 Advances in Pure Mathematics, 2011, 1, 267-273 doi:10.4236/apm.2011.15047 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM A Common Fixed Point Theorem for Compatible Mappings of Type (C) Mancha Rangamma, Swathi Mathur, Pervala Srikanth Rao Department of Mathematics, Osmania University. Hyderabad, India E-mail: mathur.swathi@gmail.com Received May 18, 2011; revised July 2, 2011; accepted July 15, 2011 Abstract We establish a common fixed-point theorem for six self maps under the compatible mappings of type (C) with a contractive condition , which is independent of earlier contractive conditions. Keywords: Fixed Point, Compatible Mappings of Type (C), Complete Metric Space 1. Introduction The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research activity during the last two decades. Re-searchers like R. P. Pant et al. [2,3] have shown that how the three types of contractive conditions (Banach, Meir keeler and contractive gauge function/φ contractive con-dition) hold simultaneously or independent of each other and as a result of this study they have proved a fixed point theorem using Lipschitz type contractive condition  and gauge function . In this paper we generalize the result of K. Jha, R. P. Pant, S. L. Singh  and prove a fixed point theorem for six self mappings in a complete metric space. ,,,0dAxByc xyc1 (1.1) where,  11,, ,max ,,1,, ,,2nndy ydyyxykdSxTy dAxSxdByTydSxBy dAxTy or a Meir-Keeler type ,—contractive condition of the form, given ε > 0, there exists a δ > 0 such that ,xy implies ,dAxBy (1.2) or, a φ-contractive condition of the form ,dAxBy xy, (1.3) involving a contractive function φ: R+→R+ is such that φ(t) < t for each t > 0. Clearly, condition (1.1) is a special case of both conditions (1.2) and (1.3). Pant et al.  have shown the two type of contractive condition (1.2) and (1.3) are independent. The contractive conditions (1.2) and (1.3) hold simultaneously whenever (1.2) or (1.3) is assumed with additional conditions on δ and φrespectively. It follows, therefore, that the known common fixed point theorems can be extended and gen-eralized if instead of assuming one of the contractive condition (1.2) or (1.3) with additional conditions on δ and φ. we assume contractive condition  which is condition (1.2) together with the following condition of the form   1212,max[,,,,,, 2for 01, 12,dAxBykdSxTy dAxSxkdByTydSxBy dAxTykk  (1.4) instead of assuming one of the contractive conditions (1.2) or (1.3) with additional conditions on δ and φ. Definition: Two self mappings A and S of a metric space (X, d) are said to be compatible (see Jungck ) if, lim ,0nnnd ASxSAx whenever nx is a sequence in X such that lim limnnnnAxSx t for some tX. Definition: Two self mappings A and S of a metric space (X, d) are said to be compatible mappings of type (A) (See ) if lim ,0nnndASxSSx and lim ,0nnnd SAxAAx whenever nx is a sequence M. RANGAMMA ET AL. 268 in X such that li limnnnnmAxSxt for some t. XDefinition: Two self mappings A and S of a metric space (X,d) are said to be compatible mappings of type (B) (See) if, ,nx At1li,limlim ,2nn nnnnASxSSxdASdAtAAx md and ,nx St1li,limlim ,2nn nnnnSAxAAxdSAdSt SSx md whenever nx is a sequence in X such that li limnnnnmAxSxt for some t. XDefinition: Two self mappings A and S of a metric space (X,d) are said to be compatible mappings of type (C) (see ) if, lilim,nnddAt1lim, m,3lim,nn nnnnnd ASxSSxASxAtdAtAAx SSx  and limm,nnddSt A1lim ,,3lim, linn nnnnndSAxAAxSAxStdSt SSxAx   whenever