Oscillation Theory of $h$-fractional Difference Equations

In this paper, we initiate the oscillation theory for $h$-fractional difference equations of the form \begin{equation*} \begin{cases} _{a}\Delta^{\alpha}_{h}x(t)+r(t)x(t)=e(t)+f(t,x(t)),\ \ \ t\in\mathbb{T}_{h}^{a},\ \ 1<\alpha<2,\x(a)=c_{0},\ \ \Delta_{h}x(a)=c_{1},\ \ \ c_{0}, c_{1}\in\mathbb{R}, \end{cases} \end{equation*} where $_{a}\Delta^{\alpha}_{h}$ is the Riemann-Liouville $h$-fractional difference of order $\alpha,$ $\mathbb{T}_{h}^{a}:=\{a+kh, k\in\mathbb{Z^{+}}\cup\{0\}\},$ and $a\geqslant0,$ $h>0.$ We study the oscillation of $h$-fractional difference equations with Riemann-Liouville derivative, and obtain some sufficient conditions for oscillation of every solution. Finally, we give an example to illustrate our main results.     

Oscillation criteria for certain even order quasilinear difference equations

Some new oscillation theorems for even-order difference equations of the form $$\Delta(|\Delta^{k-1}x_{n}|^{\alpha-1}\Delta ^{k-1}x_{n})+f(n,x_{\sigma(n)})=0$$ are obtained. Our results generalize some well-known results for second order difference equations.     

Oscillation of 2nd-order Nonlinear Noncanonical Difference Equations with Deviating Argument

The purpose of this paper is to establish some new criteria for the oscillation of the second-order nonlinear noncanonical difference equations of the form \[ \Delta \left( a\left( n\right) \Delta x\left( n\right) \right) +q(n)x^{\beta }\left( g(n)\right) =0,\text{ \ \ }n\geq n_{0} \] under the assumption \[ \sum_{s=n}^{\infty }\frac{1}{a\left( s\right) }<\infty \text{.} \] Corresponding difference equations of both retarded and advanced type are studied. A particular example of Euler type equation is provided in order to illustrate the significance of our main results.     

COMPARISON AND OSCILLATION THEOREMS FOR AN ADVANCED TYPE DIFFERENCE EQUATION

ＣＯＭＰＡＲＩＳＯＮ　ＡＮＤ　ＯＳＣＩＬＬＡＴＩＯＮ　ＴＨＥＯＲＥＭＳ　ＦＯＲ　ＡＮ　ＡＤＶＡＮＣＥＤ　ＴＹＰＥ　ＤＩＦＦＥＲＥＮＣＥ　ＥＱＵＡＴＩＯＮＣＯＭＰＡＲＩＳＯＮＡＮＤＯＳＣＩＬＬＡＴＩＯＮＴＨＥＯＲＥＭＳＦＯＲＡＮＡＤＶＡＮＣＥＤＴＹＰＥ...     

Oscillation criteria for nonlinear neutral functional dynamic equations on time scales

In this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation $$\left[ {r\left( t \right)\left[ {m\left( t \right)y\left( t \right) + p\left( t \right)y\left( {\tau \left( t \right)} \right)} \right]^\Delta } \right]^\Delta + q\left( t \right)f\left( {y\left( {\delta \left( t \right)} \right)} \right) = 0$$ on a time scale $\mathbb{T}$ which is unbounded above, where m, p, q, r, T and δ are real valued rd-continuous positive functions defined on $\mathbb{T}$ . The main investigation of the results depends on the Riccati substitutions and the analysis of the associated Riccati dynamic inequality. The results complement the oscillation results for neutral delay dynamic equations and improve some oscillation results for neutral delay differential and difference equations. Some examples illustrating our main results are given.     

带有极大值项的中立型差分方程的振动性和非振动性

本文考虑带有极大值项的一阶中立型差分方程,获得了所有解振动的充分条件,同时也研究了非振动解的渐近性质。     

Asymptotic behavior for non-oscillatory solutions of difference equations with several delays in the neutral term

In this paper, we first consider difference equations with several delays in the neutral term of the form * $$\Delta\left(y_{n}+\sum_{i=1}^{L}p_{i}y_{n-{k_{i}}}-\sum_{j=1}^{M}r_{j}y_{n-{\rho_{j}}}\right)+q_{n}y_{n-\tau}=0\quad \mbox{for}\ n\in\mathbb{Z}^{+}(0),$$ study various cases of coefficients in the neutral term and obtain the asymptotic behavior for non-oscillatory solution of (*) under some hypotheses. Moreover, we consider reaction-diffusion difference equations with several delays in the neutral term of the form $$\begin{array}{l}\Delta_{1}\left(u_{n,m}+\displaystyle \sum_{i=1}^{L}p_{i}u_{n-{k_{i}},m}-\displaystyle \sum_{j=1}^{M}r_{j}u_{n-{\rho_{j}},m}\right)+q_{n,m}u_{n-\tau,m}\\[18pt]\quad {}=a^{2}\Delta_{2}^{2}u_{n+1,m-1}\end{array}$$ for (n,m)∈?⁺(0)×Ω, study various cases of coefficients in the neutral term and obtain the asymptotic behavior for non-oscillatory solution under some hypotheses.     

