Fixed-point theorems in infinite-dimensional spaces: Difference between revisions

Revision as of 08:50, 7 May 2025

Revision as of 08:51, 7 May 2025

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<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.

<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>