Towards a theory of the asymptotic method of Krylov-Bogolyubov for differential-functional equations (Q1221042)

The authors study the problem of constructing an asymptotic solution of the equation \[ \dot x(t)=\int_{-r}^0 [d\eta(\theta)]\,x(t+\theta)+\int_{-r}^0 [d\xi(\theta)]\,\dot x(t+\theta)+\varepsilon F(\nu t,x_t,\dot x_t),\tag{1} \] where \(x(t)\in E^n\), \([\eta(\theta)]\) and \([\xi(\theta)]\) are \(n\times n\)-matrices whose entries are functions of bounded variation, and \(F(\nu t,x_t,\dot x_t)\) is a nonlinear functional of \(x_t=x(t+\theta)\) and \(\dot x_t=\dot x(t+\theta)\), \(-r\leq\theta\leq 0\), which is periodic in \(\nu t\) and \(\varepsilon\) is a small positive parameter. They construct bounded solutions of (1) in the form of formal series in powers of \(\varepsilon\) which have an asymptotic character as \(\varepsilon\to 0\).