Content deleted Content added
No edit summary |
Added a categorical variation on fixed-point failure in composite functors. Tag: Reverted |
||
Line 10:
<blockquote>'''Browder fixed-point theorem:''' Let ''K'' be a nonempty closed bounded convex set in a [[uniformly convex Banach space]]. Then any non-expansive function ''f'' : ''K'' → ''K'' has a fixed point. (A function <math>f</math> is called non-expansive if <math>\|f(x)-f(y)\|\leq \|x-y\| </math> for each <math>x</math> and <math>y</math>.)</blockquote>
Other results include the [[Markov–Kakutani fixed-point theorem]] (1936-1938) and the [[Ryll-Nardzewski fixed-point theorem]] (1967) for continuous affine self-mappings of compact convex sets, as well as the [[Earle–Hamilton fixed-point theorem]] (1968) for holomorphic self-mappings of open domains. Also, Aniki & Rauf (2019) presented some interesting results on the stability of partially ordered metric spaces for coupled fixed point iteration procedures for mixed monotone mappings. A formulation referred to as [[Algebraic fixed-point traps]] addresses fixed-point obstructions in the presence of entropy-reducing collapse morphisms, where composite functors <math>F = E \circ \phi</math> with generative <math>\phi</math> and collapse functor <math>E</math> admit no object <math>\mu_F</math> such that <math>\mu_F \cong F(\mu_F)</math>. Thus, <math>\mu_F \nexists</math> and the symbolic identity <math>\phi^\infty</math> cannot emerge. This breakdown contrasts with conditions like those in the Kakutani fixed-point theorem, where convexity and upper semi-continuity guarantee existence.
<blockquote>'''[[Kakutani fixed-point theorem]]:''' Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.</blockquote>
| |||