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Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps. (English) Zbl 1206.54047
Summary: We introduce a notion of cyclic orbital Meir-Keeler contraction and give sufficient conditions for the existence of fixed points and best proximity points of such a map. Our main result is a generalization of a best proximity point result due to C. Di Bari, T. Suzuki and C. Vetro [Nonlinear Anal., Theory Methods Appl. 69, No. 11, A, 3790–3794 (2008; Zbl 1169.54021)].

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
References:
[1]Kirk, W. A.; Srinivasan, P. S.; Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions, Fixed point theory 4, 79-89 (2003) · Zbl 1052.54032
[2]Eldred, A. A.; Veeramani, P.: Existence and convergence of best proximity points, J. math. Anal. appl. 323, 1001-1006 (2006) · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081
[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 · doi:10.1016/j.na.2007.10.014
[4]Meir, A.; Keeler, E.: A theorem on contraction mappings, J. math. Anal. appl. 28, 326-329 (1969) · Zbl 0194.44904 · doi:10.1016/0022-247X(69)90031-6
[5]Karpagam, S.; Agrawal, Sushama: Best proximity point theorems for p-cyclic Meir–Keeler contractions, Fixed point theory appl. 2009 (2009) · Zbl 1172.54028 · doi:10.1155/2009/197308
[6]Al-Thagafi, M. A.; Shahzad, N.: Convergence and existence results for best proximity points, Nonlinear anal. 70, 3665-3671 (2009) · Zbl 1197.47067 · doi:10.1016/j.na.2008.07.022
[7]Eldred, A. A.; Kirk, W. A.; Veeramani, P.: Proximal normal structure and relatively non expansive mappings, Studia math. 171, No. 3, 283-293 (2005) · Zbl 1078.47013 · doi:10.4064/sm171-3-5
[8]Espinola, R.: A new approach to relatively nonexpansive mappings, Proc. amer. Math. soc. 136, No. 6, 1987-1995 (2008) · Zbl 1141.47035 · doi:10.1090/S0002-9939-08-09323-4
[9]T. Suzuki, M. Kikkawa, C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal. Available online. · Zbl 1178.54029 · doi:10.1016/j.na.2009.01.173