Perturbed Fredholm boundary value problems for delay differential systems (Q1773472)

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.