On some boundary value problems for systems of linear functional differential equations

R. Hakl, Masaryk University, Brno, Czech Republic

E. J. Qualitative Theory of Diff. Equ., No. 10. (1999), pp. 1-16.

Abstract: In this paper on the segment $I=[a,b]$ we will consider the system of linear functional differential equations
\begin{equation}\label{1}
x'_i(t)=\sum\limits_{k=1}^n\ell_{ik}(x_k)(t)+q_i(t)\qquad (i=1,\dots,n)
\end{equation}
and its particular case
$$x'_i(t)=\sum\limits_{k=1}^n p_{ik}(t)x_k(\tau_{ik}(t))+q_i(t)\qquad (i=1,\dots,n)\eqno{(1')}$$
with the boundary conditions
\begin{equation}\label{2}
\int_a^b x_i(t)d\varphi_i(t)=c_i\qquad (i=1,\dots,n).
\end{equation}
Here $\ell_{ik}:C(I;\Bbb R)\to L(I;\Bbb R)$ are linear bounded operators, $p_{ik}$ and $q_i\in L(I;\Bbb R)$, $c_i\in\Bbb R$ $(i,k=1,\dots,n)$, $\varphi_i:I\to\Bbb R$ $(i=1,\dots,n)$ are the functions with bounded variations, and $\tau_{ik}:I\to I$ $(i,k=1,\dots,n)$ are measurable functions. The optimal, in some sense, conditions of unique solvability of the problems $(\ref{1})$, $(\ref{2})$ and $(1')$, $(\ref{2})$ are established.

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