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 [GRAPHICS] with nu(-) < nu(+) given constants, where a(x) is a generic continuous positive function and f is a Caratheodory 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 vertical bar x'vertical bar -> 0 and f (., x, x') as vertical bar t vertical bar -> +infinity. 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. (c) 2007 Elsevier Inc. All rights reserved.