On the Cauchy problem for singular functional differential equations

Bravyi, E.; Plaksina, I. M.

doi:10.1186/s13662-017-1149-7

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…Note, the results can be used in the study of applied, in particular, computational problems such as in, for example [34,35]. The statements obtained in Theorems 1-4 improve all results known to the author (see [10,[19][20][21][22][23][28][29][30]).…”

Section: Preliminariessupporting

confidence: 62%

“…The point t 31 is the minimum point of P − on the interval t 3 ∈ [0, t 1 ]. After substituting t 3 = t 31 into the function P − we obtain Corollary 1 (which was also obtained in [30] in another way). Now let us finish the proof of Theorem 3.…”

Section: Conflicts Of Interestsupporting

confidence: 57%

“…Note that only conditions (3) [27] (it corresponds to pointwise restrictions) and conditions (6) [20][21][22][23]25,28,29] (corresponds to integral restrictions) were studied in earlier works (primarily in the case of the first order equations). The author is almost unaware of any works where mixed constraints (4) or (5) were used to obtain conditions for the solvability of the Cauchy problem (the only exception is the work [30], published during the preparation of this article). Definition 1.…”

Section: Preliminariesmentioning

confidence: 99%

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On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

Bravyi

2023

Mathematics

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Cauchy problems are considered for families of, generally speaking, non-Volterra functional differential equations of the second order. For each family considered, in terms of the parameters of this family, necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family are obtained. Such necessary and sufficient conditions are obtained for the following four kinds of families: integral restrictions are imposed on positive and negative functional operators, namely, operator norms are specified; pointwise restrictions are imposed on positive and negative functional operators in the form of values of operators’ actions on the unit function; an integral constraint is imposed on a positive functional operator, a pointwise constraint is imposed on a negative functional operator; a pointwise constraint is imposed on a positive functional operator, an integral constraint is imposed on a negative functional operator. In all cases, effective conditions for the solvability of the Cauchy problem for all equations of the family are obtained, expressed through some inequalities regarding the parameters of the families. The set of parameters of families of equations for which Cauchy problems are uniquely solvable can be easily calculated approximately with any accuracy. The resulting solvability conditions improve the solvability conditions following from the Banach contraction principle. An example of the Cauchy problem for an equation with a coefficient changing sign is given. Taking into account various restrictions for the positive and negative parts of functional operators allows us to significantly improve the known solvability conditions.

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Section: Preliminariessupporting

confidence: 62%

Section: Conflicts Of Interestsupporting

confidence: 57%

Section: Preliminariesmentioning

confidence: 99%

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On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

Bravyi

2023

Mathematics

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“…Авторы статьи выражают благодарность И.М. Плаксиной, работы которой по сингулярным краевым задачам (например, [20], [21]), инициировали данное исследование, В.В. Малыгиной за ряд существенных замечаний, а также участникам Пермского семинара по функционально-дифференциальным уравнениям за внимание к работе и плодотворное обсуждение ее результатов.…”

Section: признак однозначной разрешимости для уравнения с распределенunclassified

Solvability of inhomogeneous autonomous differential equation with aftereffect on negative half-axis

Balandin¹,

Sabatilina²

2019

Iz. VUZ. Matematika

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“…In contrast to the case of ordinary differential equations, significantly less is known for singular equations with argument deviations. [9][10][11][12][13][14][15][16] The recent works [17][18][19][20] contain a thorough investigation of linear scalar functional differential equations with time singularities based on the study of the solution space of a model equation. 21 Singular initial value problems for non-linear equations have been studied mainly in the Volterra case.…”

Section: Introduction and Problem Settingmentioning

confidence: 99%

A singular problem for functional differential equations with decreasing non‐linearities

Pylypenko

Rontó

2021

Math Methods in App Sciences

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On the Cauchy problem for singular functional differential equations

Cited by 5 publications

References 22 publications

On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

Solvability of inhomogeneous autonomous differential equation with aftereffect on negative half-axis

A singular problem for functional differential equations with decreasing non‐linearities

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