Generalized Green Operator Research Articles

Linear differential-algebraic boundary value problem with singular pulse influence

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.

Linear differential-algebraic boundary value problem with nonsingular pulse influence

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 nonsingular 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 neralizes 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.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

A Generalized Green Operator for a Linear Noetherian Differential-Algebraic Boundary Value Problem

We find solvability conditions and describe the construction of a generalized Green operator for the Cauchy problem. For a linear Noetherian differential-algebraic boundary value problem, we describe the construction of a generalized Green operator, find existence conditions for equilibrium states, and describe their construction.

Nonlinear Boundary-Value Problems for the Lyapunov Equation in the Space Lp

We study boundary-value problems for a Lyapunov-type equation in the space Lp.(I, ℒ(ℋ)): Necessary and sufficient conditions for the solvability of the corresponding boundary-value problem are established both in linear and nonlinear cases. The solutions of the linear boundary-value problem are constructed by using the generalized Green operator. It is proposed to find the approximate solutions of the nonlinear equation by using iterative Newton–Kantorovich-type algorithms.

Linear Noetherian boundary value problem for a matrix difference-algebraic Lyapunov equation

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.

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

About the equilibrium positions of a matrix differential-algebraic boundary value problem

In the article we found the solvability conditions and the construction of the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem. We obtained sufficient conditions of transformationsof the matrix differential-algebraic equation to a traditional differential-algebraic equation with an unknown in the form of a column vector. The problem that reviewed in the article continues the study of solvability conditions for the linear Noetherian boundary value problems given in the monographs of M.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. We investigated the general case when the linear bounded operator corresponding to the homogeneous part of the linear Cauchy problem for the matrix differential-algebraic system does not have the reverse operator. We introduced the definition of the equilibrium positions of the matrix differential-algebraic system and the matrix differential-algebraic boundary-value problem to solve the matrix differential-algebraic boundary-value problem. We proposed sufficient conditions of existence and constructive schemes for finding the equilibrium positions of the matrix differential-algebraic system and the matrix differential-algebraic boundary value problem. The cases~of equilibrium positions of the matrix differential-algebraic system, which are constant matrices, and equilibrium positions depending on an independent variable are considered separately. To solve the matrix differential-algebraic boundary-value problem, we used the original solvability conditions and~the construction of the general solution of the Sylvester-type matrix equation, while the Moore-Penrose matrix pseudoinverse technique was essentially used. In the article we constructed the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem. The proposed solvability conditions and the construction of the generalized Green operator of the linear Noetherian matrix differential-algebraic boundary value problem, were illustrated in detail with examples.

Matrix Differential-Algebraic Boundary-Value Problem with Pulsed Action

We establish necessary and sufficient conditions for the existence of solutions of a linear matrix boundaryvalue problem for a system of matrix differential-algebraic equations with pulsed action. We also construct a generalized Green operator of linear boundary conditions for a system of matrix differentialalgebraic equations with pulse action. To solve the matrix differential-algebraic boundary problem with pulsed action, we use the original conditions of solvability and the structure of the general solution of the linear matrix equation.

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

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.

On a regularization method for solving linear Noetherian boundary value problem for difference system

The article proposes unusual regularization conditions as well as a scheme for finding bounded solutions of the linear Noetherian boundary value problem for a system of difference equations in the critical case, significantly using the Moore-Penrose matrix pseudo-inversion technology. The problem posed in the article continues the study of the a sufficient condition for solvability and regularization conditions for linear Noetherian boundary value problems in the critical case given in the monographs by A.N. Tikhonov, V.Ya. Arsenin, S.G. Krein, A.M. Samoilenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boichuk. The general case is studied in which a linear bounded operator corresponding to a homogeneous part of a linear Noetherian boundary value problem has no inverse. The noninvertibility of the operators corresponding to a homogeneous part of a linear Noetherian boundary value problem is a consequence of the fact that the number of boundary conditions does not coincide with the number of unknown variables of the difference equations. Using the theory of generalized inverse operators and Moore-Penrose pseudoinverse matrix in the article, a generalized Green operator is constructed and the type of a linear perturbation of a regularized linear Noether boundary value problem for a system of difference equations in the critical case is found. The proposed regularization conditions, as well as the scheme for finding of bounded solutions to linear Noetherian boundary value problems for a system of difference equations in the critical case, are illustrated in details with examples. In contrast to the earlier articles of the authors, the regularization problem for a linear Noether boundary value problem for a system of difference equations in the critical case has been resolved constructively, and sufficient conditions has been obtained for the existence of a bounded solution to the regularization problem.

