Global Journal of Pure an Applie Mathematics. ISSN 0973-1768 Volume 13 Number (017) pp. 43-49 Research Inia Publications http://www.ripublication.com Common Fixe Point Theorem for Six Mappings Qamrul Haque Khan Department of Mathematics A.M.U. Aligarh 000 Inia. Abstract Using the notion of reciprocally continuous we obtaine a result common fixe point theorem for six mappings which generalize the results of M.Kulkarni an V.H. Bashah Bijenra Singh an others. Keywors: Compatible mappings weakly compatible reciprocally continuous fixe point. Subject Classification: Primary 47H10 Seconary 54H5. 1. INTRODUCTION In this note we prove a common fixe point theorem for the help of compatible mappings [3] an reciprocally continuous mappings [8]. Here we enlist the some efinitions which are uses in our result. Definition 1.1 [3] Two self- mappings S an T of a metric spaces (X ) is sai to be compatible iflimn (STxn TSxn) = 0 whenever {xn} is a sequence in X such that limnsxn = limntxn = z for some z in X.
44 Qamrul Haque Khan Definition 1.[8] Two self -mappings S an T of a metric spaces (X ) is sai to be reciprocally continuous if limnstxn =Sz an limntsxn = Tz whenever {xn} is a sequence in X such that limnsxn = limntxn = z for some z in X. Definition 1.3[5] Two self- mappings S an T of a metric spaces (X ) is sai to be compatible of type (A) if limn (STxn TTxn) = 0 an limn (TSxn SSxn) = 0 whenever {xn} is a sequence in X such that limnsxn = limntxn = z for some z in X. Definition 1.4[9] Two self- mappings S an T of a metric spaces (X ) is sai to be compatible of type (B) if limn (STxn TTxn) ½ [limn (STxn Sz) + limn (Sz SSxn)] an limn (TSxn SSxn) ½ [limn (TSxn Tz) + limn (Tz TTxn)] whenever {xn} is a sequence in X such that limnsxn = limntxn = z for some z in X. Definition 1.5[10] Two self- mappings S an T of a metric spaces (X ) is sai to be compatible of type (p) on X if limn (SSxn TTxn) = 0 whenever {xn} is a sequence in X such that limnsxn = limntxn = z for some z in X. Definition 1.6 [] Two self- mappings S an T of a metric spaces (X ) is sai to be weakly compatible of type(p) limn (SSxn TTxn) (SzTz) limn (Tz TTxn) limn (SSxn TTxn) (SzTz) limn (Sz SSxn) whenever {xn} is a sequence in X such that limnsxn = limntxn = z for some z in X. Definition 1.7[4] A pair of self- mappings (ST) of a metric space (X) is sai to weakly compatible if (ST) commutes only at their coincience points of S an T. Kulkarni [6] given the followings result.
Common Fixe Point Theorem for Six Mappings 45 Theorem 1.8 [6] Let S A an T be the three continuous self-mappings of a complete metric space (X ) such that the following conition are satisfy. ST = TS SA = AS S(X) A(X) S(X) T(X) Sx Sy a Ax Sx Ty Sx Ax Sy Ty Sx b Ax Sx Ty Sy Ax Sy Ty Sy for all x y in X where a b are non- negative reals satisfying a+ b < 1 a 0 b 0. Thus S A an T have a unique common fixe point.. MAIN RESULT Theorem.1. Let A B S T I an J be the self-mappings of a complete metric space (X ) satisfying AB(X) J (X) ST(X) I (X) an for each xy X where a b are non- negative reals satisfying a+b < 1 a 0 b 0 either ABx STy a Ix ABx Jy ABx Ix STy Jy ABx b Ix ABx Jy STy Ix STy Jy STy] (.1.1) (a) Moreover if {ABI} are compatible (AB I) is reciprocally continuous an (ST J) are weakly compatible or (a') If {ST J} are compatible (ST J) is reciprocally continuous an (AB I) are weakly compatible. Then AB ST I an J have a unique common fixe point. Furthermore if the pairs (AB) (AI) (BI) (SJ) an (TJ) commute at the earlier common fixe point then A B S T I an J also have the same unique common fixe point. Proof. Let x0 be an arbitrary point in X. Since AB(X) J(X) we can fin a point x1 in X such that ABx0 = Jx1. Also since ST (X) I(X) we can choose a point x with STx1 = Ix. Using this argument repeately one can construct a sequence {zn} such that zn = ABxn = Jxn+1 zn+1 = STxn+ for n = 01.. for the sake of brevity let us put z z STx ABx n1 n n1 n
46 Qamrul Haque Khan a b Ix ABx Jx ABx Ix STx Jx ABx n Ix ABx Jx Ix STx Jx STx n a n n n1 n1 n n n n1 n1 n1 n1 n1 z z z z b z z z z z n n1 n n n n1 n n1 a b 1 a z n zn 1 n1 zn Similarly one can show that z a b 1 a z n zn 1 n1 zn Thus for every n we have z z k z z n n1 n1 n n which shows that {zn} is a Cauchy sequence in the complete metric space (X) an so has a limit point z in X. Now it follows that the sequence ABx n Jxn 1 an STx n1 Ix n which are subsequence also converge to the point z. Since the mappings (AB I) are reciprocally continuous therefore lim n ABIx n ABz an lim n IABx n Iz. Compatibility of (ABI) yiels that lim ( ABIx IABx ) 0 i.e. (ABz Iz) = 0. Hence ABz=Iz. Now n n n ABz STx a Iz ABz Jx ABz Iz STx Jx ABz n1 n1 n1 n1 b Iz ABz Jx STx Iz STx Jx STx taking limit n reuces to n1 n1 n1 n1 n1 ABz z a ABz z yieling thereby ABz = z = Iz.
