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Unbounded perturbations of forced second order ordinary differential equations at resonance. (English) Zbl 0627.34008
In der vorliegenden Arbeit wird die erzwungene nichtlineare gewöhnliche Differentialgleichung der Form ẍ(t)\(+m^ 2x(t)+g(t,x(t)=e(t)\) mit den Bedingungen \(x(0)-x(2\pi)=\dot x(0)-\dot x(2\pi)=0\) betrachtet, wobei \(m\in {\mathbb{Z}}^+\cup \{0\}\), \(e\in L^ 1(0,2\pi)\) und g eine nichtlineare Funktion ist, die die “Carathéodory-Bedingungen” erfüllt. Durch einen Satz wird unter Aufstellung bestimmter Voraussetzungen die Existenz mindestens einer Lösung der Gleichung im Falle der Resonanz gewährleistet. Der hier angegebene Beweis geht von einen topologischen Standpunkt aus. Ein wesentliches Hilfsmittel beim Beweis ist die Technik von Leray-Schauder. Aus den Bemerkungen des Autors wird deutlich, daß seine ergänzenden Ergebnisse zur Verallgemeinerung bestehender Aussagen führte.
Reviewer: M.L.Mehra

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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