On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case

Keywords: boundary-value problems, differential-algebraic equations, noncritical case, pseudoinverse matrices

Abstract

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works 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 are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of nonlinear boundary-value problems for ordinary differential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk.

The study of the nonlinear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the nonlinear boundary value problems for the differential algebraic equations, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the nonlinear differential algebraic boundary value problems.

In this article we found the conditions of the existence and constructed the iterative scheme for finding the solutions of the weakly nonlinear Noetherian differential-algebraic boundary value problem. The proposed scheme of the research of the nonlinear differential-algebraic boundary value problems in the article can be transferred to the nonlinear matrix differential-algebraic boundary value problems. On the other hand, the proposed scheme of the research of the nonlinear Noetherian differential-algebraic boundary value problems in the critical case in this article can be transferred to the autonomous seminonlinear differential-algebraic boundary value problems.

Downloads

Download data is not yet available.

References

S.L. Campbell. Singular Systems of differential equations. - San Francisco - London - Melbourne. - Pitman Advanced Publishing Program, 1980. - 178 p.

A.A. Boichuk, A.M. Samoilenko. Generalized inverse operators and Fredholm boundary-value problems; 2-th edition. 2016. De Gruyter, Berlin; Boston, 298 p.

S.M. Chuiko. On a Reduction of the Order in a Differential-Algebraic System, Journal of Mathematical Sciences., - 2018. - 235. V. 1. - P. 2-14.

O.V. Nesmelova. Nonlinear boundary value problems for nondegenerate differential-algebraic system, Proceedings of Institute of Applied Mathematics and Mechanics of NAS of Ukraine, - 2018. - V. 32. - P. 78-91.

O.V. Nesmelova. Seminonlinear boundary value problems for nondegenerate differential-algebraic system, Vysnyk of V.N. Karazin Kharkiv National University. Ser. “Mathematics, Applied Mathematics and Mechanics”, - 2019. - V. 89. - P. 10-20.

S.M. Chuiko, E.V. Chuiko, I.A. Boichuk. On the reduction of a Noetherian boundary-value problem to a first-order critical case, Journal of Mathematical Sciences (N.Y.), - 2015. - 208. V. 5, - P. 607-619.

А.S. Chuiko. The convergence region of an iterative procedure for a weakly nonlinear boundary value problem, Nonlinear oscillation, - 2005. - 8. V. 2. - P. 278-288.

D.K. Lika, Yu.A. Ryabov. Iteration methods and majorizing Lyapunov equations in the theory of nonlinear oscillations. 1974. Chisinau: Shtynica, 292 p.

O.B. Lykova, A.A. Boichuk. Construction of periodic solutions of nonlinear systems in critical cases, Ukrainian Mathematical Journal, - 1988. - 40. V. 1. - P. 62-69.

S. Chuiko. Weakly nonlinear boundary value problem for a matrix differential equation, Miskolc Mathematical Notes, - 2016. - 17. V. 1. - P. 139-150.

I.G. Malkin. Some problems of the theory of nonlinear oscillations. 1956. Gostekhizdat, Moscow, 491 p.

A. Boichuk, S. Chuiko. Autonomous Weakly Nonlinear Boundary Value Problems in Critical Cases, Differential Equations, - 1992. - V. 10. - P. 1353-1358.

Published
2019-12-23
Cited
0 article
How to Cite
Chuiko, S., & Nesmelova, O. (2019). On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 90, 60-72. https://doi.org/10.26565/2221-5646-2019-90-04
Section
Статті