Perturbed Fredholm boundary value problems for delay differential systems. (English) Zbl 1077.34069

Boichuk, Alexander A.; Grammatikopoulos, Myron K.

Abstr. Appl. Anal. 2003, No. 15, 843-864 (2003).

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 \(h_i(t)\leq t\) 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.

Reviewer: Sergei A. Brykalov (Ekaterinburg)

MSC:

34K10	Boundary value problems for functional-differential equations
34K06	Linear functional-differential equations

Keywords:

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