WEAK INEQUALITIES WITH CONTROL FUNCTIONS AND FIXED POINT RESULTS

  • Received : 2009.10.22
  • Accepted : 2010.01.08
  • Published : 2010.05.30

Abstract

In recent times control functions have been used in several problems of metric fixed point theory. Also weak inequalities have been considered in a number of works on fixed points in metric spaces. Here we have incorporated a control function in certain weak inequalities. We have established two fixed point theorems for mapping satisfying such inequalities. Our results are supported by examples.

Keywords

References

  1. Ya.I.Alber,S.Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, in :I. Gohberg,Yu. Lyubich(Eds.), New Results in Operator Theory, in : Advances and Appl., 98, Birkhuser, Basel, 1997,7-22.
  2. A. D. Arvanitakis, A proof of the generalized Banach contraction conjecture, Proc. Amer. Math. Soc., 131(12) (2003), 3647-3656. https://doi.org/10.1090/S0002-9939-03-06937-5
  3. G.V.R.Babu, B.Lalitha and M.L.Sandhya, Common fixed point theorems involving two generalized altering distance functions in four variables, Proc. Jangjeon Math. Soc., 10(1)(2007), 83-93.
  4. D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458-464. https://doi.org/10.1090/S0002-9939-1969-0239559-9
  5. I . Beg, M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory and Applications, 2006. Article ID 74503, 7 pages.
  6. C. E. Chidume, H. Zegeye, S. J. Aneke, Approximation of fixed points of weakly contractive nonself maps in Banach spaces, J.Math. Anal. Appl., 270(1) (2002), 189-199. https://doi.org/10.1016/S0022-247X(02)00063-X
  7. B.S.Choudhury, P.N. Dutta, A unified fixed point result in metric spaces involving a two variable function, Filomat, 14 (2000), 43-48.
  8. B.S. Choudhury, A common unique fixed point result in metric spaces involving generalized altering distances, Mathematical Communications, 10 (2005), 105-110.
  9. B.S. Choudhury, P.N. Dutta, Common fixed points for fuzzy mappings using generalized altering distances, Soochow J. Math., 31(1) (2005), 71-81.
  10. B.S. Choudhury, K.Das, A new contraction principle in Menger spaces, Acta Mathematica Sinica, 24(8)(2008), 1379-1386. https://doi.org/10.1007/s10114-007-6509-x
  11. B.S. Choudhury, K.Das, P.N. Dutta, A fixed point result in Menger spaces using a real function, Acta Math. Hungar., 2008, DOI: 10.1007/s10474-008-7242-3.
  12. B.S. Choudhury, P.N. Dutta, A.kundu, Fixed point results using altering distance function and $\phi$- function in metric spaces, IJMA 5-8, (1-12), 2007, 281-287.
  13. P.N. Dutta and B.S.Choudhury, A Generalisation of Contraction Principle in Metric Spaces, Fixed Point Theory and Applications, 2008, (2008), Article ID 406368,8 pages.
  14. D.Ilic, V.Rakocevic, Common fixed points for maps on cone metric space, J.Math. Anal. Appl., 341 (2008), 876-882. https://doi.org/10.1016/j.jmaa.2007.10.065
  15. M.S, Khan, M. Swaleh, S. Sessa, Fixed points theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), 1-9. https://doi.org/10.1017/S0004972700001659
  16. J. Merryfield, B. Rothschild, J. D. Stein, Jr., An application of Ramsey's theorem to the Banach contraction principle, Proceedings of the American Mathematical Society, 130 (4) (2002), 927-933. https://doi.org/10.1090/S0002-9939-01-06169-X
  17. S.V.R. Naidu, Some fixed point theorems in Metric spaces by altering distances, Czechoslovak Mathematical Journal, 53(1) (2003), 205-212. https://doi.org/10.1023/A:1022991929004
  18. B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis: Theory, Methods and Application, 47(4) (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1
  19. K.P.R. Sastry, G.V.R. Babu, Some fixed point theorems by altering distances between the points, Ind. J. Pure. Appl, Math., 30(6) (1999), 641-647.
  20. K.P.R. Sastry, S.V.R.Naidu, G.V.R. Babu, G.A.Naidu, Generalization of common fixed point theorems for weakly commuting map by altering distances, Tamkang Jr. Math., 31(3) (2000), 243-250.
  21. Y.Song, Coincidence points for noncommuting f-weakly contractive mappings, Int. J. Comput. Appl. Math., (IJCAM), 2(1) (2007), 51-57.
  22. Q. Zhang and Y.Song, Fixed point theory for generalized $\phi$- weak contractions, Applied Mathematics Letters, 22(1) (2009), 75-78. https://doi.org/10.1016/j.aml.2008.02.007