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Strongly unique minimization of functionals in Banach spaces with applications to theory of approximation and fixed points. (English) Zbl 0593.49004
The paper proves a theorem on the existence of a strong minimum point for a real valued functional defined on a closed convex set and fulfilling some growth conditions. A strong minimum for a functional means that every minimizing sequence converges to the minimum point (in other papers the property is called well-posedness, see e.g. T. Zolezzi [Appl. Math. Optimization 4, 209-223 (1978; Zbl 0381.90105)]. Moreover, there are some estimates for the best approximation problem in Hilbert and \(L^ p\)-spaces and an existence theorem for (strongly unique) centers in a Banach space. The last result gets as a corollary an existence theorem of fixed points for a uniformly Lipschitz mapping on a Hilbert space.
Reviewer: R.Lucchetti

49J27 Existence theories for problems in abstract spaces
41A50 Best approximation, Chebyshev systems
90C48 Programming in abstract spaces
46B99 Normed linear spaces and Banach spaces; Banach lattices
46C99 Inner product spaces and their generalizations, Hilbert spaces
47H10 Fixed-point theorems
49K40 Sensitivity, stability, well-posedness
Zbl 0381.90105
Full Text: DOI
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