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…     

非线性中立型时滞差分方程解的渐近性

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OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF A CLASS OF FIRST ORDER NEUTRAL DIFFERENTIAL EQUATION WITH PIECEWISE CONSTANT DELAY

The oscillatory and asymptotic behavior of a class of first order nonlinear neutral differential equation with piecewise constant delay and with diverse deviating arguments are considered. We prove that all solutions of the equation are nonoscillatory and give sufficient criteria for asymptotic behavior of nonoscillatory solutions of equation.     

半线性高阶微分方程振动解的渐近性

研究了半线性高阶微分方程(m(t)[r(t)■p(y′(t))]~((n-1)))~((n))+q(t)■p(t))=f(t)的振动解的渐进性.利用H(o|¨)lder不等式给出了方程(1)的振动解渐进趋向于零的充分条件.     

Concentrated terms and varying domains in elliptic equations: Lipschitz case

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H¹an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A nonlinear elliptic problem with terms concentrating in the boundary

In this paper, we investigate the behavior of a family of steady‐state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a ε‐neighborhood of a portion Γ of the boundary. We assume that this ε‐neighborhood shrinks to Γ as the small parameter ε goes to zero. Also, we suppose the upper boundary of this ε‐strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood. Copyright © 2012 John Wiley & Sons, Ltd.     

二阶差分方程解的振动性与渐近性

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OSCILLATION OF FIRST ORDER NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH MIXED ARGUMENT

This paper devotes to study the oscillatory behavior of solutions of a first order nonlinear impulsive differential equation with mixed argument. First, without assuming the deviating argument to be retarded or advanced, a sufficient condition is established for all solutions of the differential equation to be oscillatory. Next, a sufficient condition for the differential equation to have nonoscillatoty solution is given. Finally, a sufficient and necessary condition for all solutions of the differential equation to be oscillatory is obtained.     

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)     

时滞抛物型与双曲型微分方程解的振动性的讨论

对一类线性以及非线性抛物型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,给出了解振动的一些结论.并且对一类线性以及带强迫项的非线性双曲型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,也给出了一些结论.     

Oscillatory traveling waves of polytropic filtration equation with generalized Fisher–KPP sources

The polytropic filtration equation with generalized Fisher–KPP sources is considered. We will show that the equation may have finite times oscillatory traveling waves, and try to give a complete classification by virtue of a singular exponent in the source according to the finiteness of the oscillatory times of traveling waves.     

Periodic traveling waves and locating oscillating patterns in multidimensional domains

We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.

     

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.     

一类分段常数变量线性中立型泛函微分方程解的振动性质

本研究分段常数变量线性中立型泛函微分方程的振动性。用不同的方法研究所考虑的方程,获得存在振动解的两个判定定理。     

Oscillation of fourth order sub-linear differential equations

In this paper, we deal with oscillatory and asymptotic properties of solutions of a fourth order sub-linear differential equation with the oscillatory operator. We establish conditions for the nonexistence of positive and bounded solutions and an oscillation criterion.     

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.     

一阶中立型泛函微分方程解的渐近性与振动性

本文研究具有变系数和多偏差的一阶中立型微分方程，讨论了方程的非振动解的渐近性，得到了方程振动的充分判据，其中有些是变号系数的情形．     

Moving gap solitons in periodic potentials

We address the existence of moving gap solitons (traveling localized solutions) in the Gross–Pitaevskii equation with a small periodic potential. Moving gap solitons are approximated by the explicit solutions of the coupled‐mode system. We show, however, that exponentially decaying traveling solutions of the Gross–Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled‐mode formalism and are estimated by using techniques of spatial dynamics and local center‐stable manifold reductions. Existence of bounded traveling solutions of the Gross–Pitaevskii equation with a single bump surrounded by oscillatory tails on a large interval of the spatial scale is proven by using these techniques. Copyright © 2008 John Wiley & Sons, Ltd.