FIXED POINT THEOREM USING WEAK COMPATIBILITY IN MENGER SPACE

Transcription

1 ISSN International Journal of Advance Research, IJOAR.org Volume 1, Issue 10, October 2013, Online: ISSN FIXED POINT THEOREM USING WEAK COMPATIBILITY IN MENGER SPACE RAMESH BHINDE Govt.Post Graduate College Alirajpur, India ltrbhinde3@yahoo.in Abstract In this paper, I present a fixed point theorem for six self mappings in Menger space under the condition of weak compatibility, which is generalize the result of Pant and Chauhan[2]. Keywords Triangular norm, Menger space, common fixed point, weakly compatible mappings

2 ISSN Introduction Menger[7] introduced the notion of probabilistic metric space, which is a generalization of metric space.the theory of probabilistic metric space has developed in many direction especially in nonlinear analysis and application. The idea of Menger was to use distribution function F x,y instead of nonnegative real numbers as value of metric. The important development of fixed point theory in Menger space was due to Sehgal and Bharucha Reid [11].Sessa[14] introduced weakly commuting maps in metric space. Jugnck [6] enlarged this concept to compatible maps. The notion of compatible maps in Menger space has been introduced by Mishra[11].Singh and Jain [15] generalized the result of Mishra[11] using the concept of weak compatibility. In this paper, I generalize the result of B.D Pant and Sunny Chouhan [2]. 2 Preliminaries Definition 2.1 A triangular norm (shorty t-norm) is a binary operation on the unit interval [0, 1] such that for all a, b, c, d [0, 1] the following conditions are satisfied: (a) a 1 = a (b) a b = b a (c) a b c d whenever a c and b d; (d) a (b c) = (a b) c. Examples of t-norms are a b = max {a + b 1, 0} and a b = min {a, b}. Definition 2.2 (Schweizer and Sklar [16]) The ordered pair (X, F) is called a probabilistic metric space (shortly PM-space) if X is a nonempty set and F is a probabilistic distance satisfying the following conditions: for all x, y, z X and t, s > 0, (FM-0) Fx,y(t) = 1 x = y (FM-1) Fx,y(0) = 0 (FM-2) Fx,y = Fy,x; (FM-3) Fx,z(t) = 1, Fz,y(s) = 1 Fx,y(t + s) = 1. The ordered triple (X, F, ) is called Menger space if (X, F) is a PM-space, is a t-norm and the following condition is also satisfies: for all x, y, z X and t, s > 0, (FM-4) Fx,y(t + s) Fx,z(t) Fz,y(s). Definition 2.3 (Singh and Jain [15]) Self maps A and B of a Menger space (X, F, ) are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if Ax = Bx for some x X then ABx = BAx. Lemma 2.4(Singh and Jain [15]) Let {xn} be a sequence in a Menger space (X, F, ) with continuous t-norm and t t t. If there exists a constant k (0, 1) such that Fxn,xn+1(kt) Fxn 1,xn(t) for all t > 0 and n = 1, 2..., then {xn} is a Cauchy sequence in X.

3 ISSN Lemma 2.5 (Singh and Jain [15]) Let (X, F, ) be a Menger space. If there exists k (0, 1) such that Fx,y(kt) Fx,y(t) for all x, y X and t > 0, then x = y. 3 Main Result Theorem 3.1. Let A, B, S, T, L and M be self mappings on a Menger space (X, F, ) where is the min t-norm and satisfying: (3.1.1) AB(X) M(X) and ST(X) L(X) (3.1.2) either AB(X) or M(X) or ST(X) or L(X) is a complete subspace of X (3.1.3) The pairs {ST,M} and {AB, L} are weakly compatible; (3.1.4) ST = TS and AB = BA (3.1.5) either MT = TM or MS = SM and either LA = AL or LB = BL (3.1.6) there is a k (0, 1) such that ABx,ST y (ku) ABx,Lx (u) STy,My (u) Lx,My (u) FABx,My(αu) FSTy,Lx ((2 α)u) for all x, y X, for all u > 0, for all α (0, 2) and for some positive integer m. Then A, B, S, T, L and M have a unique common fixed point in X. Proof: Let x0 X. From the condition (3.1.1) x1, x2 X such that ABx0 = Mx1= y0 and STx1= Lx2 = y1 Inductively, we construct sequences {xn} and {yn} in X such that ABx2n = Mx2n+1 = y2n and STx2n+1 = Lx2n+2 = y2n+1 for n = 0, 1, 2, Using result of lemma 2.4 we can be shown that {yn} is a Cauchy sequence in X and so the subsequences {y2n} and {y2n+1} are also Cauchy in X. Case I : Suppose that either AB(X) or M(X) is a complete subspace of X. Since {y2n} AB(X) M(X), there is z X such that y2n z as n and y2n+1 z as n. Clearly, z M(X). So, there is a v X such that z = Mv. Now, taking x = x2n (where n 1), y = v and α = 1 in (3.1.6), we get that y2n,st v (ku) y2n,y2n 1(u) STv,z (u) y2n-1,z(u) Fy2n,z(u) FSTv,y2n 1(u). Now, as n,then z,stv(ku) STv,z(u) FSTv,z(u) z,stv(ku) STv,z(u) this is true for all u > 0. STv = z= Mv. Since {ST,M} is weakly compatible, it follows that MSTv = STMv i.e, Mz = STz.