nx is a sequence in X such that li limnnnnmAxSxt for some t. X0nnAxDefinition: Two self mappings A and S of a metric space (X,d) are said to be compatible mappings of type (P) (see ), if whenever lim ,ndSSx A nx is a sequence in X such that lim limnnnnAxSxt for some . From the propositions given in [4-8] all compatibility conditions are equivalent when A and S are continuous. We observe that they are independent if the functions are discontinuous. tXWe give an example which is compatible mapping of type (C) but is neither compatible nor compatible map-ping of type (A), compatible mapping of type (B) and compatible mapping of type (P). Example: Let X = [1,10] with ,dxdyxy De-fine self maps S and A of X by  1 11 15,103 15 an 1 154 510ifxifxSxif xAxifxxifx Let 15nxn for be a sequence in X. Hence 1nfor such a sequence nx both , nSx nAx converge to 1 as . nLet t = 1. Now, , , , as . The pair (S, A) is not compatible, compatible of type (A), compatible of type (B), compati-ble of type (P) but is only compatible of type (C). 1nSAx 2nASx 3nSSx 1nAAx n2. K. Jha, R. P. Pant and S. L. Singh  Proved the Following Common Fixed Point. 2.1. Theorem Let (A, S) and (B, T) be compatible pairs of self map-pings of a complete metric space (X, d) such that AXTX and (2.1.1) BX SXgiven 0 there exist 0 such that for all ,xyX ,xy implies ,dAxBy (2.1.2)  1212,max[,,,, ,,2for 01, 12.dAxBykdSxTy dAxSxkdByTydSxBy dAxTykk (2.1.3) If one of the mappings A, B, S and T is continuous then A, B, S and T have a unique common fixed point. We generalise this theorem by extending four self maps to six self maps and replacing the condition of compatibility of self maps by the compatible mapping of type (C). To prove our theorem we shall use the following lemma. 2.2. Lemma Let A, B, S, T, L and M be self mappings of (X,d) such that  ,LXSTXMXABX. (2.2.1) Assume further that given ε > 0 there exists a δ > 0 such that for all ,xyX ,Mxy implies ,dLxMy (2.2.2) where  ,max ,,1,, ,,2MxydABx STydLxABxdMySTydABxMy dLxSTy (2.2.3) If 0xX and the sequence ny in X defined by Copyright © 2011 SciRes. APM M. RANGAMMA ET AL.269 n21nthe rule 2121 22nnySTxLx (2.2.4) and 22nnyABxMx for 1, 2,3nThen we have the following for every 0, ,pqdy y implies ,pqdy y11where p and q are of opposite parity.  (2.2.5) 1lim ,0nnndy y  (2.2.6) ny is a cauchy sequence in X. (2.2.7) Proof: Since from (2.2.2) for every0 max,,, ,1(,),2dABx STydLxABxdMy STydABxMydLxSTy implies ,dLxMy for all ,xyX suppose that ,ypqdy. Putting p = 2n and in the above inequality, we have 2qm1112122 21,,,pqn mnmdyydyydLxMx  and 2212 212212 221 21221221,, ,max ,,1,, ,2,pqn mnmnnn nnn nnnndy ydyydABxSTxdABxSTxd LxABxd MxSTxdABxMxdLx STx   which implies that 112 21,,pqn mdyydLxMx  Now, for 0xX, by (2.2.3), we have Similarly, we have 21 22221,,nn nndyydyy . Thus the sequence 1,nndy y is non increasing and converges to the greatest lower bound of its range . Now we prove that t = 0 0tIf 0t, (2.2.2) implies that 12,mmdy yt whenever 1,mmtdyyt. But since ,dy y1mm converges to t, there exists a k such that 1,mmdy yt so that y t1,kktdy which by (2.2.5) implies 12,kkdy yt, which contradicts the infri-mum nature of t. Therefore, we have . 