格上的迭代与迭代方程
Iteration and iterative equation on lattices
摘要：
本文研究了格上映射的迭代及相应的类多项式迭代方程。由于格不需要具有度量空间结构，因此Schauder不动点定理和Banach不动点定理都不成立。利用Tarski的不动点定理，证明了Riesz空间的凸完备子格上保序解的存在性.进一步，在Riesz空间的特殊情形$\mathbb{R}^n$和$\mathbb{R}$中，我们分别讨论了上半连续解和可积解.最后，我们给出了Riesz空间的更多特殊情况，以供讨论迭代方程.
英文摘要：
In this paper we investigate iteration of maps on lattices and the corresponding polynomial-like iterative equation. Since a lattice need not have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point theorem is available. Using Tarski's fixed point theorem, we prove the existence of order-preserving solutions on convex complete sublattices of Riesz spaces. Further, in $\mathbb{R}^n$ and $\mathbb{R}$, special cases of Riesz space, we discuss upper semi-continuous solutions and integrable solutions respectively. Finally, we indicate more special cases of Riesz space for discussion on the iterative equation.
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