Some Common Fixed Point Theorems in Menger Space

Affiliation(s)

Departement of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Sonepat, Haryana, India.

Department of Applied Sciences, B. M. Institute of Engineering and Technology, Sonepat, Haryana, India.

Departement of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Sonepat, Haryana, India.

Department of Applied Sciences, B. M. Institute of Engineering and Technology, Sonepat, Haryana, India.

ABSTRACT

This paper consists four sections. First section is central to the text. In second section, we generalize the results of Kohli and Vashistha [1] for pairs of mappings using weakly compatible maps. Third section deals the results for pair of weakly compatible maps along with property (E.A.) using different types of control functions, which generalize the results of Kohli and Vashistha [1] and Kubiaczyk and Sharma [2]. Fourth section is concerned with results for occasionally weakly compatible maps and generalizes, extends and unifies several well known comparable results in literature.

This paper consists four sections. First section is central to the text. In second section, we generalize the results of Kohli and Vashistha [1] for pairs of mappings using weakly compatible maps. Third section deals the results for pair of weakly compatible maps along with property (E.A.) using different types of control functions, which generalize the results of Kohli and Vashistha [1] and Kubiaczyk and Sharma [2]. Fourth section is concerned with results for occasionally weakly compatible maps and generalizes, extends and unifies several well known comparable results in literature.

KEYWORDS

Weakly Compatible Maps; Occasionally Weakly Compatible Maps; Property (E.A.); Common Property (E.A.)

Weakly Compatible Maps; Occasionally Weakly Compatible Maps; Property (E.A.); Common Property (E.A.)

Cite this paper

S. Kumar and A. Rani, "Some Common Fixed Point Theorems in Menger Space,"*Applied Mathematics*, Vol. 3 No. 3, 2012, pp. 235-245. doi: 10.4236/am.2012.33038.

S. Kumar and A. Rani, "Some Common Fixed Point Theorems in Menger Space,"

References

[1] J. K. Kohli and S. Vashistha, “Common Fixed Point Theorems in Probabilistic Metric Spaces,” Acta Mathematica Hungarica, Vol. 115, No. 1-2, 2007, pp. 37-47.

[2] I. Kubiaczyk and S. Sharma, “Some Common Fixed Point Theorems in Menger Space under Strict Contractive Conditions,” Southeast Asian Bulletin of Mathematics, Vol. 32, No. 1, 2008, pp. 117-124.

[3] K. Menger, “Statistical Metrices,” Proceedings of the National Academy of Sciences USA, Vol. 28, No. 12, 1942, pp. 535-537.

[4] V. M. Sehgal and A. T. Bharucha-Reid, “Fixed Points of Contraction Mappings on Probabilistic Metric Spaces,” Mathematical Systems Theory, Vol. 6, No. 1, 1972, pp. 97-102.

[5] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” North Holland, Amsterdam, 1983.

[6] S. Sessa, “On a Weak Commutativity Condition of Mappings in Fixed Point Considerations,” Publications DeL’Institut Mathematique, Nouvelle Serie Tome 32, 1982, pp. 149-153.

[7] G. Jungck, “Compatible Mappings and Common ?xed Points,” International Journal of Mathematics and Mathematical Sciences, Vol. 9, No. 4, 1986, pp. 771-779.

[8] S. N. Mishra, “Common Fixed Points of Compatible Mappings in PM-Spaces,” Mathematica Japonica, Vol. 36, No. 2, 1991, pp. 283-289.

[9] G. Jungck, “Common Fixed Points for Non-Continuous Non-Self Maps on Non-Metric Spaces,” Far East Journal of Mathematical Sciences, Vol. 4, No. 2, 1996, pp. 199215.

[10] D. Mihet, “A Note on a Common Fixed Point Theorem in Probabilistic Metric Spaces,” Acta Mathematica Hungarica, Vol. 125, No. 1-2, 2009, pp. 127-130.

[11] M. Aamri and D. El Moutawakil, “Some New Common ?xed Point Theorems under Strict Contractive Conditions,” Journal of Mathematical Analysis and Application, Vol. 27, 2002, pp. 181-188.

[12] M. Abbas, I. Altun and D. Gopal, “Common Fixed Point Theorems for Non Compatible Mappings in Fuzzy Metric Spaces,” Bulletin of Mathematical Analysis and Applications, Vol. 1, No. 2, 2009, pp. 47-56.

[13] M. Imdad, M. Tanveer and M. Hasan, “Some Common Fixed Point Theorems in Menger PM Spaces,” 2011, in Press.

[14] M. Imdad Javid Ali, “Jungck’s Common Fixed Point Theorem and E.A. Property,” Acta Mathematica Sinica, Vol. 24, No. 1, 2008, pp. 87-94.

[15] M. A. Al-Thagafi and N. Shahzad, “Generalized I-Nonexpansive Selfmaps and Invariant Approximations,” Acta Mathematica Sinica, English Series, Vol. 24, No. 5, 2008, pp. 867-876.

[16] G. V. R. Babu and G. N. Alenmayehu, “A Common Fixed Point Theorem for Weakly Contractive Mappings Satisfying Property (E.A.),” Applied Mathematics E-Notes, Vol. 10, 2010, pp. 167-174.

[17] H. K. Pathak, R. Rodr?guez-Lopez and R. K. Verma, “A Common Fixed Point Theorem Using Implicit Relation and Property (E.A.) in Metric Spaces,” Filomat, Vol. 21, No. 2, 2007, pp. 211-234.

