AI 中文总结
该文扩展拟一致空间的$U$-不动点与$U$-起点框架，通过滤子公理重构存在性框架，给出双边收缩条件，证明Banach型不动点定理并推广至多值映射等场景，还去除了对应不动点定理的$T_0$假设。
AI 中文摘要
我们扩展了拟一致空间的$U$-不动点与$U$-起点框架，通过滤子公理重新构建原存在性框架，并针对拟伪度量生成族给出双边收缩条件。针对该条件，我们在$T_0$双完备拟一致空间上证明了Banach型不动点定理，得到Picard型误差界，还推导了Boyd–Wong伴随结果与稳定性估计。共轭拟一致性导出了终点形式，选择器论证将结果推广到多值映射与交换族的公共起点，Knaster–Tarski变体覆盖序完备空间上的单调选择器，直接拟Hausdorff路径处理弱收缩多值映射。此外，$T_0$商将对应的基于对合的不动点定理中的$T_0$假设去除。
英文摘要
We extend the $\U$-fixed-point and $\U$-startpoint framework for quasi-uniform spaces. We reformulate the original existence framework through the filter axioms and supply a two-sided contraction condition on a generating family of quasi-pseudometrics. For this condition we prove a Banach-style fixed-point theorem on $T_0$ bicomplete quasi-uniform spaces, with a Picard-type error bound; a Boyd--Wong companion and a stability estimate follow. The conjugate quasi-uniformity yields an endpoint version, a selector argument extends the result to multivalued maps and to common startpoints of commuting families, a Knaster--Tarski variant covers monotone selectors on order-complete spaces, and a direct quasi-Hausdorff route handles weakly contractive multivalued maps. Moreover, the $T_0$-quotient removes the $T_0$ assumption from the corresponding involution-based fixed-point theorem.
