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Convergence and existence results for best proximity points. (English) Zbl 1197.47067
Authors’ summary: We provide a positive answer to a question raised by {\it A. A.\thinspace Eldred} and {\it P. Veeramani} [J. Math. Anal. Appl. 323, No. 2, 1001--1006 (2006; Zbl 1105.54021)] about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. Moreover, we introduce a new class of maps, called cyclic $\varphi$-contractions, which contains the cyclic contraction maps as a subclass. Convergence and existence results of best proximity points for cyclic $\varphi $-contraction maps are also obtained.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
41A50Best approximation, Chebyshev systems
Full Text: DOI
[1] 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
[2] Kirk, W. A.: Contraction mappings and extensions, Handbook of metric fixed point theory, 1-34 (2001) · Zbl 1019.54001
[3] 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
[4] Petrusel, G.: Cyclic representations and periodic points, Studia univ. Babes--bolyai, math. 50, 107-112 (2005) · Zbl 1117.47046
[5] Rus, I. A.; Petrusel, A.; Petrusel, G.: Fixed point theorems for set-valued Y-contractions, Banach center publ. 77, 227-237 (2007) · Zbl 1126.47047 · http://journals.impan.gov.pl/bc/Cont/bc77-0.html