We generalize the famous Tarski result by showing that: if X is a complete lattice, and $f:X\to X$ is an increasing and continuous mapping, then for all points $x^0\in X$, the limits of sequences ${(f^n(\lim {\sup }_k{f}^k(x^0)))}_{n=1}^{\infty }$ and ${(f^n(\lim {\inf }_k{f}^k(x^0)))}_{n=1}^{\infty }$ are fixed points of f. These limits are the tight fixed-point bounds between which sufficiently large iterations $f^k(x^0)$ are located. We provide an application of this result to studying best-response dynamics.

中文翻译：

---

关于完整格上递增和连续映射的迭代序列

通过证明以下内容，我们可以概括出著名的Tarski结果：如果X是一个完整的晶格，并且$F：X\to X$ 是一个不断增加的连续映射，那么对于所有点 $X^0\in X$，序列的限制 ${（F^{ñ}（\mathrm{林}{\mathrm{SUP}}_{ķ}{F}^{ķ}（X^0）））}_{ñ=\mathrm{1个}}^{\infty }$ 和 ${（F^{ñ}（\mathrm{林}{\mathrm{信息}}_{ķ}{F}^{ķ}（X^0）））}_{ñ=\mathrm{1个}}^{\infty }$是f的不动点。这些限制是严格的定点范围，在它们之间有足够大的迭代次数$F^{ķ}（X^0）$位于。我们将此结果应用于研究最佳响应动力学。