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On Well Posed Generalized Best Approximation Problems

โœ Scribed by Chong Li

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
143 KB
Volume
107
Category
Article
ISSN
0021-9045

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โœฆ Synopsis

Let C be a closed bounded convex subset of X with 0 being an interior point of C and p C be the Minkowski functional with respect to C. Let G be a nonempty closed, boundedly relatively weakly compact subset of a Banach space X. For a point x # X, we say the minimization problem min C (x, G) is well posed if there exists a unique point zร„ such that p C (zร„ &x)=* C (x, G) and every sequence [z n ]/G satisfying lim n ร„ p C (z n &x)=* C (x, G) converges strongly to the point zร„ , where * C (x, G)=inf z # G p C (z&x). Under the assumption that C is both strictly convex and Kadec, we prove that the set X o (G) of all x # X such that the problem min C (x, G) is well posed is a residual subset of X extending the results in the case that the modulus of convexity of C is strictly positive due to Blasi and Myjak. In addition, we also prove these conditions are necessary.

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