1lim ,0nnndy y We shall prove that ny is a cauchy sequence in X. In virtue of (2.2.6), it is sufficient to show that 2ny is a cauchy sequence. Suppose that 2ny is not a cauchy sequence. Then there is an 0 such that for each integer 2k, there exists even integers 2m(k) and 2n(k) with 2m(k) > 2n(k) 2k such that  22,mk nkdy y (2.2.9) For each even integer 2k, let 2m(k) be the least even integer exceeding 2n(k) satisfying (2.2.9), that is  222,nk nkdy y (2.2.10) and  22,nk mkdy y. Then for each even integer 2k, we have     22 2222221 212,,,,nk mknk nknknknk nkdy ydy ydyydyy  From (2.2.6) and (2.2.10), it follows that 22,mk nkdy y from which , we have  2212122212212 221 21221221221 22121 22221 2121 2,,,max ,,1,, ,2,max ,,1,, ,,2,nnn nnnnnn nnn nnnnnn nnnnnn nnnndyydyydLx MxdABxSTxd LxABxd MxSTxdABxMxdLx STxdy ydy ydyydy ydyydy y  22, as mk nkdy yk  . (2.2.11) From the triangle inequality, we have (2.2.8)    221 222122122 21,,,,,nk mknkmkmk mkmkmknk nkdy ydy ydy ydyydy y From (2.2.6) and (2.2.10), as k221,nk mkdy y (2.2.12) and  ,dy y2121mk mk Therefore by (2.2.2) and (2.2.4), we have Copyright © 2011 SciRes. APM M. RANGAMMA ET AL. 270    22221 2122()2()12()2 ()1,,,,,mk nknk mknkmknk nknkmkdy ydy ydyydyydLxMx (2.2.13) (Since by (2.2.5) and 112 21,,pqn mdyydLxMx  we have  22 221,,mk nknk nkdyydy y From (2.2.5), (2.2.6) and (2.2.12) as , we get k, which is a contradiction. Therefore, n2y is a cauchy sequence in X and so is ny. 2.3. Main Theorem Let A, B, S, T, L and M be self mappings of a complete metric space (X, d) satisfying (2.3.1) ,LXSTXM XABX (2.3.2) given 0 there exists a 0 such that for all ,xyX (2.3.3) ,Mxy implies ,dLxLy where ,Mxy is defined as in (5.2.3)  1212,max , ,,, ,,2for 01.dLxMyk dABxSTydLxABxdMy STykdABx MydLx STykk， (2.3.4) The pair (L, AB) and (M, ST ) be compatible mappings of type (C) (2.3.5) AB(X) is complete one of the mappings AB,ST,L and M is continuous. (2.3.6) Then AB, ST, L and M have a unique common fixed point. Further if the pairs (A, B), (A, L), (B, L), (S, T), (S, M) and (T, M) are commuting mappings then A, B, S, T, L and M have a unique common fixed point. Proof: Let 0x be any point in X. Define sequences nx and in X given by the rule n(2.3.7) 1 and y22 2nnxSTxyMxABxn for yL21 2122nn n This can be done by virtue of (2.3.2). since the con-tractive condition (2.3.3) of the theorem implies the con-tractive condition (2.2.2) and (2.2.3) of the lemma 2.2.1 so by using the lemma 2.2.1 we conclude that {yn} is a Cauchy sequence in X, but by (2.3.6) AB(X) is complete, it converges to a point z = ABu for some u in X. 0,1,2,nHence nyzX. Also its subsequences converge as follows 21nMxz and 21nSTxz; and 2nLx z22nABx z as . (2.3.8) nNow we will prove the theorem by different cases . Case (i): AB is continuous then from (2.3.8) we have 22nABABx and 2nABLx converges ABz as . (2.3.9) nSince (AB, L) are compatible mappings of type (C), we have from (2.3.9),  2222222lim,lim,1lim ,3lim, lim,1,,31lim, lim,3nnnnnnnnnnnnnnd LLxABzd LLxABLxd ABLxABzdABzABABxdABz LLxdABz ABzdABz ABzdABzLLxdABzLLx   2n (2.3.10) which shows LLx2n converges to ABz as . nNow, we show that z is the fixed point of AB. In view of (2.3.10), (2.3.8), (2.3.4) and (2.3.9) 221122122 212122212 2112,lim ,lim max,,,,,2max,,, ,,,2nnnnnnn nnnnn ndABzzdLLx Mxkd ABLxSTxd LLxABLxd MxSTxkdABLxMx dLLxSTxkdABzzd ABzzdzzkdABzz dABz,(2.3.11) ,dABz z a contradiction if ABz z yielding there-fore ABzz. Now, we show that z is also a fixed point of L. In view of (2.3.8), (2.3.4) and (2.3.11)  2112122121212111,lim ,lim max,,,, ,2,max,,, ,,,2nnnnnn nndLzz dLzMxkdABz STxdLzABzkdMx STxdABzMxdLzSTxkdzzdLzzd zzkdzz dLz  (2.3.12) Copyright © 2011 SciRes. APM M. RANGAMMA ET AL.271 ,dLzz, a contradiction if Lz zImplying there by . Lz zThus ABz Lz z. Since there exist  LX STXyX