First-order ordinary differential equations with several bounded separate solutions
The author deals with the equation (1) \(x'= F(t,x)+\lambda\), where \(F: \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is a continuous function, \(F(t,x)\to -\infty\) as \(| x|\to \infty\) uniformly on \(t\in\mathbb{R}\), and \(F(t,x)\) is either bounded on \(\mathbb{R}\times K\) for each compact set \(K\subset \mathbb{R}\) or \(T\)-periodic in \(t\) for some \(T>0\). The existence and the number of bounded or separate or singular solutions to (1) are discussed according to \(\lambda\in\mathbb{R}\). The obtained results are compared with those of \textit{J. Mawhin} [Z. Angew. Math. Phys. 38, 257-265 (1987; Zbl 0644.34035)].
scientific article; zbMATH DE number 1218226