Nonmonotone equations with large almost periodic forcing terms
This paper deals with the equation \[ \dot{u}=j(u)+g(u)+ch(t), \] where \(j:\mathbb R\to\mathbb R\) is an increasing homeomorphism of the type \(j(u)=\alpha_+u^+ -\alpha_- u^-\), \(g:\mathbb R \to \mathbb R\) is a bounded function with derivative vanishing at infinity, \(c\) is a parameter and \(h:\mathbb R \to\mathbb R\) is an almost periodic function. The main result gives the existence of a unique almost periodic solution if \(c\) is large enough and \(h\) satisfies a condition related to the size of the set of zeros. More precisely, \[ ||g'||_{\infty} \lambda (H^{-1}(0))<\min \{ \alpha_+ ,\alpha_- \}, \] where \(H:\Omega \to \mathbb R\) is the extension of \(h\) to a compact topological group \(\Omega\) and \(\lambda\) is the corresponding Haar measure.
scientific article; zbMATH DE number 6117521