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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).

##### MSC:
 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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##### References:
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