Fixed-point theorems in infinite-dimensional spaces

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a number of <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed-point</a> theorems in infinite-dimensional spaces generalise the <a href="/facts/Brouwer_fixed-point_theorem/xP2p03lQ">Brouwer fixed-point theorem</a>. They have applications, for example, to the proof of <a href="/facts/Existence_theorem/Au8x1FSt">existence theorems</a> for <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>. 
The first result in the field was the <a href="/facts/Schauder_fixed-point_theorem/vfgwV8Ab">Schauder fixed-point theorem</a>, proved in 1930 by <a href="/facts/Juliusz_Schauder/VSpCQfY1">Juliusz Schauder</a> (a previous result in a different vein, the <a href="/facts/Banach_fixed-point_theorem/TRygeoCP">Banach fixed-point theorem</a> for <a href="/facts/Contraction_mapping/VSu4qNqa">contraction mappings</a> in complete <a href="/facts/Metric_spaces/gnBBRXtc">metric spaces</a> was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, first proved for finite <a href="/facts/Simplicial_complex/fF2282CF">simplicial complexes</a>, to spaces of infinite dimension. For example, the research of <a href="/facts/Jean_Leray/MyAEDyc7">Jean Leray</a> who founded <a href="/facts/Sheaf_theory/MM3hcRw4">sheaf theory</a> came out of efforts to extend Schauder's work.

<blockquote><a href="/facts/Schauder_fixed-point_theorem/vfgwV8Ab">Schauder fixed-point theorem</a>: Let C be a <a href="/facts/Nonempty/ffEk3eA6">nonempty</a> <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Convex_set/vdAuJRJl">convex</a> subset of a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> V. If f : C → C is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> with a <a href="/facts/Compact_set/d0cgXJH7">compact</a> image, then f has a fixed point.</blockquote>
<blockquote>Tikhonov (Tychonoff) fixed-point theorem: Let V be a <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex topological vector space</a>. For any nonempty compact convex set X in V, any continuous function f : X → X has a fixed point.</blockquote>
<blockquote>Browder fixed-point theorem: Let K be a nonempty closed bounded convex set in a <a href="/facts/Uniformly_convex_Banach_space/H9gGXdy7">uniformly convex Banach space</a>. Then any non-expansive function f : K → K has a fixed point. (A function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is called non-expansive if 
 
 
 
 ‖
 f
 (
 x
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 −
 f
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 ‖
 ≤
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 x
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 {\displaystyle \|f(x)-f(y)\|\leq \|x-y\|}
 
 for each 
 
 
 
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 {\displaystyle x}
 
 and 
 
 
 
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.)</blockquote>
Other results include the <a href="/facts/Markov%25E2%2580%2593Kakutani_fixed-point_theorem/wAha0RPr">Markov–Kakutani fixed-point theorem</a> (1936-1938) and the <a href="/facts/Ryll-Nardzewski_fixed-point_theorem/DCP0rVo9">Ryll-Nardzewski fixed-point theorem</a> (1967) for continuous affine self-mappings of compact convex sets, as well as the <a href="/facts/Earle%25E2%2580%2593Hamilton_fixed-point_theorem/V1yRgsM4">Earle–Hamilton fixed-point theorem</a> (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><a href="/facts/Kakutani_fixed-point_theorem/qMQNEjzU">Kakutani fixed-point theorem</a>: 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>

Fixed-point theorems in infinite-dimensional spaces open-in-new

Fixed-point theorems in infinite-dimensional spaces