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

This article is cited in 25 scientific papers (total in 25 papers)

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
Received: 28.03.1985

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  mathscinet  zmath  isi
    2. Bong C., “On Some Conditions of Reversibility of C-Continuity Differential-Functional Operators”, Dokl. Akad. Nauk, 329:3 (1993), 278–280  mathnet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  isi
    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  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi
    13. V. E. Slyusarchuk, “Bounded and periodic solutions of nonlinear functional differential equations”, Sb. Math., 203:5 (2012), 743–767  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  isi
    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  isi
    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  crossref  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. V. E. Slyusarchuk, “Conditions for almost periodicity of bounded solutions of non-linear differential-difference equations”, Izv. Math., 78:6 (2014), 1232–1243  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  isi
    22. V. E. Slyusarchuk, “Almost-periodic solutions of discrete equations”, Izv. Math., 80:2 (2016), 403–416  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  crossref  isi
    25. V. E. Slyusarchuk, “To Favard's theory for functional equations”, Siberian Math. J., 58:1 (2017), 159–168  mathnet  crossref  crossref  isi  elib  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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