# Boundary value problems for systems of non-degenerate difference-algebraic equations

### 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 boundary value problems for difference equations, initiated in A.A. Markov, S.N. Bernstein, Ya.S. Besikovich, A.O. Gelfond, S.L. Sobolev, V.S. Ryaben'kii, V.B. Demidovich, A. Halanay, G.I. Marchuk, A.A. Samarskii, Yu.A. Mitropolsky, D.I. Martynyuk, G.M. Vayniko, A.M. Samoilenko, O.A. Boichuk and O.M. Stanzhitsky. Study of nonlinear singularly perturbed boundary value problems for difference equations in partial differences is devoted to the work of V.P. Anosov, L.S. Frank, P.E. Sobolevskii, A.L. Skubachevskii and A. Asheraliev.

Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A.M. Samoilenko and O.A. Boichuk on linear boundary value problems for difference-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 linear boundary value problem for a difference-algebraic equation.

The solvability conditions are found in the paper, as well as the construction of a generalized Green operator for the Cauchy problem for a difference-algebraic system. The solvability conditions are found, as well as the construction of a generalized Green operator for a linear Noetherian difference-algebraic boundary value problem. An original classification of critical and noncritical cases for linear difference-algebraic boundary value problems is proposed.

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### References

A.A. Boichuk. Boundary-value problems for systems of difference equations, Ukrainian Mathematical Journal. - 1997. - 6. V. 49. - P. 832-835.

S.L. Campbell.Limit behavior of solutions of singular difference equations, Linear algebra and its appl. - 1979. - V. 23. - P. 167-178.

S.M. Chuiko. On a reduction of the order in a differential-algebraic system, Journal of Mathematical Sciences. - 2018. - 1. - V. 235. - P. 2-18.

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

V.K. Romanko. Difference equations. 2014. Bean, Moscow, 112 p.

A generalized matrix differential-algebraic equation, Journal of Mathematical Sciences (N.Y.). - 2015. - 1. V. 210. - P. 9-21.

А.А. Boichuk, V.F. Zhuravlev, A.M. Samoilenko. Normally solvable boundary value problems. 2019. Scientific Opinion, Kiev, 628 p.

S.M. Chuiko. A Generalized Green operator for a boundary value problem with impulse action, Differential Equations. - 2001. - 8. V. 37. - P. 1189-1193.

V.Ya. Gutlyanskii, V.I. Ryazanov, A.S. Yefimushkin. On the boundary-value problems for quasiconformal functions in the plane, Journal of Mathematical Sciences. - 2016. - V. 214. - P. 200-219.

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

I.I.Skrypnik. Removability of isolated singularities for anisotropic elliptic equations with gradient absorption, Israel Journal of Mathematics, - 2016. - 1. V. 217. - P. 163-179.

S.M.Chuiko. Nonlinear matrix differential-algebraic boundary value problem, Lobachevskii Journal of Mathematics. - 2017. - 2. V. 38. - P. 236-244.

A.N. Tikhonov, V.Ya. Arsenin. Solution of Ill-Posed Problems. 1986. Winston, Washington, 288 p.

S.M. Chuiko. On the regularization of a linear Fredholm boundary-value problem by a degenerate pulsed action, Journal of Mathematical Sciences. - 2014. - 1. V. 197. - P. 138-150.

S.M.Chuiko, Ya.V. Kalinichenko. On the question of the regularization of the Cauchy problem for a system of linear difference equations, Visnyk of V.N.Karazin Kharkiv National University Ser. "Mathematics, Applied Mathematics and Mechanics". - 2018. - V. 28. - P. 27-34.

V.I. Korobov, M.O. Bebiya. Stabilization of one class of nonlinear systems, Automation and Remote Control. - 2017. - 1. V. 78. - P. 20-25.

*Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics*,

*90*, 26-41. https://doi.org/10.26565/2221-5646-2019-90-02

Copyright (c) 2019 Sergey Chuiko, Yaroslav Kalinichenko, Mykyta Popov

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