Oscillations caused by several retarded and advanced arguments

We consider the equation $\text{u}{}_{\mathrm{t}}\text{+}\text{▽}\text{· [}\text{v}\text{(x)?]=0, x ∈}$R², with $\text{v}$ defined as a divergenceless vector valued function (in the generalized sense) and subject to initial data u₀. We prove the existence of a generalized global solution to this equation under assumptions of bounded variation for $\text{v}$ and u₀, and smoothness for ?.     

Oscillations of difference equations with several deviated arguments

We find the solutions ${f,g,h \colon G \to X,\,\varphi \colon G \to \mathbb{K}}$ of each of the functional equation $$\sum\limits_{\lambda \in K} f(x + \lambda y) = |K| \varphi (y) g(x) + |K|h(y), \quad x, y \in G$$ , where (G, + ) is an abelian group, K is a finite, abelian subgroup of the automorphism group of G, X is a linear space over the field ${\mathbb{K} \in \{ \mathbb{R},\mathbb{C}\}}$ .     

含扩散项的多时滞Nicholson苍蝇模型的振动性

研究了含扩散项的多时滞Nicholson苍蝇模型在Neumann边值条件下的振动性，通过对多元函数解曲线或曲面的结构的分析和构造上下解，得到了含扩散项的多时滞Nicholson苍蝇模型在Neumann边值条件下振动的充分条件：当e^δτi-aN^＊ Piτi（aiN^＊-1）〉1/e，i=1，2，…，n时，方程的所有正解N（t，x）都关于正平衡点N^＊振动。     

Oscillations and nonoscillations caused by delays

This note is devoted to the study of the dependence upon the delays, of the oscillatory beha vior of all solutions of a scalar linear retarded functional differential equation.     

Oscillations of first-order differential inequalities with deviating arguments

Summary This paper contains some new results on the oscillatory behaviours of differential inequalities (N_z) and (L_z) caused by retarded, advanced and general deviating arguments g_i(t) (i=1,...,n).     

Oscillation of first-order differential equations with retarded arguments

In this paper, we provide a test under which every solution of a first-order delay differential equation oscillates. An example is given to illustrate the significance of the result.     

Oscillation of nonlinear differential systems with retarded arguments

In this work we investigate some oscillatory properties of solutions of non-linear differential systems with retarded arguments. We consider the system of the form

分段常数变元的中立型方程的振动性

给出了一类含连续和分段常数变元的中立型方程振动的两个充分条件,改进和推广了已知文献中的一些结果.     

Stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type

This paper is concerned with the numerical properties of θ-methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments. Using two θ-methods, namely the one-leg θ-method and the linear θ-method, the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained, and the conditions of oscillations for the θ-methods are also obtained. It is proved that oscillations of the analytic solution are preserved by the θ-methods. Furthermore, the relationships between stability and oscillations are revealed. Some numerical experiments are presented to illustrate our results.     

A collocation approach for the numerical solution of certain linear retarded and advanced integrodifferential equations with linear functional arguments

This article presents a Taylor collocation method for the approximate solution of high‐order linear Volterra‐Fredholm integrodifferential equations with linear functional arguments. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Some numerical examples, which consist of initial and boundary conditions, are given to show the properties of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Oscillations of a class of difference equations with continuous arguments

This paper is concerned with the difference equations of the form

Sufficient conditions for all solutions of this equations to be oscillatory are obtained.     

On exponential dichotomy and stability of a differential system with retarded arguments

Questions concerning exponential dichotomy for systems of differential equations with delay are considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1418–1429, October, 1991.     

Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method

This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu [1].     

Regularly varying solutions of half-linear functional differential equations with retarded arguments

We consider half-linear retarded functional differential equations of the form $$ (|x'(t)|^\alpha \operatorname{sgn} x'(t))' = q(t)|x(g(t))|^\alpha \operatorname{sgn} x(g(t)), \alpha > 0, g(t) < t $$ and obtain a necessary and sufficient condition to possess a slowly varying solution and a regularly varying solution of index 1 at the same time.