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Zbl 1197.47067
Al-Thagafi, M.A.; Shahzad, Naseer
Convergence and existence results for best proximity points. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 10, A, 3665-3671 (2009). ISSN 0362-546X

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.
[Ioan A. Rus (Cluj-Napoca)]

MSC 2000:: *47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
54H25 Fixed-point theorems in topological spaces
41A50 Best approximation

Keywords: best proximity point; fixed point; cyclic $\varphi$-contraction map; metric space; Banach space

Citations: Zbl 1105.54021

Cited in: Zbl 1229.54066 Zbl 1205.47052 Zbl 1197.47071

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