On bounded solutions of ordinary nonlinear differential equations of order \(n\) (Q633566)

Consider the nonlinear nth-order differential equation \[ a_0 x^{(n)} + a_1 x^{(n - 1)} + \cdots + a_n x = f(t,x,\dot {x},\ddot {x},\dots,x^{(n - 1)}), \tag{1} \] where \(a_0 ,a_1 ,\dots,a_n\) are constants and the function \(f\) is continuous and satisfies a Lipschitz condition with respect to the space variables. The author proves the existence and uniqueness or only the existence of a bounded solution of (1) by both the contraction mapping principle and the Tikhonov fixed-point theorem. Then, the author obtains a qualitative estimate for the smallness of a nonlinear perturbation preserving the basic characteristics of the behavior of a linear equation (absolute stability or exponential dichotomy) under the passage to the nonlinear equation (1).