Solvability of a Nonhomogeneous Boundary Value Problem for a Differential System with Measures

Abstract

We consider linear nonhomogeneous systems of differential equations containing distributions as coefficients. Descriptions of such systems typically contain some form of products of discontinuous functions by generalized derivatives of functions of bounded variation. These products do not always exist in the sense of distribution theory, and accordingly, different approaches to the definition of a solution may give different results. A detailed review of the literature on this topic can be found in [1], where the main approaches to the definition of a solution of the corresponding equations are also described. We also note the papers [2–6] dealing with the analysis of classes of linear and quasilinear differential systems with measures whose solutions are treated in the sense of distribution theory, whereby their definition (under some restrictions on the coefficients) is independent of the interpretation of the product of a measure by a function of locally bounded variation. Here we study the solvability of a nonhomogeneous boundary value problem for a differential system with measures assuming that the value of a parameter λ occurring linearly in the equation coincides with some eigenvalue of the corresponding homogeneous problem. The simpler case in which λ does not coincide with any eigenvalue was considered in [4].