Differential equations

Abstract

Let I be an interval contained in R, not reducing to a single point, E atopological vector space over R, and A and B two open subsets of E. Let (x, y, t) ↦ g(x, y, t) be a continuous map of A × B × I into E; to every differentiable map u of I into A whose derivative takes its values in B we associate the map t ↦ g(u(t), u’(t), t) of I into E, and denote it by $\tilde g$(u); so $\tilde g$is defined on the set D(A, B) of differentiable functions of I into B whose derivatives have their values in B. We shall say that the equation $\tilde g$(u) = 0 is a differential equation in u (relative to the real variable t); a solution of this equation is also called an integral of the differential equation (on the interval I); it is a differentiable map of I into A, whose derivative takes values in B, such that g(u(t), u’(t), t) = 0 for every t ∈ I. By abuse of language we shall write the differential equation $\tilde g$(u) = 0 in the form

$$g\left( {x,x',t} \right) = 0, $$

on the understanding that x belongs to the set D(A, B).