Asymptotic method in the theory of second-order Volterra integrodifferential equations (Q1064764)

The method of Krylov-Bogoljubov [cf. \textit{N. N. Bogoljubov} and \textit{Yu. A. Mitropol'skij}, Asymptotic methods in the theory of nonlinear oscillations (1963; Zbl 0111.088)] for constructing asymptotic approximate solutions in the case of resonance is extended to integrodifferential equations of the form \[ (Dx)(t)=\epsilon (g_ 1(\nu t,x(t),\frac{dx(t)}{dt})+\int^{\infty}_{0}g_ 2(n(t-s),x(t- s),\frac{dx(t-s)}{dt})dk(s), \] where D is the linear integrodifferential operator \[ (Dx)(t)=(\frac{d}{dt^ 2}+\alpha_ 1\frac{d}{dt}+\alpha_ 2)x(t)+\int^{\quad \infty}_{0}(\beta_ 1\frac{d}{dt}+\beta_ 2)x(t- s)dk(s), \] k has bounded variation on [0,\(\infty)\), \(g_ i(\nu t,x,y)\), \(i=1,2\) are functions which are smooth on \(R\times R^ 2\), \(2\pi\)- periodic in \(\nu\) t, \(\alpha_ i\), \(\beta_ i\in R\), \(i=1,2\), \(\epsilon\) is a small positive parameter.