Unbounded Operator Coefficients Research Articles

Existence Results for Some Higher-Order Evolution Equations with Time-Dependent Unbounded Operator Coefficients

Abstract In this paper we establish the existence of C(n)-pseudo almost automorphic mild solutions to some n-order evolution equations involving not only time-dependent operator coefficients but also pseudo almost automorphic forcing terms. To illustrate our abstract results, the existence of C(2)-pseudo almost automorphic mild solutions to some damped second-order nonautonomous differential equations, which among other things model the nonautonomous damped beam equation, will be discussed.

Exponentially Convergent Method for the First-Order Differential Equation in a Banach Space with Integral Nonlocal Condition

For the first-order differential equation with unbounded operator coefficient in a Banach space, we study the nonlocal problem with integral condition. An exponentially convergent algorithm for the numerical solution of this problem is proposed and justified under the assumption that the operator coefficient A is strongly positive and certain existence and uniqueness conditions are satisfied. The algorithm is based on the representations of operator functions via the Dunford–Cauchy integral along a hyperbola covering the spectrum of A and the quadrature formula containing a small number of resolvents. The efficiency of the proposed algorithm is illustrated by several examples.

Well-posed solvability of functional-differential equations with unbounded operator coefficients

We study functional-differential equations with unbounded variable operator coefficients and variable delays in a Hilbert space. We prove the well-posed solvability of initial-boundary value problems for the above-mentioned equations in Sobolev spaces of vector functions on the positive half-line.

Analysis of Operator Models Arising in Problems of Hereditary Mechanics

The correct solvability of initial-value problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces is determined. Numerous problems of hereditary mechanics and thermal physics have motivated the study of such equations. Models taking into account a Kelvin–Voight friction are considered. A spectral analysis of the operator functions, which are the symbols of integro-differential equations considered in this paper, is made.

On Generalized Regularized Trace of a Fourth-Order Differential Operator with Operator Coefficient

We deduce a formula for the trace of a boundary-value problem with unbounded operator coefficient and boundary conditions depending on the parameter.

The stability analysis of abstract Takagi–Sugeno fuzzy impulsive system

In this paper based on the comparison method the sufficient conditions of stability in terms of the two measures of abstract locally-linear impulsive differential equation with unbounded operator coefficients (an abstract Takagi–Sugeno fuzzy impulsive system) have been derived. As an application of the obtained results the locally-linear impulsive system (hybrid T–S fuzzy system) has been considered. An example of impulsive nonperiodic finite-dimension T–S fuzzy system with unstable structural matrices was constructed. It is shown that zero solution of such T–S fuzzy system could be asymptotically stable in cone. The simulations results of example demonstrate the effectiveness of proposed approach.

Spectral analysis of differential operators with unbounded operator coefficients in weighted spaces of functions

The spectral properties of operators constructed from a family of evolution operators are studied in Banach spaces of vector functions defined by a weight function of pre-exponential growth. Necessary and sufficient conditions for the invertibility of such operators are obtained in terms of exponential dichotomy. We prove a spectrum mapping theorem for semigroups of Howland difference operators generated by the operator under study.

О состояниях обратимости линейных дифференциальных операторов с неограниченными периодическими коэффициентами

О состояниях обратимости линейных дифференциальных операторов с неограниченными периодическими коэффициентами

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.

Regularization by a modified quasi-boundary value method of the ill-posed problems for differential-operator equations of the first order

In this paper, we consider the differential-operator equation du(t)dt+Au(t)=0, with A a self-adjoint unbounded operator coefficient, which does not have a fixed sign. The Cauchy problem for the equation above with conditions of the form u(0)=foru(T)=f, is known to be an ill-posed problem. In this work, we will use a modified quasi-boundary value method; we obtain an approximate non-local problem depending on a small parameter α∈]0,1[. We show that the approximate problems are well-posed and that their solutions converge if the original problem has a classical solution. We also obtain a convergence result for these solutions.

Stability and Regularization of Difference Schemes in Banach and Hilbert Spaces

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.

