Discrete Volterra Equations Research Articles

Exponential Stability of Integro-Differential Volterra Equation on Time Scales

Abstract We study the Volterra integro-differential equation on time scales and provide sufficient conditions for boundness of all solutions of considered equation. Using that result, we present the conditions for exponential stability of considered equation. All the results proved on the general time scale include results for both integral and discrete Volterra equations.

Asymptotic properties of solutions to discrete Volterra type equations

We study higher order nonlinear discrete Volterra equations of the form We discuss the problem of the existence of solutions with prescribed asymptotic behavior. In the proof of the main theorem we use the iterated remainder operator and the Schauder's type fixed point theorem. This approach allows us to control the degree of approximation. Our results are also new for the special cases of the above equation such as linear Volterra equation, ordinary difference equation and discrete Emden–Fowler type equation.

Asymptotic Properties of Solutions to Discrete Volterra Monotone Type Equations

We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use o(un) as a measure of approximation.

On the oscillation of discrete Volterra equations with positive and negative nonlinearities

On the oscillation of discrete Volterra equations with positive and negative nonlinearities

Qualitative approximation of solutions to discrete Volterra equations

Qualitative approximation of solutions to discrete Volterra equations

Asymptotic behavior of solutions of discrete Volterra equations

We consider the nonlinear discrete Volterra equations of non-convolution type \[\Delta^m x_n=b_n+\sum\limits_{i=1}^{n}K(n,i)f\left(i,x_i\right), \quad n\geq 1.\] We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior, especially asymptotically polynomial and asymptotically periodic solutions. We use \(\operatorname{o}(n^s)\), for a given nonpositive real \(s\), as a measure of approximation. We also give conditions under which all solutions are asymptotically polynomial.

Bounded solutions of nonlinear discrete volterra equations

AbstractWe give sufficient conditions, under which for every real constant, there exists a solution of the nonlinear discrete Volterra equationconvergent to this constant. We give also conditions under which all solutions are asymptotically constant. Sufficient conditions for the existence of asymptotically periodic solutions of the above equation are also derived.

A new method to generate non-autonomous discrete integrable systems via convergence acceleration algorithms

In this paper, we propose a new algebraic method to construct non-autonomous discrete integrable systems. The method starts from constructing generalizations of convergence acceleration algorithms related to discrete integrable systems. Then the non-autonomous version of the corresponding integrable systems are derived. The molecule solutions of the systems are also obtained. As an example of the application of the method, we propose a generalization of the multistep ϵ-algorithm, and then derive a non-autonomous discrete extended Lotka–Volterra equation. Since the convergence acceleration algorithm from the lattice Boussinesq equation is just a particular case of the multistep ϵ-algorithm, we have therefore arrived at a generalization of this algorithm. Finally, numerical experiments on the new algorithm are presented.

Boundedness and stability of discrete Volterra equations

In this paper, using the fixed point theorem, we investigate the boundedness and asymptotic stability of the zero solution of the discrete Volterra equation

STABILITY PROPERTIES IN NONLINEAR DISCRETE VOLTERRA EQUATIONS WITH UNBOUNDED DELAY

We study some stability properties in discrete Volterra equations by employing to change Yoshizawa's results in [13] for the nonlinear equations into results for the nonlinear discrete Volterra equations with unbounded delay.

Explicit asymptotic limits for a class of discrete Volterra equations

This paper presents explicit asymptotic limits for some solutions of a special kind of Volterra difference equations and exhibits a class of equations having asymptotic equilibrium.

Asymptotic periodicity of nonlinear discrete Volterra equations and applications

Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.

On the boundedness of the solutions in nonlinear discrete Volterra difference equations

In this article, we investigate the boundedness property of the solutions of linear and nonlinear discrete Volterra equations in both convolution and non-convolution case. Strong interest in these kind of discrete equations is motivated as because they represent a discrete analogue of some integral equations. The most important result of this article is a simple new criterion, which unifies and extends several earlier results in both discrete and continuous cases. Examples are also given to illustrate our main theorem.

Non-negative convergent solutions of discrete Volterra equations

The existence of non-negative convergent solutions of discrete Volterra equations is obtained, by using a variety of fixed-point theorems. Examples are given to illustrate the results.

A numerical investigation on the asymptotic behavior of discrete Volterra equations with two delays

We describe a numerical approach to the solution of two-delay Volterra integral equations, and we carry out a nonlinear stability analysis on an interesting test equation by means of a parallel investigation both on the continuous and the discrete problem.

On admissibility of the resolvent of discrete Volterra equations

Suppose that a pair of sequence spaces is admissible with respect to a discrete linear Volterra operator. This paper gives sufficient conditions for the same pair of spaces to be admissible with respect to the associated resolvent operator. The spaces considered include spaces of weighted bounded and weighted convergent sequences. Classes of discrete kernels are discussed for which the appropriate weight sequences are not purely exponential, but the product of an exponential and a slowly decaying sequence.

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.

Sharp conditions for boundedness in linear discrete Volterra equations

If all solutions of a discrete linear nonconvolution Volterra equation satisfying some positivity conditions are bounded, it is found that the kernel must obey certain restrictions. These are closely related to known sufficient conditions for boundedness of solutions. Applications to systems of Volterra difference equations are provided.

On one approach to the solution of the problem of guaranteeing estimation for Volterra equations

The problem of minimax filtration for systems described by discrete Volterra equations is considered. It is suggested that a simpler estimation algorithm instead of an optimal one be used. With the aid of the theory of duality, the quality estimate of such an approximation is obtained.

Total Stability in Nonlinear Discrete Volterra Equations with Unbounded Delay

We study the total stability in nonlinear discrete Volterra equations with unbounded delay, as a discrete analogue of the results for integrodifferential equations by Y. Hamaya (1990).