基于偏差泛函的Banach型不动点定理
A Banach-Type Fixed Point Theorem via Discrepancy Functionals
AI总结：
本文建立了基于偏差泛函的Banach型不动点定理，该定理以非负泛函收缩替代度量收缩，可推导多种特例，证明对称性与三角不等式非必要，为非度量收缩量提供通用框架。
AI中文摘要：
本文针对收缩一般非负泛函而非度量的映射，建立了Banach型不动点定理。主要假设为该泛函通过简单下界支配基础度量，且在极限处满足相容性条件。所得结果提供了一个通用框架，可将扰动度量空间中的近期不动点定理及多项式收缩定理作为特例推导出来，同时表明对称性和三角不等式并非收缩论证的必要条件。因此，收缩量可被解释为一般偏差或散度度量而非度量。该结果强调，Banach收缩原理背后的关键要素并非度量结构本身，而是收缩泛函控制基础空间几何的能力。这一视角为获得广泛非度量收缩量类的不动点结果提供了简单灵活的框架。
英文摘要：
In this paper we establish a Banach-type fixed point theorem for mappings that contract a general nonnegative functional rather than a metric. The principal assumption is that the functional dominates the underlying metric through a simple lower bound, together with a compatibility condition at limits. Our result provides a common framework that recovers recent fixed point theorems in perturbed metric spaces and for polynomial contractions as particular cases, while showing that neither symmetry nor the triangle inequality is essential for the contraction argument. Consequently, the contractive quantity may be interpreted as a general discrepancy or divergence measure rather than a metric. The results highlight that the essential ingredient behind Banach's contraction principle is not the metric structure itself, but the ability of the contracted functional to control the geometry of the underlying space. This perspective provides a simple and flexible framework for obtaining fixed point results for broad classes of non-metrical contractive quantities.