such that ZLZ STySTy . we prove. MyIn view of (2.3.12) and (2.3.4) 1212,,max ,,,,,,2max, ,,,,2dSTyMydLzMykdABzSTy dLzABzdMySTykdABzMy dLzSTykdSTySTy dzzdMySTykdSTyMydzz (2.3.13) ,dSTyMy a contradiction if STy MyTherefore MySTy. Hence we have MyLzABzzSTy Now, taking a sequence {zn} in X such that zn = y  n 1, it follows that nMzMyz and as n. nSTzSTy z since the pair (M, ST) is compatible of type (C) 1lim ,lim ,3lim ,lim ,nn nnnnnnnd STMzMMzdSTMzSTzdSTz MMzdSTz STSTz    (2.3.14) That is 1,,3,,dSTMyMMy dSTMySTzdSTz MMydSTz STSTy  which implies in view of the fact that My = z = ST y 1,,3,1,,3dSTzMzdSTzSTz dSTzMzdSTzSTzdSTzMz dSTzMz, Therefore . STz MzHence we have ; STz MzABz Lz z. (2.3.15) Now,we show that z is a fixed point of ST. In view of (2.3.15) and (2.3.4) 1212,,max ,,,,2max ,,,,,2dzSTz dzMzkdABzSTz dLzABzkdMz STzdABz MzdLz STzkdzSTz dzz dSTzSTzkdzSTz dzSTz  ,(2.3.16) ,dzSTz a contradiction if STz zTherefore z = STz. Hence zSTzABzLzMz which shows that z is a -common fixed point of AB, ST, L and M. Case(ii): L is continuous From (2.3.8) we have Since 2nLLx and 22nLABx converges to Lz as . (2.3.17) nsince (L,AB) are compatible mappings of type (C),we have from (2.3.17)   2222222lim,lim,1lim ,3lim ,lim ,1lim ,lim,3lim ,1lim ,3nnnnnnnnnnnnnnnndABABx LzdABABx LABxd LABxLzdLzLLx dLzABABxdLzLz dLzLzdLz ABABxdLz ABABx   2n(2.3.18) which shows 2nABABx Lz as . nNow, we show that z is a fixed point of L. In view of (2.3.17), (2.3.8), (2.3.4) and (2.3.18) 221122122212122212 2112,lim ,lim max,,,,,2max,,, ,,,2nnnnnnnn nnnnn ndLzzdLABx Mxkd ABABxSTxd LABxABABxdMxSTxkdABABxMxd LABxSTxkdLz zdLzLzdz zkd Lzzd Lzz, (2.3.19) ,dLzzLz a contradiction if yielding therefore Lz. Since LX STX there exist u such that XzLzSTu. Copyright © 2011 SciRes. APM M. RANGAMMA ET AL. 272 We prove that . STu MuNow, In view of (2.3.8) and (2.3.4)  212 2222212,lim ,max ,,,,,,2max,,, ,,,2nnnnnndzMudLx MukdABx STudLx ABxdMuSTukd ABxMud LxSTukdzMu dzz dMuSTukdzMu dzMun (2.3.20) ,dzMu a contradiction if zMuThus Muz. Therefore ,we have . zLzSTuMu Now, taking a sequence in X such that nznzu , it follows that 1nnMzMuz and as nSTzSTu z nsince (M, ST) are compatible mappings of type (C), we get 1lim ,lim ,3lim ,lim ,nn nnnnnnndSTMzMMzdSTMzSTzdSTz MMzdSTz STSTz   (2.3.21) That is 1,,3,,dSTMu MMudSTMu STzdSTz MMudSTz STSTu ] which implies in view of the fact that MuzSTu 1,,3,,1,,3dSTzMzdSTzSTzdSTzMzdSTzSTzdSTzMzdSTzMz  which shows that. STz MzNow, we show that z is also a fixed point of M In view of (2.3.8) and (2.3.4)  ,dzMz a contradiction if zMzwhich shows that zMz. Since  MxABx there exist a vX such that zMzABv. We prove zLv, from (2.3.4) we have  1212,,max ,,,, ,2,max ,,,,,,2dLvz dLvMzkdABvSTzdLvABvkdMzSTzv dABvMzdLvSTzkdzzdLvz dzzkdzz dLvz (2.3.23) ,dLvz a contradiction if LvzThus Lvz Now, taking a sequence nv in X such that nvv 1n, it follows that and nnLv zLvABvABv z as , since (L, AB) are com-patible mappings of type (C), we have n1lim ,lim,3lim ,lim ,nn nnnnnnnd ABLvLLvd ABLvABzd ABzLLvdABzABABv    (2.3.24) That is 1,,3,,ndABLv LLvdABLv ABzdABz LLvdABz ABABv  which implies in view of the fact that (L)v = z = (AB)v d(ABz, Lz) 1,,3dABz ABzABz Lzd 1,,3d ABzABzdABzLz 212 2222212,lim ,max ,,,, ,,2max,,,,,2nnnnnnndzMzdLx MzkdABx STzdLxABxkdMzSTzdABx MzdLx STzkd zMzdzzdMzMzkdzMz dzMzwhich shows ABz = Lz Since ABz = Lz = z also z = Mz = STz. ,(2.3.22) If the mappings M or ST is continuous instead of L or AB then the proof that z is a Common fixed point of L, M, AB, and ST is similar. Uniqueness: Let w be another common fixed point of L, M, AB, and ST then Lw = Mw = ABw = STw = w. From (2.3.4) we have Copyright © 2011 SciRes. APM M. RANGAMMA ET AL. Copyright © 2011 SciRes. APM 273  1212,,max ,,,,,2,max,,, ,,,2dzwdLzMwkdABzSTw dLzABzkdMwSTwdABzMwdLzSTwkdzw dzzdwwkdzwdzw (2.3.25) Let 11nxn for 1nThen as . 