On the oscillatory behaviour of a class of nonlinear delay difference equations of second order

In this paper, sufficient conditions have been obtained for oscillation of all solutions of a class of nonlinear neutral delay difference equations of the form

Interior Calderón–Zygmund estimates for solutions to general parabolic equations of <Emphasis Type="Italic">p</Emphasis>-Laplacian type

We study general parabolic equations of the form \(u_t = \text{ div }\,\mathbf {A}(x,t, u,D u) +\text{ div }\,(|\mathbf {F}|^{p-2} \mathbf {F})+ f\) whose principal part depends on the solution itself. The vector field \(\mathbf {A}\) is assumed to have small mean oscillation in x, measurable in t, Lipschitz continuous in u, and its growth in Du is like the p-Laplace operator. We establish interior Calderón–Zygmund estimates for locally bounded weak solutions to the equations when \(p>2n/(n+2)\). This is achieved by employing a perturbation method together with developing a two-parameter technique and a new compactness argument. We also make crucial use of the intrinsic geometry method by DiBenedetto (Degenerate parabolic equations, Springer, New York, 1993) and the maximal function free approach by Acerbi and Mingione (Duke Math J 136(2):285–320, 2007).     

超线性时滞微分方程解的振动性

研究一阶超线性时滞微分方程x′(t) p(t)[x(t—γ)]^α＝0(α＞1)解的振动性及非振动性,获得了保证其所有解振动的“almost sharp”准则,并应用所得结果于混合型时滞微分方程x′（t） ∑^ni=1pi（t）[x（t-γi]^αi=0,得到一族振动准则。     

Rate of convergence of finite difference approximations for degenerate ordinary differential equations

In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations:

where is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain conditions, this scheme converges with the first order rate and that such a rate is sharp. To the best of our knowledge, this is the first sharp result in the literature. Moreover, by using the connection between our scheme and the standard upwind finite difference scheme, we get the rate of convergence of the latter, which is also new.

     

Oscillation of certain difference equations

Some new criteria for the oscillation of difference equations of the form

On the oscillation of third order nonlinear difference equations

Some new criteria for the oscillation of third order nonlinear difference equations $$\begin{array}{l}\Delta^{2}\bigl(\frac{1}{a(k)}(\Delta x(k))^{\alpha}\bigr)+q(k)f(x[g(k)])=0\quad\mbox{and}\\[6pt]\Delta^{2}\bigl(\frac{1}{a(k)}(\Delta x(k))^{\alpha}\bigr)=q(k)f(x[g(k)])+p(k)h(x[\sigma(k)])\end{array}$$ are established.     

New Kamenev-type oscillation criteria for half-linear partial differential equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping under quite general conditions. These results are extensions of the recent results developed by Sun [Y.G. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.     

Oscillation criteria of second order half linear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation

$${\left( {r\left( t \right)g\left( {{x^\Delta }\left( t \right)} \right)} \right)^\Delta } + p\left( t \right)f\left( {x\left( {\tau \left( t \right)} \right)} \right) = 0,$$

on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=?, difference equations when T=? but can be applied on different types of time scales such as when T=q^? for q > 1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

     

Bounded Oscillation of Nonlinear Neutral Differential Equations of Arbitrary Order

The paper is concerned with oscillation properties of n-th order neutral differential equations of the form

Oscillation of neutral delay difference equations of second order with positive and negative coefficients

This paper is concerned with a class of neutral difference equations of second order with positive and negative coefficients of the forms

Philos-type oscillation theorems for second order damped elliptic equations

Using general means, we establish several new Philos-type oscillation theorems for the second order damped elliptic differential equation
Σi,j=1 N Di[aij（x）Djy]＋Σi=1 N bi（x）Diy＋c（x）f（y）=0
under quite general assumptions. The obtained results are extensions of the well-known oscillation results due to Kamenev, Philos, Yan for second order linear ordinary differential equations and improve recent results of Xu, Jia and Ma.     

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     

Parabolic Equations with Measurable Coefficients

We investigate the unique solvability of second order parabolic equations in non-divergence form in , p ≥ 2. The leading coefficients are only measurable in either one spatial variable or time and one spatial variable. In addition, they are VMO (vanishing mean oscillation) with respect to the remaining variables. The second author was partially supported by NSF Grant DMS-0140405.