Almost-periodic solutions to the matrix Riccati equation (Q2703981)

The author considers the equation NEWLINE\[NEWLINEX'+ Xg(t) X+ f(t)X+ l(t)= 0\quad\text{for }t\in\mathbb{R},\tag{\(*\)}NEWLINE\]NEWLINE where the values of \(X\), \(g\), \(f\), \(l\) are complex \(m\times m\)-matrices. It is claimed that, for sufficiently small \(l\), \((*)\) has an (even infinitely many) almost-periodic solution \(X\), provided the entries of \(g\), \(f\), \(l\) are almost-periodic functions given by certain absolutely convergent series \(\sum P_k\) satisfying conditions on the frequencies which seem contradictory.NEWLINENEWLINENEWLINEThe proof proceeds by induction on \(m\), using the almost-periodicity of the indefinite integral of an almost-periodic function as in the entries of \(g\) above.