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Volume : VI, Issue : I, February - 2016

SOME FIXED POINT THEOREMS IN MENGER SPACE

Arvind Bhore, Rambabu Dangi

By : Laxmi Book Publication

Abstract :

In this paper, we define a property called M.S. property and using this property, we obtain a unique common fixed point for weakly compatible self-mappings of a Menger space. In this manuscript we extend very recent fixed point theorems in the setting of Menger spaces in three senses: on the one hand, we introduce the notion of Menger probabilistic metric-like space by avoiding a non-necessary constraint (at least, for our purposes) in the properties that define a Menger space; on the other hand, we consider a more general class of auxiliary functions in the contractivity condition; finally, we show that the function t→1/t−1 (which appears in many fixed point theorems in the fuzzy context) can be replaced by more appropriate and general functions. We illustrate our main statements with an example in which previous results cannot be applied.

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    Cite This Article :

    Arvind Bhore, Rambabu Dangi(2016). SOME FIXED POINT THEOREMS IN MENGER SPACE. Indian Streams Research Journal, Vol. VI, Issue. I, http://isrj.org/UploadedData/7831.pdf

    References :

    1. G. Jungck, Compatible mappings and fixed points, Amer. Math. Monthly,83(1976), 261-263.
    2. K. Menger, Statistical metric, Proc. Nat. Acad., 28(1942), 535-537.
    3. Rhoades, B. E. (2001), “Some theorems on weakly contractive maps.” Nonlinear Analysis, 47 , 2683-2693.
    4. S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, icspaces, Math. Japon., 36(1991), 283-289.
    5. S. Sessa, On a weak commutative condition in fixed point consideration,Publ. Inst. Math., 32(1982), 146-153.
    6. V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping onPM spaces, Math. Systems Theory, 6(1972), 97-100.
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    8. B. Singh, S. Jain, A fixed point theorem in Menger Space through weakcompatibility, J. Math. Anal. Appl., 301(2005), 439-448.
    9. Vetro, C., Gopal, D., and Imdad, M. (2010), “Common fixed point theorems for weak contractions in fuzzy metric
    10. G. Jungck, Compatible mappings and fixed points, Amer. Math. Monthly,83(1976), 261-263.
    11. K. Menger, Statistical metric, Proc. Nat. Acad., 28(1942), 535-537.
    12. Rhoades, B. E. (2001), “Some theorems on weakly contractive maps.” Nonlinear Analysis, 47 , 2683-2693.
    13. S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, icspaces, Math. Japon., 36(1991), 283-289.
    14. S. Sessa, On a weak commutative condition in fixed point consideration,Publ. Inst. Math., 32(1982), 146-153.
    15. V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping onPM spaces, Math. Systems Theory, 6(1972), 97-100.
    16. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland,Amsterdam, 1983.
    17. B. Singh, S. Jain, A fixed point theorem in Menger Space through weakcompatibility, J. Math. Anal. Appl., 301(2005), 439-448.
    18. Vetro, C., Gopal, D., and Imdad, M. (2010), “Common fixed point theorems for weak contractions in fuzzy metric
    19. G. Jungck, Compatible mappings and fixed points, Amer. Math. Monthly,83(1976), 261-263.
    20. K. Menger, Statistical metric, Proc. Nat. Acad., 28(1942), 535-537.
    21. Rhoades, B. E. (2001), “Some theorems on weakly contractive maps.” Nonlinear Analysis, 47 , 2683-2693.
    22. S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, icspaces, Math. Japon., 36(1991), 283-289.
    23. S. Sessa, On a weak commutative condition in fixed point consideration,Publ. Inst. Math., 32(1982), 146-153.
    24. V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping onPM spaces, Math. Systems Theory, 6(1972), 97-100.
    25. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland,Amsterdam, 1983.
    26. B. Singh, S. Jain, A fixed point theorem in Menger Space through weakcompatibility, J. Math. Anal. Appl., 301(2005), 439-448.
    27. Vetro, C., Gopal, D., and Imdad, M. (2010), “Common fixed point theorems for weak contractions in fuzzy metric
    28. V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping onPM spaces, Math. Systems Theory, 6(1972), 97-100.
    29. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland,Amsterdam, 1983.
    30. B. Singh, S. Jain, A fixed point theorem in Menger Space through weakcompatibility, J. Math. Anal. Appl., 301(2005), 439-448.
    31. Vetro, C., Gopal, D., and Imdad, M. (2010), “Common fixed point theorems for weak contractions in fuzzy metric
    32. G. Jungck, Compatible mappings and fixed points, Amer. Math. Monthly,83(1976), 261-263.
    33. K. Menger, Statistical metric, Proc. Nat. Acad., 28(1942), 535-537.
    34. Rhoades, B. E. (2001), “Some theorems on weakly contractive maps.” Nonlinear Analysis, 47 , 2683-2693.
    35. S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, icspaces, Math. Japon., 36(1991), 283-289.
    36. S. Sessa, On a weak commutative condition in fixed point consideration,Publ. Inst. Math., 32(1982), 146-153.
    37. V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping onPM spaces, Math. Systems Theory, 6(1972), 97-100.
    38. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland,Amsterdam, 1983.
    39. B. Singh, S. Jain, A fixed point theorem in Menger Space through weakcompatibility, J. Math. Anal. Appl., 301(2005), 439-448.
    40. Vetro, C., Gopal, D., and Imdad, M. (2010), “Common fixed point theorems for weak contractions in fuzzy metric
    41. G. Jungck, Compatible mappings and fixed points, Amer. Math. Monthly,83(1976), 261-263.
    42. K. Menger, Statistical metric, Proc. Nat. Acad., 28(1942), 535-537.
    43. Rhoades, B. E. (2001), “Some theorems on weakly contractive maps.” Nonlinear Analysis, 47 , 2683-2693.
    44. S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, icspaces, Math. Japon., 36(1991), 283-289.
    45. S. Sessa, On a weak commutative condition in fixed point consideration,Publ. Inst. Math., 32(1982), 146-153.
    46. G. Jungck, Compatible mappings and fixed points, Amer. Math. Monthly,83(1976), 261-263.
    47. K. Menger, Statistical metric, Proc. Nat. Acad., 28(1942), 535-537.
    48. Rhoades, B. E. (2001), “Some theorems on weakly contractive maps.” Nonlinear Analysis, 47 , 2683-2693.
    49. S. N. Mishra, Common fixed points of compatible mappings in PM-spaces, icspaces, Math. Japon., 36(1991), 283-289.
    50. S. Sessa, On a weak commutative condition in fixed point consideration,Publ. Inst. Math., 32(1982), 146-153.
    51. V. M. Sehgal, A. T. Bharucha-Reid, Fixed point of contraction mapping onPM spaces, Math. Systems Theory, 6(1972), 97-100.
    52. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland,Amsterdam, 1983.
    53. B. Singh, S. Jain, A fixed point theorem in Menger Space through weakcompatibility, J. Math. Anal. Appl., 301(2005), 439-448.
    54. Vetro, C., Gopal, D., and Imdad, M. (2010), “Common fixed point theorems for weak contractions in fuzzy metric

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