[18] G. Jungck and B. E. Rhoades, “Fixed Points for Set-Valued Functions without Continuity,” Indian Journal of Pure and Applied Mathematics, Vol. 29, No. 3, 1998, pp. 227-238.

[19] M. A. Al-Thagafi and N. Shahzad, “A Note on Occasionally Weakly Compatible Maps,” International Journal of Mathematical Analysis, Vol. 3, No. 2, 2009, pp. 55-58.

[20] G. V. R. Babu and G. N. Alenmayehu, “Points of Coincidence and Common Fixed Points of a Pair of Generalized Weakly Contractive Mappings,” Journal of Advanced Research in Pure Mathematics, Vol. 2, No. 2, 2010, pp. 89-106.

[21] G. Jungck and B. E. Rhoades, “Fixed Point Theorems for Occasionally Weakly Compatible Mappings,” Fixed Point Theory, Vol. 7, 2006, pp. 286-296.

[1] J. K. Kohli and S. Vashistha, “Common Fixed Point Theorems in Probabilistic Metric Spaces,” Acta Mathematica Hungarica, Vol. 115, No. 1-2, 2007, pp. 37-47.

[2] I. Kubiaczyk and S. Sharma, “Some Common Fixed Point Theorems in Menger Space under Strict Contractive Conditions,” Southeast Asian Bulletin of Mathematics, Vol. 32, No. 1, 2008, pp. 117-124.

[3] K. Menger, “Statistical Metrices,” Proceedings of the National Academy of Sciences USA, Vol. 28, No. 12, 1942, pp. 535-537.

[4] V. M. Sehgal and A. T. Bharucha-Reid, “Fixed Points of Contraction Mappings on Probabilistic Metric Spaces,” Mathematical Systems Theory, Vol. 6, No. 1, 1972, pp. 97-102.

[5] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” North Holland, Amsterdam, 1983.

[6] S. Sessa, “On a Weak Commutativity Condition of Mappings in Fixed Point Considerations,” Publications DeL’Institut Mathematique, Nouvelle Serie Tome 32, 1982, pp. 149-153.

[7] G. Jungck, “Compatible Mappings and Common ?xed Points,” International Journal of Mathematics and Mathematical Sciences, Vol. 9, No. 4, 1986, pp. 771-779.

[8] S. N. Mishra, “Common Fixed Points of Compatible Mappings in PM-Spaces,” Mathematica Japonica, Vol. 36, No. 2, 1991, pp. 283-289.

[9] G. Jungck, “Common Fixed Points for Non-Continuous Non-Self Maps on Non-Metric Spaces,” Far East Journal of Mathematical Sciences, Vol. 4, No. 2, 1996, pp. 199215.

[10] D. Mihet, “A Note on a Common Fixed Point Theorem in Probabilistic Metric Spaces,” Acta Mathematica Hungarica, Vol. 125, No. 1-2, 2009, pp. 127-130.

[11] M. Aamri and D. El Moutawakil, “Some New Common ?xed Point Theorems under Strict Contractive Conditions,” Journal of Mathematical Analysis and Application, Vol. 27, 2002, pp. 181-188.

[12] M. Abbas, I. Altun and D. Gopal, “Common Fixed Point Theorems for Non Compatible Mappings in Fuzzy Metric Spaces,” Bulletin of Mathematical Analysis and Applications, Vol. 1, No. 2, 2009, pp. 47-56.

[13] M. Imdad, M. Tanveer and M. Hasan, “Some Common Fixed Point Theorems in Menger PM Spaces,” 2011, in Press.

[14] M. Imdad Javid Ali, “Jungck’s Common Fixed Point Theorem and E.A. Property,” Acta Mathematica Sinica, Vol. 24, No. 1, 2008, pp. 87-94.

[15] M. A. Al-Thagafi and N. Shahzad, “Generalized I-Nonexpansive Selfmaps and Invariant Approximations,” Acta Mathematica Sinica, English Series, Vol. 24, No. 5, 2008, pp. 867-876.

[16] G. V. R. Babu and G. N. Alenmayehu, “A Common Fixed Point Theorem for Weakly Contractive Mappings Satisfying Property (E.A.),” Applied Mathematics E-Notes, Vol. 10, 2010, pp. 167-174.

[17] H. K. Pathak, R. Rodr?guez-Lopez and R. K. Verma, “A Common Fixed Point Theorem Using Implicit Relation and Property (E.A.) in Metric Spaces,” Filomat, Vol. 21, No. 2, 2007, pp. 211-234.

[18] G. Jungck and B. E. Rhoades, “Fixed Points for Set-Valued Functions without Continuity,” Indian Journal of Pure and Applied Mathematics, Vol. 29, No. 3, 1998, pp. 227-238.

[19] M. A. Al-Thagafi and N. Shahzad, “A Note on Occasionally Weakly Compatible Maps,” International Journal of Mathematical Analysis, Vol. 3, No. 2, 2009, pp. 55-58.

[20] G. V. R. Babu and G. N. Alenmayehu, “Points of Coincidence and Common Fixed Points of a Pair of Generalized Weakly Contractive Mappings,” Journal of Advanced Research in Pure Mathematics, Vol. 2, No. 2, 2010, pp. 89-106.

[21] G. Jungck and B. E. Rhoades, “Fixed Point Theorems for Occasionally Weakly Compatible Mappings,” Fixed Point Theory, Vol. 7, 2006, pp. 286-296.