First order ordinary differential equations with several periodic solutions (Q1102427)

The solvability and the number of solutions of the periodic boundary value problem \(u'(x)+f(x,u(x))=s,\) \(u(0)=u(2\pi)\) is discussed where \(s\in {\mathbb{R}}\) and \(f: I\times {\mathbb{R}}\to {\mathbb{R}}\) is continuous and such that f(x,u)\(\to \infty\) as\(| u| \to \infty\) uniformly in \(x\in I=[0,2\pi]\).