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

- Journal title : Communications of the Korean Mathematical Society
- Volume 24, Issue 1, 2009, pp.41-55
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2009.24.1.041

Title & Authors

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

Ali, Javid; Imdad, Mohammad;

Ali, Javid; Imdad, Mohammad;

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

implicit functions;weakly compatible mappings;coincidence and fixed points;

Language

English

Cited by

1.

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.

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.

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.

13.

G. Jungck, Compatible mappings and common fixed points (2), Internat. J. Math. Math. Sci. 11 (1988), 285–288.

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.

18.

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.