A Theory of the Infinite Horizon LQ-Problem for Composite Systems of PDEs with Boundary Control

We study the infinite horizon linear-quadratic (LQ) problem and the associated algebraic Riccati equations for systems with unbounded control actions. The operator-theoretic context is motivated by composite systems of partial differential equations (PDEs) with boundary or point control. Specific focus is placed on systems of coupled hyperbolic/parabolic PDE with an overall “predominant” hyperbolic character, such as, e.g., some models for thermoelastic or fluid-structure interactions. While unbounded control actions lead to Riccati equations with unbounded (operator) coefficients, unlike in the parabolic case solvability of these equations becomes a major issue, owing to the lack of sufficient regularity of the solutions to the composite dynamics. In the present case, even the more general theory appealing to estimates of the singularity displayed by the kernel which occurs in the integral representation of the solution to the control system fails. A novel framework which embodies possible hyperbolic com...

Bounded solutions of an abstract differential equation of special form with delayed argument

We obtain sufficient conditions for the existence and uniqueness of a solution bounded on the real axis for an abstract linear first-order differential equation with delayed argument in the case of an unbounded operator coefficient.

Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.

Integrodifferential equations in viscoelasticity theory

We prove the correct solvability of the initial problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Such equations occur in many problems of the theory of viscoelasticity with memory and the heat transfer theory.

On conditions for invertibility of difference and differential operators in weight spaces

We obtain necessary and sufficient conditions for the invertibility of the difference operator , , , whose domain is given by the condition , where , , is the Banach space of sequences (of vectors in a Banach space ) summable with weight for and bounded with respect to for , is a bounded linear operator, and is a closed -invariant subspace of . We give applications to the invertibility of differential operators with an unbounded operator coefficient (the generator of a strongly continuous operator semigroup) in weight spaces of functions.

The regularized trace of a self adjoint differential operator of higher order with unbounded operator coefficient

Let L 0 and L be operators which are formed by the differential expressions. ℓ 0 ( y ) = ( - 1 ) m y ( 2 m ) ( x ) + Ay ( x ) and ℓ ( y ) = ( - 1 ) m y ( 2 m ) ( x ) + Ay ( x ) + Q ( x ) y ( x ) respectively, in the space H 1 = L 2(0, π; > H), with same boundary condition y (2 i−1) (0) = y (2 i−1) ( π) = 0, ( i = 1, 2, … , m) where H is an infinite dimensional separable Hilbert space. Here, A is an unbounded self adjoint operator in H and, for every x ∈ [0, π], Q( x) is a self-adjoint trace class operator in H. Assuming the operator A and the operator function Q( x) satisfy some additional conditions, the following formula has been found. lim p → ∞ m q = 1 n p λ q - μ q - 1 π ∫ 0 π ( Q ( x ) φ j q , φ j q ) dx = 1 4 [ trQ ( 0 ) + trQ ( π ) ] - 1 2 π ∫ 0 π trQ ( x ) dx for the regularized trace of L. Here, n 1 < n 2 < ⋯and j 1, j 2, …are sequences of natural numbers with a particular property. Furthermore, μ 1 ⩽ μ 2 ⩽ ⋯and λ 1 ⩽ λ 2 ⩽ ⋯are the eigenvalues of the operators L 0 and L, respectively; and φ 1, φ 2, …is a complete orthonormal sequence consisting of eigenvectors of the operator A.

Approximation of generalized bounded solutions of evolution equations with unbounded operator

We substantiate a parametrization method for a differential equation in a Banach space with an unbounded operator coefficient. We propose an algorithm for finding bounded generalized solutions with an arbitrary order of accuracy.

Estimates for the embedding operator of a sobolev space of periodic functions and for the solutions of differential equations with periodic coefficients

We obtain estimates for the embedding operator of a Sobolev space in the space of continuous periodic functions and use them to estimate the solutions of differential equations with periodic coefficients. We prove a theorem on a necessary and sufficient condition for the invertibility of a differential operator with unbounded operator coefficients.