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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

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
Full Text: DOI
[1] Amann, H; Ambrosetti, A; Mancini, G, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z., 158, 179-194, (1978) · Zbl 0368.35032
[2] Amann, H; Mancini, G, Some applications of monotone operator theory to resonance problems, Nonlinear anal. TMA, 3, 6, 815-830, (1979) · Zbl 0418.47028
[3] Berestycki, H; De Figueiredo, D.G, Double resonance in semi-linear elliptic problems, Comm. in partial differential equations, 6, 1, 91-120, (1981) · Zbl 0468.35043
[4] Brezis, H; Nirenberg, L, Characterizations of the ranges of some nonlinear operators and application to boundary value problems, Ann. scuola norm. sup. Pisa, 5, 225-326, (1978) · Zbl 0386.47035
[5] Cesari, L; Kannan, R, Qualitative study of a class of nonlinear boundary value problems at resonance, J. differential equations, 56, 63-81, (1985) · Zbl 0554.34009
[6] Conti, G; Iannacci, R; Nkashama, M.N, Periodic solutions of Liénard systems at resonance, Ann. mat. pura appl., 139, 313-328, (1985), (4) · Zbl 0577.34035
[7] Cronin, J, Equations with bounded nonlinearities, J. differential equations, 14, 581-586, (1973) · Zbl 0251.34049
[8] De Figueiredo, D.G, Semilinear elliptic equations at resonance: higher eigenvalues and unbounded nonlinearities, (), 89-99
[9] De Figueiredo, D.G; Ni, W.N, Perturbations of second order linear elliptic problems by nonlinearities without landesman-lazer condition, Nonlinear anal. TMA, 5, 1, 57-60, (1981)
[10] Ding, T.R, Some fixed point theorems and periodically perturbed nondissipative systems, Chinese ann. math., 2, 3, 281-300, (1981) · Zbl 0484.34025
[11] Ding, T.R, Nonlinear oscillations at a point of resonance, Sci. sinica ser. A, 25, 9, 918-931, (1982) · Zbl 0509.34043
[12] Ding, T.R, Unbounded perturbations of forced harmonic oscillations at resonance, (), 59-66, (1) · Zbl 0538.34028
[13] Drabek, P, Solvability of nonlinear problems at resonance, Comment. math. univ. carolin., 23, 2, 359-368, (1982) · Zbl 0506.47044
[14] Fabry, C; Franchetti, C, Nonlinear equations with growth restrictions on the non-linear term, J. differential equations, 20, 283-291, (1976) · Zbl 0326.47060
[15] Fucik, S; Hess, P, Nonlinear perturbations of linear operators having nullspace with strong unique continuation property, Nonlinear anal. TMA, 3, 2, 271-277, (1979) · Zbl 0426.47042
[16] Giusti, E, Analisi matematica 1, (1983), Editore Boringhieri società per azioni Torino
[17] Kannan, R, Perturbation methods for nonlinear problems at resonance, (), 209-225
[18] Landesman, E.M; Lazer, A.C, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. math. mech., 19, 609-623, (1970) · Zbl 0193.39203
[19] Lazer, A.C, On Schauder’s fixed point theorem and forced second order nonlinear oscillations, J. math. anal. appl., 21, 21-425, (1968) · Zbl 0155.14001
[20] Lazer, A.C; Leach, D.E, Bounded perturbations of forced harmonic oscillators at resonance, Ann. mat. pura appl., 82, 49-68, (1969), (4) · Zbl 0194.12003
[21] Mawhin, J, Landesman-Lazer’s type problems for nonlinear equations, (), 147 · Zbl 0436.47050
[22] Mawhin, J, Compacité, monotonie et convexité dans l’étude des problèmes aux limites semi-linéaires, () · Zbl 0497.47033
[23] {\scJ. Mawhin}, The dual least action principal and nonlinear differential equations, to appear. · Zbl 0627.47034
[24] Mahwin, J; Ward, J.R, Periodic solutions of some forced Liénard differential equations at resonance, Arch. math., 41, 337-351, (1983) · Zbl 0537.34037
[25] Nkashama, M.N, Conditions de résonance ou de non-résonance non-uniformes et solutions périodiques d’equations différentielles non-linéaires, () · Zbl 0618.34018
[26] {\scP. Omari and F. Zanolin}, Existence results for forced nonlinear periodic BVP’s at resonance, Ann. Mat. Pura Appl., in press. · Zbl 0589.34005
[27] Reissig, R, Continua of periodic solutions of the Liénard equation, (), 126-133, ISNM
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