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Keywords:
Dufiing's equation; damping; forced autonomous Duffing’s equation; Leray-Schauder continuation theorem; Wirtinger type inequalities; uniqueness; boundary value problem
Dufiing's equation; damping; forced autonomous Duffing’s equation; Leray-Schauder continuation theorem; Wirtinger type inequalities; uniqueness; boundary value problem
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 $e\in L^2[0,1]$.
References:
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