Operator analogs of WKB-type estimates and the solvability of boundary- value problems (Q1311135)

The differential equation (1) \(d^ 2y/dt^ 2-k^ 2Q(t)=F(t)\) where \(t \in \mathbb{R}\) is considered in the Banach space \(B\). It is assumed that \(Q\) is a sufficiently smooth operator function, \(Q(t)\) are bounded and uniformly invertible and there exists a continuous branch of the root \(Q^{1/2}(t)\), such that for all \(t \in \mathbb{R}\), the operators \(- Q^{1/2} (t)\) generate contraction semigroups. It is proved that if the spectra of the limit operators \(Q(\pm \infty)\) do not contain negative numbers then for sufficiently large \(k\) the equation will have a unique bounded solution for any bounded \(F(t)\). For the problem (1), (2) where (2) \(\lim_{t \to \pm \infty} \| dy/dt \pm kQ^{1/2} (\pm \infty) y \|=0\) and \(F \in L_ 1 (\mathbb{R},B)\) the existence of a solution is proved and a formula which constitutes a generalization of an estimation for the Green's function is given.