Best proximity points for cyclic Meir-Keeler contractions. (English) Zbl 1169.54021

Summary: We introduce a notion of cyclic Meir-Keeler contractions and prove a theorem which assures the existence and uniqueness of a best proximity point for cyclic Meir-Keeler contractions. This theorem is a generalization of a recent result due to A. A. Eldred and P. Veeramani [J. Math. Anal. Appl. 323, No. 2, 1001–1006 (2006; Zbl 1105.54021)].


54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems


Zbl 1105.54021
Full Text: DOI


[1] Ćirić, Lj. B., A new fixed-point theorem for contractive mappings, Publ. Inst. Math. (Beograd), 30, 25-27 (1981) · Zbl 0493.54028
[2] Eldred, A. A.; Kirk, W. A.; Veeramani, P., Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171, 283-293 (2005) · Zbl 1078.47013
[3] Eldred, A. A.; Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323, 1001-1006 (2006) · Zbl 1105.54021
[4] Goebel, K.; Kirk, W. A., (Topics in metric fixed point theory. Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28 (1990), Cambridge University Press) · Zbl 0708.47031
[5] Jachymski, J., Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl., 194, 293-303 (1995) · Zbl 0834.54025
[6] Lim, T. C., On characterizations of Meir-Keeler contractive maps, Nonlinear Anal., 46, 113-120 (2001) · Zbl 1009.54044
[7] Matkowski, J., Fixed point theorems for contractive mappings in metric spaces, Časopis Pěst. Mat., 105, 341-344 (1980) · Zbl 0446.54042
[8] Meir, A.; Keeler, E., A theorem on contraction mappings, J. Math. Anal. Appl., 28, 326-329 (1969) · Zbl 0194.44904
[9] Proinov, P. D., Fixed point theorems in metric spaces, Nonlinear Anal., 64, 546-557 (2006) · Zbl 1101.54046
[10] Suzuki, T., Fixed point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces, Nonlinear Anal., 64, 971-978 (2006) · Zbl 1101.54047
[11] Suzuki, T., Some notes on Meir-Keeler contractions and \(L\)-functions, Bull. Kyushu Inst. Technol., 53, 1-13 (2006) · Zbl 1104.54020
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.