Existence results for three-point boundary value problems for systems of linear functional differential equations (Q2919611)

By using a successive approximation scheme, the authors give conditions for the solvability of the three-point boundary value problem for the system of \(n\) linear functional-differential equations NEWLINE\[NEWLINE\left\{ \begin{aligned} & x'(t)=(lx)(t)+f(t),\;t\in [a,b],\\ & Ax(a)+Bx(\xi)+\bar C x(b)=d, \end{aligned}\right.NEWLINE\]NEWLINE where \(f\in L_1([a,b],\mathbb R^n)\), \(d={\text{col}}(d_1,\dotsc,d_n)\), \(\xi\in (a,b)\), the \((n\times n)\)-matrices \(A,\,B\) and \(\bar C\) are singular and \(\bar C\) has the form NEWLINE\[NEWLINE\bar C=\left( \begin{matrix} V & W\\ 0_{n-q,q} & 0_{n-q} \end{matrix} \right),NEWLINE\]NEWLINE \(V\) is a non-singular square matrix of dimension \(q<n\), \(W\) is a matrix of dimension \(q\times (n-q)\), and \(0_{ij}\) is the zero matrix of dimension \(i\times j\).NEWLINENEWLINE A numerical example for a system of three linear differential equations with argument deviations subject to three-point boundary conditions is finally addressed.