# zbMATH — the first resource for mathematics

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.
##### MSC:
 34B05 Linear boundary value problems for ordinary differential equations
Full Text: