Asymptotic decomposition of boundary value problems for differential equations in Banach spaces (Q1337902)

The author considers the boundary-value problem \[ \frac{du}{dt} = kA(t)u+ F,\;(t\in \mathbb{R}),\qquad \lim_{t\to\pm\infty} V_\pm u(t)= 0, \tag{P} \] where \(k\) is a major asymptotic parameter, \(u, F: \mathbb{R}\to B\) (\(B\) a Banach space), \(A(t): D(A(t))\subset B\to B\) are closed linear operators which are stabilized as \(t\to \pm\infty\). He assumes that \(B= B_1(t)+ B_2(t)\), where \(B_i(t)\) are invariant under \(A(t)\) and the spectra of \(A(t)/B_i(t)\), \(i= 1,2\), are uniformly separated for \(t\in \mathbb{R}\) and \(V_\pm\) are linear bounded operators for which \(B_i(\pm \infty)\) are invariant, \(i= 1,2\). Under some additional assumptions, the author shows that the asymptotic behaviour of the solutions of (P) as \(k\to \infty\) can be described by means of the asymptotic behaviour of some auxiliary problems in \(B_i(t)\), \(i= 1,2\), even though the system cannot be decoupled.