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The existence of best proximity points in metric spaces with the property UC. (English) Zbl 1178.54029

Authors’ abstract: A. A. Eldred and P. Veeramani [J. Math. Anal. Appl. 323, No. 2, 1001–1006 (2006; Zbl 1105.54021)] proved a theorem which ensures the existence of a best proximity point of cyclic contractions in the framework of uniformly convex Banach spaces. In this paper we introduce a notion of the property UC and extend the Eldred and Veeramani theorem to metric spaces with the property UC.

MSC:

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

Citations:

Zbl 1105.54021
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References:

[1] Eldred, A. A.; Kirk, W. A.; Veeramani, P., Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171, 283-293 (2005), MR2188054 · Zbl 1078.47013
[2] Eldred, A. A.; Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323, 1001-1006 (2006), MR2260159 · Zbl 1105.54021
[3] Di Bari, C.; Suzuki, T.; Vetro, C., Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal., 69, 3790-3794 (2008) · Zbl 1169.54021
[4] Goebel, K.; Kirk, W. A., Topics in metric fixed point theory, (Cambridge Studies in Advanced Mathematics, vol. 28 (1990), Cambridge University Press), MR1074005 · Zbl 0708.47031
[5] Prus, S., Geometrical background of metric fixed point theory, (Kirk, W. A.; Sims, B., Handbook of Metric Fixed Point Theory (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 93-132, MR1904275 · Zbl 1018.46010
[6] Suzuki, T., Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340, 1088-1095 (2008), MR2390912 · Zbl 1140.47041
[7] 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
[8] Edelstein, M., On fixed and periodic points under contractive mappings, J. London Math. Soc., 37, 74-79 (1962), MR0133102 · Zbl 0113.16503
[9] Meir, A.; Keeler, E., A theorem on contraction mappings, J. Math. Anal. Appl., 28, 326-329 (1969), MR0250291 · Zbl 0194.44904
[10] Caristi, J., Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215, 241-251 (1976), MR0394329 · Zbl 0305.47029
[11] Caristi, J.; Kirk, W. A., Geometric fixed point theory and inwardness conditions, (Lecture Notes in Math., vol. 490 (1975), Springer: Springer Berlin), 74-83, MR0399968 · Zbl 0315.54052
[12] Subrahmanyam, P. V., Remarks on some fixed point theorems related to Banach’s contraction principle, J. Math. Phys. Sci., 8, 445-457 (1974), MR0358749 · Zbl 0294.54033
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