Mather sets for sublinear Duffing equations
The paper deals with the Duffing equation \(x'' + g(x) = p(t)\) with sublinear growth conditions, where \(p(t)\) is periodic and \(g\) satisfies some regularity conditions. The author studies the existence of so-called Mather sets, which are sets of generalized quasiperiodic and unlinked periodic solutions. The main theorem gives conditions under which there exists a \(W > 0\) such that for any \(0 < w < W\), the equation possesses a solution \(z(w)\) of Mather type with rotation number \(w\). It follows that (i) if \(w\) is a rational number \(p/q\), there are \(q\) periodic solutions \(\{z(w)\}\) of period \(q\); (ii) if \(w\) is an irrational number then the solution \(z(w)\) is either a usual quasiperiodic solution or a generalized one exhibiting a Denjoy's minimal set \(M(w)\).
scientific article; zbMATH DE number 720561