Linear differential-algebraic boundary value problem with singular pulse influence
Abstract
The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action generalizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.Downloads
References
A. M. Samoilenko, N. A. Perestyuk. Impulsive Differential Equations. 1987. Vishcha shkola, Kiev, 287 p. (in Russian).
A. A. Boichuk, A. M. Samoilenko. Generalized inverse operators and Fredholm boundary-value problems (2-th edition). 2016. De Gruyter, Berlin; Boston, 298 pp. https://doi.org/10.1515/9783110378443.
S. L. Campbell. Singular systems of differential equations. 1980. Pitman Advanced Publishing Program, San Francisco–London–Melbourne, 178 p.
Yu. E. Boyarintsev, V. F. Chistyakov. Algebro-Differential Systems. Methods for Solving and Studying. 1998. Nauka, Novosibirsk, 224 p. (in Russian).
A. A. Boichuk, L. M. Shehda. Degenerate Fredholm boundary-value problems, Nonlinear oscillation, — 2007. — V. 10, 3. — P. 306-314. https://doi.org/10.1007/s11072-007-0024-y.
A. M. Samoilenko, M. I. Shkil’, V. P. Yakovets. Linear systems of differential equations with degeneration. 2000. Vyshcha Shkola, Kiev, 296 p. (in Russian).
S. M. Chuiko. On a reduction of the order in a differential-algebraic system, Journal of Math. Sciences, — 2018. — V. 235, 1. — P. 2-14. https://doi.org/10.1007/s10958-018-4054-z.
S. M. Chuiko. A generalized Green operator for a linear Noetherian differential-algebraic boundary value problem, Siberian Advances in Mathematics, — 2020. — V. 30, 4. — P. 177-191. https://doi.org/10.3103/S1055134420030037.
D. Wexler. On Boundary Value Problems for an Ordinary Linear Differential Systems, Ann. Vft. Pura et Appl., — 1968. — V. 80. — P. 123-136.
A. A Boichuk, E. V. Chuiko, S. M. Chuiko. Generalized Green operator of a boundary-value problem with degenerate pulse influence, Ukrainian Mathematical Journal, — 1996. — V. 48, 5. — P. 588-594 (in Russian).
A. A. Boichuk, N. A. Perestyuk, A. M. Samoilenko. Periodic solutions of impulse differential systems in critical cases, Differents. Uravn., — 1991. — V. 27, 9. — P. 1516-1521 (in Russian).
S. M. Chuiko. Noether boundary value problems for degenerate differentialalgebraic systems with linear pulse conditions, Dynamical systems, — 2014. — V. 4 (32), 1–2. — P. 89-100 (in Russian).
A. A. Boichuk, A. A. Pokutnyi, V. F. Chistyakov. Application of perturbation theory to the solvability analysis of differential algebraic equations, Computational Mathematics and Mathematical Physics, — 2013. — V. 53, 6. — P. 777-788. https://doi.org/10.1134/S0965542513060043.
S. M. Chuiko. A generalized Green operator for a boundary value problem with impulse action, Differential Equations, — 2001. — V. 37, 8. — P. 1189-1193. https://doi.org/10.1023/A:1012440123296.
S. M. Chuiko. A Green operator for boundary value problems with an impulsive effect, Doklady Mathematics, — 2001. — V. 64, 1. — P. 41–43.
S. M. Chuiko, O. V. Nesmelova. Nonlinear boundary-value problems for degenerate differential-algebraic systems, Journal of Mathematical Sciences, — 2021. — V. 252, 4. — P. 463-471. https://doi.org/10.1007/s10958-020-05174-5.
V. I. Korobov, M. O. Bebiya. Stabilization of one class of nonlinear systems, uncontrollable in the first approximation, Rep. NAS Ukraine, — 2014. — V. 2. — P. 20-25. https://doi.org/10.15407/dopovidi2014.02.020.
![Creative Commons License](http://i.creativecommons.org/l/by-nc-nd/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
The copyright holder is the author.
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal. (Attribution-Noncommercial-No Derivative Works licence).
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (see The Effect of Open Access).