Resonance Problems for Some Non-autonomous Ordinary Differential Equations

Abstract

Recent years have seen a lot of activity in the study of quasilinear non-autonomous ordinary differential equations or systems of the form (ϕ(y′))′ = f(t, y, y′), where $\phi : A \subset {\mathbb{R}}^{n} \rightarrow B \subset {\mathbb{R}}^{n}$ is some homeomorphism such that $\phi (0) = 0$ between the open sets A and B. The situation generalizes the classical case where $A = B = {\mathbb{R}}^{n}$ and $\phi$ is the identity, and the well-studied case of the p-Laplacian (p > 1) where $\phi (s) =\| {s\|}^{p-2}s$. Contemporary researches concern less standard situations where $\phi : B(a) \rightarrow {\mathbb{R}}^{n}$ (singular homeomorphism) and $\phi : {\mathbb{R}}^{n} \rightarrow B(a)$ (bounded homeomorphism), where B(a) is the open ball of centre 0 and radius a. For n = 1, a model for the first case, namely $\phi (s) = \frac{s} {\sqrt{1-{s}^{2}}}$, corresponds to acceleration in special relativity, and a model for the second situation, namely $\phi (s) = \frac{s} {\sqrt{1+{s}^{2}}}$, corresponds to problem with curvature satisfying various conditions. In those case, both topological and variational methods, and sometimes combination of them give new complementary existence and multiplicity results. We will describe some of them. Some attention will be given to the generalized forced pendulum equation $(\phi (y^\prime))^\prime + A\sin y = h(t)$ when ϕ is singular or bounded.