Invertibility of nonautonomous functional-differential operators

V. E. Slyusarchuk

Sbornik. Mathematics, Tome 58 (1987) no. 1, pp. 83-100

Résumé

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 $$ \lim_{\omega\to+\infty}\sup_{\|x\|_{C_\omega^{(m)}}=1,|t|}\|(Ax)(t)-(A_\omega x)(t)\|_E=0\qquad\forall\,T>0 $$ 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.