g- reciprocal Continuity in Probabilistic Metric Spaces

Arvind Bhatt
1.290 546

Abstract


In the present paper we obtain a common fixed point theorem by
employing the notion of g- reciprocal continuity in probabilistic metric space.
We demonstrate that g- reciprocal continuity ensures the existence of common
xed point under strict contractive conditions, which otherwise do not ensure
the existence of fixed points.


Keywords


Probabilistic metric space, fixed point theorem, g- reciprocal continuity, noncompatible mappings, g- compatible.

Full Text:

PDF

References


Menger, K. Statistical metrices, Nat. Acad.Sci.USA. 28(1942) 535-537.

Schweizer, B. and Skalar, A. Probabilistic metric spaces, Pacic J. of Math.10(1960)

-324.

Sehgal, V.M. Some fixed point theorems in functional analysis and probability. Ph.D.

Dissertation, Wayne State Univ. Michigan, (1966).

Kannan, R. Some results on fixed points, Bull. Calcutta Math. Soc. 60(1968) 71-76.

Egbert, R.J. Products and quotients of probabilistic metric spaces, Pacic J.Math.,

(1968) 437-455.

Sehgal, V.M. and Bharucha-Reid, A.T. Fixed points of contraction mappings on Prob-

abilistic metric spaces, Math. Systems Theo. 06(1972) 97-102.

Bharucha-Reid, A.T. Fixed point theorems in Probabilistic analysis, Bull.Amer. Math.

Soc. 82(1976) 641-657.

Jungck, G. Commuting mappings and xed points, Amer. Math. Month. 73(1976) 261-

Sessa, S. On a weak commutativity condition of mappings in xed point considerations,

Publ. Inst. Math. 32(1982) 149-153.

Schweizer, B. and Skalar, A. Statistical metric spaces, North Holland Amsterdam,

(1983).

Jungck, G. Compatible mappings and common xed points, Int.J.Math.and Math.Sci.

(1986) 771-779.

Jungck, G. Commuting mappings and xed points, Amer.Math.Monthly, 83(1976)261-

Mishra, S.N. Common fixed points of compatible mappings in PM-spaces, Math. Japon.

(1991) 283-289.

Jungck, G. and Pathak, H.K., Fixed points via biased maps, Proc. Amer. Math. Soc.

(1995) 2049-2060.

Jungck, G., Common xed points for noncontinuous nonself maps on nonmetric spaces,

Far East J. Math. Sci. 04(1996) 199-215.

Pathak, H.K. and Khan, M.S., A comparison of various types of compatible maps and

common xed points, Indian J. Pure Appl. Math. 28 no. 4,(1997), 477-485.

Pant, R.P., Common fixed points of four maps, Bull. Calcutta Math.Soc. 90(1998)

-286.

Naschie, MS.EL., On the uncertainty of Cantorian geometry and two-slit experiment,

Chaos Solitons and Frac. 09(03)(1998) 517-529.

Pant, R.P., Discontinuity and xed points, J. Math. Anal. Appl. 240 (1999), 284-289.

Naschie, MS.EL., On the verications of heterotic string theory and 1, Chaos, Solitons

and Frac. 232(2000) 397-407.

Pant, R.P.and Pant, V., Common xed points under strict contractive conditions,

J.Math.Anal.Appl.248(2000) 327-332.

Hadzic, O. and Pap, E.,Fixed Point Theory in Probabilistic Metric Spaces, Kluwer

Academic Publishers, 2001.

Aamri, M. and Moutawakil, D. El., Some new common xed point theorems under

strict contractive conditions, J. Math. Anal. Appl. 270(2002) 181-188.

Chandra, H. and Bhatt, A., Fixed point theorems for occasionally weakly compat-

ible maps in probabilistic semi-metric space, International Journal of Mathematical

anal.03(2009) 563-570.

R. P. Pant and R. K. Bisht, Common fixed points of pseudo compatible mappings, Re-

vista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas,

DOI 10.1007/s 13398-013-0119-5.