On some boundary value problems for systems of linear functional differential equations. (English) Zbl 0948.34040

Hakl, Robert

Electron. J. Qual. Theory Differ. Equ. 1999, Paper No. 10, 16 p. (1999).

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

Keywords:

system of linear functional-differential equations; linear differential system with deviating arguments; boundary value problem