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 Mat. Sb. (N.S.), 1986, Volume 130(172), Number 1(5), Pages 86–104 (Mi msb1851)

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 $\|x\|_{C^{(m)}}=\sup_{t\in R,k=\overline{0,m}}\|\frac{d^kx(t)}{dt^k}\|_E$. 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 $L(C^{(m)},C^{(0)})$ $(m\geqslant0);$
$2) \operatorname{Ker}(\frac{d^m}{dt^m}+A)=0;$
$3)$ 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
$$\varlimsup_{\omega\to+\infty}\inf_{\|x\|_{C_\omega^{(m)}}=1}\max_{t\in[-\frac\omega2,\frac\omega2]}\|\frac{d^mx(t)}{dt^m}+(A_\omega x)(t)\|_E>0.$$

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.

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English version:
Mathematics of the USSR-Sbornik, 1987, 58:1, 83–100

Bibliographic databases:

UDC: 517.9
MSC: 34K30, 47E05

Citation: V. E. Slyusarchuk, “Invertibility of nonautonomous functional-differential operators”, Mat. Sb. (N.S.), 130(172):1(5) (1986), 86–104; Math. USSR-Sb., 58:1 (1987), 83–100

Citation in format AMSBIB
\Bibitem{Sly86} \by V.~E.~Slyusarchuk \paper Invertibility of nonautonomous functional-differential operators \jour Mat. Sb. (N.S.) \yr 1986 \vol 130(172) \issue 1(5) \pages 86--104 \mathnet{http://mi.mathnet.ru/msb1851} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=847344} \zmath{https://zbmath.org/?q=an:0646.34016} \transl \jour Math. USSR-Sb. \yr 1987 \vol 58 \issue 1 \pages 83--100 \crossref{https://doi.org/10.1070/SM1987v058n01ABEH003093} 

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This publication is cited in the following articles:
1. V. G. Kurbatov, “On the invertibility of almost periodic operators”, Math. USSR-Sb., 67:2 (1990), 367–377
2. Bong C., “On Some Conditions of Reversibility of C-Continuity Differential-Functional Operators”, Dokl. Akad. Nauk, 329:3 (1993), 278–280
3. Slyusarchuk V., “Necessary and Sufficient Conditions for Existence and Uniqueness of Bounded and Almost-Periodic Solutions of Nonlinear Differential Equations”, Acta Appl. Math., 65:1-3 (2001), 333–341
4. V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of the non-linear difference operator $(\mathscr Dx)(t)=x(t+1)-f(x(t))$ in the space of bounded continuous functions on the real axis”, Sb. Math., 192:4 (2001), 565–576
5. V. E. Slyusarchuk, “Necessary and Sufficient Conditions for the Lipschitzian Invertibility of the Nonlinear Differential Mapping $d/dt-f$ in the Spaces $L_p({\mathbb R},{\mathbb R})$, $1\le p\le\infty$”, Math. Notes, 73:6 (2003), 843–854
6. Slyusarchuk V.Yu., “Generalization of the Mukhamadiev Theorem on the Invertibility of Functional Operators in the Space of Bounded Functions”, Ukr. Math. J., 60:3 (2008), 462–480
7. V. E. Slyusarchuk, “Conditions for the invertibility of the nonlinear difference operator $(\mathscr Rx)(n)=H(x(n),x(n+1))$, $n\in\mathbb Z$, in the space of bounded number sequences”, Sb. Math., 200:2 (2009), 261–282
8. Slyusarchuk V.Yu., “Method of Local Linear Approximation in the Theory of Bounded Solutions of Nonlinear Differential Equations”, Ukr. Math. J., 61:11 (2009), 1809–1829
9. Slyusarchuk V.Yu., “Method of Local Linear Approximation in the Theory of Bounded Solutions of Nonlinear Difference Equations”, Nonlinear Oscil., 12:3 (2009), 380–391
10. Perestyuk M.O., Slyusarchuk V.Yu., “Green-Samoilenko Operator in the Theory of Invariant Sets of Nonlinear Differential Equations”, Ukr. Math. J., 61:7 (2009), 1123–1136
11. V. E. Slyusarchuk, “The method of local linear approximation in the theory of nonlinear functional-differential equations”, Sb. Math., 201:8 (2010), 1193–1215
12. Slyusarchuk V.Yu., “Conditions for the Existence of Bounded Solutions of Nonlinear Differential and Functional Differential Equations”, Ukr. Math. J., 62:6 (2010), 970–981
13. V. E. Slyusarchuk, “Bounded and periodic solutions of nonlinear functional differential equations”, Sb. Math., 203:5 (2012), 743–767
14. Slyusarchuk V.Yu., “Method of Local Linear Approximation of Nonlinear Differential Operators by Weakly Regular Operators”, Ukr. Math. J., 63:12 (2012), 1916–1932
15. Chaikovs'kyi A.V., “On Solutions Defined on an Axis for Differential Equations with Shifts of the Argument”, Ukr. Math. J., 63:9 (2012), 1470–1477
16. Slyusarchuk V.Yu., “Conditions for the Existence of Almost Periodic Solutions of Nonlinear Differential Equations in Banach Spaces”, Ukr. Math. J., 65:2 (2013), 341–347
17. V. E. Slyusarchuk, “The study of nonlinear almost periodic differential equations without recourse to the $\mathscr H$-classes of these equations”, Sb. Math., 205:6 (2014), 892–911
18. V. E. Slyusarchuk, “Conditions for almost periodicity of bounded solutions of non-linear differential-difference equations”, Izv. Math., 78:6 (2014), 1232–1243
19. Slyusarchuk V.Yu., “Conditions For Almost Periodicity of Bounded Solutions of Nonlinear Differential Equations Unsolved With Respect To the Derivative”, Ukr. Math. J., 66:3 (2014), 432–442
20. V. E. Slyusarchuk, “Conditions for the Existence of Almost-Periodic Solutions of Nonlinear Difference Equations in Banach Space”, Math. Notes, 97:2 (2015), 268–274
21. Slyusarchuk V.Yu., “a Criterion For the Existence of Almost Periodic Solutions of Nonlinear Differential Equations With Impulsive Perturbation”, Ukr. Math. J., 67:6 (2015), 948–959
22. V. E. Slyusarchuk, “Almost-periodic solutions of discrete equations”, Izv. Math., 80:2 (2016), 403–416
23. V. E. Slyusarchuk, “Necessary and sufficient conditions for the existence and uniqueness of a bounded solution of the equation $\dfrac{dx(t)}{dt}=f(x(t)+h_1(t))+h_2(t)$”, Sb. Math., 208:2 (2017), 255–268
24. Slyusarchuk V.Yu., “Favard-Amerio Theory For Almost Periodic Functional-Differential Equations Without Using the a"i-Classes of These Equations”, Ukr. Math. J., 69:6 (2017), 916–932
25. V. E. Slyusarchuk, “To Favard's theory for functional equations”, Siberian Math. J., 58:1 (2017), 159–168
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