Coincidences and Common Fixed Points of Weakly Compatible Mappings in Menger Spaces


  • Government Degree College, Uttarakhand, 262523, India
  • Lahore University of Management Sciences, Department of Mathematics, Lahore, 54792, Pakistan
  • R. H. Government Postgraduate College, Kashipur, Uttarakhand, 244713, India


In this paper, we prove common fixed point theorems for weakly compatible mappings in Menger spaces. An example is also given which demonstrates the validity of our result.


Triangle Norm, Menger Space, Weakly Compatible Mappings, Fixed Point.

Subject Discipline

Mathematical Sciences

Full Text:


J. Ali, M. Imdad and D. Bahuguna, Common fixed point theorems in Menger spaces with common property (E.A.), Comput. Math. Appl., 60(12) (2010), 3152–3159. MR2739482 (2011g:47124).

J. Ali, M. Imdad, D. Mihet¸ and M. Tanveer, Common fixed points of strict contractions in Menger spaces, Acta Math. Hungar., 132(4) (2011), 367–386. MR2820123.

I. Beg and M. Abbas, Common fixed points of weakly compatible and noncommuting mappings in Menger spaces, Int. J. Mod. Math., 3(3) (2008), 261–269. MR2463941 (2010d:54056)

G. L. Cain and R. H. Kasreil, Fixed and periodic points of local contraction mappings on probabilistic metric spaces, Math. Syst. Theory, 9(4) (1975/1976), 289–297. MR0458397 (56 #16600)

S. S. Chang, Y. J. Cho and S. M. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers, Inc., Huntington, New York, 2001. ISBN: 156072 MR2018691 (2004j:47143)

S. S. Chang, Y. J. Cho, B. S. Lee and G. M. Lee, Fixed degree and fixed point theorems for fuzzy mappings in probabilistic metric spaces, Fuzzy Sets and Systems, 87(3) (1997), 325–334. MR1449485 (97m:54015)

S. Chauhan and B. D. Pant, Common fixed point theorem for weakly compatible mappings in Menger space, J. Adv. Res. Pure Math., 3(2) (2011), 107–119. MR2800793

B. S. Choudhury and K. P. Das, A new contraction principle in Menger spaces, Acta Math. Sinica (English Series), 24(8) (2008), 1379–1386. MR2438308 (2009f:54053)

J. -X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Analysis, 71(5-6) (2009), 1833–1843. MR2524396 (2010g:54045)

J. -X. Fang and Y. Gao, Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Analysis, 70(1) (2009), 184–193. MR2468228 (2009k:47164)

T. L. Hicks, Fixed point theory in probabilistic metric spaces, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 13 (1983), 63–72. MR0786431 (86h:54052)

M. Imdad, J. Ali and M. Tanveer, Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces, Chaos, Solitons & Fractals, 42(5) (2009), 3121–3129. MR2562820 (2010j:54064)

G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9(4) (1986), 771–779. MR0870534 (87m:54122)

G. Jungck and B. E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29 (3) (1998), 227–238. MR1617919

K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A., 28 (1942), 535–537. MR0007576 (4,163e)

S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon., 36(2) (1991), 283–289. MR1095742

D. O’Regan and R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput., 195(1) (2008), 86–93. MR2379198

B. D. Pant and S. Chauhan, A contraction theorem in Menger space, Tamkang J. Math., 42(1) (2011), 59–68. MR2815806

B. D. Pant and S. Chauhan, Common fixed point theorems for two pairs of weakly compatible mappings in Menger spaces and fuzzy metric spaces, Sci. Stud. Res. Ser. Math. Inform., 21(2) (2011), 81-96.

B. D. Pant, S. Chauhan and Q. Alam, Common fixed point theorem in probabilistic metric space, Kragujevac J. Math., 35(3) (2011), 463-470.

A. Razani and M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos, Solitons & Fractals, 32(1) (2007), 26–34. MR2271101 (2007g:54067)

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Ser. Prob. Appl. Math. North-Holland Publ. Co., New York, 1983. ISBN: 0-444-00666-4 MR0790314 (86g:54045).

V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory, 6(1972), 97–102. MR0310858 (46 #9956)

S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.), 32(46) (1982), 149–153. MR0710984 (85f:54107)

S. Shakeri, A contraction theorem in Menger probabilistic metric spaces, J. Nonlinear Sci. Appl., 1(3) (2008), 189–193. MR2486960 (2009k:54058)

H. Sherwood, Complete probabilistic metric spaces, Z. Wahr. Verw. Gebiete, 20 (1971), 117–128. MR0317290 (47 #5837)

B. Singh and S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301 (2) (2005), 439–448. MR2105684

S. L. Singh and B. D. Pant, Common fixed points of weakly commuting mappings on non-Archimedean Menger spaces, Vikram Math. J., 6 (1985/1986), 27–31. MR0937506 (88m:54061)


  • There are currently no refbacks.