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Theory of singular perturbations in case of spectral singularities of the limit operator. (Russian) Zbl 0624.34043
The authors consider the Cauchy problem in a Banach space for the differential equation \(\epsilon y'-A(t)y=h(t)\), t in (0,T), as \(\epsilon \to 0^+\) under the principal assumptions that (1) the operator A(t) has the spectral representation \(A(t)=\sum^{n}_{k=1}\lambda_ k(t)P_ k(t),\) t in [0,T], for projections \(P_ k\), and (2) the spectrum of A(t) is such that for \(i\neq j\), \(\lambda_ i(t)\neq \lambda_ j(t)\); \(\lambda_ 1(t)=t^{k_ 0}(t-t_ 1)^{k_ 1}...(t-t_ r)^{k_ r}a(t),\) a(t)\(\neq 0\); \(k_ 0+k_ 1+...+k_ r=m\geq 1\); Re \(\lambda\) \({}_ i(t)\geq 0\) for \(i=1,...,n\). Using the method of regularization they construct a uniformly valid asymptotic approximation of the solution.
Reviewer: F.A.Howes

MSC:
34D15 Singular perturbations of ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34G10 Linear differential equations in abstract spaces
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