Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

UNIFYING A MULTITUDE OF COMMON FIXED POINT THEOREMS EMPLOYING AN IMPLICIT RELATION

  • Ali, Javid (DEPARTMENT OF MATHEMATICS ALIGARH MUSLIM UNIVERSITY) ;
  • Imdad, Mohammad (DEPARTMENT OF MATHEMATICS ALIGARH MUSLIM UNIVERSITY)
  • Published : 2009.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

A general common fixed point theorem for two pairs of weakly compatible mappings using an implicit function is proved without any continuity requirement which generalizes the result due to Popa [20, Theorem 3]. In process, several previously known results due to Fisher, Kannan, Jeong and Rhoades, Imdad and Ali, Imdad and Khan, Khan, Shahzad and others are derived as special cases. Some related results and illustrative examples are also discussed. As an application of our main result, we prove an existence theorem for the solution of simultaneous Hammerstein type integral equations.

Keywords

References

  1. A. Ahmad and M. Imdad, A common fixed point theorem for four mappings satisfying a rational inequality, Publ. Math. Debrecen 41 (1992), no. 3-4, 181–187.
  2. R. Chugh and S. Kumar, Common fixed points for weakly compatible maps, Proc. Indian Acad. Sci.(Math. Sci.) 111 (2001), no. 2, 241–247. https://doi.org/10.1007/BF02829594
  3. B. C. Dhage, A fixed point theorem in Banach algebras involving three operators with applications, Kyungpook Math. J. 44 (2004), 145–155.
  4. B. C. Dhage, Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem, Nonlinear Anal.(TMA), in press. https://doi.org/10.1016/j.na.2008.03.033
  5. B. Fisher, Common fixed point and constant mappings satisfying rational inequality, Math. Sem. Notes 6 (1978), 29–35.
  6. D. H. Griffel, Applied Functional Analysis, Ellis Horwood Limited, Chichester, 1981.
  7. G. E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 (1973), 201–206.
  8. M. Imdad and Q. H. Khan, Six mappings satisfying a rational inequality, Rad. Mat. 9 (1999), 251–260.
  9. M. Imdad and Javid Ali, Pairwise coincidentally commuting mappings satisfying a rational inequality, Italian J. Pure Appl. Math. 20 (2006), 87–96.
  10. G. S. Jeong and B. E. Rhoades, Some remarks for improving fixed point theorems for more than two maps, Indian J. Pure Appl. Math. 28 (1997), no. 9, 1177–1196.
  11. M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, Wiley Eastern Limited, New Delhi, 1985.
  12. G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976), 261–263. https://doi.org/10.2307/2318216
  13. G. Jungck, Compatible mappings and common fixed points (2), Internat. J. Math. Math. Sci. 11 (1988), 285–288. https://doi.org/10.1155/S0161171288000341
  14. G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci. 4 (1996), no. 2, 199–215.
  15. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76.
  16. T. I. Khan, A study of common fixed point theorems under certain weak conditions of commutativity, Ph. D. thesis, Aligarh Muslim University, Aligarh, India, 2002.
  17. P. P. Murthy, Important tools and possible applications of metric fixed point theory, Nonlinear Anal.(TMA) 47 (2001), 3479–3490. https://doi.org/10.1016/S0362-546X(01)00465-5
  18. R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), 436–440. https://doi.org/10.1006/jmaa.1994.1437
  19. R. P. Pant, Common fixed points of four mappings, Bull. Calcutta Math. Soc. 90 (1998), 281–286.
  20. V. Popa, Some fixed point theorems for weakly compatible mappings, Rad. Mat. 10(2001), 245–252.
  21. S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. 32 (1982), no. 46, 149–153.
  22. N. Shahzad, Invariant approximations, generalized I-contractions and R-subweakly commuting maps, Fixed Point Theory Appl. 2 (2005), no. 1, 79-86.
  23. S. L. Singh and S. N. Mishra, Remarks on Jachymski's fixed point theorems for compatible maps, Indian J. Pure Appl. Math. 28 (1997), no. 5, 611–615.

Cited by

  1. Unified relation-theoretic metrical fixed point theorems under an implicit contractive condition with an application vol.2016, pp.1, 2016, https://doi.org/10.1186/s13663-016-0531-6
  2. Common fixed points of mappings satisfying implicit contractive conditions vol.2012, pp.1, 2012, https://doi.org/10.1186/1687-1812-2012-105