Asymptotic stability and asymptotic solutions of second-order differential equations (Q860548)

Consider the second-order differential equation \[ \frac{d^2x(t)}{dt^2}+2\frac{d}{dt}\big(f(t)x(t)+g(t,x(t),x'(t)\big)x(t)=0,\; t>T>0, \eqno(1) \] where the coefficients \(f(t)\) and \(g(t,x(t),x'(t)\) are complex-valued continuous functions of time \(t\). The asymptotic stability for the equation (1) has been widely studied by using energy functions and Lyapunov stability and instability theorems. In this paper the author uses a different approach based on integral representations of solutions via different asymptotic solutions and an error estimates technique. This work is a continuation of the previous article by the author [Elec. J. Diff. Equ. 85, 1--20 (2004; Zbl 1050.34072)].