Constant Operator Coefficients Research Articles

Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients

We study the question of the existence of a unique bounded in the mean solution for the second-order difference equation with piecewise constant operator coefficients and of the stationary solution of the corresponding difference equation with constant operator coefficients. The case is considered when the corresponding “algebraic” operator equations have separated roots.

Bounded Solutions of a Difference Equation with Piecewise Constant Operator Coefficients

Bounded Solutions of a Difference Equation with Piecewise Constant Operator Coefficients

Bounded Solutions of Difference Equations in a Banach Space with Asymptotically Constant Operator Coefficient

We study the problem of existence of a unique bounded solution for a difference equation on the half axis with asymptotically constant operator coefficients in a Banach space. Necessary and sufficient conditions for the existence and uniqueness of bounded solutions are obtained for equations with and without initial condition.

Bounded Solutions of a Differential Equation with Piecewise Constant Operator Coefficients

Bounded Solutions of a Differential Equation with Piecewise Constant Operator Coefficients

On the Invertibility Conditions of Finite-Difference Operators

The invertibility of second-order finite-difference operators with constant operator coefficients acting in the Banach space of two-sided vector sequences is proved under the condition of their uniform injectivity (in particular, left invertibility), or surjectivity (in particular, right invertibility), or the Fredholm property.

Exponentially Dichotomous Difference Equations with Piecewise Constant Operator Coefficients

We establish necessary and sufficient conditions for the exponential dichotomy of the solutions of linear difference equations with piecewise constant operator coefficients.

Bounded solutions of a difference equation with finite number of jumps of operator coefficient

We study the problem of existence of a unique bounded solution of a difference equation with variable operator coefficient in a Banach space. There is well known theory of such equations with constant coefficient. In that case the problem is solved in terms of spectrum of the operator coefficient. For the case of variable operator coefficient correspondent conditions are known too. But it is too hard to check the conditions for particular equations. So, it is very important to give an answer for the problem for those particular cases of variable coefficient, when correspondent conditions are easy to check. One of such cases is the case of piecewise constant operator coefficient. There are well known sufficient conditions of existence and uniqueness of bounded solution for the case of one jump. In this work, we generalize these results for the case of finite number of jumps of operator coefficient. Moreover, under additional assumption we obtained necessary and sufficient conditions of existence and uniqueness of bounded solution.

Експоненціально дихотомічні різницеві рівняння з кусково-сталими операторними коефіцієнтами

УДК 517.988.6 Отримано необхiднi i достатнi умови експоненцiальної дихотомiї розв’язкiв лiнiйних рiзницевих рiвнянь iз кусковосталими операторними коефiцiєнтами.

Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.

Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

В работе обобщается известный метод Лагранжа для линейного неоднородного дифференциального уравнения на случай уравнений второго порядка с постоянными операторными коэффициентами в локально выпуклых пространствах. Решения представляются равномерно сходящимися функциональными векторнозначными рядами, порожденными парой элементов локально выпуклого пространства. Найдены достаточные условия непрерывной зависимости решений от порождающей пары. Также найдено решение задачи Коши для рассматриваемых уравнений и указаны условия его существования и единственности. Кроме того, при определенных условиях получено так называемое общее решение рассматриваемых уравнений (функция самого общего вида, из которой можно получить любое частное решение). Исследование проводится с помощью характеристик (порядка и типа) оператора и последовательности операторов. Также в исследовании применяется сходимость операторных рядов относительно равностепенно непрерывной борнологии. Библиография: 25 названий.

Lyapunov transformation of differential operators with unbounded operator coefficients

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established.

Spectral analysis of differential operators with unbounded periodic coefficients

To a linear differential operator (respectively, equation) with unbounded periodic coefficients in a Banach space of vector functions defined on the entire real line, we assign a difference operator (respectively, a difference equation) with a constant operator coefficient defined in the corresponding Banach space of two-sided vector sequences. For the differential and difference operators, we prove the coincidence of the dimensions of their kernels and coranges, the simultaneous complementability of kernels and ranges, the simultaneous invertibility, and a relationship between the spectra.

Spectral analysis of difference and differential operators in weighted spaces

This paper is concerned with describing the spectrum of the difference operator with a constant operator coefficient , which is a bounded linear operator in a Banach space . It is assumed that acts in the weighted space , , of two-sided sequences of vectors from . The main results are obtained in terms of the spectrum of the operator coefficient and properties of the weight function.Applications to the study of the spectrum of a differential operator with an unbounded operator coefficient (the generator of a strongly continuous semigroup of operators) in weighted function spaces are given.Bibliography: 23 titles.

Chebyshev orthogonal polynomials and the Neumann problem

In a Banach space E, we consider the Neumann problem for a second-order homogeneous equation with a constant operator coefficient densely defined in E. Under a weakened strong positivity condition on the operator coefficient (we use the approach due to V.I. Gorbachuk and A.I. Knyazyuk to the study of a-positive operators), we present a criterion for the well-posed solvability of the problem. We obtain a representation of the solution via Chebyshev operator polynomials of the first and second kind.

The cayley transform and the solution of an initial value problem for a first order differential equation with an unbounded operator coefficient in hilbert space

An initial value problem for a first order differential equation with an unbounded constant operator coefficient A in Hilbert space is considered. We give the definition of a σ-solution and using the Cayley transform we deduce an explicitformula for the solution in case the operator –A is self-adjoint and positiv definite. On the basis of this formula we propose a numerical algorithm for the approximate solution of the initial value problem and give an error estimate. It turns out that, contrary to the case of a bounded operator A, the rate of convergence is not exponential but only polynomial and depends on the smoothness of the initial data. It is proved that the approximate solution is a best approximation in some Hilbert subspace. An example concerning the homogeneous heat equation is given.

A Method for Solving Initial Value Problems for Linear Differential Equations in Hilbert Space Based on The Cayley Transform

A first-order differential equation with a bounded operator coefficient in Hilbert space is considered. In case the coefficient is constant we establish a one-to-one correspondence between the continuous initial value problem and some discrete initial value problem. Using the Cayley transform we give explicit formulas for their solutions as well as formulas connecting corresponding continuous and discrete semigroups. On the basis of these formulas we propose a numerical algorithm for solving initial value problems with a bounded constant operator coefficient which has an exponential rate of convergence. If the operator coefficient is variable we use its piecewise constant approximation and the previously cited algorithm. Error estimates are given.

FIRST ORDER HYPERBOLIC EQUATIONS WITH CONSTANT OPERATOR COEFFICIENTS

This paper considers the Cauchy problem for a hyperbolic equation with constant operator coefficients in Hilbert space: where and are selfadjoint operators and is semibounded.As an example we consider ultraparabolic systems.Bibliography: 10 items.

ON THE EXISTENCE AND ASYMPTOTICS OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH UNBOUNDED OPERATOR COEFFICIENTS IN A BANACH SPACE

In this work theorems are proved concerning existence and uniqueness of solutions of differential equations of arbitrary order with unbounded constant operator coefficients in a Banach space. Asymptotic formulas are given for solutions of equations with variable coefficients.