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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

MSC:
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
Citations:
Zbl 0381.90105
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References:
[1] Baillon, J.B, Quelques aspects de la theorie des pointes fixes dans LES espaces Banach I, (), expose no. 7 · Zbl 0414.47039
[2] Clarkson, J.A, Uniformly convex spaces, Trans. amer. math. soc., 40, 396-414, (1936) · Zbl 0015.35604
[3] Dunham, C.B, Problems in best approximation, () · Zbl 0243.41015
[4] Edelstein, M, Fixed point theorems in uniformly convex Banach spaces, (), 369-374 · Zbl 0286.47035
[5] Garkavi, A.L, The best possible net and the best possible cross-section of a set in a normed space, Izv. akad. nauk. SSSR, 26, 87-106, (1962) · Zbl 0108.10801
[6] Goebel, K; Kirk, W.A, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia math., 47, 135-140, (1973) · Zbl 0265.47044
[7] Köthe, G, Topological vector spaces, (1969), Springer-Verlag Berlin/Heidelberg/New York
[8] Leżański, T, Sur le minimum des fonctionnelles dans LES espaces de Banach, Studia math., 68, 49-66, (1980) · Zbl 0441.49014
[9] Lifschitz, E.A, Fixed point theorems for operators in strongly convex spaces, (), 23-28
[10] Lim, T.C, On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. math., 32, 421-430, (1980) · Zbl 0454.47045
[11] Lim, T.C, Fixed point theorems for uniformly Lipschitzian mappings in L^{p}-spaces, Nonlinear anal., 7, 555-563, (1983) · Zbl 0533.47049
[12] Mach, J, On the existence of best simultaneous approximation, J. approx. theory, 25, 258-265, (1979) · Zbl 0422.41022
[13] Newman, D.J; Shapiro, H.S, Some theorems on Chebyshev approximation, Duke math. J., 30, 673-681, (1963) · Zbl 0116.04502
[14] Nürnberger, G, Unicity and strong unicity in approximation theory, J. approx. theory, 26, 54-70, (1979) · Zbl 0411.41010
[15] Prus, B, On a minimization of functionals in Banach spaces, Ann. univ. mariae Curie-sklodowska sect. A, 33, 165-188, (1979) · Zbl 0478.49008
[16] Smarzewski, R, Strongly unique best approximation in Banach spaces, J. approx. theory, 47, (1986), in press · Zbl 0615.41027
[17] Traub, J.F; Woźniakowski, H, A general theory of optimal algorithms, (1980), Academic Press New York · Zbl 0441.68046
[18] Ward, J.D, Chebyshev centers in spaces of continuous functions, Pacific J. math., 52, 283-287, (1974) · Zbl 0267.46017
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