuLuGdttfdttfdttf,,0,,0,,0)()()( i.e.     uuLuGuuLuGuuLuGdttfdttfdttf,,0,,0,,0)()()( i.e.   uuLuGuuLuGdttfdttf,,0,,0)()(  , which is a contradiction.  we get uLu  uPuLu  i.e. u is fixed point of L and P.Now , we prove that u is fixed point of M .Consider ,       RwQuPuGRwQuPuGNwMuLuGMuuuGdttfdttfdttfdttf,,0,,0,,0,,0)()()()(      uMuuGuMuuGMuuuGdttfdttfdttf,,0,,0,,0)()()( 
  • 6. Unique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfyingwww.irjes.com 27 | Pagei.e.   MuuuGMuuuGdttfdttf,,0,,0)()(  , which is a contradiction.  we get uMu Hence uQuMu  i.e. u is fixed point of M and Q.At last we prove that u is fixed point of N.Consider ,       RuQuPuGRuQuPuGNuMuLuGNuuuGdttfdttfdttfdttf,,0,,0,,0,,0)()()()( i.e.     RuuuGRuuuGNuuuGdttfdttfdttf,,0,,0,,0)()()( i.e.   RuuuGNuuuGdttfdttf,,0,,0)()(  , which means   NuuuGNuuuGdttfdttf,,0,,0)()(  as RuNu  .Which implies that uNu  . Hence we get uRuNu  .i.e. u is fixed point of N and R .Thus u is the common fixed point of QPNML ,,,, and R.Now , we prove that u is the unique common fixed point of QPNML ,,,, and R.If possible , let us assume that  is another fixed point of QPNML ,,,, and R.       RQuPuGRQuPuGNMuLuGuuGdttfdttfdttfdttf,,0,,0,,0,,0)()()()(    ,,0,,0)()(uuGuuGdttfdttfi.e.   ,,0,,0)()(uuGuuGdttfdttf , which is again a contradiction.Hence finally we will have u .Thus u is the unique common fixed point of QPNML ,,,, and R.Corollary 2.2: Let ),( GX be a complete G-metric space and XXPNML :,,,be mappings such thati) )()( XPXL  , )()( XPXM  , )()( XPXN ii)),,(0),,(0),,(0)()()(PzPyPxGPzPyPxGNzMyLxGdttfdttfdttf for all Xzyx ,, where     ,0,0: is a continuous and non-decreasing function ,    ,0,0: is a lower semi continuous and non-decreasing function such that0)()(  tt  if and only if 0t , also     ,0,0:f is a Lebesgue integrable functionwhich is summable on each compact subset ofR , non negative and such that for each 0 ,00)( dttfiii) The pairs ),( PL , ),( PM , ),( PN are weakly compatible.Then PNML ,,, have a unique common fixed point in X.Proof : By taking RQP  in Theorem 2.1 we get the proof.Corollary 2.3: Let ),( GX be a complete G-metric space and XXPL :, bemappings such that
  • 7. Unique Common Fixed Point Theorem for Three Pairs of Weakly Compatible Mappings Satisfyingwww.irjes.com 28 | Pagei) )()( XPXL ii)),,(0),,(0),,(0)()()(PzPyPxGPzPyPxGLzLyLxGdttfdttfdttf for all Xzyx ,, where     ,0,0: is a continuous and non-decreasing function ,    ,0,0: is a lower semi continuous and non-decreasing function such that0)()(  tt  if and only if 0t , also     ,0,0:f is a Lebesgue integrable functionwhich is summable on each compact subset ofR , non negative and such that for each 0 ,00)( dttfiii) The pair ),( PL is weakly compatible.Then PL, have a unique common fixed point in X.Proof: By substituting NML  and RQP  in Theorem 2.1 we get the proof.Remark: The Corollary 2.3 is the result proved by Vishal Gupta and Naveen Mani [5] in complete metricspace.IV. References[1] Alber Ya.I. & Guerre-Delabriere S. (1997). Principle of weakly contractive maps in Hilbert spaces, New Results in Operatortheory and its applications in I.Gohberg and Y.Lyubich (Eds.), 98, Operator Theory: Advances and Applications,(7-22).Birkhauser, Basel, Switzerland.[2] Beg I. & Abbas M.,Coincidence point and invariant approximation for mappings satisfying generalized weak contractivecondition. Fixed point theory and Appl., article ID 74503, 1-7.[3] Dutta P.N. & Choudhury B.S. (2008). A generalization of contraction principle in metric spaces. Fixed point theory and Appl.,article ID406368, 1-8.[4] Fisher B., Common Fixed Point of Four Mappings, Bull. Inst. of .Math.Academia. Sinicia, 11(1983), 103-113.[5] Gupta V. & Mani N. , A Common Fixed Point Theorem for Two Weakly Compatible Mappings Satisfying a New ContractiveCondition of Integral Type, Mathematical Theory and Modeling , Vol.1, No.1, 2011[6] Jungck G. (1976). Commuting mappings and fixed points. Amer.Math.Monthly, 83, 261-263.[7] Jungck G. (1986). Compatible mappings and common fixed points. Internat. J. Math. Sci ., 9, 43-49.[8] Jungck, G. & Rhoades, B.E. (1998). Fixed points for set valued functions without continuity. Indian J.Pure.Appl.Math., 29, No. 3,227-238.[9] Khan M.S., Swaleh M.&Sessa S. (1984). Fixed point theorems by altering distances between the points, Bull.Austral.Math.Soc.,30, 1-9.[10] Mustafa Z., Sims B., A new approach to generalized metric spaces, J.Nonlinear Convex Anal. 7 (2006), 289-297.[11] Pant R.P. (1994). Common fixed points of non commuting mappings. J.Math.Anal.Appl., 188, 436-440.[12] Rhoades B.E. (2001). Some theorems on weakly contractive maps, Nonlinear Analysis:Theory. Methods&Applications,47 (4),2683-2693.[13] Sastry K.P.R., Naidu S.V.R., Babu, G.V.R. & Naidu, G.A. (2000). Generalizations of common fixed point theorems for weaklycommuting mappings by altering distances. Tamkang J.Math., 31, 243-250.[14] Sessa S. (1982). On weak commutative condition of mappings in fixed point considerations. Publ.Inst.Math. N.S., 32, no.46, 149-153.[15] Suzuki T. (2008). A generalized Banach contraction principle that characterizes metric completeness. Proc.Amer.Math.soc.,136(5), 1861-1869.

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