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Tytuł artykułu

Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex space

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According to Mickael's selection theorem any surjective continuous linear operator from one Prechet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if E is a Frechet space and T : E -> E is a continuous linear operator such that the Cauchy problem x = T x, x(0) = X0 is solvable in [0,1] for any X06 E, then for anyf zawiera się C([0, 1],E), there exists a continues map S : [0,1] x E -> E, (t x) ->o StX such that for any X0 zawiera się w E, the function x(t) = StX0 is a solution of the Cauchy problem x(t) = Tx(t) +- f(t), x(0) = X0 (they call S a fundamental system of solutions of the equation x = Tx + f). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Frechet spaces and strong duals of Frechet-Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.
Wydawca
Rocznik
Strony
611--626
Opis fizyczny
Bibliogr. 28 poz.
Twórcy
  • Moscow State University, Faculty of Mathematics and Mechanics, Vorobjovy gory, 119899 Moscow, Russia
Bibliografia
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  • [10] G. Herzog, On row-finite systems of differential equations in Banach spaces, Demonstratio Math. 31 (1998), 835-839.
  • [11] G. Herzog, R. Lemmert, Nonlinear fundamental systems for linear differentia equations in Prechet spaces, Demonstratio Math. 33 (2000), 313-318.
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  • [15] S. G. Lobanov, An example of a non-normable Fréchet space in which every continuous linear operator has an exponential, Russian Math. Surveys, 34 (1979), 201-202.
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  • [18] E. Mickael, A selection theorem, Proc. Amer. Math. Soc. 17 (1966), 1404-1406.
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  • [23] S. A. Shkarin, Differential equations and smooth functions in locally convex spaces, (in Russian) Ph. D. Thesis, 1991, Moscow.
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  • [25] S. A. Shkarin, Compact perturbations of linear differential equations in locally convex spaces, Studia Mathematica [submitted],
  • [26] S. A. Shkarin, On solvability of linear ordinary differential equations in Archiv der Mathematik [submitted].
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Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA3-0007-0016
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