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

New results for the Nonoscillatory Asymptotic Behavior of High Order Differential Equations of Poincar\'e type