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On ordinary differential equations in locally convex spaces. (English) Zbl 0612.34056
The paper contains existence and uniqueness results for ordinary differential equations \(u'(t)=f(t,u(t)),\) \(u(0)=u_ 0\) in locally convex Hausdorff spaces E. First, existence and comparison criteria are given for the linear autonomous case, if \(E={\mathbb{R}}^ J\) (J a general index set) with the product topology. In the general case, a dissipativity condition for f in terms of the system of seminorms that generate the topology of E is assumed, implying that successive iterations converge by the results for the linear case.
Reviewer: H.Engler

MSC:
34G10 Linear differential equations in abstract spaces
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34G20 Nonlinear differential equations in abstract spaces
34A45 Theoretical approximation of solutions to ordinary differential equations
46A03 General theory of locally convex spaces
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[1] Deimling, K., Ordinary differential equations in Banach spaces, (1977), Springer Berlin · Zbl 0364.34030
[2] Hille, E., Pathology of infinite systems of linear first order differential equations with constant coefficients, Annali mat. pura appl., 55, 133-148, (1961) · Zbl 0113.06905
[3] Lemmert, R.; Weckbach, Ä., Charakterisierungen zeilenendlicher matrizen mit abzählbarem spektrum, Math. Z., 188, 119-124, (1984) · Zbl 0554.47015
[4] Schaefer, H.H., Topological vector spaces, (1971), Springer Berlin · Zbl 0212.14001
[5] Schröder, J., Das iterationsverfahren bei allgemeinerem abstandsbegriff, Math. Z., 66, 111-116, (1956) · Zbl 0073.33503
[6] Volkmann, P., Gewöhnliche differentialungleichungen mit quasimonoton wachsenden funktionen in topologischen vektorräumen, Math. Z., 127, 157-164, (1972) · Zbl 0226.34058
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