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Results on the existence and convergence of best proximity points. (English) Zbl 1205.47052
Summary: We first consider a cyclic $\varphi$-contraction map on a reflexive Banach space $X$ and provide a positive answer to a question raised by M. A. Al-Thagafi and N. Shahzad on the existence of best proximity points for cyclic $\varphi$-contraction maps in reflexive Banach spaces in [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 10 (A), 3665–3671 (2009; Zbl 1197.47067)]. In the second part of the paper, we discuss the existence of best proximity points in the framework of more general metric spaces. We obtain some new results on the existence of best proximity points in hyperconvex metric spaces as well as in ultrametric spaces.
MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces
References:
 [1] [2] [3] [4]