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

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.