IJSER Home >> Journal >> IJSER
International Journal of Scientific and Engineering Research
ISSN Online 2229-5518
ISSN Print: 2229-5518 3    
Website: http://www.ijser.org
scirp IJSER >> Volume 1, Issue 3, December 2010
On Common Fixed Point For Compatible mappings in Menger Spaces
Full Text(PDF, 3000)  PP.  
M. L. Joshi, Jay G. Mehta
Common fixed point, menger space, compatible maps, weakly compatible maps.
In this paper the concept of compatible map in menger space has been applied to prove common fixed point theorem. A fixed point theorem for self maps has been established using the concept of compatibility of pair of self maps.In 1942 Menger has introduced the theory of prob-abilistic metric spaces in which a distribution function was used instead of non-negative real number as value of the metric. In 1966, Sehgal initiated the study of contraction mapping theorems in probabilistic metric spaces. Since then several generalizations of fixed point Sehgal and Bharucha-Reid, Sherwood, and Istrat-escu and Roventa have obtained several theorems in probabilistic metric space. The study of fixed point theorems in probabilistic metric spaces is useful in the study of existence of solutions of operator equations in probabilistic metric space and probabilistic functional analysis. In 2008, Altun and Turkoglu proved two common fixed point theorems on complete PM-space with an implicit relation.
[1] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 535–537.

[2] V.M. Sehgal, Some fixed point theorems in function analysis and probability, Ph.D dissertation, Wayne State Univ. Michigan (1966).

[3] V.M. Sehgal and A.T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory 6 (1972), 97–102.

[4] H. Sherwood, Complete probabilistic metric spaces, Z. wahrscheinlichkeits theorie and verw. Grebiete 20 (1971), 117–128.

[5] V.I. Istratescu and I. Sacuiu, Fixed point theorem for contraction mappings on probabilistic metric spaces, Rev. Roumaine Math. Pures. Appl. 18 (1973), 1375–1380. [

6] I. Altun and D. Turkoglu, Some fixed point theorems on fuzzy metric spaces with implicit relations, Commun. Korean Math. Soc., 23(1)(2008), 111-124.

[7] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland(Amsterdam, 1983).

[8] S.L. Singh, B.D. Pant and R. Talwar, Fixed points of weakly commuting mappings on Menger spaces, Jnanabha 23 (1993),115–122.

[9] S. Kumar and R. Chugh, Common fixed point theorems using minimal commutativity and reciprocal continuity conditions in metric spaces, Sci. Math. Japan 56 (2002), 269–275.

[10] D. Mihet, A generalization of a contraction principle in probabilistic metric spaces, Part II, Int. J. Math. Math. Sci. 2005 (2005), 729–736.

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

[12] A. Jain and B. Singh, Common fixed point theorem in Menger space through compatible maps of type (A), Chh. J. Sci. Tech. 2 (2005), 1-12.

[13] B. Singh and S. Jain, Semi-compatibility and fixed point theorem in Menger space, Journal of the Chungcheong Mathematical Society 17 (1), (2004), 1-17.

[14] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Trend in Logic 8, Dordrecht (2000).

[15] R. A. Rashwan & A. Hedar: On common fixed point theorems of compatible mappings in Menger spaces. Demonstratio Math. 31 (1998), no. 3, 537–546.

[16] G. Jungck & B. E. Rhoades: Fixed points for set valued functions without continuity. Indian J. Pure Appl. Math. 29 (1998), no. 3, 227–238.

Untitled Page