Homoclinic solutions for linear and linearizable ordinary differential equations (Q1608758)

Let \(F: \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}^N\) be a continuous function. The authors consider the problem \[ \dot x= F(t,x),\quad x(-\infty)= x(+\infty). \] They investigate the existence of solutions for the above-mentioned problem in the case when \(F(t,x)= A(t)x+ b(t)\); their main result are determined by the specific properties of the matrix \(A(t)\). They also give two existence theorems in the case that \(F\) is linearizable.