论文标题
非线性perron-frobenius理论中迭代的融合
Convergence of iterates in nonlinear Perron-Frobenius theory
论文作者
论文摘要
让$ c $是一个封闭的圆锥体，在Banach空间中具有非空内部$ C^\ Circ $。令$ f：c^\ circ \ rightarrow c^\ circ $为订单的订单下均匀函数，$ c^\ circ $中的固定点。我们引入了一个条件，该条件可以确保迭代$ f^k（x）$收敛到c^\ circ $中所有$ x \的固定点。此条件概括了$ \ Mathbb {r}^n _ {> 0} $上地图的类型k订单保留的概念。我们还证明，当迭代收敛到固定点时，在两种特殊情况下，收敛速率始终是R线性：用于分段仿射图，以及用于订购，均匀的，均匀的，分析性的，多重性凸的功能上的$ \ Mathbb {r}^n _ {> 0} $。以后的类别包括与非负张量的均匀特征值问题相关的地图。
Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that the iterates $f^k(x)$ converge to a fixed point for all $x \in C^\circ$. This condition generalizes the notion of type K order-preserving for maps on $\mathbb{R}^n_{>0}$. We also prove that when iterates converge to a fixed point, the rate of convergence is always R-linear in two special cases: for piecewise affine maps and also for order-preserving, homogeneous, analytic, multiplicatively convex functions on $\mathbb{R}^n_{>0}$. This later category includes the maps associated with the homogeneous eigenvalue problem for nonnegative tensors.
