Estimates of Integrally Bounded Solutions of Linear Differential Inequalities

Abstract

We study integrally bounded solutions of the differential equation $\mathscr {A}(x)=z$, where $\mathscr {A}$ is a linear differential operator of order $l$ defined on functions $x\colon \mathbb {R} \to H$ ($\mathbb {R}=(-\infty ,\infty$) and $H$ is a finite-dimensional Euclidean space). The right-hand side $z$ is an integrally bounded function on $\mathbb {R}$ ranging in $H$ and satisfying the inequality $(\psi (t), z(t))\geq \delta |z(t) |$, $t\in \mathbb {R}$, $\delta >0$. Conditions are given on the operator $\mathscr {A}$ and the function $\psi \colon \mathbb {R}\to H$ that guarantee an inverse inequality of the following form for the solutions $x$ under consideration:

$$ \int _{\tau }^{\tau +1}\big |x^{(l)}(t)\big |\thinspace dt\leq c\int _{\tau -1}^{\tau +2}\big |x(t)\big |\thinspace dt,$$

where the constant $c$ is independent of the choice of a real number $\tau$ and function $x$.