On the characteristic exponent function for a first order ordinary differential equation
Consider the equation \(\dot x = f(t,x)\), \((t,x) \in I \times \mathbb{R}\), where \(I \supset [0,1]\) is an open interval and \(f(t,x)\) is a continuous function with a continuous first derivative \(f_ x'(t,x)\). Let \(\Lambda\) be the set of all \(\xi \in \mathbb{R}\) for which the maximal solution \(x(t, \xi)\) of this equation through the point \((0, \xi)\) is defined on [0,1]. For any \(\xi \in \Lambda\) the characteristic exponent of the solution \(x(t,\xi)\) is a number \(\mu (\xi) = \int^ 1_ 0 f_ x'(t,x (t, \xi))dt\). The main result of the paper is: \(\mu (\xi)\) is a weakly convex function if \(f_ x' (t,x)\) is weakly convex in \(x\) for all fixed \(t\). For the proof of this theorem there are established two formulae: the first one is the expression for the sum \(\mu (\xi_ 1) + \mu (\xi_ 2)\), and the second formula gives an expression for the value \(\mu (\xi_ 2)\) in terms of arbitrary values \(\xi_ 1\) and \(\xi_ 3\) such that \(\xi_ 1 < \xi_ 2 < \xi_ 3\), \(\xi_ i \in \Lambda\). These formulae yield two consequences: \(1^ 0\) the Riccati equation is characterized inside the class of equations \(\dot x = f(t,x)\), where \(f_ x'(t,x)\) is weakly convex in \(x\), \(2^ 0\) for equations of the same class satisfying an additional condition, it is given some upper estimation of the number of closed solutions; in this case the stability of closed solutions is discussed.
scientific article; zbMATH DE number 716349