Consider the system of ordinary differential equations \[ u'=-\lvert u\rvert ^2u+h(t), \tag {1} \] where \[ h:[0,1]\to \mathbb {R}^N \] is a continuous function and \(\lvert \cdot \rvert \) is the Euclidean norm in \(\mathbb {R}^N\). Use Brouwer’s fixed point theorem to show that (1) has a solution \(u\in C^1[0,1]\) such that \[ u(0)=u(1). \] Use this result to conclude that if \[ h:\mathbb {R}\to \mathbb {R}^N \] is continuous and periodic of period 1, then (1) has a solution of period 1.
(Hint: Consider equation (1) subject to initial data \(u_0:\lvert u_0\rvert \le r\), where \(r\) is large (in relation to \(h\)) and examine solution curves by computing \(\frac {d}{dt}\lvert u\rvert ^2\) for solutions \(u\).) In relation to the above, what can you say about the equation \[ u'=\lvert u\rvert ^2u+h(t). \tag {2} \]