## Linear ordinary differential equations of class $$L^ pS$$.(English)Zbl 0718.34070

Let $${\mathbb{R}}^ n$$ denote the real euclidean space of n-vectors x and let $$L({\mathbb{R}}^ n)$$ denote the real euclidean space of $$n\times n$$ matrices A. Let $$J=(\alpha,+\infty)$$, -$$\infty \leq \alpha$$, and let $$A: t\to A(t),$$ $$t\in J$$, $$A(t)\in L({\mathbb{R}}^ n)$$, be a continuous function. Let I denote the identity of $$L({\mathbb{R}}^ n)$$ and let $E_ A(t,s)=\lim_{k}[\int^{t}_{s}A(t_ 1)dt_ 1+...+\int^{t}_{s}...\int^{t_{k-1}}_{s}A(t_ 1)...A(t_ k)dt_ k...dt_ 1]$ represent the so called Cauchy (or evolution) matrix associated with A. Then the solutions of the linear ordinary differential equation $$(A)\quad \dot x=A(t)x$$ are given by $$x: t\to x(t)=E_ A(t,s)x(s),$$ t,s$$\in J$$. Equation (A) is said to be asymptotically stable if $(1)\quad \lim_{t\to +\infty}E_ A(t,\tau)=0,\quad \alpha <\tau,$ (uniformly) exponentially stable if for every $$\vartheta >\alpha$$ there exist $$\gamma$$ ($$\vartheta$$)$$\geq 1$$, $$\mu (\vartheta)>0$$ such that $(2)\quad | E_ A(t,\tau)| \leq \gamma (\vartheta)e^{-\mu (\vartheta)(t-\tau)},\quad \vartheta \leq \tau \leq t.$ We denote by AS, ES the classes of matrix functions A for which (1), (2) hold, respectively. Obviously, ES$$\subset AS$$ and we consider a family, depending on a parameter $$p>0$$, of classes intermediate between ES and AS.

### MSC:

 34D05 Asymptotic properties of solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems 34B05 Linear boundary value problems for ordinary differential equations

Zbl 0718.34071