On the cross-ratio of four solutions of a first order ordinary differential equation
The authors generalize the classical result concerning the Riccati equation \[ \dot x(t)= a_ 2(t)x^ 2+ a_ 1(t)x+ a_ 0(t),\quad t\in I\subset R:\tag{1} \] If \(\varphi_ i(t)\), \(i=1,2,3,4\), are different solutions of (1) defined on the same interval \(J\subset I\), then their cross-ratio \[ {\varphi_ 3(t)- \varphi_ 1(t)\over \varphi_ 4(t)- \varphi_ 1(t)}: {\varphi_ 3(t)- \varphi_ 2(t)\over \varphi_ 4(t)- \varphi_ 2(t)}= R(t)\tag{2} \] is constant on \(J\). The authors consider the equation (3) \(\dot x(t)= f(t,x)\), \((t,x)\in I\times R\), where \(I\) is an open interval, \(f(t,x)\) and partial derivative \(f_ x(t,x)\) are continuous. They prove two theorems. Theorem 1. If \(f_ x(t,x)\) is weakly convex (weakly concave) in \(x\) for any fixed \(t\in I\), and if \(\varphi_ 1(t)< \varphi_ 2(t)< \varphi_ 3(t)< \varphi_ 4(t)\), \(t\in J\subset I\), are four solutions of (3) on the same open interval \(J\), then the cross-ratio \(R(t)\) is weakly increasing (weakly decreasing) on \(J\). It is constant if and only if (3) is a Riccati equation in the domain \(t\in J\), \(\varphi_ 1(t)\leq x\leq \varphi_ 4(t)\). Theorem 2. Let \(f(t,x)\), \((t,x)\in R^ 2\) be \(T\)-periodic in \(t\) and let \(f_ x(t,x)\) be weakly convex (weakly concave) in \(x\) for all fixed \(t\in R\). Then (3) has either a finite number of \(T\)-periodic solutions -- and then at most three -- or it has infinitely many \(T\)-periodic solutions -- and they form a simply connected domain in the \((t,x)\)-plane. In the last case (3) is a Riccati equation in the mentioned domain. The paper also contains the relations to the methods and results of other authors, namely D. J. Allwright.
scientific article; zbMATH DE number 554303