If f is a continuous seminorm, we prove two f-best approximation theorems for functions Φ not necessarily continuous as a consequence of our version of Glebov's fixed point theorem. Moreover, we obtain another fixed point theorem that improves a recent result of [4]. In the last section, we study continuity-type properties of set valued parametric projections and our results improve recent theorems due to Mabizela [11].
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1041,
author = {Tiziana Cardinali},
title = {Best approximations, fixed points and parametric projections},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {22},
year = {2002},
pages = {243-260},
zbl = {1029.41016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1041}
}
Tiziana Cardinali. Best approximations, fixed points and parametric projections. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 22 (2002) pp. 243-260. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1041/
[000] [1] B. Brosowsky and F.N. Deutsch, Radial continuity of valued metric projections, J. Approx. Theory 11 (1974), 236-253. | Zbl 0283.41014
[001] [2] B. Brosowsky, F.N. Deutsch and G. Nurnberger, Parametric approximation, J. Approx. Theory 29 (1980), 261-277. | Zbl 0483.41033
[002] [3] T. Cardinali and F. Papalini, Sull'esistenza di punti fissi per multifunzioni a grafo debolmente chiuso, Riv. Mat. Univ. Parma (4) 17 (1991), 59-67.
[003] [4] T. Cardinali and F. Papalini, Fixed point theorems for multifunctions in topological vector spaces, J. Math. Anal. Appl. 186 (3) (1994), 769-777. | Zbl 0829.47045
[004] [5] F. Deutsch, Theory of best approximation in normed linear spaces, Mimeographed notes, 1972.
[005] [6] K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z. 112 (1969), 234-240. | Zbl 0185.39503
[006] [7] N.I. Glebov, On a generalization of the Kakutani fixed point theorem, Soviet Math. Dokl. 10 (1969), 446-448. | Zbl 0187.07503
[007] [8] P. Govindarajulu and D.V. Pai, On properties of sets related to f-projections, J. Math. Anal. Appl. 73 (1980), 457-465. | Zbl 0476.46012
[008] [9] G. Köthe, Topological vector spaces (I), Springer-Verlag, Berlin, Heidelberg, New York 1969. | Zbl 0179.17001
[009] [10] T.C. Lin, A note on a theorem of Ky Fan, Canad. Math. Bull. 22 (1979), 513-515. | Zbl 0429.47019
[010] [11] S. Mabizela, Upper semicontinuity of parametric projection, Set Valued Analysis 4 (1996), 315-325. | Zbl 0870.41022
[011] [12] T.D. Narang, On f-best approximation in topological spaces, Archivum Mathematicum, Brno 21 (4) (1985), 229-234. | Zbl 0585.41029
[012] [13] T.D. Narang, A note of f-best approximation in reflexive Banach space, Matem. Bech. 37 (1985), 411-413. | Zbl 0635.41031
[013] [14] E.V. Oshman, On the continuity of metric projection in Banach space, Math. USSR Sb. 9 (1969), 171-182. | Zbl 0198.45906
[014] [15] S. Reich, Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. Appl. 62 (1978), 104-113. | Zbl 0375.47031
[015] [16] V.M. Seghal and S.P. Singh, A theorem on the minimization of a condensing multifunction and fixed points, J. Math. Anal. Appl. 107 (1985), 96-102. | Zbl 0602.47039
[016] [17] I. Singer, The theory of best approximation and functional analysis, Regional Conference Series in Applied Math., Philadelphia 1974. | Zbl 0291.41020
[017] [18] C. Waters, Ph. D. thesis, Univ. of Wyoming 1984.
[018] [19] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer-Verlag 1990. | Zbl 0684.47028
[019] [20] E. Zeidler, Variational methods and optimization III, Springer-Verlag 1990. | Zbl 0684.47029