Orbital Continuity and Common Fixed Points in Menger PM-Spaces

Download PDF


DOI: 10.46793/KgJMat2307.1011N


In this paper, we prove that if a pair of semi R-commuting self-mappings defined on Menger PM-spaces with a nonlinear contractive condition posses a unique common fixed point, then these mappings are orbitally continuous. Also, we investigate whether this assertion and it converse holds if we replace semi R-commutativity with some other concept of commutativity in the weaker sense.


Orbital continuity, probabilistic metric spaces, common fixed point, nonlinear contractive condition.


[1]   S. Banach, Sur la opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math. 3 (1922), 133–181.

[2]   D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464.

[3]   Y. J. Cho, P. P. Murthy and M. Stojaković, Compatible mappings of type (A) and common fixed points in Menger spaces, Comm. Korean Math. Soc. 7(2) (1992), 325–339.

[4]   L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math. 12(26) (1971), 19–26.

[5]   J. X. Fang and Y. Gao, Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Anal. 70 (2009), 184–193.

[6]   K. Goebel, A coincidence theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 733–735.

[7]   S. N. Ješić, D. O’Regan and N. A. Babačev, A common fixed point theorem for R-weakly commuting mappings in probabilistic spaces with nonlinear contractive conditions, Appl. Math. Comput. 201(1-2) (2008), 272–281.

[8]   S. N. Ješić, R. M. Nikolić and R. P. Pant, Common fixed point theorems for self-mappings in Menger PM-spaces with nonlinear contractive condition, J. Fixed Point Theory Appl. 20(90) (2018), 2–11.

[9]   G. Jungck, Commuting maps and fixed points, Amer. Math. Monthly 83 (1976), 261–263.

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

[11]   G. Jungck and B. E. Rhoades, Fixed point for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (1996), 227–238.

[12]   G. Jungck, Common fixed points for noncontinuous nonself mappings on noncomplete spaces, Far East J. Math. Sci. 4(2) (1996), 199–212.

[13]   R. Machuca, A coincidence theorem, Amer. Math. Monthly 74 (1967), 569–572.

[14]   K. Menger, Statistical metric, Proc. Nat. Acad. Sci. USA 28 (1942), 535–537.

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

[16]   R. M. Nikolić, R. P. Pant, V. T. Ristić and A. Šebeković, Common fixed points theorems for self-mappings in Menger PM-spaces, Mathematics 10(14) (2022), Paper ID 2449, 11 pages. https://doi.org/10.3390/math10142449

[17]   R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), 436–440.

[18]   R. P. Pant and A. Pant, Fixed point theorems under new commuting conditions, J. Int. Acad. Phys. Sci. 17(1) (2013), 1–6.

[19]   A. Pant and R. P. Pant, Orbital continuity and fixed points, Filomat 31(11) (2017), 3495–3499.

[20]   H. K. Pathak, Y. J. Cho and S. M. Kang, Remarks on R-weakly commuting mappings and common fixed point theorems, Bull. Korean Math. Soc. 34 (1997), 247–257.

[21]   H. K. Pathak and S. M. Kang, A comparison of various types of compatible maps and common fixed points, Indian J. Pure Appl. Math. 28(4) (1997), 477–485.

[22]   B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 415–417.

[23]   B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, Elsevier, New York, USA, 1983.

[24]   S. Sessa, On a weakly commutativity condition in a fixed point considerations, Publ. Inst. Math. 32(46) (1986), 149–153.

[25]   K. P. R. Shastri, S. V. R. Naidu, I. H. N. Rao and K. P. R. Rao, Common fixed points for asymptotically regular mappings, Indian J. Pure Appl. Math. 15(8) (1984), 849–854.

[26]   S. L. Singh, A. Tomar Weaker forms of commuting maps and existence of fixed points, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 10(3) (2003), 145–161.