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- W2080207910 abstract "Two commonly used boundary conditions which imply existence of fixed points for local strong pseudocontractions in Banach spaces are compared, a previous fixed-point theorem for this class of mappings is improved, and an almost fixed-point result is obtained for local pseudocontractions. Throughout this paper we suppose X is a Banach space and D an open subset of X. We use B(x; r) to denote the closed ball centered at x E X with radius r > 0 and D and AD to denote, respectively, the closure and boundary of D. An operator T: D -* X is said to be a local k-pseudocontraction (k > 0) (see [6]) if each point x E D has a neighborhood N for which (Xk)lluvll k. (1) For k 1, and the assumption (used in [1], [3], [4]) that the mapping I T not assume its infimum on aD. We show in fact, that for local strong pseudocontractions these conditions are in some sense equivalent. In the process we are able to sharpen Theorem 1 of [2] by showing that boundedness of the domain D is not an essential assumption. This fact is a consequence of the implication (iii) =X (ii) in Theorem 1 below. We also obtain (Theorem 2) a localization of a result of [3]. Received by the editors August 24, 1979. 1980 Mathematics Subject ClassificationL Primary 47H10." @default.
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- W2080207910 date "1981-01-01" @default.
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- W2080207910 title "On the fixed-point theory for local $k$-pseudocontractions" @default.
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- W2080207910 doi "https://doi.org/10.1090/s0002-9939-1981-0589138-4" @default.
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