Common Fixe Point Theorem for Six Mappings 47 Since AB (X) J(X) always exists a point z' in X such that Jz' = z. Now ABz aiz ABz Jz' ABz Iz Jz' ABz b[ Iz ABz Jz' Iz Jz' z a b z STz ' yieling thereby = Jz'. But since (STJ) are weakly compatible therefore Now STz = ST(Jz') = J () = Jz. ABz STz a Iz ABz Jz ABz Iz STz Jz ABz b Iz ABz Jz STz Iz STz Jz STz yieling thereby STz = Jz = z. z STz a b z STz Let v be another common fixe point of AB ST I an J. ABz STv a Iz ABz Jv ABz Iz STv Jv ABz b Iz ABz Jv STv Iz STv Jv STv. yieling thereby z = v. Finally we nee to show that z is also a common fixe point of ABSTI an J.For this let z be a common fixe point of the pair (AB I) then Az = A (ABz) = A (BAz) = AB (Az) Az = A (Iz) = I (Az) Bz = B (ABz) = BA (Bz) = AB (Bz) Bz = B (Iz) = I (Az) Which shows that Az an Bz is a common fixe point of (AB I) yieling thereby Az = Bz = Iz = ABz = z in view of the uniqueness of the common fixe point of the pair (ABI). Similarly we can show by using commutativity of (ST) (SJ) (TJ) it can show that Sz = z = Tz = Jz = STz. Thus z is the unique common fixe point of A B S T I an J. Theorem..Theorem.1 remains true if we replace compatibility conition by (a) Compatibility of type (A) or (b) Compatibility of type (B) or
48 Qamrul Haque Khan (c) Compatibility of type (P) or () Weak compatibility of type (P). Proof. The proof is ientical except minor changes where the particular compatibility conition is be use which is also straightforwar by using particular efinition. Hence we omit the proof. Relate Example. We construct example to emonstrate the valiity of the hypotheses an egree of generality of our main result. Example: Consier X = [01] with the usual metric. Define self-mappings ABSTI an J as Ax = x/3.bx = 3x/4Sx = x/4tx = 4x/3Ix = x/4 an Jx= 3x/4.Clearly AB(X) = [01/] J(X) = [03/4]ST(X) = [01/5] IX) = [01/4].Also the pairs of mappings (AB)(ST)(AB)(BI)(SJ) an(tj) are commuting hence compatible or weak compatible. For all x y in X(x> y) with a = 1/0 an b = 1/5 ( x y 5 ) a [( x 4 x ) (3y 4 x ) + (x 4 y 5 ) (3y 4 x )] +b [( x 4 x ) (3y 4 x ) + (x 4 y 5 ) (3y 4 x )] a ( 3y 4 x ) (x y 5 ) + b (3y 5 y 5 ) (x y 5 ) x/> x/4 > y/5 an 3y /4 > y/5 one can get ( x y 5 ) (a + b) ( x y 5 ) which verifies the contraction conition.1.1. Clearly 0 is the unique common fixe point of A B S T I an J. REFERENCES [1] V.H.Bashah an Bijenra Singh On common fixe points of commuting mappings Vikram Mathematical Journal Vol. V.(1984) 13-16. [] M. Ima an Q.H. Khan A General common fixe point theorem for six mappings.the Aligarh Bulletin of Mathematics Vol 19(000) 3-47. [3] G.Jungck Compatible mappings an common fixe points Internet J. Math. Math. Sci. 9 (4) (1986) 771-779.
Common Fixe Point Theorem for Six Mappings 49 [4] G.Jungck an B.E.Rhoaes Fixe point for set-value function without continuity Inian. J. Pure Appl. Math. 30 () (1999) 147-15. [5] G. Jungck P.P. MurthyY. J. Cho Compatible mappings of type (A) an common fixe point theorems Math. Japan 38 (1993) 381-390. [6] M. Kulkarni Commuting fixe point theorem for three mappings The Mathematical Eucation vol. no. June (1993) 14-131. [7] S.N. Lal P.P. Murthy an Y.J. Cho An extension of TELCI TAS an Fisher s Theorem. J. Korean. Math. Soc. 33 (1996) 891-908. [8] R.P. Pant A common fixe point theorem uner a new conition Inian J. Pure Appl. Math. 30 () (1999) 147-15. [9] H.K. Pathak an M.S. Khan Compatible mappings of type (B) an common fixe point theorems of Gregus type Czech Math. Jour 45 (10) (1995) 685-698. [10] H.K. Pathak Y.J. Cho S.S. Chang an S.M. Kang Compatible mappings of type (P) an fixe point theorem in metric spaces an probabilistic metric spaces Novi Sa. J. Math. 6 () (1996) 87-109.
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