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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

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
Full Text: EuDML