A new approach to the method of nonlinear variation of parameters for a perturbed nonlinear neutral functional differential equation (Q1118747)

Suppose that \(\Omega \subset R\times C([-r,0],E^ n)\) where \(E^ n\) is a real or complex n-dimensional linear vector space and \(f,g: \Omega\to E^ n\). The paper deals with the systems \((i)\quad (d/dt)Dx_ t=f(t,x_ t)\) and \((ii)\quad (d/dt)Dy_ t=f(t,y_ t)+g(t,y_ t)\) where D is a continuous linear function and \(x_ t(s)=x(t+s)\), \(s\in [- r,0]\). Suppose that the system (i) admits a unique solution \(x_ t(\sigma,\phi)\) through \((\sigma,\phi)\in \Omega\) for \(t\geq \sigma\). Suppose also that \(\psi (t,\sigma,\phi)=(\partial /\partial \phi)x_ t(\sigma,\phi)\) exists and is continuous for all \(t\geq \sigma\) and that \(\psi^{-1}(t,\sigma,\phi)\) exists for all \(t\geq \sigma\). Under natural conditions on D any solution \(y_ t(\sigma,\phi)\) of (ii) satisfies \(y_ t(\sigma,\phi)=x_ t(\sigma,\phi +\int^{t}_{\sigma}\psi^{- 1}(s,\sigma,z_ s)Y_{\sigma}g(s,y_ s(\sigma,\phi))ds)\) as far as \(z_ t(\sigma,\phi)\) exists for \(t\geq \sigma\) as solution of \((d/dt)z_ t(\sigma,\phi)=\psi^{-1}(t,\sigma,z_ t)Y_{\sigma}g(t,x_ t(\sigma,z_ t))\) and Y is the \(n\times n\) matrix of bounded variation on compact sets, continuous from the right. An application is made to the study of the asymptotic behavior of solutions of perturbed equations.