International Journal of Scientific & Technology Research

IJSTR@Facebook IJSTR@Twitter IJSTR@Linkedin
Home About Us Scope Editorial Board Blog/Latest News Contact Us

IJSTR >> Volume 2- Issue 2, February 2013 Edition

International Journal of Scientific & Technology Research  
International Journal of Scientific & Technology Research

Website: http://www.ijstr.org

ISSN 2277-8616

Common Fixed Point Theorems For Finite Number Of Mappings Without Continuity And Compatibility In Menger Spaces

[Full Text]



Dr. Aradhana Sharma





Theorem 1 : Let A, B, S, T, I, J, L, U, P and Q be self maps on a Menger space (X, F, t) with t(a, a) ≥ a for all a  [0, 1], satisfying (1.1) P(X)  ABIL(X), Q(X)  STJU(X), (1.2) {P, STJU} or {Q, ABIL} satisfies the property (S-B), (1.3) there exists k  (0, 1) such that FPx,Qy(ku) ≥ min {FABILy,STJUx(u),FPx,STJUx(u), FQy,ABILy(u), FQy,STJUx(u), FPx,ABILy(u)} for all x, y  X and u > 0, (1.4) if one of P(X), ABIL(X), STJU(X) or Q(X) is a closed subspace of X then (i) P and STJU have a coincidence point and (ii) Q and ABIL have a coincidence point. Further if (1.5) AB = BA, AI = IA, AL = LA, BI = IB, BL = LB, IL = LI, QL = LQ, QI = IQ, QB = BQ, ST = TS, SJ = JS, SU = US, TJ = JT,TU = UT, JU = UJ, PU = UP, PJ = JP, PT = TP, (1.6) the pairs {P, STJU} and {Q, ABIL} are weakly compatible. Then A, B, S, T, I, J, L, U, P and Q have a unique common point in X.



[1]. Dedeic, R. and Sarapa, N.: A common fixed point theorem for three mappings on Menger spaces, Math. Japonica, 34(1989), 919-923.

[2]. Jungck, G.: Commuting mappings and fixed points, Amer. Math. Monthly, 83(1976), 261-263.

[3]. Jungck, G.: Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9(1986), 771-779.

[4]. Jungck G., and Rhoades, B. E.: Fixed point for set valued functions without continuity, Ind. J. Pure Appl. Maths., 29(3) (1998), 227-238.

[5]. Menger, K.: Statistical Metric, Proc. Nat. Acad. Sci. U. S. A., 28(1942), 535-537.

[6]. Mishra, S.N.: Common fixed points of compatible mappings in PM spaces, Math. Japonica, 36(2)(1991),283-289

[7]. Schweizer, B. and Sklar, A.: Statistical metric spaces, Pacific. J. Math., 10(1960), 313-334.

[8]. Schweizer, B. and Sklar, A.: Probabilistic metric spaces, Amsterdam; North Hollend, 1983.

[9]. Sessa, S.: On weak commutativity condition of mappings in a fixed points considerations, Publ. Inst. Mat., 32(46) (1982), 149-153.

[10]. Sharma, Sushil, Deshpande, B. and Tiwari, R.: Common fixed point theorems for finite number of mappings without continuity and compatibility in Menger spaces, J. Korean Soc. Math. Edu.,Pure Appl.Math. Vol.15, No. 2(2008), 135-151.

[11]. Sharma, Sushil and Bamboria, D.: Some new common fixed point theorems in fuzzy metric space under strict contractive conditions, J. Fuzzy Math., Vol. 14, No.2 (2006),1-11.