  In this paper we study the asymptotic behavior of nonoscillatory solutions
for high order differential equations of Poincar\'e type. We introduce two new
and more weak than classical hypotheses on the coefficients, which implies a
well posedness result and a characterization of asymptotic behavior for the
solution of the Poincar\'e equation. The proof of the main results is based on
the application of the scalar method. We define a change of variable to reduce
the order of Poincar\'e type equation and deduce that the new variable
satisfies a nonlinear differential equation. We apply the variation of
parameters method and the Banach fixed point theorem to get the well posedness
and asymptotic behavior of the nonlinear equation. Then, by rewritten the
results in terms of the original variable we establish the existence of a
fundamental system of solutions for and we precise the formulas for the
asymptotic behavior of the Poincar\'e type equation.