1nxn,0 iff 1nn ndABLxLLx x ,0 iff nnd ABLxABtx1 ,0 iff dABt ABABxx1nn ,0 iff dABt LLxx1nnn, ,,nnnABx STxnLx Mxt converges to 1 as . XThe pairs (L, AB) and (M, ST) are compatible map-pings of type (c) and also satisfies the conditions (2.3.2), (2.3.3), (2.3.4), (2.3.5) and (2.3.6) ,dzw a contradiction if zwyielding there by zw. Remarks: Main theorem remains true if we replace condition compatible mappings of type (C) by Finally we need to show that z is a common fixed point of L, M, A, B, S and T. 1) compatible mappings of type (A) or For this let z is the unique common fixed point of (AB, L) and (ST, M). 2) compatible mappings of type (B) or 3) compatible mappings of type (P) Since (A, B), (A, L), (B, L) are commutative ;AzAABzA BAzABAzAzALzLAz 3. References ;Bz BABzBABzABAzBz BLz LBz .  K. Jha, R. P. Pant and S. L. Singh, “On the Existence of Common Fixed Point for Compatible Mappings,” Journal of Mathematics, Vol. 37, 2005, pp. 39-48. which shows that Az, Bz are common fixed points of (AB, L) yielding there by Az Z Bz LzABz  in the view of uniqueness of common fixed point of the pairs (AB, L) .  R. P. Pant, P. C. Joshi and V. Gupta, “A Meir-Keelar Type Fixed Point Theorem,” Indian Journal of Pure & Applied Mathematics, Vol. 32, No. 6, 2001, pp. 779-787. Similarly using the, commutativity of (S,T), (S,M) and (T,M) it can be shown that Sz = z = Tz = Mz = STz.  R. P. Pant, “A Common Fixed Point Theorem for Two Pairs of Maps Satisfying the Condition (E.A),” Journal of Physical Sciences, Vol. 16, No. 12, 2002, pp. 77-84. Now, we need to show that Az = Sz (Bz = Tz) also re-mains a common fixed point of both the pairs (AB, L) and (ST, M).  G. Jungck, “Compatible Mappings and Common Fixed Points,” International Journal of Mathematics and Mathe- matical Sciences, Vol. 9, 1986, pp. 771-779. From (2.3.4) we have  1212,,max ,,,, ,,2max,,,,,,02dAzSz dLzMzkdABz STzdLzABzkdMzSTzdABzMzdLzSTzkd zzd zzdzzkdzz dzz   G. Jungck, P. P. Murthy and Y. J. cho, “Compatible Map-pings of Type(A) and Common Fixed Point Theorems,” Mathematica Japonica, Vol. 38, No. 2, 1993 pp. 381-390.   H. K. Pathak and M. S. Khan, “Compatible Mappings of Type (B) and Common Fixed Point Theorems of Gregus Type,” Czechoslovak Mathematical Journal, Vol. 45, No. 120, 1995, pp. 685-698  H. K. Pathak, Y. J. cho, S. M. Kang and B. Madharia, “Compatible Mappings of Type (C) and Common Fixed Point Theorems of Gergus Type,” Demonstratio Mathe-matica, Vol. 31, No. 3, 1998, pp. 499-518. implies that Az = Sz.  H. K. Pathak, Y. J. Cho, S. S. Chang, et al., “Compatible Mappings of Type (P) and Fixed Point Theorem in Metric Spaces and Probabilistic Metric Spaces,” Novisad. Jour-nal of Mathematics, Vol. 26, No. 2, 1996, pp. 87-109. similarly it can be shown that Bz = Tz. Thus z is the unique common fixed point of A, B, S, T, L and M. This establishes the theorem. Now we give an example to claim our result.  J. Jachymski, “Common Fixed Point Theorem for Some Families of Mappings,” Indian Journal of Pure & Ap-plied Mathematics, Vol. 25, 1994, pp. 925-937. Example: Let 1,X with,dxyxy. De-fine self maps A, B, S, T, L and :MXX by Lx x, 2Mxx, , 221xSTAB 421x1, x