Linear Volterra Difference Equations Research Articles

Asymptotically ordinary linear Volterra difference equations with infinite delay

• A linear Volterra difference equation with infinite delay and sufficiently small coefficients is shown to be asymptotically equivalent to a linear ordinary difference equation. • The coefficient matrix of the corresponding ordinary difference equation can be written as a limit of successive approximations. • The eigenvalues of the approximating matrices converge to the characteristic roots of the Volterra difference equation at an exponential rate. A class of linear Volterra difference equations with infinite delay is considered. It is shown that if the coefficient matrices are sufficiently small, then the Volterra difference equation is asymptotically equivalent to a linear ordinary difference equation at infinity. The coefficient matrix of the ordinary difference equation is a solution of an associated matrix equation which can be obtained by successive approximations. The eigenvalues of the approximating matrices converge exponentially to the characteristic roots of the Volterra difference equation. As a corollary, an efficient new method is obtained for the computation of the characteristic roots with an explicit error estimate.

Long-time behavior of numerical solutions to nonlinear fractional ODEs

In this work, we study the long time behavior, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By means of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grünwald–Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including convolution quadrature schemes based on classical second order BDF and product integration schemes based on quadratic interpolation approximation, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

Stability for Linear Volterra Difference Equations in Banach Spaces

This paper is devoted to studying the existence and stability of implicit Volterra difference equations in Banach spaces. The proofs of our results are carried out by using an appropriate extension of the freezing method to Volterra difference equations in Banach spaces. Besides, sharp explicit stability conditions are derived.

New criteria for global exponential stability of linear time-varying volterra difference equations

Abstract Linear time-varying Volterra difference equations are considered. By a novel approach, we get some new explicit criteria for global exponential stability. Some examples are given to illustrate the obtained results. To the best of our knowledge, the obtained results are new.

Subexponential Solutions of Linear Volterra Difference Equations

AbstractWe study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.

Existence of bounded solution of Volterra difference equations via Darbo's fixed-point theorem

A linear Volterra difference equation of non-convolution type of the form where , , and , is considered. Sufficient conditions for the existence of a bounded solution of this equation are given. Using this result, an asymptotic equivalence of some solution and given sequence, depending on terms of sequence b, is presented. Finally, the existence of a solution of the considered Volterra equation estimated by a given sequence is obtained.

On the existence of almost automorphic solutions of Volterra difference equations

Given a complex summable sequence and a discrete almost automorphic function with values in a complex Banach space X, we give criteria for the existence of discrete almost automorphic solutions of the linear Volterra difference equation , , for λ in a distinguished subset of the complex plane, determined by . We also prove the existence of a discrete almost automorphic solution of the nonlinear Volterra difference equation , , where is a discrete almost automorphic function in n for each that satisfies a global Lipschitz condition and takes values in X. Finally, we treat the same type of problem for Volterra functional difference equations.

Stability properties and existence theorems of pseudo almost periodic solutions of linear Volterra difference equations

The purpose of this article is to discuss the existence of pseudo almost periodic solutions of linear Volterra equation: , n ∈ Z, by using an exponentially stable of the zero solution, which is equivalent to the exponential behaviors of the resolvent matrix G(n, m) as n → ∞ and of some summability of the kernel. AMS (MOS) 2000 Subject classifications: 39A11.

On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence

Schauder’s fixed point technique is applied to asymptotical analysis of solutions of a linear Volterra difference equationx(n+1)=a(n)+b(n)x(n)+∑i=0nK(n,i)x(i)where n∈N0, x:N0→R, a:N0→R, K:N0×N0→R, and b:N0→R⧹{0} is ω-periodic. In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant c∈R, there exists a solution x=x(n) such thatx(n)∼c+∑i=0n-1a(i)β(i+1)β(n),where β(n)=∏j=0n-1b(j), for n→∞ and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.

Formal adjoint equations and asymptotic formula for solutions of Volterra difference equations with infinite delay

For linear Volterra difference equations with infinite delay, we obtain an explicit asymptotic representation formula of solutions by means of the formal adjoint equation associated with a certain bilinear form.

ASYMPTOTIC EQUIVALENCE OF VOLTERRA DIFFERENCE SYSTEMS

In this paper, sufficient conditions are obtained for theasymptotic relationships between the solutions of a linear Volterra difference equations and its perturbed nonlinear Volterra equation.

Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

A linear Volterra difference equation of the form where x : ℕ0 → ℝ, a : ℕ0 → ℝ, K : ℕ0 × ℕ0 → ℝ and b : ℕ0 → ℝ∖{0} is ω‐periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on is assumed. The results generalize some of the recent results.

Stability and robust stability of positive linear Volterra difference equations

AbstractWe first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron–Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. Copyright © 2008 John Wiley & Sons, Ltd.

Stability and asymptoticity of Volterra difference equations: A progress report

We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Z -transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried out in Section 6. Finally, we state some problems.

Uniform Asymptotic Stability and Robust Stability for Positive Linear Volterra Difference Equations in Banach Lattices

For positive linear Volterra difference equations in Banach lattices, the uniform asymptotic stability of the zero solution is studied in connection with the summability of the fundamental solution and the invertibility of the characteristic operator associated with the equations. Moreover, the robust stability is discussed and some stability radii are given explicitly.

Asymptotic Representation of the Solutions of Linear Volterra Difference Equations

This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. We give examples showing the accuracy of our results.

On the asymptotic stability of linear Volterra difference equations of convolution type

We show that the condition is not necessary, though sufficient, for the asymptotic stability of . We prove the existence of a class of Volterra difference equations (VDEs) that violate this condition but whose zero solutions are asymptotically stable.

On linear Volterra difference equations with infinite delay

Linear neutral, and especially non-neutral, Volterra difference equations with infinite delay are considered and some new results on the behavior of solutions are established. The results are obtained by the use of appropriate positive roots of the corresponding characteristic equation.

Decomposition of Phase Space for Linear Volterra Difference Equations in a Banach Space

We study the spectrum of the solution operator for a linear Volterra difference equation in a Banach space, and obtain the decomposition of the phase space, together with that of the variation of constants formula in the phase space. As an application, we establish the existence of almost periodic solutions for forced linear Volterra difference equations with an almost periodic forcing term.

Stability properties and asymptotic almost periodicity for linear Volterra difference equations in a Banach space

Stability properties and asymptotic almost periodicity for linear Volterra difference equations in a Banach space