Fixed fuzzy points of generalized Geraghty type fuzzy mappings on complete metric spaces

Document Type: Research Paper

Authors

1 Department of Mathematics and Applied Mathematics, University of Pre- toria, Hatfield, Pretoria, South Africa

2 Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0002, Pretoria South Africa

Abstract

Generalized Geraghty type fuzzy mappings on
complete metric spaces are introduced and a fixed point theorem that
generalizes some recent comparable results for fuzzy mappings in
contemporary literature is obtained. Example is provided to show the
validity of obtained results over comparable classical results for fuzzy
mappings in fixed point theory. As an application, existence of coincidence
fuzzy points and common fixed fuzzy points for hybrid pair of single valued
self mapping and a fuzzy mapping is also established.

Keywords


[1] R. P. Agarwal, M. Meehan and D. O'Regan, Fixed point theory and applications, Cambridge
University Press, 2001.
[2] A. Amini-Harandi and H. Emami, A xed point theorem for contraction type maps in partially
ordered metric spaces and application to ordinary di erential equations, Nonlinear Anal., 72
(2010), 2238-2242.
[3] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations
integrales, Fund. Math., 3 (1922), 133{181.
[4] R. Baskaran and P. V. Subrahmanyam, A note on the solution of a class of functional
equations, Applicable Anal., 22 (1986), 235{241.
[5] R. Bellman, Methods of nonliner analysis, vol. II, 61 of Mathematics in Science and Engi-
neering, Academic Press, New York, NY, USA, 1973.

[6] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20
(1969), 458{464.
[7] D. Dukic , Z. Kadelburg and S. Radenovic, Fixed points of Geraghty type mappings in various
generalized metric spaces, Abstract Appl. Anal., Article ID 561245 (2011), 13 pages.
[8] M. Edelstein, On xed and periodic points under contractive mappings, J. London Math.
Soc., 37 (1962), 74{79.
[9] M. Edelstein, An extension of Banach contraction principle, Proc. Amer. Math. Soc., 12 (1)
(1961), 7{10.
[10] V. D. Estruch and A. Vidal, A note on xed fuzzy points for fuzzy mappings, Rend Istit.
Univ. Trieste., 32 (2001), 39-45.
[11] M. Geraghty, On contractive mappings, Proc Amer Math Soc., 40 (1973), 604{608.
[12] M. E. Gordji, M. Ramezani, Y. J. Cho and S. Pirbavafa, A generalization of Geraghty's
theorem in partially ordered metric spaces and applications to ordinary di erential equations,
Fixed Point Theory Appl., 1 (74) (2012), pages 9.
[13] M. E. Gordji, H. Baghani, H. Khodaei and M. Ramezani, Geraghty's xed point theorem for
special multivalued mappings, Thai J. Math., 10 (2012), 225{231.
[14] R. H. Haghi, Sh. Rezapour and N. Shahzad, Some xed point generalizations are not real
generalizations, Nonlinear Anal., 74 (2011), 1799{1803.
[15] S. Heilpern, Fuzzy mappings and fuzzy xed point theorems, J. Math. Anal. Appl., 83 (1981),
566{569.
[16] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces,
Nonlinear Analysis: Theory, Methods Appl., 3 (74), (2011), 768{774.
[17] G. Jungck, Commuting mappings and xed points, Amer. Math Monthly, 83 (1976), 261{263.
[18] B. S. Lee and S. J. Cho, A xed point theorem for contractive type fuzzy mappings, Fuzzy
Sets and Systems, 61 (1994), 309{312.
[19] S. B. Nadler, Multivalued contraction mappings, Paci c J. Math., 30 (1969), 475{488.
[20] J. J. Nieto and R. R. Lopez, Contractive mapping theorems in partially ordered sets and
applications to ordinary di erential equations, Order, 22 (3) (2005), 223{239.
[21] S. Park, Fixed points of fô€€€contractive maps, Rocky Mountain J. Math., 8 (4) (1978), 743{
750.
[22] E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc., 13 (1962), 459{465.
[23] B. E. Rhoades, A comparison of various de nitions of contractive mappings, Transaction.
Amer. Math. Soc., 226 (1977), 257{290.
[24] V. M. Sehgal, A xed point theorem for mappings with a contractive iterate, Proc. Amer.
Math. Soc., 23 (3) (1969), 631{634.
[25] C. S. Sen, Fixed degree for fuzzy mappings and a generalization of Ky Fan's theorem, Fuzzy
Sets and Systems, 24 (1987), 103{112.
[26] T. Suzuki, Mizoguchi and Takahashi's xed point theorem is a real generalization of Nadler's,
J. Math. Anal. Appl., 340 (2008), 752{755.
[27] D. Turkoglu and B. E. Rhoades, A xed fuzzy point for fuzzy mapping in complete metric
spaces, Math. Commun., 10 (2005), 115{121.
[28] J. S. W. Wong, Mappings of contractive type on abstract spaces, J. Math. Anal. Appl., 37
(1972), 331-340.
[29] L. A. Zadeh, Fuzzy Sets, Informations and Control, 8 (1965), 103-112.
[30] E. H. Zarantonello, Solving functional equations by contractive averaging, Mathematical Re-
search Center, Madison, Wisconsin, Technical Summary Report No. 160, June 1960.
[31] E. Zeidler, Nonlinear functional analysis and its applications I: Fixed Point Theorems,
Springer{Verlag, Berlin, 1986.