Solvability of a forced autonomous Duffing’s equation with periodic boundary conditions in the presence of damping. (English) Zbl 0785.34024

The author investigates the existence of a solution for the forced autonomous Duffing’s equation (1) \(u''+cu'+g(u)=e(x)\), \(0<x<1\), \(u(0)=u(1)\), \(u'(0) = u'(1)\), where \(g:R \to R\) is a continuous function, \(e:[0,1] \to R\) is a function in \(L^ 2[0,1]\) and \(c \in R\), \(c \neq 0\) is given. It is proved that Duffing’s equation (1) in the presence of the damping term has at least one solution providing that there exists an \(R>0\) that \(g(u) \cdot u \geq 0\) for \(| u | \geq R\) and \(\int^ 1_ 0e(x)dx=0\). It is further proved that if \(g\) is strictly increasing on \(R\) with \(\lim g(u)=-\infty\), \(\lim g(u)=+\infty\) and is Lipschitz continuous with Lipschitz constant \(\alpha<4^ 2+e^ 2\), then Duffing’s equation given above has exactly one solution for every \(e\in L^ 2[0,1]\). The author uses Mawhin’s version of the Leray-Schauder continuation theorem, and Wirtinger type inequalities to get the needed estimates. Moreover, he presents some uniqueness results for the boundary value problem (1).


34B15 Nonlinear boundary value problems for ordinary differential equations
47J05 Equations involving nonlinear operators (general)
34C25 Periodic solutions to ordinary differential equations
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