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Solvability of a nonhomogeneous boundary value problem for a differential system with measures. (English. Russian original) Zbl 1070.34024
Differ. Equ. 39, No. 3, 353-361 (2003); translation from Differ. Uravn. 39, No. 3, 328-336 (2003).
The authors consider nonhomogeneous linear boundary value problems of the form \[ JY'=(B'+\lambda A')Y+F' \] on an interval \([a,b]\) with certain inhomogeneous boundary conditions in \(a\) and \(b\). Here, \(A\) and \(B\) are Hermitian \(n\times n\)-matrix functions, \(J\) is a constant \(n\times n\)-matrix satisfying \(J^\ast=-J\), \(J^\ast J= E\), and \(F\) is an \(n\times 1\)-vector function. The functions \(A\), \(B\) and the inhomogeneity \(F\) are only assumed to be right continuous and locally of bounded variation over \([a,b]\) so that their (generalized) derivatives appearing in the differential equation are Stieltjes measures. The main result of this paper is a solvability condition for the above boundary value problem and a formula for the solution \(Y\) in the case that the complex parameter \(\lambda\) is an eigenvalue of the corresponding homogeneous boundary eigenvalue problem (with \(F'=0\)); the simpler case where \(\lambda\) is an eigenvalue of the latter has been considered in the previous work of the references. To this end, the problem is reduced to an equivalent loaded integral equation of Fredholm-Stieltjes type with matrix kernel.
34B05 Linear boundary value problems for ordinary differential equations
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