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

Bibliographie

[1] Mukhamadiev E., “Ob obratimosti differentsialnykh operatorov v prostranstve nepreryvnykh i ogranichennykh na osi funktsii”, DAN SSSR, 196:1 (1971), 47–49 | Zbl

[2] Mukhamadiev E., “Ob obratimosti funktsionalnykh operatorov v prostranstve ogranichennykh na osi funktsii”, Matem. zametki, 11:3 (1972), 269–274 | Zbl

[3] Mukhamadiev E., “Issledovaniya po teorii periodicheskikh i ogranichennykh reshenii differentsialnykh uravnenii”, Matem. zametki, 30:3 (1981), 443–460 | MR | Zbl

[4] Kolesov Yu. S., “Neobkhodimye i dostatochnye usloviya eksponentsialnoi dikhotomii reshenii lineinykh pochti periodicheskikh uravnenii s posledeistviem”, Vestn. Yaroslavskogo universiteta, 5 (1973), 28–62 | MR

[5] Kolesov Yu. S., “Obzor rezultatov po teorii ustoichivosti reshenii differentsialno-raznostnykh uravnenii s pochti periodicheskimi koeffitsientami”, Issledovaniya po ustoichivosti i teorii kolebanii, Yaroslavl, 1977, 82–141 | MR

[6] Nosov V. R., “Periodicheskie resheniya sistem lineinykh uravnenii obschego vida s otklonyayuschimsya argumentom”, Differents. uravneniya, 7:4 (1971), 639–650 | MR | Zbl

[7] Nosov V. R., “O suschestvovanii periodicheskikh reshenii u lineinykh sistem obschego vida s raspredelennym otkloneniem argumenta”, Differents. uravneniya, 7:12 (1971), 2168–2175 | MR | Zbl

[8] Antonevich A. B., Ryvkin V. B., “O normalnoi razreshimosti zadachi o periodicheskikh resheniyakh lineinogo differentsialnogo uravneniya s otklonyayuschimsya argumentom”, Differents. uravneniya, 10:8 (1974), 1347–1353 | MR | Zbl

[9] Kurbatov V. G., “O lokalnoi fredgolmovosti raznostnogo operatora”, DAN SSSR, 274:3 (1984), 534–536 | MR | Zbl

[10] Slyusarchuk V. E., “Obratimost funktsionalno-differentsialnykh operatorov”, DAN UkrSSR, ser. A, 1980, no. 9, 29–32 | MR

[11] Slyusarchuk V. E., “Obratimost pochti periodicheskikh $c$-nepreryvnykh funktsionalnykh operatorov”, Matem. sb., 116(158) (1981), 483–501 | MR | Zbl

[12] Slyusarchuk V. E., “Ogranichennye resheniya lineinykh funktsionalno-differentsialnykh uravnenii”, Problemy sovremennoi teorii periodicheskikh dvizhenii, no. 5, Izhevsk, 1981, 5–12 | MR

[13] Slyusarchuk V. E., “O razreshimosti funktsionalnykh i funktsionalno-differentsialnykh uravnenii v prostranstve ogranichennykh na osi funktsii”, Priblizhennye i kachestvennye metody issledovaniya differentsialnykh i differentsialno-funktsionalnykh uravnenii, Izd-vo IM AN USSR, Kiev, 1983, 83–88 | MR

[14] Daletskii Yu. L., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970 | MR

[15] Krasnoselskii M. A., Burd V. Sh., Kolesov Yu. S., Nelineinye pochti periodicheskie kolebaniya, Nauka, M., 1970 | MR

[16] Massera Kh., Sheffer Kh., Lineinye differentsialnye uravneniya i funktsionalnye prostranstva, Mir, M., 1970 | MR | Zbl

[17] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[18] Zhikov V. V., Levitan B. M., “Teoriya Favara”, UMN, 32:2 (1977), 123–171 | MR | Zbl

[19] Levitan B. M., Zhikov V. V., Pochti periodicheskie funktsii i differentsialnye uravneniya, Izd-vo MGU, M., 1978 | MR | Zbl

[20] Slyusarchuk V. E., “Integralnoe predstavlenie $c$-nepreryvnykh operatorov”, DAN UkrSSR, ser. A, 1981, no. 8, 34–37 | MR

[21] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968 | MR | Zbl

[22] Krein S. G., Lineinye uravneniya v banakhovom prostranstve, Nauka, M., 1971 | MR