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COMMON COUPLED FIXED POINT FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION

  • Deshpande, Bhavana (Department of Mathematics, Govt. P. G. Arts and Science College) ;
  • Handa, Amrish (Department of Mathematics, Govt. P. G. Arts and Science College)
  • Received : 2014.01.22
  • Accepted : 2014.12.09
  • Published : 2015.01.31

Abstract

We establish a coupled coincidence and common coupled fixed point theorem for hybrid pair of mappings under generalized non-linear contraction. An example supporting to our result has also been cited. We improve, extend and generalize several known results.

Keywords

References

  1. M. Abbas, D. Ilic and M. A. Khan, Coupled coincidence point and coupled fixed point theorems in partially ordered metric spaces with u-distance, Fixed Point Theory Appl. 2010 (2010) Article ID 134897, 11 pages.
  2. M. Abbas, L. Ciric, B. Damjanovic and M. A. Khan, Coupled coincidence point and common fixed point theorems for hybrid pair of mappings, Fixed Point Theory Appl. doi:10.1186/1687-1812-2012-4 (2012).
  3. S. Banach, Sur les Operations dans les Ensembles Abstraits et leur. Applications aux Equations Integrales, Fund. Math. 3 (1922), 133-181. https://doi.org/10.4064/fm-3-1-133-181
  4. V. Berinde, Coupled fixed point theorems for ${\varphi}$ - contractive mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal. 75 (2012), 3218-3228. https://doi.org/10.1016/j.na.2011.12.021
  5. T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), no. 7, 1379-1393. https://doi.org/10.1016/j.na.2005.10.017
  6. B. S. Choudhury and A. Kundu, A coupled coincidence point results in partially ordered metric spaces for compatible mappings, Nonlinear Anal. 73 (2010), 2524-2531. https://doi.org/10.1016/j.na.2010.06.025
  7. B. Deshpande, Common fixed point for set and single valued functions without conti-nuity and compatibility, Mathematica Moravica 11 (2007), 27-38.
  8. B. Deshpande and R. Pathak, Fixed point theorems for noncompatible discontinuous hybrid pairs of mappings on 2- metric spaces, Demonstratio Mathematica, XLV (2012), no. 1, 143-154.
  9. B. Deshpande and S. Chouhan, Common fixed point theorems for hybrid pairs of map-pings with some weaker conditions in 2-metric spaces, Fasciculi Mathematici, 46 (2011), 37-55.
  10. B. Deshpande and S. Chouhan, Fixed points for two hybrid pairs of mappings satisfying some weaker conditions on noncomplete metric spaces, Southeast Asian Bull. Math. 35 (2011), 851-858.
  11. B. Deshpande, S. Sharma and A. Handa, Tripled fixed point theorem for hybrid pair of mappings under generalized nonlinear contraction, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 21 (2014), no. 1, 23-38. https://doi.org/10.7468/jksmeb.2014.21.1.23
  12. H. S. Ding, L. Li and S. Radenovic, Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces, Fixed Point Theory Appl. 2012, 2012:96.
  13. I. Kubiaczyk and B. Deshpande, Coincidence point for noncompatible multivalued map-pings satisfying an implicit relation, Demonstratio Mathematica XXXIX (2006), no. 4, 555-562.
  14. I. Kubiaczyk and B. Deshpande, A common fixed point theorem for multivalued mappings through T-weak commutativity, Mathematica Moravica 10 (2006), 55-60.
  15. I. Kubiaczyk and B. Deshpande, Common fixed point of multivalued mappings without continuity, Fasciculi Mathematici 37 (2007), no. 9, 19-26.
  16. I. Kubiaczyk and B. Deshpande, Noncompatibility, discontinuity in consideration of common fixed point of set and single valued mappings, Southeast Asian Bull. Math. 32 (2008), 467-474.
  17. V. Lakshmikantham and L. Ciric, Coupled fixed point theorems for nonlinear contrac-tions in partially ordered metric spaces, Nonlinear Anal. 70 (2009), no. 12, 4341-4349. https://doi.org/10.1016/j.na.2008.09.020
  18. N. V. Luong and N. X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011), 983-992. https://doi.org/10.1016/j.na.2010.09.055
  19. M. Jain, K. Tas, S. Kumar and N. Gupta, Coupled common fixed point results involving a ${\varphi}$ - ${\psi}$ contractive condition for mixed g-monotone operators in partially ordered metric spaces, J. Inequal. Appl. 2012, 2012:285. https://doi.org/10.1186/1029-242X-2012-285
  20. J. T. Markin, Continuous dependence of fixed point sets, Proceedings of the American Mathematical Society, 38 (1947), 545-547.
  21. S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475
  22. B. Samet, Coupled fixed point theorems for generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517. https://doi.org/10.1016/j.na.2010.02.026
  23. B. Samet and H. Yazidi, Coupled fixed point theorems in partially ordered F-chainable metric spaces, TJMCS 1 (2010), 142-151.
  24. B. Samet and C. Vetro, Coupled fixed point, F-invariant set and fixed point of N-order, Ann. Funct. Anal. 1 (2010), 46-56. https://doi.org/10.15352/afa/1399900586
  25. S. Sharma and B. Deshpande, Compatible multivalued mappings satisfying an implicit relation, Southeast Asian Bull. Math. 30 (2006), 535-540.
  26. S. Sharma and B. Deshpande, Fixed point theorems for set and single valued mappings without continuity and compatibility, Demonstratio Mathematica XL (2007), no. 3, 649-658.
  27. S. Sharma, B. Deshpande and R. Pathak, Common fixed point theorems for hybrid pairs of mappings with some weaker conditions, Fasciculi Mathematici 39 (2008), 71-84.

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