On the Norm with Respect to Vector Measures of the Solution of an Infinite System of Ordinary Differential Equations

In the present paper we give some necessary conditions that satisfy the solutions of an infinite system of ordinary differential equations. We investigate the behavior of the solutions of a general system of equations, regarding the norm of a Banach function space based on a vector measure. To this aim we construct a vector measure by an standard procedure. Assuming that the solution of each individual equation of the system belongs to a Banach function space based on scalar measures we deduce, with natural conditions, that a solution of such system belongs to a Banach function space based on a vector measure. We also give an example of a system of non-linear Bernoulli equations and show the relation with an equation involving the integral operator