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Zbl 1077.34069
Boichuk, Alexander A.; Grammatikopoulos, Myron K.
Perturbed Fredholm boundary value problems for delay differential systems.
(English)
[J] Abstr. Appl. Anal. 2003, No. 15, 843-864 (2003). ISSN 1085-3375; ISSN 1687-0409/e

The considered linear boundary value problem with a small parameter $\varepsilon$ has the form $$\dot z(t)=\sum^k_{i=1} A_i(t)z(h_i(t))+\varepsilon\sum^k_{i=1}B_i(t)z(h_i(t))+g(t),\quad t\in [a,b];\ z(s)=\psi(s), \ s<\alpha; \ \ell z=\alpha.$$ The unknown solution $z$ takes values in a finite-dimensional space. The functions $h_i(t)\le t$ are measurable. In case $h_i(t)<\alpha$, it is assumed that $z(h_i(t))=\psi(h_i(t))$. The boundary conditions are described by the bounded linear operator $\ell$. The Fredholm properties of the boundary value problem are obtained in the form of power series in $\varepsilon$. Examples are given.
[Sergei A. Brykalov (Ekaterinburg)]
MSC 2000:
*34K10 Boundary value problems for functional-differential equations
34K06 Linear functional-differential equations

Keywords: linear boundary value problem; linear functional-differential equation; small parameter

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