zbMATH — the first resource for mathematics

Strongly unique best approximations and centers in uniformly convex spaces. (English) Zbl 0617.41046
The basic result of the paper is Lemma 2.1: Let X be a uniformly convex space with the modulus of convexity of power type \(q\geq 2\). Then there is a constant \(d>0\) such that \[ \| tx+(1-t)y\|^ q\leq t\| x\|^ q+(1-t)\| y\|^ q-dw_ q(t)\| x-y\|^ q \] for all x,y\(\in X\) and \(t\in (0,1)\), where \(w_ q(t)=t(1-t)^ q+(1-t)t^ q.\) Using this result we prove that if X satisfies the assumptions of Lemma 2.1 and \(M\subset X\) is a closed convex nonempty subset of X then for every \(x\in X\) there exists a unique \(m\in M\) such that \(\| x- m\|^ q\leq \| x-y\|^ q-d\| m-y\|^ q\) for all \(y\in M\), where \(d>0\) is as in Lemma 2.1. In particular we show that it is true when X is the Lebesgue space \(L_ p\) or the Sobolev space \(H^{m,p}\) \((m\geq 0,1<p<\infty)\). We establish similar results for relative centers and asymptotic centers and we apply them to derive a fixed point theorem for uniformly Lipschitzian mappings.

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text: DOI
[1] Angelos, J.R; Egger, A, Strong uniqueness in Lp spaces, J. approx. theory, 42, 14-26, (1984) · Zbl 0551.41044
[2] Barros-Neto, J, An introduction to the theory of distributions, (1973), Dekker New York · Zbl 0273.46026
[3] Cheney, E.W, Introduction to approximation theory, (1966), Mc Graw-Hill New York · Zbl 0161.25202
[4] Diestel, J; Uhl, J.J, Vector measures, (1977), Providence · Zbl 0369.46039
[5] Dunham, C.B, Problems in best approximation, () · Zbl 0243.41015
[6] Figiel, T, Example of an infinite dimensional Banach space non-isomorphic to its Cartesian square, Studia math., 42, 295-306, (1972) · Zbl 0213.12801
[7] 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
[8] Hanner, O, On the uniform convexity of Lp and lp, Ark. mat., 3, 239-244, (1958)
[9] Lim, T.C, On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. math., 32, 421-430, (1980) · Zbl 0454.47045
[10] Lim, T.C, Fixed point theorems for uniformly Lipschitzian mappings in L^{p}-spaces, Nonlinear anal., 7, 555-563, (1983) · Zbl 0533.47049
[11] Lindenstrauss, J; Tzafiri, L, Classical Banach spaces. II. function spaces, (1979), Springer-Verlag Berlin
[12] Pisier, G, Martingales with values in uniformly convex spaces, Israel J. math., 20, 326-350, (1975) · Zbl 0344.46030
[13] Smarzewski, R, Strongly unique best approximation in Banach spaces, J. approx. theory, 47, 184-194, (1986) · Zbl 0615.41027
[14] Smarzewski, R, Strongly unique minimization of functionals in Banach spaces with applications to theory of approximation and fixed points, J. math. anal. appl., 115, 155-172, (1986) · Zbl 0593.49004
[15] Smarzewski, R, The classical and extended strong unicity of approximations in Banach spaces, (1986), Annales UMCS Lublin, (in Polish)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.