Fixed-point theorems for set-valued mappings and existence of best approximants |
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Authors: | João B Prolla |
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Institution: | Departamento de Matemática , Universidade Estadual de Campinas , Campinas, SP, Brasil |
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Abstract: | The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlin's extension to normed spaces of Kakutani's Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space. |
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