一类具有偏差变元的二阶泛函微分方程周期解

利用重合度理论,研究一类具有偏差变元的二阶微分方程x″+f(t,x′(t))+g(t,x(t-τ(t)))=p(t)的周期解的存在性问题.其中,f,g∈C(R×R,R),且对任意的x∈R,g(t+ω,x)=g(t,x),p∈C(R,R),τ∈C(R,R)是ω-周期的.在不要求对所有的y∈R,函数f(t,y)≤0(f(t,y)≥0),t∈R的情况下,得到该类方程至少存在一个ω-周期解的充分条件.     

具偏差变元的p—Laplacian方程的周期解问题

利用重合度拓展理论研究了一类具偏差变元的p-Laplacian方程（φp（x″（t））″）＋f（x″（t））＋g（t,x（t-τ（t））=e（t）的周期解问题,得到了其解的存在性．     

一类三阶非线性中立型时滞差分方程的振动性

运用分析的方法研究一类具有连续变量的三阶非线性中立型时滞差分方程的振动性,给出该类方程所有有界解振动和方程振动的几个充分条件,推广并改进已有结论.     

Convergence analysis of a fast second-order time-stepping numerical method for two-dimensional nonlinear time–space fractional Schrödinger equation

In this article, we consider the two-dimensional nonlinear time–space fractional Schrödinger equation with space described by the fractional Laplacian. A second-order fractional backward difference formula in the temporal direction while Fourier spectral method in the spatial direction is proposed to solve the model numerically. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. By using the generalized discrete Gronwall inequality developed by Dixon and McKee and the temporal–spatial error splitting argument, the convergence of the fast time-stepping numerical method is also proved in a simple manner without imposing the Courant-Friedrichs-Lewy (CFL) condition. Finally, some numerical results are provided to support the theoretical analysis.