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.


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