Solutions Of Volterra Difference Equations Research Articles

Generalized almost periodic solutions of Volterra difference equations

In this paper, we investigate several new classes of generalized $\rho$-almost periodic sequences in the multi-dimensional setting. We specifically analyze the class of Levitan $\rho$-almost periodic sequences and the class of remotely $\rho$-almost periodic sequences. We provide many important applications of the established theoretical results to the abstract Volterra difference equations.

Almost Periodic Solutions of Nonlinear Volterra Difference Equations with Unbounded Delay

In order to obtain the conditions for the existence of periodic and almost periodic solutions of Volterra difference equations, \( x(n+1)=f(n,x(n))+\sum_{s=-\infty}^{n}F(n,s, {x(n+s)},x(n)) \), we consider certain stability properties, which are referred to as (K, \( \rho \))-weakly uniformly-asymptotic stability and (K, \( \rho \))-uniformly asymptotic stability. Moreover, we discuss the relationship between the \( \rho \)-separation condition and the uniformly-asymptotic stability property in the \( \rho \) sense.

On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type

A Volterra difference equation of the form$$x(n+2)=a(n)+b(n)x(n+1)+c(n)x(n)+\sum\limits^{n+1}_{i=1}K(n,i)x(i)$$where $a, b, c, x \colon\mathbb{N} \to\mathbb{R}$ and $K \colon \mathbb{N}\times\mathbb{N}\to \mathbb{R}$ is studied. For every admissible constant $C \in \mathbb{R}$, sufficient conditions for the existence of a solution $x \colon\mathbb{N} \to\mathbb{R}$ of the above equation such that\[x(n)\sim C \, n \, \beta(n),\]where $\beta(n)= \frac{1}{2^n}\prod\limits_{j=1}^{n-1}b(j)$, are presented. As a corollary of the main result, sufficient conditions for the existence of an eventually positive, oscillatory, and quickly oscillatory solution $x$ of this equation are obtained. Finally, a conditions under which considered equation possesses an asymptotically periodic solution are given.

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.

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.

Estimation of the solutions of Volterra difference equations

Volterra equations, whose solution is defined by the whole previous history, are widely used in the modelling of the processes in continuous mechanics and biomechanics, problems of control and estimation and also some schemes of numerical solutions of integral and integral–differential equations [1–8]. In this connection, there is an essential interest in such properties of the solutions as stability, limiting periodicity, boundedness and various estimates of the solutions defined by the acting perturbations. In [9], such estimates were obtained for the solutions of nonlinear scalar integral Volterra equations of convolution type

Periodic solutions of Volterra difference equations and attractivity

Periodic solutions of Volterra difference equations and attractivity