4 ISSN Again,taking x = x2n, y = Mz, α = 1 and using the fact that STz = Mz in (3.1.6), we get that y2n,mz(ku) y2n,y2n 1(u) Mz,Mz(u) y2n 1,Mz(u) Fy2n,Mz(u) FMz,y2n-1(u). Now, as n,we get that z,mz(ku) z,mz(u) Fz,Mz(u) z,mz(u) this is true for all u > 0. STz =Mz = z. Since ST = TS, we have ST(Tz) = TS(Tz) = T(STz) = Tz. Now, taking x = x2n (n 1), y = Tz and α = 1 in (3.1.6), we get that y2n,t z(ku) y2n,y2n 1(u) Tz,M(Tz)(u) y2n 1,M(Tz)(u) Fy2n,M(Tz)(u) FTz,y2n 1(u). Now, suppose that MT = TM, so we have y2n,t z(ku) y2n,y2n 1(u) Tz,T z(u) y2n 1,T z(u) Fy2n,T z(u) FTz,y2n 1(u). As n,we get that z,tz(ku) z,tz(u) this is true for all u > 0. So,Tz = z. Thus Mz = Sz = Tz = z. Similar is the case when MS = SM. There we first show that Sz = z. Since ST(X) L(X), there is a w X such that z = Lw. Similarly, by taking x = w, y = x2n+1 and α = 1 in (3.1.6), we get that ABw,y2n+1(ku) ABw,z(u) y2n+1,y2n(u) z,y2n(u) FABw,y2n(u) Fy2n+1,z(u). Now, as n,we get that ABw,z(ku) ABw,z(u), for all u > 0. ABw = z= Lw. Since {AB, L} is weakly compatible, AB(Lw) = L(ABw) = Lz; i.e, ABz = Lz. Now, taking x = z, y = x2n+1, α = 1 in (3.1.6) and using ABz = Lz, we get that Lz,y2n+1(ku)

5 ISSN Lz,Lz(u) y2n+1,y2n(u) Lz,y2n(u) FLz,y2n(u) Fy2n+1,Lz(u). Now, as n,we get that, Lz,z(ku) Lz,z(u), for all u > 0. Lz = z.thus (ABz =)Lz = z. Since AB = BA, we have AB(Az) = A(BA)z = A(ABz) = Az. Suppose LA = AL, so L(Az) = (LA)z = (AL)z = A(Lz) = Az. Now, taking x = Az, y = x2n+1, α = 1 in (3.1.6) and using AB(Az) = Az, we get that Az,y2n+1(ku) Az,Az(u) y2n+1,y2n(u) Az,y2n(u) FAz,y2n(u) Fy2n+1,Az(u). Now, as n,we get that Az,z(ku) Az,z(u), for all u > 0. Az = z. Since AB = BA, we have z = (AB)z = B(Az) = Bz. Thus Az = Bz = Lz =z. Hence, Az = Bz = Lz = Mz = Sz = Tz = z. Similar is the case when LB = BL. There we first show that Bz = z. Case II : Suppose that either ST(X) or L(X) is a complete subspace of X. we first get that Az = Bz = Lz = z and then Mz = Sz = Tz = z. Hence, Az = Bz = Lz = Mz = Sz = Tz = z. i.e, z is a common fixed point for A, B, S, T, L and M. References [1]Servet Kutukcu,A fixed point theorem in Menger Space,International Mathematical Forum, 1, 2006, no. 32, [2] S. N. Mishra, Common fixed points of compatible mappings in PMspaces,ic spaces, Math. Japon., 36 (1991), [3] B. Singh, S. Jain, A fixed point theorem in Menger Space through weak compatibility, J. Math. Anal. Appl., 301 (2005), [4] G. Jungck and B. E. Rhoades, Fixed point for set valued functions without continuity, Indian. J. Pure Appl. Math.,29(3), (1988), [5] B. D. Pant and Sunny Chauhan, Fixed Point Theorems in Menger Space Using Semi- Compatibility, Int. J. Contemp. Math. Sciences,5(19) (2010), [6] K. Menger, Statistical metric, Proc. Nat. Acad. (USA), 28 (1942), [7] B. Schweizer and A. Sklar, Statistical metric space, North-Halland Seriesin Probability and Applied Math 5, North-Holland, Amsterdam(1983). [8] V.M Sehgal and Bharucha Reid, Fixed points of contraction mappings in PM-spaces, Math. Systems Theory, 6 (1972), 97{102.