Uniqueness of periodic solutions for asymptotically linear Duffing equations with strong forcing
The author deals with equations of the form \[ x''+cx'+ax+g(x)=\lambda p(t),\tag{1} \] where \(g:\mathbb{R}\to \mathbb{R}\) is \(C^1\) and asymptotically linear, that is \[ \lim_{|x|\to\infty}{g(x)\over x}=0.\eqno{(2)} \] The forcing term \(p\) is \(T\)-periodic and the existence and uniqueness of \(T\)-periodic solutions are investigated. The main result is the following theorem: Suppose that (i) \(a\neq 0\) and, if \(c=0\) then \(a\neq(2\pi m/T)^2\) for all integers \(m\), (ii) \(g:\mathbb{R}\to \mathbb{R}\) is \(C^1\), satisfies (2), and \(g'\) is bounded \(\|g(x)\|\leq L\) for all \(x\in \mathbb{R}\), (iii) \(p:\mathbb{R}\to \mathbb{R}\) is continuous and \(T\)-periodic, and let \(u_0\) be the unique \(T\)-periodic solution to \(u''+cu'+au=p(t) \), then the set of critical points of \(u_0\) is of measure 0. Then there exists a \(\lambda_0\geq 0\) such that for \(|\lambda|\geq\lambda_0\) equation (1) has a unique \(T\)-periodic solution. Assuming \(c>0\), then, for \(|\lambda|\) sufficiently large, this solution is asymptotically stable if \(a>0\), and unstable if \(a<0\).
scientific article; zbMATH DE number 1346176