Weakly nonlinear impulsive boundary value problems for systems of integrodifferential equations

Бондар, І. А.; Gromyak, M.; Kozlova, N. O.

doi:10.18514/mmn.2016.1897

Cited by 4 publications

(3 citation statements)

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“…де будемо використовувати припущення i позначення з робiт [1]- [3]: A(t), B(t), Φ(t), K(t, s) (m×n), (m×n), (n×m), (n×n) вимiрнi матрицi, компоненти яких належать простору…”

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“…Зокрема, умови iснування розв'язкiв слабконелiнiйних крайових задач для систем iнтегро-диференцiальних рiвнянь дослiджувалися у роботi [2] (критичний випадок першого порядку) та був запропонований iтерацiйнi алгоритми побудови розв'язкiв таких задач. Також цiкавий випадок розглядався у роботi [3], коли у нерозв'язну слабконелiнiйну крайову задачу вводили iмпульс, щоб звести її до розв'язної (при цьому iмпульси задавалися не по всiх копонентах невiдомої n-вимiрної вектор-функцiї x(t) = col(x 1 (t), x 2 (t), ..., x k i (t), ..., x n (t)), а лише по k i ). Логiчно при подальших дослiдженнях виникає питання розв'язностi слабконелiнiйної крайової задачi у критичному випадку другого порядку.…”

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Weakly Nonlinear Boundary-Value Problems for Systems of Integrodifferential Equations. Critical Case of the Second Order,

Бондар¹

2019

BMJ

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This research is based on the general theory of effective methods of finding solutions was developed of autors for weakly nonlinear systems of integrodifferential equations and boundaryvalue problems for them in the critical case of the first order. The previously obtained statements for a weakly nonlinear boundary-value problem define the conditions of solvability at rank B 0 = r, that is, a first-order critical case. In this paper we are interested in the question of what to do when a sufficient condition is not fulfilled? After all, then the weak nonlinear boundary-value problem is not solvable and in the proposed form the solution does not exist. The answer to this question is that rank B 0 < r, hence P B0 = 0. Then a sufficient condition for the solution of weakly nonlinear boundary-value problem in the critical case of the first order cannot be applied.In the given research paper by means of Moore-Penrose pseudoinverse matrices and constructive methods nonlinear systems analysis there were investigated conditions for the existence and there were suggested iterative algorithms of constructing solutions of boundary-value problems for weakly nonlinear system of integrodifferential equations. The existence conditions and the structure of solutions of the weakly nonlinear boundary value problem in the critical case of the second order are obtained. It was shown that the existence of a solution depends on the conditions obtained by means of nonlinearities and the second approximation to the desired solution.We seek the solution of this problem in the class of vector functions x = x(t, ε) such that x(•, ε) ∈ D 2 ([a, b]),ẋ(•, ε) ∈ L 2 [a, b], x(t, •) ∈ C[0, ε 0 ].

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Weakly Nonlinear Boundary-Value Problems for Systems of Integrodifferential Equations. Critical Case of the Second Order,

Бондар¹

2019

BMJ

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“…To investigate the existence of solutions to such problems, as will be shown below, you can use the apparatus of the theory of pseudo-inverse matrices and operators, which was developed in the works of A.M. Samoilenko, O.A. Boichuk [1,11] and actively developed for the case of weakly perturbed boundary value problems for systems of integro-differential equations [4], impulse boundary value problems [6,8] and BVPs for integro-dynamic equations on time scales [7].…”

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Weakly perturbed boundary-value problems for systems of integro-differential equations with impulsive action

Бондар¹

2015

Tatra Mountains Mathematical Publications

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The weakly perturbed BVP's for impulsive integro-differential systems are considered. Under the assumption that the generating problem (for ε = 0) does not have solutions on the space W12[a,b] for some inhomogeneity and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these problems on the space D2([a,b]{τi}I) in the form of a Laurent series in powers of small parameter ε with finitely many terms with negative powers of ε, and we suggest an algorithm of construction of these solutions.

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fWeakly Nonlinear Boundary-Value Problems for Systems of Impulsive Integrodifferential Equations. Critical Case of the Second Order

Бондар

2020

J Math Sci

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Weakly nonlinear impulsive boundary value problems for systems of integrodifferential equations

Cited by 4 publications

References 11 publications

Weakly Nonlinear Boundary-Value Problems for Systems of Integrodifferential Equations. Critical Case of the Second Order,

Weakly Nonlinear Boundary-Value Problems for Systems of Integrodifferential Equations. Critical Case of the Second Order,

Weakly perturbed boundary-value problems for systems of integro-differential equations with impulsive action

fWeakly Nonlinear Boundary-Value Problems for Systems of Impulsive Integrodifferential Equations. Critical Case of the Second Order

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