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On some boundary value problems for systems of linear functional differential equations. (English) Zbl 0948.34040
Summary: The author studies the system of linear functional-differential equations $x'_i(t)=\sum_{k=1}^n\ell_{ik}(x_k)(t)+q_i(t),\qquad i=1,\dots,n,\tag{1}$ and its particular case $x'_i(t)=\sum_{k=1}^n p_{ik}(t)x_k(\tau_{ik}(t))+q_i(t),\qquad i=1,\dots,n,\tag{$$1'$$}$ on the segment $$I= [a,b]$$ with the boundary conditions $\int_a^b x_i(t)d\varphi_i(t)=c_i,\qquad i=1,\dots,n. \tag{2}$ Here, $$\ell_{ik}:C(I;\mathbb{R})\to L(I;\mathbb{R})$$ are linear bounded operators, $$p_{ik}$$ and $$q_i\in L(I;\mathbb{R})$$, $$c_i\in\mathbb{R}$$, $$i,k=1,\dots,n$$, $$\varphi_i:I\to\mathbb{R}$$, $$i=1,\dots,n$$, are functions with bounded variations, and $$\tau_{ik}:I\to I$$, $$i,k=1,\dots,n$$, are measurable functions. Optimal conditions (in some sense) of the unique solvability of the problems (1), (2) and $$(1')$$, (2) are established.

##### MSC:
 34K10 Boundary value problems for functional-differential equations
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