Consider the periodic Riccati equation $y'(x)=y(x)^2+q(x)$ on the real line $\mathbb{R}$, where $q\in C^\infty(\mathbb{R})$ is a periodic function with period $T=1$. Suppose $q(x)$ can take both positive and negative values but $\int\_0^1 q(x)dx<0$. Does a periodic solution to this Riccati equation exist? Any references? Thanks a lot.



Existence of periodic solutions to scalar Riccati equations
The answer is negative.
The MSN review says that this is the contents of


MR1466035
Tang, Fenjun
The periodic solutions of Riccati equation with periodic coefficients.
Ann. Differential Equations 13 (1997), no. 2, 165–169.


I looked at this paper and the argument there makes no sense.


Setting $y=-w'/w$ we obtain $w''+qw=0$, and periodicity
of $y$ means $w(x+T)=\lambda w(x)$ for some $\lambda\neq 0$, so that $w$ is an eigenfunction of the shift operator. For generic $q$ there will be two such eigenfunctions (with distinct eigenvalues), and the question is whether at least one of them is free of zeros on $[0,T]$.


Here is a simple counterexample. Let $T=2\pi$, and
$q(x)=1$ for $0\leq x\leq 3\pi/2$, and then extend $q$ to a smooth
$2\pi$ periodic function arbitrarily, but so that
$$\int\_0^{2\pi}q(x)dx<0.$$
Then the general solution $w$ of our linear equation will
be of the form $w(x)=A\cos(x-\alpha),\; 0<x<3\pi/2$, so every solution
has at least one zero. This means that the corresponding Riccati equation has no smooth real solutions $y=-w'/w$ defined on the real line, thus no
periodic solutions.


