Necessary and sufficient conditions for invertibility of nonautonomous functional-differential operators

Slyusarchuk, V. Yu.

doi:10.1007/bf01240454

Cited by 25 publications

(11 citation statements)

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“…Therefore, according to Theorem 5, the operator L Q : l ∞ (Z, E) → l ∞ (Z, E) has a continuous inverse. Conversely, if the operator L Q : l ∞ (Z, E) → l ∞ (Z, E) has a continuous inverse, then, according to Theorem 5, for every number H > 0 there exists a number r > 0 and an element A = (A n ) n∈Z ∈ E 1 for which inequality (17) [and, hence, (18)] is also true. Inequality (18) yields (24).…”

Section: Theorem 5 For Every Number H > 0 There Exist a Number R > mentioning

confidence: 94%

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Method of local linear approximation in the theory of bounded solutions of nonlinear difference equations

Slyusarchuk

2009

Nonlinear Oscill

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Section: Theorem 5 For Every Number H > 0 There Exist a Number R > mentioning

confidence: 94%

“…These operators were studied in [14][15][16][17][18][19][20]. The definition of c-continuous operator that uses locally convergent sequences was proposed by the author (see, e.g., [21,22]).…”

Section: Auxiliary Statementsmentioning

confidence: 99%

Method of local linear approximation in the theory of bounded solutions of nonlinear difference equations

Slyusarchuk

2009

Nonlinear Oscill

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“…These operators were studied in [2][3][4][5][6][7][8][9], etc. The definition of c-continuous operator that uses locally convergent sequences was proposed by the author (see, e.g., [10][11][12]).…”

Section: Main Functional Spaces and C-continuous Operatorsmentioning

confidence: 99%

“…The function G.t / is called the Green function [15,21] of the operator L defined by relation (6). With the use of this function, for every h 2 C 0 .R; E/; the unique bounded solution of the equation…”

Section: Representation Of Bounded Solutions Of Linear Inhomogeneous mentioning

confidence: 99%

Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Slyusarchuk

2010

Ukr Math J

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Let E be a finite-dimensional Banach space, let C 0 .R; E/ be a Banach space of functions continuous and bounded on R and taking values in E; let KW C 0 .R; E/ ! C 0 .R; E/ be a c-continuous bounded mapping, let AW E ! E be a linear continuous mapping, and let h 2 C 0 .R; E/: We establish conditions for the existence of bounded solutions of the nonlinear equation Main Functional Spaces and c-Continuous OperatorsLet R be the set of all real numbers, let C be the set of all complex numbers, let E be a real or complex Banach space with norm k k E ; and let L.E; E/ be the Banach algebra of all linear continuous operators AW E ! E with the normBy C 0 .R; E/ we denote the Banach space of functions x D x.t/ continuous and bounded on R and taking values in E with the normNote that kak E D jaj if E D R or E D C: Let C 1 .R; E/ denote the Banach space of functions x 2 C 0 .R; E/ each derivative of which is an element of the space C 0 .R; E/ with the normWe say that a sequence .x n / n 1 of elements of the space C 0 .R; E/ converges locally to an element x 2 C 0 .R; E/ and write x n loc: C 0 .R;E / ! x as n ! 1 if this sequence is bounded and, for every number p > 0; one has National University of Water Management and Nature Resources Use, Rivne, Ukraine.

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“…Следовательно, для H = 0 выполняется неравенство (10). Поэтому уравне-ние (45) имеет хотя бы одно решение x ∈ C 1 (R, E).…”

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Ограниченные И Периодические Решения Нелинейных Дифференциально-Функциональных Уравнений

Слюсарчук¹,

Slyusarchuk²

2012

Матем. сб.

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МАТЕМАТИЧЕСКИЙ СБОРНИКТом 203, № 5 УДК 517.988.6 В. Е. СлюсарчукОграниченные и периодические решения нелинейных дифференциально-функциональных уравненийПриведены условия существования ограниченных и периодических ре-шений нелинейного дифференциально-функционального уравненияиспользующие локальные линейные аппроксимации оператора F . Библиография: 23 названия.Ключевые слова: ограниченные и периодические решения, нелиней-ные дифференциально-функциональные уравнения, обратимость нели-нейных операторов.

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Necessary and sufficient conditions for invertibility of nonautonomous functional-differential operators

Cited by 25 publications

References 1 publication

Method of local linear approximation in the theory of bounded solutions of nonlinear difference equations

Method of local linear approximation in the theory of bounded solutions of nonlinear difference equations

Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Ограниченные И Периодические Решения Нелинейных Дифференциально-Функциональных Уравнений

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