AI 中文总结
本文引入弱于经典性质的集值映射方向正则性概念，建立基于方向极小时间函数的分类框架，推导不动点定理并应用于重合点等问题，提供统一方向视角并推广经典结果。
AI 中文摘要
受集值映射度量正则性理论发展的启发，本文引入了若干弱于对应经典正则性性质的方向正则性概念。我们研究了这些概念之间的关系，并基于方向极小时间函数建立了一个分类框架。特别关注方向轨道正则性和方向轨道Aubin连续性，它们被证明为不动点现象的分析提供了自然的框架。利用这些概念，我们在方向假设下推导了集值映射的近似和精确不动点定理，所得结果用极小时间函数和轨道逼近过程表示，允许各向异性和非对称行为，而这些行为是经典非方向框架无法捕捉的。我们进一步将发展的理论应用于方向重合点结果、耦合不动点定理以及Milyutin型扰动稳定性性质。所提出的方法为不动点理论和变分分析提供了统一的方向视角，特别是，它将文献中的若干已知结果作为特殊情况恢复，并在比经典压缩或全局正则性条件弱得多的假设下得到新的推广。
英文摘要
Motivated by developments in metric regularity theory for set-valued mappings, this paper introduces several directional regularity notions that are weaker than the corresponding classical regularity properties. We investigate the relationships among these concepts and establish a classification framework based on directional minimal time functions. Particular attention is devoted to directional orbital regularity and directional orbital Aubin continuity, which are shown to provide a natural setting for the analysis of fixed point phenomena. Using these notions, we derive approximate and exact fixed point theorems for set-valued mappings under directional assumptions. The obtained results are expressed in terms of minimal time functions and orbital approximation procedures, allowing for anisotropic and asymmetric behaviors that are not captured by the classical nondirectional framework. We further apply the developed theory to directional coincidence point results, coupled fixed point theorems, and Milyutin-type perturbation stability properties. The proposed approach provides a unified directional perspective on fixed point theory and variational analysis. In particular, it recovers several known results from the literature as special cases and yields new extensions under substantially weaker assumptions than classical contraction or global regularity conditions.
