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Fixed Point Theorems on Closed Sets Through Abstract Cones

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Fixed Point Theorems on Closed Sets Through Abstract Cones

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Title: Fixed Point Theorems on Closed Sets Through Abstract Cones
Author: Lakshmikantham, V.; Eisenfeld, Jerome
Abstract: Let D be a closed subset of a complete metric space (X,p). We seek (i) conditions upon which a map T : D -> X has a fixed point in D and (ii) the construction of an iterative sequence whose limit is a fixed point in D. If X is a Banach space then a classical approach is to set G = I - T and use a numerical search method to minimize ||GX|| in D. Another approach, which does not require a Banach space structure, was recently introduced by Caristi and Kirk ([1],[2]). They prove that a metrically inward contractor map T has a fixed point. Both methods assume conditions which guarantee that for arbitrary x in D there exists y in D such that p(y,Ty) < p(x,TX). This condition is the basis of our study.
URI: http://hdl.handle.net/10106/2492
Date: 1976-03

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