×

Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces. (English) Zbl 1101.54047

Motivated essentially by the work of W. A. Kirk [J. Math. Anal. Appl. 277, No. 2, 645–650 (2003; Zbl 1022.47036)] and T.-C. Lim [Nonlinear Anal., Theory Methods Appl. 46, No. 1(A), 113–120 (2001; Zbl 1009.54044)], the author introduces the concept of asymptotic contraction of Meir-Keeler (\(ACMK\)) type on a metric space and obtains fixed point theorems fo such maps. The main result states that if \(T\) is an \(ACMK\) on a complete metric space \(X\) and if \(T^l\) is continuous for some natural number \(l\), then \(T\) has a unique fixed point.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arandelović, I. D., On a fixed point theorem of Kirk, J. Math. Anal. Appl., 301, 384-385 (2005) · Zbl 1075.47031
[2] Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3, 133-181 (1922) · JFM 48.0201.01
[3] Boyd, D. W.; Wong, J. S.W., On nonlinear contractions, Proc. Amer. Math. Soc., 20, 458-464 (1969) · Zbl 0175.44903
[4] Jachymski, J. R.; Jóźwik, I., On Kirk’s asymptotic contractions, J. Math. Anal. Appl., 300, 147-159 (2004) · Zbl 1064.47052
[5] Kirk, W. A., Fixed points of asymptotic contractions, J. Math. Anal. Appl., 277, 645-650 (2003) · Zbl 1022.47036
[6] Lim, T. C., On characterizations of Meir-Keeler contractive maps, Nonlinear Anal., 46, 113-120 (2001) · Zbl 1009.54044
[7] Meir, A.; Keeler, E., A theorem on contraction mappings, J. Math. Anal. Appl., 28, 326-329 (1969) · Zbl 0194.44904
[8] Suzuki, T., Several fixed point theorems in complete metric spaces, Yokohama Math. J., 44, 61-72 (1997) · Zbl 0882.47039
[9] Suzuki, T., Several fixed point theorems concerning \(\tau \)-distance, Fixed Point Theory Appl., 2004, 195-209 (2004) · Zbl 1076.54532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.