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Title: Solvability of a forced autonomous Duffing's equation with periodic boundary conditions in the presence of damping (English)
Author: Gupta, Chaitan P.
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 38
Issue: 3
Year: 1993
Pages: 195-203
Summary lang: English
Category: math
Summary: Let $g$: $\bold R\rightarrow \bold R$ be a continuous function, $e$: $[0,1]\rightarrow \bold R$ a function in $L^2[0,1]$ and let $c \in \bold R$, $c\neq 0$ be given. It is proved that Duffing's equation $u'' + cu' + g(u)=e(x)$, $0<x<1$, $u(0)=u(1)$, $u'(0)=u'(1)$ in the presence of the damping term has at least one solution provided there exists an $\bold R > 0$ such that $g(u)u\geq 0$ for $|u|\geq \bold R$ and $\int^{1}_{0}e(x)dx=0$. It is further proved that if $g$ is strictly increasing on $\bold R$ with $\lim_{u\rightarrow -\infty} g(u)=-\infty$, $\lim_{u\rightarrow \infty} g(u)=\infty$ and it Lipschitz continuous with Lipschitz constant $\alpha<4\pi^2+c^2$, then Duffing's equation given above has exactly one solution for every $e\in L^2[0,1]$. (English)
Keyword: Dufiing's equation
Keyword: damping
Keyword: forced autonomous Duffing’s equation
Keyword: Leray-Schauder continuation theorem
Keyword: Wirtinger type inequalities
Keyword: uniqueness
Keyword: boundary value problem
MSC: 34B15
MSC: 34C25
MSC: 47H15
MSC: 47J05
MSC: 47N20
idZBL: Zbl 0785.34024
idMR: MR1218025
DOI: 10.21136/AM.1993.104546
Date available: 2008-05-20T18:45:30Z
Last updated: 2020-07-28
Stable URL:
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Reference: [5] Mawhin J.: Compacitè, Monotonie et Convexitè dans l'etude de problèmes aux limites semilinèaires.Sem. Anal. Moderne Université de Sherbrooke 19 (1981).
Reference: [6] Mawhin J.: Landesman-Lazer type Problems for Non-linear Equations.Confer. Sem. Mat. Univ. Bari 147 (1977). MR 0477923
Reference: [7] Mawhin J.: Topological Degree Methods in Nonlinear Boundary Value Problems.CBMS Regional Conf. Math. Ser. Math, vol. 40, American Math. Society, Providence, RI, 1979. Zbl 0414.34025, MR 0525202
Reference: [8] Nieto J. J , and Rao V. S. H.: Periodic Solutions for Scalar Lienard Equations.Acta Math. Hung. 57 (1991), 15-27. MR 1128836, 10.1007/BF01903798


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