On a Reduction of the Order in a Differential-Algebraic System

The conditions of solvability and the structure of a generalized Green operator of the Cauchy problem for a linear differential-algebraic system are found. The sufficient conditions of reducibility of a differential-algebraic equation to a sequence of systems joining differential and algebraic equations are constructed. An original classification and a single scheme of construction of the solutions of differentialalgebraic equations are proposed.

On the Solvability of a Matrix Boundary-Value Problem

We found solvability conditions and a construction of the generalized Green operator for a linear matrix boundary-value problem; we present an operator that reduces a linear matrix equation to the conventional linear Noether boundary-value problem. To solve a linear matrix system, we use the operator that reduces a linear matrix equation to a linear algebraic equation with a rectangular matrix.

On the regularization of the Cauchy problem for a system of linear difference equations

The article proposes unusual regularization conditions as well as a scheme for finding solutions of the linear Cauchy problem for a system of difference equations in the critical case, significantly using the Moore-Penrose matrix pseudo-inversion technology. The problem posed in the article continues the study of the regularization conditions for linear Noetherian boundary value problems in the critical case given in the monographs by S.G. Krein, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko and A.A. Boichuk. The general case is studied in which a linear bounded operator corresponding to a homogeneous part of a linear Cauchy problem has no inverse. In the article, a generalized Green operator is constructed and the type of a linear perturbation of a regularized linear Cauchy problem for a system of difference equations in the critical case is found. The proposed regularization conditions, as well as the scheme for finding solutions to linear Cauchy problems for a system of difference equations in the critical case, are illustrated in details with examples. In contrast to the earlier articles of the authors, the regularization problem for a linear Cauchy problem for a system of difference equations in the critical case has been resolved constructively, and sufficient conditions has been obtained for the existence of a solution to the regularization problem.

To the issue of a generalization of the matrix differential-algebraic boundary-value problem

We pose a linear matrix differential-algebraic boundary-value problem generalizing the traditional linear boundary-value problems for differential-algebraic equations. We have found the constructive conditions of existence and an algorithm of construction of the solutions of a linear matrix differentialalgebraic boundary-value problem. We propose the construction of a generalized Green operator for the determination of solutions of the linear differential-algebraic boundary-value problem and give some examples of construction of such solutions.

Generalized green operator of Noetherian boundary-value problem for matrix differential equation

We find necessary and sufficient conditions for solvability and the construction of the generalized Green operator for Noetherian linear boundary-value problem for a linear matrix differential equation. We propose an operator, which leads a linear matrix algebraic equation to the traditional linear algebraic system with a rectangular matrix. We use pseudoinverseMoore–Penrose matrices and orthogonal projections for solving a linear algebraic system.

A generalized matrix differential-algebraic equation

The conditions of solvability and the structure of the generalized Green operator of a linear Noetherian matrix differential-algebraic boundary-value problem are found. The sufficient conditions for reducibility of the generalized matrix differential-algebraic equation to the traditional differential-algebraic equation with the unknown function in the form of a vector-column are obtained. In the solution of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix Sylvester-type equation are used.

Linear boundary value problems for normally solvable operator equations in a Banach space

We consider linear boundary value problems for operator equations with generalized-invertible operator in a Banach or Hilbert space. We obtain solvability conditions for such problems and indicate the structure of their solutions. We construct a generalized Green operator and analyze its properties and the relationship with a generalized inverse operator of the linear boundary value problem. The suggested approach is illustrated in detail by an example.

Solutions of the Schrödinger equation in a Hilbert space

Necessary and sufficient conditions for the existence of a solution of a boundary-value problem for the Schrödinger equation are obtained in the linear and nonlinear cases. Analytic solutions are represented using the generalized Green operator.

ЛИНЕЙНЫЕ КРАЕВЫЕ ЗАДАЧИ ДЛЯ НОРМАЛЬНО РАЗРЕШИМЫХ ОПЕРАТОРНЫХ УРАВНЕНИЙ В БАНАХОВОМ ПРОСТРАНСТВЕ

The paper deals with linear boundary value problems for the operator equations with generally inverses operator with functions in Banach or Hilbert spaces. The authors also get the solvability conditions and formulae for resenting the solutions of such boundary value problem. The authors have construct a generalized Green operator, investigated its characters and links generally inverses operator of the linear boundary value problem. The example is in details considered.