A note on a boundary value problem for linear differential difference equations of mixed type

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. Standzhitsky. 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. 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 linear boundary value problems for the differential-algebraic boundary value problem for a matrix Lyapunov equation, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear differential-algebraic boundary value problem for a matrix Lyapunov 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 matrix Lyapunov equation. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation in the critical case in this article can be transferred to the seminonlinear differential-algebraic boundary value problem for a matrix Lyapunov equation.

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Positive solutions of an integral boundary value problem for singular differential equations of mixed type with p -Laplacian

The Journal of Nonlinear Sciences and Applications ◽

10.22436/jnsa.009.12.12 ◽

2016 ◽

Vol 09 (12) ◽

pp. 6048-6057

Author(s):

Changlong Yu ◽

Jufang Wang ◽

Yanping Guo

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Positive Solutions ◽

Mixed Type ◽

Boundary Value ◽

Integral Boundary ◽

Singular Differential Equations ◽

Integral Boundary Value Problem ◽

Equations Of Mixed Type

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On the smoothness of the solution of a nonlocal boundary value problem for the multidimensional second-order equation of the mixed type of the second kind in Sobolev space

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.21.201901.24-33 ◽

2019 ◽

Vol 21 (1) ◽

pp. 24-33

Author(s):

Sirojiddin Z. Dzamalov

Keyword(s):

Boundary Value Problem ◽

Unique Solvability ◽

Infinite Number ◽

Mixed Type ◽

Boundary Value ◽

Nonlocal Boundary ◽

Second Order ◽

Nonlocal Boundary Value Problem ◽

Equations Of Mixed Type ◽

Second Order Equations

In this paper we prove the unique solvability and smoothness of the solution of a nonlocal boundary-value problem for a multidimensional mixed type second-order equation of the second kind in Sobolev space Wℓ2(Q), (2≤ℓ is an integer). First, we have studied the unique solvability of the problems in the space W22(Q). Solution uniqueness for a nonlocal boundary-value problem for a mixed-type equation of the second kind is proved by the methods of a priori estimates.Further, to prove the solution existence in the space W22(Q), the Fourier method is used. In other words, the problem under consideration is reduced to the study of unique solvability of a nonlocal boundary value problem for an infinite number of systems of second-order equations of mixed type of the second kind. For the unique solvability of the problems obtained, the ``ε-regularization'' method is used, i.e, the unique solvability of a nonlocal boundary-value problem for an infinite number of systems of composite-type equations with a small parameter was studied by the methods of functional analysis. The necessary a priori estimates were obtained for the problems under consideration. Basing on these estimates and using the theorem on weak compactness as well as the limit transition, solutions for an infinite number of systems of second-order equations of mixed type of the second kind are obtained. Then, using Steklov-Parseval equality for solving an infinite number of systems of second-order equations of mixed type of the second kind, the unique solvability of original problem was obtained. At the end of the paper, the smoothness of the problem's solution is studied.

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