Existence of closed solutions of an equation \(\dot x=f(t,x)\), where \(f_x'(t,x)\) is weakly convex or concave in \(x\) (Q1279945)

Consider the scalar differential equation \((*) \;dx/dt = f(t,x)\) where \(f\) is assumed to be sufficiently smooth in \(x\). A solution \(x(t,0,x_0)\) to \((*)\) is said to be closed if \(x(0;0,x_0)=x_0=x(1;0,x_0)\). The authors estimate the maximal number of closed solutions to \((*)\) and their stability under the assumption that \(f_x (t,x)\) is weakly convex (or weakly concave) in \(x\) for each fixed \(t \in [0,1]\). If \(f_x\) is weakly convex then there exist two functions \(\alpha\) and \(\beta\) such that \(f_x (x,t)\) is strictly decreasing for \(x \leq \alpha (t)\) and strictly increasing for \(x \geq \beta (t)\) and - when \(\alpha (t) < \beta (t)\) - constant for \(\alpha (t) \leq x \leq \beta (t)\). Under some assumptions on the functions \(\alpha\) and \(\beta\), the authors prove results on the existence of closed solutions and determine their stability. The results are illustrated by means of Abel's equation \(dx/dt = \sum^3_{i=0} a_i (t)x^i\).