Non-degeneracy and uniqueness of periodic solutions for \(2n\)-order differential equations
This paper investigates the non-degeneracy of the \(T\)-periodic boundary value problem for the linear \(2n\)-order ordinary differential equation \[ u^{(2n)} + \sum_{m=1}^{2n-1} a_m u^{(m)} =q(t)u, \] where \(a_m \in \mathbb{R}\) and \(q\) is a \(T\)-periodic \(L^p\) potential (\(1 \leq p \leq \infty\)) (we recall that non-degeneracy means that \(u(t) \equiv 0\) is the only \(T\)-periodic solution of the given equation). As pointed out by the authors, in spite of the large bibliography dealing with the case \(n=1\) (that is, second order equations), the higher-order problem is less investigated. The main result of the paper gives a non-degeneracy condition, which is expressed as an algebraic relation involving the coefficients \(a_m\) and suitable Lebesgue norms of the positive or the negative part of the potential \(q(t)\). To this aim, a crucial role is played by the use of sharp Sobolev and Wirtinger inequalities. As an application of this non-degeneracy criterion, some existence/uniqueness results for related nonlinear \(T\)-periodic problems are given, as well, dealing with the semilinear and the superlinear case. Topological degree theory is used in this second part of the paper.
scientific article; zbMATH DE number 6157297