Oscillation of certain partial difference equations

In this paper we consider a class of nonlinear delay partial difference equations and a class of linear delay partial difference equations with variable coefficients, which may change sign. We obtain oscillation criteria for these equations. There are no results for the oscillation of these equations up to now.     

A pair of difference differential equations of Euler-Cauchy type

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.

     

First-order advanced difference equations

We investigate the existence of solutions to nonlinear first-order difference problems with advanced arguments. Sufficient conditions when such problems have solutions (extremal or unique) are given. Linear advanced difference inequalities are also discussed. According to my knowledge, it is a first paper when a monotone iterative method is applied to nonlinear boundary value problems for first-order difference equations with advanced arguments. An example illustrates the theoretical results.     

Regularization of linear difference equations with analytic coefficients and their applications

We propose a regularization method for four-element linear difference equations with analytic coefficients. We study these equations in the class of functions which are holomorphic in the complex plane with a cruciform cut and vanish at infinity. We give several examples illustrating the dependence of the solvability properties of equations on the choice of periodic coefficients. We describe various applications.     

Delay difference equations in Banach spaces

We derive explicit stability conditions for semilinear delay difference equations in a Banach space. It is assumed that the nonlinearities of the considered equations satisfy the local Lipschitz condition. By virtue of the new estimates for the norm of functions of quasi-Hermitian operators, explicit stability and boundedness conditions are given. Applications to infinite dimensional delay difference systems are discussed.     

Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.     

Recursion operators,conservation laws,and integrability conditions for difference equations

We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.     

Minimal universal denominators for linear difference equations

We provide minimal universal denominators for linear difference equations with fixed leading and trailing coefficients. In the case of first-order equations, they are factors of Abramov's universal denominators. While in the case of higher order equations, we show that Abramov's universal denominators are minimal.     

Differential Galois theory of linear difference equations

We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric equations.     

Mathematical induction,difference equations and divisibility

Many exercises in mathematical induction require the student to prove a divisibility property of a function of the integers. Such problems are generally presented as being independent of each other. However, many of these problems can be presented in terms of difference equations, and the theory of difference equations can be used to provide a uniform method for creating such divisibility problems. This article shows how a multitude of such problems can be created, and how standard problems from textbooks can be analysed in terms of difference equations.     

Peak effects in stable linear difference equations

We consider asymptotically stable scalar difference equations with unit-norm initial conditions. First, it is shown that the solution may happen to deviate far away from the equilibrium point at finite time instants prior to converging to zero. Second, for a number of root distributions and initial conditions, exact values of deviations or lower bounds are provided. Several specific difference equations known from the literature are also analysed and estimates of deviations are proposed. Third, we consider difference equations with non-random noise (ie bounded-noise autoregression) and provide upper bounds on the solutions. Possible generalizations, eg to the vector case are discussed and directions for future research are outlined.     

Linear difference equations mod 2 with applications to nonlinear difference equations1

Known results for linear difference equations mod 2 with T-periodic solutions are extended and compiled for applications to the semicycle analysis of nonlinear difference equations. For the calculation of T, four methods are presented. A further application concerns rational functions in the field of integers mod 2.     

Oscillation analysis of neutral difference equations with delays

In this paper, we obtain some oscillation criteria for a class of neutral difference equations with time delays. We also investigate the behavior of the eventually positive solutions of these equations. To verify our results we give various numerical simulations by using the MATLAB programming.     

Symmetry analysis and conservation laws of some third-order difference equations

We adopt a procedure for determining the continuous (Lie) symmetries for third-order difference equations and utilize these to reduce the order of the equations – this reduction leads to some solutions of the equations under investigation. We further investigate the existence of first integrals of the third order equations using a, now, established, procedure.     

Dichotomy and trichotomy of difference equations

Dedicated to Professor Gerry Ladas on his sixtieth birthday in acknowledgement of his leading role of difference equations. We extend the notions of dichotomy and trichotomy to nonlinear ordinary difference equations. This is accomplished by using two completely different approaches. In the first approach we use a notion, independent of the nature of our difference equation, called tracking. In the second approach we introduce a discrete analogue of dichotomy and trichotomy in variation.     

Oscillation of some nonlinear difference equations

We investigate the oscillatory behavior of all solutions of a new class of first order nonlinear neutral difference equations. Several explicit oscillation criteria are established. Our main results are supported by illustrative examples.     

On nonlinear difference and differential equations

A generalization of the logarithmic norm to nonlinear operators, the Dahlquist constant is introduced as a useful tool for the estimation and analysis of error propagation in general nonlinear first-order ODE's. It is a counterpart to the Lipschitz constant which has similar applications to difference equations. While Lipschitz constants can also be used for ODE's, estimates based on the Dahlquist constant always give sharper results.The analogy between difference and differential equations is investigated, and some existence and uniqueness results for nonlinear (algebraic) equations are given. We finally apply the formalism to the implicit Euler method, deriving a rigorous global error bound for stiff nonlinear problems.Dedicated to my teacher and friend, Professor Germund Dahlquist, on the occasion of his 60th birthday.     

Exponential dichotomy and boundedness for retarded functional difference equations

We characterize the exponential dichotomy of difference equations with infinite delay. We apply the results to study the robustness of exponential dichotomy. This kind of dichotomy gives us relevant information about boundedness of solutions for several perturbed quasi linear systems with infinite delay. Applications to Volterra difference equations are shown.     

Oscillation of solutions of second-order linear differential equations and corresponding difference equations

In this paper, we present conditions ensuring that solutions of linear second-order differential equations oscillate, provided solutions of corresponding difference equations oscillate. We also establish the converse result, namely, when oscillation of solutions of difference equations implies oscillation of solutions of corresponding differential equations.     

Asymptotic behavior of solutions of functional difference equations

For linear functional difference equations, we obtain some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations. We also apply our results to Volterra difference equations with infinite delay.