Heteroclinic connections for fully non-linear non-autonomous second-order differential equations

We investigate the solvability of the following strongly non-linear non-autonomous boundary value problem with ν−<ν+ given constants, where a(x) is a generic continuous positive function and f is a Carathéodory non-linear function. We show that the solvability of (P) is strictly connected to a sharp relation between the behaviors of f(t,x, ) as x′→0 and f( ,x,x′) as t→+∞. Such a relation is optimal for a wide class of problems, for which we prove that (P) is not solvable when it does not hold.