Let G be a piecewise constant $n\times n$ matrix function which is defined on a smooth closed curve $Γ$ in the complex sphere and which has m jumps. We consider the problem of determining the partial indices of the factorization of the matrix function G in the space $L^p(Γ)$. We show that this problem can be reduced to a certain problem for systems of linear differential equations.
Studying this related problem, we obtain some results for the partial indices for general n and m. A complete answer is given for n=2, m=4 and for n=m=3. One has to distinguish several cases. In some of these cases, the partial indices can be determined explicitly. In the remaining cases, one is led to two possibilities for the partial indices. The problem of deciding which is the correct possibility is equivalent to the description of the monodromy of n-th order linear Fuchsian differential equations with m singular points.
Levinson Theorem for Differential Equations with Piecewise Constant Argument Generalized
在这项工作中，它提出了一个适应的渐近定理N。Levinson（1948）提出的分段常变元微分方程，推广到了M. 2007年的艾哈迈德。为了简单起见，不失一般性，考虑了参数延迟的情况。了N. Levinson定理是M. S. P. Eastham在他的工作，这是目前在这个书目。更相关的假设，这一定理的突出，它是建立一个版本的这个定理与这些假设的常微分方程。这种形式是推广到具有分段常数变元的微分方程的形式。这种适应性是通过利用压缩算子构造的Banach不动点来证明的。
Limit cycles by perturbing quadratic isochronous centers inside piecewise smooth polynomial differential systems
本文研究了分段光滑多项式微分系统的一阶Melnikov函数的零点个数，估计了在具有间断直线$x=0$的$n$次分段光滑多项式微分系统类内，由二次等距中心的周期环分支出的极限环的个数.给出了一阶Melnikov函数关于二次等距中心$S_1, S_2$和$S_3$的零点个数的一个精确上界.对于二次等距中心$S_4$，由于一阶Melnikov函数的复杂性，我们给出了其零点个数的一个粗略估计.进一步，当$S_4$在所有$n$次光滑多项式微分系统内扰动时，我们将$S_4$的上界从$14n+11$ in \cite{LLLZ}，$12n-1$ in \cite{SZ}改进为$[(5n-5)/2]$.此外，还得到了分段光滑多项式微分系统的一阶Melnikov函数与一阶平均函数等价的一些证据。
