Lie symmetry analysis and explicit formulas for solutions of some third-order difference equations

Abstract

Lie group theory is applied to rational difference equations of the form

where (a_n)n∈?₀, (b_n)n∈?₀ are non-zero real sequences. Consequently, new symmetries are derived and exact solutions, in unified manner, are constructed. Based on some constraints in the expression of the symmetry generators, we split these solutions into different categories. This work generalises a recent result by Ibrahim [9].     

具分段常变元的时滞微分方程的振动性

The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+b(t)x([t-k])=0 is considered,where a(t) and b(t) are continuous functions on [-k,∞),b(t)≥0,k is a positive integer and [·] denotes the greatest integer function.Some new oscillation and nonoscillation conditions are obtained.     

Oscillation Criteria for First Order Delay Difference Equations

Oscillation criteria for all solutions of the first order delay difference equation of the form where {p_n} is a sequence of nonnegative real numbers and k is a positive integer are established especially in the case that the well-known oscillation conditions are not satisfied. Dedicated to Professor Y.G. Sficas on the occasion of his 60h birthday     

Book Review

Linear difference equations with variable delay are considered. The most important result of this paper is a new oscillation criterion, which should be looked upon as the discrete analogue of a well-known oscillation criterion for first order linear delay differential equations. This criterion constitutes a substantial improvement of an oscillation result due to the first author (Funkcial. Ekvac. 34 (1991), pp. 157–172). The results obtained extend the ones by the authors and Stavroulakis (J. Differ. Equ. Appl. 10 (2004), pp. 419–435) concerning the special case of linear difference equations with constant delay.     

An asymptotic result and a stability criterion for linear nonautonomous delay difference equations

Linear delay difference equations with variable coefficients and constant delays are considered. By the use of an appropriate solution of the so called generalized characteristic equation, an asymptotic result is obtained and a stability criterion is established.     

Eigenvalues of discrete Sturm-Liouville problems with nonlinear eigenparameter dependent boundary conditions

We consider the discrete right definite Sturm-Liouville problems with nonlinear eigenparameter dependent boundary conditions

,

where T > 1 is an integer and λ is the spectrum parameter. We obtain the existence of the eigenvalues, the oscillation properties of the eigenfunctions and the interlacing results of the eigenvalues of the above problem with the eigenvalues of the Dirichlet problem and the Neumann problem.     

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φ_v) _v=0 ^/r−1 of vector refinement equations. The space spanned by the translates of φ_v can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸_v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.     

A stability criterion

An important technique for determining the stability of a system of ordinary differential equations is to determine whether there are any roots in the positive half-plane of a certain polynomial P(z). Cesari has given a criterion for this in terms of the topological degree of the mapping described by P(z). It is shown here that Cesari's criterion can be reformulated as the problem of approximating the real roots of polynomials which are the real and imaginary parts of the P(z) on certain lines in the z-plane. The roots need only be approxi¬mated closely enough so that their magnitudes can be compared. The derivation of this criterion uses the notion of topological degree but the criterion itself is stated entirely in elementary terms     

Initial-value problems for linear partial difference equations

The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functions _i f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made.     

Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r₂(t)((r₁(t)x^Δ(t))^Δ)^γ)^Δ+q(t)f(x(t))=0     

Relations de dépendance entre les équations fonctionnelles de Cauchy

Résumé Afin d'examiner les relations entre les différentes équations de Cauchy, nous résolvons, sans aucune hypothèse de régularité, l'équation fonctionnellea f(xy) + b f(x)f(y) + c f(x + y) + d (f(x) + f(y)) = 0, pour des fonctionsf, définies sur un anneau unifère divisible par deux et prenant leurs valeurs dans un corps, Les coefficientsa, b, c, etd appartiennent au centre de ce corps. Entre autres applications, nous en déduisons qu'une seule équation, à savoirf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), caractérise les endomorphismes des corps dont la caractéristique est différente de 2. En introduisant la notion d'équations fonctionnelles étrangères et d'équations fonctionnelles fortement étrangères, nous concluons à l'indépendance, au sens de cette notion, des équations classiques de Cauchy.

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Summary In order to study the inter-relations between the four Cauchy functional equations, we solve the functional equationa f(xy) + b f(x) f(y) + c f(x + y) + d(f(x) + f(y)) = 0. The functionf is defined over a ring which is divisible by 2 and which possesses a unit, while the values off are in a(skew)-field. The constantsa, b, c andd belong to this field and commute with all elements of thes-field. No regularity assumption is made onf. Among other applications, we show that the single equationf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), is enough to characterize field endormophisms in fields of characteristic different from 2. We introduce the notion of alien functional equations and that of strongly alien functional equations, to conclude that for such notions, Cauchy equations are indeed largely independent.

Dédié avec nos meilleurs voeux à Monsieur le Professeur Otto Haupt à l'occasion de son centenaire     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

Tests for the Oscillation of Autonomous Differential Equations with Bounded Aftereffect

Siberian Mathematical Journal - Considering autonomous delay functional differential equations, we establish some oscillation criterion that reduces the oscillation problem to computing the only...     

The concept of spectral dichotomy for linear difference equations II

This paper is a continuation of a previous one (J. Math. Anal. Appl. 185 (1994), 275–287) in which the concept of spectral dichotomy has been introduced. This new notion of dichotomy has proved to be useful since it allows to apply the well known theory of linear operators to study dynamic properties of nonautonomous linear difference equations. In the present paper we extend our result on the equivalence of the spectral dichotomy and the well known exponential dichotomy to the class of linear differenc equations whose right-hand sides are not necessarily invertible. We furthermore investigate equations on the set of positive integers for which we establish necessary and sufficient conditions for exponential and unifrom stability.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

具有振动系数时滞差分方程解的振动性

本文研究了一类具振动系数时滞差分方程解的振动性,给出了一个新的振动准则,改进了已有文献中许多相关的结果．     

Mean-value type functional equations

Summary In this work the following two conjectures concerning mean-value type functional equations are proved: then-dimensional octahedron and cube equations are equivalent (conjectured by D. Z. Djokovi and H. Haruki), and the continuous solutions of these equations on ⁿ are linear combinations of a given harmonic polynomial (conjectured by H. Haruki).