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.

NONSINGULAR INTEGRO-DIFFERENTIAL BOUNDARY VALUE PROBLEM NOT SOLVED WITH RESPECT TO THE DERIVATIVE

Bukovinian Mathematical Journal ◽

10.31861/bmj2020.02.12 ◽

2020 ◽

Vol 8 (2) ◽

pp. 127-138

Author(s):

S. Chuiko ◽

O. Chuiko ◽

V. Kuzmina

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Systematic Study ◽

Critical Case ◽

Boundary Value ◽

Sufficient Conditions ◽

Linear Boundary ◽

Actual Problem ◽

Linear Differential ◽

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. Boi- chuk, 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 linear boundary-value problems for ordinary di- ﬀerential 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 linear diﬀerential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear osci- llations, 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 linear boundary value problems for the integro-diﬀerential boundary value problem not solved with respect to the derivative, in parti- cular, ﬁnding the necessary and suﬃcient conditions of the existence of the desired solutions of the linear integro-diﬀerential boundary value problem not solved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for ﬁnding the solutions of the linear Noetherian integro-diﬀerential boundary value problem not solved with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian integro-diﬀerential boundary value problem not solved with respect to the derivative in the critical case in this article can be transferred to the seminonlinear integro-diﬀerential boundary value problem not solved with respect to the derivative.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-3-2 ◽

2020 ◽

Vol 17 (3) ◽

pp. 313-324

Author(s):

Sergii Chuiko ◽

Ol'ga Nesmelova

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Algebraic System ◽

Boundary Value ◽

Sufficient Conditions ◽

Weakly Nonlinear ◽

Differential Algebraic System ◽

Linear Differential ◽

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.

Necessary and Sufficient Conditions for the Solvability of Linear Boundary-Value Problems for the Fredholm Integrodifferential Equations

Ukrainian Mathematical Journal ◽

10.1007/s11253-015-1003-6 ◽

2015 ◽

Vol 66 (8) ◽

pp. 1200-1219 ◽

Cited By ~ 7

Author(s):

D. S. Dzhumabaev

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Sufficient Conditions ◽

Integrodifferential Equations ◽

Linear Boundary ◽

Solvability ofNth Order Linear Boundary Value Problems

International Journal of Differential Equations ◽

10.1155/2015/230405 ◽

2015 ◽

Vol 2015 ◽

pp. 1-19 ◽

Cited By ~ 3

Author(s):

P. Almenar ◽

L. Jódar

Keyword(s):

Integral Operator ◽

Boundary Value Problems ◽

Existence Of Solutions ◽

Boundary Value ◽

Sufficient Conditions ◽

Linear Boundary ◽

Order Linear ◽

Linear Integral ◽

This paper presents a method that provides necessary and sufficient conditions for the existence of solutions ofnth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.

V N Karazin Kharkiv National University Ser Mathematics Applied Mathematics and Mechanics ◽

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

10.26565/2221-5646-2019-90-04 ◽

2019 ◽

Cited By ~ 1

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Nonlinear Boundary ◽

Nonlinear Oscillations ◽

Boundary Value ◽

Sufficient Conditions ◽

Differential Algebraic Equations ◽

Nonlinear Boundary Value Problems ◽

Actual Problem ◽

Nonlinear Boundary Value

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.

Boundary Value Problems for Fourth Order Nonlinearp-Laplacian Difference Equations

Journal of Applied Mathematics ◽

10.1155/2014/343129 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Qinqin Zhang

Keyword(s):

Difference Equation ◽

Boundary Value Problem ◽

Boundary Value Problems ◽

Fourth Order ◽

Difference Equations ◽

Existence Of Solutions ◽

Boundary Value ◽

Sufficient Conditions ◽

Mountain Pass ◽

Mountain Pass Lemma

We consider the boundary value problem for a fourth order nonlinearp-Laplacian difference equation containing both advance and retardation. By using Mountain pass lemma and some established inequalities, sufficient conditions of the existence of solutions of the boundary value problem are obtained. And an illustrative example is given in the last part of the paper.

Conditions for the solvability of singular boundary value problems

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700017251 ◽

1996 ◽

Vol 53 (3) ◽

pp. 485-497

Author(s):

Xiyu Liu

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Boundary Value ◽

Sufficient Conditions ◽

Uniqueness Result ◽

Singular Boundary ◽

Singular Boundary Value Problem ◽

Singular Boundary Value Problems ◽

Consider the singular boundary value problem (r(x′))′ + f(t, x) = 0, 0 < t < 1. We give necessary and sufficient conditions for this problem to have solutions. In addition, a uniqueness result is obtained.

Positive Solutions of Two-Point Boundary Value Problems for Monge-Ampère Equations

Journal of Function Spaces ◽

10.1155/2015/620251 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8

Author(s):

Baoqiang Yan ◽

Meng Zhang

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Positive Solutions ◽

Boundary Value ◽

Sufficient Conditions ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Existence Of Positive Solutions ◽

Necessary And Sufficient ◽

Monge Ampere

This paper considers the following boundary value problem:((-u'(t))n)'=ntn-1f(u(t)), 0<t<1, u'(0)=0, u(1)=0, wheren>1is odd. We establish the method of lower and upper solutions for some boundary value problems which generalizes the above equations and using this method we present a necessary and sufficient condition for the existence of positive solutions to the above boundary value problem and some sufficient conditions for the existence of positive solutions.

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

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2018-32-14 ◽

2018 ◽

Vol 32 ◽

pp. 133-148

Author(s):

Sergei Chuiko ◽

Elena Chuiko ◽

Yaroslav Kalinichenko

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Difference Equations ◽

Critical Case ◽

Boundary Value ◽

Sufficient Conditions ◽

Bounded Solutions ◽

Homogeneous Part ◽

System Of Difference Equations ◽

Regularization Problem

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.

Multipoint focal boundary value problems on infinite intervals

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953392000236 ◽

1992 ◽

Vol 5 (3) ◽

pp. 283-289 ◽

Cited By ~ 4

Author(s):

S. Umamaheswaram ◽

M. Venkata Rama

Keyword(s):

Differential Equation ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Boundary Value Problems ◽

Boundary Value ◽

Sufficient Conditions ◽

Infinite Interval ◽

Existence Of A Solution ◽

Infinite Intervals ◽

For the differential equation y(n)=f(x,y), we state a set of necessary and sufficient conditions for the existence of a solution (i) on a semi-infinite interval for a k-point right focal boundary value problem and (ii) on (−∞,∞) for a (n−1)-point right focal boundary value problem. The conditions are in terms of the existence of a pair of solutions u(x), v(x) satisfying some auxiliary boundary conditions and algebraic inequatilities.