Fixed point theorems for hybrid contraction without continuity

  • Authors

    • U. C. Gairola H. N. B. Garhwal University
    • Ram Krishan H. N. B. Garhwal University
    2015-01-04
    https://doi.org/10.14419/gjma.v3i1.3772
  • Coincidence Point, Fixed Point, Control Function, Weak Contraction.
  • In this paper we establish a coincidence and fixed point theorems for hybrid contraction under generalized weakly contractive condition by using the concept of (IT)-commutativity in a complete metric space without appeal to continuity of mappings. Our results extend and generalize the results of Choudhury et al. [6] and others.

  • References

    1. M. Abbas and D. Dorić, A common End point theorem for set-valued generalized -weak contraction, Fixed Point Theory Appl. vol. 2010 (2010) Article ID 509658, 1-8.
      [2] M. Abbas and T. Nazir, Fixed point of generalized weakly contractive mappings in ordered partial metric spaces, Fixed Point Theory Appl. 2012 (2012), 1-19.
      [3] Ya. I. Alber and S. Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, in: I. Goldberg, Yu. Lyubich, (eds.) New Results in Operator Theory, in: Advances and Appl. vol. 98, Birkhäuser, Basel, (1997), 7-22.
      [4] M. Chandra, S. N. Mishra, S. L. Singh and B.E. Rhoades, Coincidences and fixed points of non-expansive type multi-valued and single-valued maps, Indian J. Pure. Appl. Math. 26(5) (1995), 393-401.
      [5] T. H. Chang, Common fixed point theorems for multi-valued mappings, Math. Japon. 41(2) (1995), 311-320.
      [6] B. S. Choudhury, P. Konar, B. E. Rhoades, N. Metiya, Fixed point theorems for generalized weakly contractive mappings, Nonlinear Anal. 74(6) (2011), 2116-2126.
      [7] B. S. Choudhury and N. Metiya, Multi-valued and Single-valued fixed point results in partially ordered metric spaces, Arab J. Math. Sci. 17 (2011), 135-151.
      [8] Lj. B. ‡irić, Fixed points for generalized multi-valued contractions, Mat. Vasnik, 9(24) (1972), 265-272.
      [9] B. Damjanović and D. Dorić, Multi-valued generalizations of the Kannan fixed point theorem, Filomat, 25(1) (2011), 125-131.
      [10] D. Delbosco, Un ‘estensione di un teorema sul puto fisso di. S. Reich, Rend. Sem. Mat. Univers. Torino, 35 (1976-77), 233-238.
      [11] P. N. Dutta and B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl. vol. 2008 (2008) Article ID 406368, 1-8.
      [12] U. C. Gairola, S. N. Mishra and S. L. Singh, Coincidence and fixed point theorems on product spaces, Demonstratio Math. 30(1) (1997), 15-24.
      [13] O. Hadžić and Lj. Gajić, Coincidence points for set-valued mappings in convex metric spaces, Univ. Novom Sadu Zb. Rad. Prirod-Mat. Fak. Ser. Mat. 16(1) (1986), 13-25.
      [14] S. Itoh and W. Takahashi, Single-valued mappings, Multi-valued mappings and fixed point theorems, J. Math. Anal. Appl. 59 (1977), 514-521.
      [15] G. Jungck, Common fixed points for non-continuous non-self maps on non- metric spaces, Far East J. Math. Sci. 4 (1996), 199-215.
      [16] G. Jungck and B. E. Rhoades, Fixed point for set-valued functions without continuity, Indian J. Pure Appl. Math. 29(3) (1998), 227-238.
      [17] H. Kaneko and S. Sessa, Fixed point theorems for compatible multi-valued and single-valued mappings, Internat. J. Math. Math. Sci. 12(2) (1989), 257-262.
      [18] M. S. Khan, Common fixed point theorems for multi-valued mappings, Pacific J. Math. 95(2) (1981), 337-347.
      [19] M. S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1984), 1-9.
      [20] A. Latif, A fixed point results for multi-valued generalized contraction maps, Filomat, 26(5) (2012), 929-933.
      [21] Y. Liu, J. Wu and Z. Li, Common fixed points of single-valued and multi-valued maps, Internat. J. Math. Math. Sci. 19(2005), 3045-3055.
      [22] S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30(2) (1969), 475-487.
      [23] R. P. Pant, Common fixed point theorems for contractive maps, J. Math. Anal. Appl. 226(1998), 251-258.
      [24] H. K. Pathak, S. M. Kang and Y. J. Cho, Coincidence and fixed point theorems for nonlinear hybrid generalized contractions, Czechoslovak Math. J. 48(123) (1998), 341-357.
      [25] O. Popescu, Fixed points for -weak contractions, Appl. Math. Lett. 24(2011), 1-4.
      [26] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47(2001), 2683-2693.
      [27] B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257-290.
      [28] S. Sessa and B. Fisher, On common fixed points of weakly commuting mappings and set-valued mappings, Internat. J. Math. Math. Sci. 9 (2) (1986), 323-329.
      [29] S. L. Singh and C. Kulshrestha, Coincidence theorems in metric spaces, Indian J. Phy. Natur. Sci. 2 (1982), 19-22.
      [30] S. L. Singh and S. N. Mishra, Coincidences and fixed points of non-self hybrid contractions, J. Math. Anal. Appl. 256(2001), 486-497.
      [31] S. L. Singh and S. N. Mishra, Fixed point theorems for single-valued and multi-valued maps, Nonlinear Anal. 74(6) (2011), 2243-2248.
      [32] S. L. Singh, S. N. Mishra, R. Chugh and R. Kamal, General common fixed point theorems and applications, J. Appl. Math. vol (2012), Article ID 902312, 1-14.
      [33] S. L. Singh, K. S. Ha and Y. J. Cho, Coincidence and fixed points of nonlinear hybrid contractions, Internat. J. Math. Math. Sci. 12(2) (1989), 247-256.
      [34] A. Singh, R. C. Dimri and U. C. Gairola, A fixed point theorem for near-hybrid contraction, J. Nat. Acad. Math. 22 (2008), 11-22.
      [35] F. Skof, Teorma di punti fisso per applicazioni negli spazi metric, Atti. Acad. Sci. Torino, 111(1977), 323-329.
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  • How to Cite

    Gairola, U. C., & Krishan, R. (2015). Fixed point theorems for hybrid contraction without continuity. Global Journal of Mathematical Analysis, 3(1), 8-17. https://doi.org/10.14419/gjma.v3i1.3772