New results for the non-oscillatory asymptotic behavior of high order differential equations of Poincaré type

This paper discusses the study of asymptotic behavior of non-oscillatory solutions for high order differential equations of Poincaré type. We present two new and weaker hypotheses on the coefficients, which implies a well posedness result and a characterization of asymptotic behavior for the solution of the Poincaré equation. In our discussion, we use the scalar method: we define a change of variable to reduce the order of the Poincaré equation and thus demonstrate that a new variable can satisfies a nonlinear differential equation; we apply the method of variation of parameters and the Banach fixed-point theorem to obtain the well posedness and asymptotic behavior of the non-linear equation; and we establish the existence of a fundamental system of solutions and formulas for the asymptotic behavior of the Poincaré type equation by rewriting the results in terms of the original variable. Moreover we present an example to show that the results introduced in this paper can be used in class of functions where classical theorems fail to be applied.