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Invertibility of nonautonomous functional-differential operators
V. E. Slyusarchuk
Abstract:
Let
$C^{(m)}$ be the Banach space of continuous and bounded functions on
$R$ that take values in a finite-dimensional Banach space
$E$ and have derivatives up to and including order
$m$. The norm in
$C^{(m)}$ is given by $\|x\|_{C^{(m)}}=\sup_{t\in R,k=\overline{0,m}}\big\|\frac{d^kx(t)}{dt^k}\big\|_E$. Let
$C^{(m)}_\omega$ be the Banach space of
$\omega$-periodic functions with the same norm as
$C^{(m)}$.
Theorem. {\it Suppose
$1)\ A$ is a
$c$-completely continuous element of the space
$L(C^{(m)},C^{(0)})$ $(m\geqslant0);$
$2)\ \operatorname{Ker}\bigl(\frac{d^m}{dt^m}+A\bigr)=0;$
$3)$ there exists a completely continuous operator
$A_\omega\in L(C_\omega^{(m)},C_\omega^{(0)})$ $(\omega>0)$ for which
$$
\lim_{\omega\to+\infty}\sup_{\|x\|_{C_\omega^{(m)}}=1,|t|<T}\|(Ax)(t)-(A_\omega x)(t)\|_E=0\qquad\forall\,T>0
$$
and
$$
\varlimsup_{\omega\to+\infty}\inf_{\|x\|_{C_\omega^{(m)}}=1}\max_{t\in[-\frac\omega2,\frac\omega2]}\bigg\|\frac{d^mx(t)}{dt^m}+(A_\omega x)(t)\bigg\|_E>0.
$$
Then the operator
$\frac{d^m}{dt^m}+A$ has a
$c$-continuous inverse.}
Using this theorem the invertibility of a large class of operators is studied, which class contains in particular Poisson stable operators.
Bibliography: 22 titles.
UDC:
517.9
MSC: 34K30,
47E05 Received: 28.03.1985
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