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Abstract: A Banach space X is said to satisfy if the set of minimizers of the function is nonempty for each integer , each and each continuous nondecreasing coercive real-valued function f on . We study stability of certain polyhedrality properties under making direct sums, in order to be able to use results from a paper by Fonf, Lindenstrauss and the author to show that if X satisfies and an appropriate polyhedrality property then the function space satisfies for every topological space T. This generalizes the authorʼs result from 1997, proved for finite-dimensional polyhedral spaces X. Moreover, under more restrictive conditions on X and f, the mappings on () are continuous in the Hausdorff metric for each compact K. [Copyright &y& Elsevier]

Abstract: Let G be a graph and be a continuous map. Denote by , and the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If is an open arc contained in an edge of G such that for some , then ; (2) Any isolated point of is also an isolated point of ; (3) If for some , then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) and J. Xiong (1983, 1986) on interval maps or tree maps. [Copyright &y& Elsevier]