Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance (Q1181695)

The paper considers the periodic problem \(-x''=f(x)x'+g(x)+h(t)\) in \([0,2\pi]\); \(x(0)=x(2\pi)\), \(x'(0)=x'(2\pi)\); \(f\) and \(g\) continuous from \(\mathbb{R}\) to \(\mathbb{R}\) and \(h\in L^ \infty(0,2\pi)\). The author proves the existence of at least one solution if \(\lim\sup g(s)/s\) and \(\lim\sup(2/s^ 2)\int^ s_ 0g(r) dr\) are appropriately bounded for \(s\to\pm\infty\) and if there exist \(A\) and \(B\) in \(\mathbb{R}\) such that \(g(A)\leq-h(t)\leq g(B)\).