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

• Autorzy: Shkarin S. S.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

differential equations; convex spaces; existence theorem; uniqueness theorem; linear operators

PL

równania różniczkowe; przestrzeń wypukła; operatory linearne

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2003
• Tom: Vol. 36, nr 3
• Strony: 611--626
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

Shkarin S. S.

• Moscow State University, Faculty of Mathematics and Mechanics, Vorobjovy gory, 119899 Moscow, Russia

Bibliografia

• [1] K. Astala, On Peano's theorem in locally convex spaces, Studia Math. 73 (1972), 213-223.
• [2] V. I. Averbukh, O. G. Smolyanov, The various definitions of the derivative in linear topological spaces, Russian Math. Surveys 23 (1968), 67-113.
• [3] C. Bessaga, A. Pełczyński, Selected Topics in Infinite Dimensional Topology, Polish Scientific Publishers, Warszawa, 1975.
• [4] V. I. Bogachev, Deterministic and stochastic differential equations in infinite dimensional spaces, Acta Appl. Math. 40 (1995), 25-93.
• [5] J. Bonet, P. Carreras, Barrelled Locally Convex Spaces, North Holland, Amsterdam, 1987.
• [6] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367.
• [7] A. N. Godunov, Linear differential equations in locally convex spaces, Moscow Univ. Math. Bul. 29 (1974), 31-39.
• [8] G. Herzog, One-sided estimates for ordinary differential equations in Frechet spaces, Demonstratio Math. 33 (2000), 313-318.
• [9] G. Herzog, On ordinary linear differential equations in CJ, Demonstratio Math. 28 (1995), 383-398.
• [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.
• [12] A. N. Kolmogorov, S. V. Fomin, Introductory Real Analysis, Dover, New York, 1979.
• [13] G. Köthe, Topological Vector Spaces, Vols. I and II, Springer, New York, 1969 and 1979.
• [14] R. Lemmert, On ordinary differential equations in locally convex spaces, Nonlinear Analysis 10 (1986), 1385-1390.
• [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.
• [16] S. G. Lobanov, Solvability of linear ordinary differential equations in locally convex spaces, Moscow Univ. Math. Bui. 35 (1980), 1-4.
• [17] S. G. Lobanov, O. G. Smolyanov, Ordinary differential equations in locally convex spaces, Russian Math. Surveys 49 (1994), 93-168.
• [18] E. Mickael, A selection theorem, Proc. Amer. Math. Soc. 17 (1966), 1404-1406.
• [19] V. M. Millionschikov, On the theory of differential equations in locally convex spaces, Russian Acad. Sci. Sb. Math. 57 (1962), 385-406.
• [20] B. S. Mityagin, Approximative dimension and bases in nuclear spaces, Russian Math. Surveys, 16 (1961), 59-128.
• [21] S. A. Shkarin, Peano's theorem is invalid in infinite dimensioanl Fréchet spaces, Funct. Anal. Appl. 27 (1993), 149-151.
• [22] S. A. Shkarin, Peano's theorem is invalid in infinite dimensional F'-spaces, Math. Notes 62, 1997, 108-115.
• [23] S. A. Shkarin, Differential equations and smooth functions in locally convex spaces, (in Russian) Ph. D. Thesis, 1991, Moscow.
• [24] S. A. Shkarin, Some results on solvability of ordinary linear differential equations in locally convex spaces, Russian Acad. Sci. Sb. Math., 71 (1992), 29-40.
• [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].
• [27] S. A. Shkarin, Whitney's type theorems for infinite dimensional spaces, Infinite Dimensional Analysis, Quantum Probabilty and Related Topics 3 (2000), 141-160.
• [28] S. Szufla, On the equation x' = f(t,x) in locally convex spaces, Math. Nachr. 118 (1984), 179-185.