A Note on the Inner-Outer Iterative Method for Solving the Linear Equation Ax = b

Recently, Tian et al. [Computers and Mathematics with Applications, 75(2018): 2710-2722] came up with the inner-outer iterative method to solve the linear equation Ax=b and studied the corresponding convergence of the method. In this paper, we improve the main results of the inner-outer method and get weaker convergence results. Moreover, the parameters can be adjusted suitably so that the convergence property of the method can be substantially improved.

GLOBAL EXISTENCE AND FINITE TIME BLOW-UP OF SOLUTIONS TO A NONLOCAL P-LAPLACE EQUATION

In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L 2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].

The optimal convergence factor of the gradient based iterative algorithm for linear matrix equations

A hierarchical gradient based iterative algorithm of [L. Xie et al., Computers and Mathematics with Applications 58 (2009) 1441-1448] has been presented for finding the numerical solution for general linear matrix equations, and the convergent factor has been discussed by numerical experiments. However, they pointed out that how to choose a best convergence factor is still a project to be studied. In this paper, we discussed the optimal convergent factor for the gradient based iterative algorithm and obtained the optimal convergent factor. Moreover, the theoretical results of this paper can be extended to other methods of gradient-type based. Results of numerical experiments are consistent with the theoretical findings.

A new oscillation criterion for two-dimensional dynamic systems on time scales

Consider the linear dynamic system on time scales\begin{equation}u^\Delta=pv, \quad\quad v^\Delta=-qu^\sigma\end{equation}where $p>0$ and $q$ are rd-continuous functions on a time scale $\mathbb T$ such that $\sup\mathbb T=\infty$. When $p(t)$ is allowed to take on negative values, we establish an oscillation criterion for system (0.1). Our result improves a main result of Fu and Lin [S. C. Fu and M. L. Lin, Oscillation and nonoscillation criteria for linear dynamic systems on time scales, Computers and Mathematics with Applications, 59(2010), 2552-2565].