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Abstrakty
We construct xo is an element of RN and a row-finite matrix T = {Ti,j(t)}i,j is an element of N of polynomials of one real variable t such that the Cauchy problem x(t) = Ttx(t), x{0) = xo in the Frechet space RN has no solutions. We also construct a row-finite matrix A = {Aij(t)}ij is an element of N of C°°(R) functions such that the Cauchy problem x{t) = Atx;(t), x(0) = xo in RN has no solutions for any xo infinity RN\ {0}. We provide some sufficient condition of solvability and unique solvability for linear ordinary differential equations x(t) = Ttx(t) with matrix elements Ti,j(t) analytically dependent on t.
Wydawca
Czasopismo
Rocznik
Tom
Strony
85--99
Opis fizyczny
Bibliogr. 19 poz.
Twórcy
autor
- Department of Mathematical, Analysis University of Seville, C/Tarfia, S/N, C.P. 41012 Seville, Spain
Bibliografia
- [1] K. Astala, On Peano's theorem in locally convex spaces, Studia Math. 73, 213-223 (1982).
- [2] V. I. Bogachev, Deterministic and stochastic differential equations in infinite dimensional spaces, Acta Appl. Math. 40, 25-93 (1995).
- [3] M. Eidelheit, Zur Theorie der System linearer Gleichungen, Studia Math. 6, 139-148 (1936).
- [4] P. Hartmann, Ordinary differential equations, Birkhauser, Boston-Basel-Stuttgart, 1982.
- [5] G. Herzog, On existence and uniqueness conditions for ordinary differential equations in Fréchet spaces, Studia Sci. Math. Hungar. 32, 367-375 (1996).
- [6] G. Herzog, On ordinary linear differential equations in CJ, Demonstratio Math. 28, 383-398 (1995).
- [7] G. Herzog, On linear time-dependent row-finite systems of differential equations, Travaux math., Fasc. VIII, 167-176, Sem. Math. Luxembourg, Centre Univ. Luxembourg, Luxembourg, 1996.
- [8] G. Herzog, On row-finite systems of differential equations in Banach spaces, Demonstratio Math. 31, 835-839 (1998).
- [9] A. N. Godunov, Linear differential equations in locally convex spaces, Moscow Univ. Math. Bull. 29, 31-39 (1974).
- [10] R. Lemmert and A. Weckbach, Characterisierungen zeiendlicher Matrizen mit abzählbarem Spectrum, Math. Zeitschrift 188, 119-124 (1984).
- [11] S. G. Lobanov, Solvability of linear ordinary differential equations in locally convex spaces, Moscow Univ. Math. Bull. 35, 1-4 (1980).
- [12] S. G. Lobanov, An example of a non-normable Fréchet space in which every continuous linear operator has an exponential, Russian Math. Surveys 34, 201-202 (1979).
- [13] S. G. Lobanov, O. G. Smolyanov, Ordinary differential equations in locally convex spaces, Russian Math. Surveys 49, 93-168 (1994).
- [14] V. M . Millionschikov, On the theory of differential equations in locally convex spaces, Russian Acad. Sci. Sb. Math. 57, 385-406 (1962).
- [15] S. A. Shkarin, Peano's theorem is invalid in infinite dimensional Fréchet spaces, Funct. Anal. Appl. 27, 149-151 (1993).
- [16] S. A. Shkarin, Peano's theorem is invalid in infinite dimensional F1-spaces, Math. Notes 62, 108-115 (1997).
- [17] S. A. Shkarin, Some results on solvability of ordinary linear differential equations in locally convex spaces, Russian Acad. Sci. Sb. Math. 71, 29-40 (1992).
- [18] S. A. Shkarin, Differential equations and smooth functions in locally convex spaces, Ph. D. Thesis, Moscow State University, Moscow, 1991.
- [19] S. Szufla, On the equation x’ = f(t,x) in locally convex spaces, Math. Nachr. 118, 179-185 (1984).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA3-0012-0011