KAMENEV-TYPE OSCILLATION CRITERIA FOR DELAY DIFFERENCE EQUATIONS

Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation △（pn（△（Xn ＋ CnXn-τ））^γ） ＋ qnX^Bn-σ = 0, n ：0, 1, 2...which extend and include several oscillation criteria in [11], and also correct a theorem and its proof in [10].

Continuous-Time Channel Prediction Based on Tensor Neural Ordinary Differential Equation

Channel prediction is critical to address the channel aging issue in mobile scenarios.Existing channel prediction techniques are mainly designed for discrete channel prediction,which can only predict the future channel in a fixed time slot per frame,while the other intra-frame channels are usually recovered by interpolation.However,these approaches suffer from a serious interpolation loss,especially for mobile millimeter-wave communications.To solve this challenging problem,we propose a tensor neural ordinary differential equation(TN-ODE)based continuous-time channel prediction scheme to realize the direct prediction of intra-frame channels.Specifically,inspired by the recently developed continuous mapping model named neural ODE in the field of machine learning,we first utilize the neural ODE model to predict future continuous-time channels.To improve the channel prediction accuracy and reduce computational complexity,we then propose the TN-ODE scheme to learn the structural characteristics of the high-dimensional channel by low-dimensional learnable transform.Simulation results show that the proposed scheme is able to achieve higher intra-frame channel prediction accuracy than existing schemes.

Oscillation of Second Order Delay Difference Equations

In this paper we study the Oscillatory behaviour of the second order delay differenceequation.（1）△（rn△An）+PnAn-k=0,n=n0,n0+1……where{Pn}（?）is a nonnegative Sequenceof real number,（?）is a positive sequence of real number with sum from n=n0 to +∞（1/rn）=+∞,K is a positive integer and △An=An+1-An we prove that each one of following conditions.imples that al solutions of Eq（1）oscillate,where Rn=sum from i=n0 to n（1/ri

RECENT DEVELOPMENTS IN THE OSCILLATION OF NEUTRAL DELAY DIFFERENCE EQUATIONS WITH SUMMABLE COEFFICIENTS

This paper surveys some new results on the oscillation of all solutions of the linear nentral delaydifference equation of the form △(yn - pnyn-k) + qnyn-l = 0, n = 0,1, 2…where { pn } and { qn } are twe real numbers sequences with qn≥0, and k and l are positive integers. These re-sults do not require the usual assumptionAlso, some interesting open problems on this topic am given.

Oscillation in a Variable Delay Logistic Difference Equation

Consider the nonautonomous delay logistic equation △yn = pnyn(1- ynk-ln), n ≥ 0, (1)where {pn}n≥0 is a sequence of nonnegative real numbers, {ln}n≥0 is a sequence of positive integers satisfying lim n→∞(n - ln) = ∞, and k is a positive constant. Only solutions which are positive for n ≥ 0 are considered. We obtain a new sufficient for all positive solutions of (1) to oscillate about k which contains the corresponding result in [2] when i = 1.

Oscillation of Nonlinear Partial Difference Equations with Continuous Variables

这份报纸与连续变量和相应双方程涉及非线性的部分差别方程的摆动。几个足够的条件为这些方程的摆动被获得。

A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.

SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.

THE UNIFORMLY CONVERGENT DIFFERENCE SCHEMES FOR A SINGULAR PERTURBATION PROBLEM OF A SELFADJOINT ORDINARY DIFFERENTIAL EQUATION

In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of lite difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.

A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.

LINEARIZED OSCILLATIONS FOR AUTONOMOUS DELAY DIFFERENTIAL EQUATIONS

The linearized oscillations of the nonlinear autonomous delay differential equation [GRAPHICS] are studied.

New Derivation of Ordinary Differential Equations for Transient Free-Surface Green Functions

Based on the Laplace transform, a direct derivation of the ordinary differential equations for the three-dimensional transient free-surface Green function in marine hydrodynamics is presented. The results for the 3D Green function and all its spatial derivatives are a set of fourth-order ordinary differential equations, which are identical with that of Clement (1998). All of these results may be used to accelerate numerical computation for the time-domain boundary element method in marine hydrodynamics.

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)))g(x′(t))=0are obtained.

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.

OSCILLATION THEOREMS FOR SECOND ORDER QUASILINEAR PERTURBED DIFFERENTIAL EQUATIONS

New oscillation criteria for the second order perturbed differential equation are presented. The special case of the results includes the corresponding results in previous papers, extends and unifies a number of known results.

NEW OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR MATRIX DIFFERENTIAL EQUATIONS

Some new oscillation criteria are established for the second-order matrix differential system （r（t）Z＇（t））＇ ＋ p（t）Z＇（t） ＋ Q（t）F（Z＇（t））G（Z（t）） = 0, t ≥ to 〉 0, are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [to,∞）, rather than on the whole half-line. The results weaken the condition of Q（t） and generalize some well-known results of Wong （1999） to nonlinear matrix differential equation.

ON THE SINGULAR PERTURBATION OF A NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH TWO PARAMETERS

In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.

OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS

In this paper,some sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formx′(t)+p(t)x(t-τ)=0are established,which improve and generalize some of the known results in the literature.

ON ENTIRE SOLUTIONS OF TWO TYPES OF SYSTEMS OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.

Oscillation for a Class of Fractional Differential Equation

We consider the oscillation of a class fractional differential equation with Robin and Dirichlet boundary conditions. By generalized Riccati transformation technique and the differential inequality method, oscillation criteria for a class of nonlinear fractional differential equation are obtained.