Necessary and sufficient conditions for the existence of periodic solutions of second-order ordinary differential equations with singular nonlinearities
The paper deals with the problem of the existence of \(T\)-periodic solutions to the equation (1) \(u'' + f(u)u' + g(u) = h(t)\), where \(g : (0, \infty) \to \mathbb{R}\) is continuous and such that \(\lim_{s \to 0 + }g (s) = - \infty\), \(\lim_{s \to 0 + } G(s) = \lim_{s \to 0 + } (\int_1 sg (\xi) d \xi) = + \infty\), \(\limsup_{s \to \infty} {g(s) \over s} \leq {\pi \over T})2\) and \(\limsup_{s \to \infty} {2G (s) \over s2} < ({\pi \over T})2\); \(f : \mathbb{R} \to \mathbb{R}\) is continuous and \(h \in L_\infty (0,T)\). The authors show that the given equation (1) possesses a \(T\)-periodic solution if and only if it possesses a lower solution.
second-order ordinary differential equation
scientific article; zbMATH DE number 775652