scholarly journals Smoothness of Generalized Solutions of the Second and Third Boundary-Value Problems for Strongly Elliptic Differential-Difference Equations

2019 ◽  
Vol 65 (4) ◽  
pp. 655-671
Author(s):  
D. A. Neverova
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Generalized Solution ◽  
Generalized Solutions ◽  
Difference Operators ◽  
The Third ◽  
Right Hand ◽  
Smoothness Of Generalized Solutions

In this paper, we investigate qualitative properties of solutions of boundary-value problems for strongly elliptic differential-difference equations. Earlier results establish the existence of generalized solutions of these problems. It was proved that smoothness of such solutions is preserved in some subdomains but can be violated on their boundaries even for infinitely smooth function on the right-hand side. For differential-difference equations on a segment with continuous right-hand sides and boundary conditions of the first, second, or the third kind, earlier we had obtained conditions on the coefficients of difference operators under which there is a classical solution of the problem that coincides with its generalized solution. Also, for the Dirichlet problem for strongly elliptic differential-difference equations, the necessary and sufficient conditions for smoothness of the generalized solution in Holder spaces on the boundaries between subdomains were obtained. The smoothness of solutions inside some subdomains except for ε-neighborhoods of angular points was established earlier as well. However, the problem of smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations remained uninvestigated. In this paper, we use approximation of the differential operator by finite-difference operators in order to increase the smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in the scale of Sobolev spaces inside subdomains. We prove the corresponding theorem.

scholarly journals Smoothness of Generalized Solutions of the Neumann Problem for a Strongly Elliptic Differential-Difference Equation on the Boundary of Adjacent Subdomains

2020 ◽  
Vol 66 (2) ◽  
pp. 272-291
Author(s):  
D. A. Neverova
Keyword(s):  
Boundary Value Problems ◽  
Neumann Problem ◽  
Difference Equations ◽  
Boundary Value ◽  
Generalized Solution ◽  
Generalized Solutions ◽  
Connected Components ◽  
Difference Operators ◽  
Right Hand ◽  
Smoothness Of Generalized Solutions

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

scholarly journals Boundary value problems for Caputo-Hadamard fractional differential inclusions with Integral Conditions

2020 ◽  
Vol 6 (1) ◽  
pp. 62-75
Author(s):  
Ahmed Zahed ◽  
Samira Hamani ◽  
Johnny Henderson
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Fractional Differential Inclusions ◽  
Integral Conditions ◽  
Right Hand ◽  
Fractional Differential

AbstractFor r ∈ (1, 2], the authors establish sufficient conditions for the existence of solutions for a class of boundary value problem for rth order Caputo-Hadamard fractional differential inclusions satisfying nonlinear integral conditions. Both cases of convex and nonconvex valued right hand sides are considered.

scholarly journals On Nonlinear Boundary Value Problems for Functional Difference Equations withp-Laplacian

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Yong Wan ◽  
Yuji Liu
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Functional Difference Equations

Sufficient conditions for the existence of solutions of nonlinear boundary value problems for higher-order functional difference equations withp-Laplacian are established by making of continuation theorems. We allowfto be at most linear, superlinear, or sublinear in obtained results.

scholarly journals Boundary Value Problems for Fourth Order Nonlinearp-Laplacian Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
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.

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

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.

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

2020 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Lyapunov Equation ◽  
Linear Boundary ◽  
Actual Problem ◽  
Linear Differential ◽  
Necessary And Sufficient

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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