On the norm with respect to vector measures of the solution of an infinite system of ordinary differential equations
The paper is devoted to the study of the function space to which the set of solutions of a (system of) differential equation(s) belongs. The vector measure technique that is used is related to the analysis of some abstract problems on optimal domains for operators acting in Banach function spaces. Consider a family of differential equations. Each of them defines an operator, and the combination of all of them can be considered as defining a vector measure in a given Banach space. Since the solutions are assumed to belong to Banach function spaces, the underlying measures have a relevant role. This is the reason why the first part of the paper is devoted to prove some results on Radon-Nikodým derivatives related to vector measures, that are interesting in themselves. After this introductory material, the author faces the problem considered in the paper using these tools. The idea is to determine the Banach function space to which the solutions of the system can be considered to belong to. In order to do that, the vector measure defined by combining all the operators related to all the differential equations is a suitable tool; the corresponding space of integrable functions is, broadly speaking, the intersection of all the spaces of solutions of each individual equation. Theorem 4.1 establishes that, under the adequate requirements on the set of measures and the Banach function space \(Y(\mu)\) to which the set of solutions of a system of differential equations belongs, there exists a vector measure such that the solutions of the system belong to the \(m\)-space generated by \(Y(\mu)\). The paper is original and brings together ideas coming from different fields; probably some of the tools proposed here can be used for obtaining further results. Some comments relating these results with some well-known classical ones are presented. Interesting examples and applications are also given.
scientific article; zbMATH DE number 6478185
scientific article