The Variation Formula for Solutions of Differential Equations

Abstract

Let E₁ be the linear space of n-dimensional functions which are defined on R × Rⁿ and satisfy the following conditions: Every function F(t, x) ∊ E₁ has a compact support and is continuously differentiable with respect to x for a fixed t ∊ R. For a fixed x, the (vector and matrix-valued) functions

$$F(t,x),\quad \frac{{\partial F(t,x)}} {{\partial x}} = {F_x}(t,x)$$

are measurable in t, and there exists a majorant m _F (t) of |F(t, x)| + |F _x (t, x)| which is summable on JR,

$$\mid F(t, x)\mid + \mid {F_x}(t, x)\mid \leqslant {m_F}(t)\quad \forall (t,x) \in R*{R^n},\quad \int\limits_R {{m_F}} (t)dt < \infty$$

((5.1))