Homogenization for semilinear hyperbolic systems with oscillatory data

The behavior of multi-dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory solutions of the discrete Boltzmann equations to the solutions of the corresponding homogenized equations is established. Moreover, we find that the weak limits of the oscillatory solutions for a model of Broadwell type are not continuous functions of the discrete velocities. Generalization of the above results to problems with multiple-scale initial data is also established.     

Oscillatory and Asymptotic Behavior for Second Order Quasilinear Difference Equations

1 IntroductionConsider the second order quasilinear difference equationA(g(Ay.--l)) + f(n,y.) = 0, for n E N(no), (l'l)where A is defined by Ay. = Vn+1--yn, n E N(no) = {no, no + 1,'' }, nO E N = {l, 2,'. }.The following hold throughout the paPer:(H0) (i) g: R-R is a continuous increasing fUnction with propertiessgng(y) = sgny) g(R) = R;(il) f: N(no) x R--+ R is continuous as a function of y E R;(iii) yf(n,y) > 0 for n E N and y / 0.By a solution of the equation (1.l) we mean a non…     

三阶非线性中立型变时滞动力方程的渐近性与振动性(英文)

考虑了时标上三阶非线性中立型变时滞动力方程的渐近性和振动性.利用广义Ric-cati技巧与完全平方技巧,获得了方程所有解渐近和振动的准则.所得结果推广了已有三阶动力方程的结果,给出了一些例子加以说明本文的主要结论.     

非线性中立型抛物微分方程解的振动性

This paper deals with the oscillatory properties of a class of nonlinear neutral parabolic partial differential equations with several delays. Sufficient criteria for the equation to be oscillatory are obtained by making use of some results of first-order functional differential inequalities. These results fully reveal the essential difference between this type and that without delays.     

OSCILLATION OF CERTAIN NONLINEAR ADVANCED DIFFERENCE EQUATIONS

This paper deals with the oscillatory properties of a class of nonlinear advanced difference equations. Sufficient criteria in the form of infinite sum for the equation to be oscillatory are obtained. In the linear cases, our results coincide with those in the literature.     

Dynamics of bidirectional associative memory networks with distributed delays and reaction–diffusion terms

In this paper, the periodic oscillatory solution and stability are investigated for a class of bidirectional associative memory neural networks with distributed delays and reaction–diffusion terms. By constructing a new Lyapunov functional, applying M-matrix theory and inequality technique, several novel sufficient conditions are derived to ensure the existence and uniqueness of periodic oscillatory solutions for bidirectional associative memory neural networks with distributed delays and reaction–diffusion terms, and all other solutions of this network converge exponentially to the unique periodic oscillatory solution. Moreover, the exponential convergence rate is estimated, which depends on the delay kernel functions and the system parameters. Two numerical examples are given to show the effectiveness of the obtained results. The results extend and improve the previously known results.     

Oscillation and nonoscillation of perturbed nonlinear equations with p-Laplacian

In this paper, we analyze oscillatory properties of perturbed half-linear differential equations (i.e., equations with one-dimensional p-Laplacian). The presented research covers the Euler and Riemann–Weber type equations with very general coefficients. We prove an oscillatory result and a nonoscillatory one, which show that the studied equations are conditionally oscillatory (i.e., there exists a certain threshold value that separates oscillatory and nonoscillatory equations). The obtained criteria are easy to use. Since the number of perturbations is arbitrary, we solve the oscillation behavior of the equations in the critical setting when the coefficients give exactly the threshold value. The results are new for linear equations as well.     

Stochastic processes associated with a symmetric oscillatory poisson process

A symmetric oscillatory Poisson process is defined and its stochastic features studied. The process represented by the symbolic integral of this oscillatory Poisson process is then discussed in detail. The results obtained are applied to the well-known stochastic problem of multiple scattering of charged particles in their passage through matter.     

Oscillatory solutions of fractional integro-differential equations

We give necessary conditions to get oscillatory solutions of a class of fractional order neutral differential equations with continuously distributed delay by means of the fractional derivative with respect to a given function. In particular, oscillatory solutions of the considered fractional equations with Caputo and Hadamard type of fractional derivatives are established. Some explicit examples are given to illustrate the main results.     

Oscillation for a second-order neutral differential equation with impulses

We consider a certain type of second-order neutral delay differential systems and we establish two results concerning the oscillation of solutions after the system undergoes controlled abrupt perturbations (called impulses). As a matter of fact, some particular non-impulsive cases of the system are oscillatory already. Thus, we are interested in finding adequate impulse controls under which our system remains oscillatory.     

非线性泛函微分方程解的性态

本文给出了一类非线性泛函微分方程解的性态的若干充要条件,推广了[1]的某些结论.     

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

The purpose of this paper is to describe the oscillatory properties of second‐order Euler‐type half‐linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non‐oscillation of the considered equations, including the so‐called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.     

On oscillatory solutions of quasilinear differential equations

Necessary and sufficient conditions for the existence of at least one oscillatory solution of a second-order quasilinear differential equation are presented. These results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions. Our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zero-counting function.     

Oscillatory and asymptotic criteria of third order nonlinear delay dynamic equations with damping term on time scales

In this paper, we are concerned with oscillatory and asymptotic behavior of third order nonlinear delay dynamic equations with damping term on time scales. By using a generalized Riccati function and inequality technique, we establish some new oscillatory and asymptotic criteria. The established results on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis as two special cases of an arbitrary time scale. We also present some applications for the established results.     

Generalized even and odd totally positive matrices

A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. Classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented.     

Weakly Nonlocal Solitary Waves in a Singularly Perturbed Nonlinear Schrödinger Equation

We consider the nonlinear Schrödinger equation perturbed by the addition of a third-derivative term whose coefficient constitutes a small parameter. It is known from the work of Wai et al. [1] that this singular perturbation causes the solitary wave solution of the nonlinear Schrödinger equation to become nonlocal by the radiation of small-amplitude oscillatory waves. The calculation of the amplitude of these oscillatory waves requires the techniques of exponential asymptotics. This problem is re-examined here and the amplitude of the oscillatory waves calculated using the method of Borel summation. The results of Wai et al. [1] are modified and extended.     

带有最大值项的高阶中立型差分方程的振动性

对一类带有最大值项的高阶中立型差分方程的振动性进行了研究,得到一些方程振动的充分条件,推广和改进了已有的一些结果.     

中立双曲型时滞偏微分方程解振动的充要条件

本文讨论一类线性中立双曲型时滞微分方程解的振动性质，获得了其一切解振动的充要条件．     

Oscillation of second‐order perturbed differential equations

We give constructive proof of the existence of vanishing at infinity oscillatory solutions for a second‐order perturbed nonlinear differential equation. In contrast to most results reported in the literature, we do not require oscillatory character of the associated unperturbed equation, monotonicity of nonlinearity, and we establish global existence of oscillatory solutions without assuming it a priori. Furthermore, as our example demonstrates, existence of bounded oscillatory solutions does not exclude existence of unbounded nonoscillatory solutions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Necessary and sufficient conditions on the asymptotic behavior of second‐order neutral delay dynamic equations with positive and negative coefficients

In this paper, we establish necessary and sufficient conditions for the solutions of a second‐order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd.