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

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{{Short description|Theorems generalizing the Brouwer fixed-point theorem}}

In [[mathematics]], a number of '''[[fixed point (mathematics)|fixed -point]] theorems in infinite-dimensional spaces''' generalise the [[Brouwer fixed -point theorem]]. They have applications, for example, to the proof of [[existence theorem]]s for [[partial differential equation]]s. The research of [[Jean Leray]] that proved influential for [[algebraic topology]] and [[sheaf theory]] was motivated by the need to go beyond the '''Schauder fixed point theorem''', proved in 1930 by [[Julius Schauder]].

The first result in the field was the '''[[Schauder fixed-point theorem]]''', proved in 1930 by [[Juliusz Schauder]] (a previous result in a different vein, the [[Banach fixed-point theorem]] for [[Contraction mapping|contraction mappings]] in complete [[metric spaces]] 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 [[algebraic topology]], first proved for finite [[simplicial complex]]es, to spaces of infinite dimension. For example, the research of [[Jean Leray]] who founded [[sheaf theory]] came out of efforts to extend Schauder's work.

The '''Schauder fixed point theorem''' states, in one version, that if ''C'' is a [[nonempty]] [[closed set|closed]] [[convex]] subset of a [[Banach space]] ''V'' and ''f'' is a continuous map from ''C'' to ''C'' whose image is [[compact|countably compact]], then ''f'' has a fixed point.

The '''Tikhonov (Tychohoff) fixed point theorem''' is now applied to any [[locally convex topological vector space]] ''V''. For any non-empty [[compact]] convex set ''X'' in ''V'', and [[continuous function]]

The <blockquote>'''[[Schauder fixed -point theorem]]:''' states, in one version, that ifLet ''C'' isbe a [[nonempty]] [[closedClosed set|closed]] [[Convex set|convex]] subset of a [[Banach space]] ''V''. andIf ''f'' is a continuous map from: ''C'' to→ ''C'' whoseis image[[continuous isfunction|continuous]] with a [[compact|countably set|compact]] image, then ''f'' has a fixed point.</blockquote>

:''f'':''X'' &rarr; ''X'',

The <blockquote>'''Tikhonov (TychohoffTychonoff) fixed -point theorem:''' isLet now''V'' appliedbe to anya [[locally convex topological vector space]] ''V''. For any non-emptynonempty [[compact]] convex set ''X'' in ''V'', andany [[continuous function]] ''f'' : ''X'' → ''X'' has a fixed point.</blockquote>

there is a fixed point for ''f''.

<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 are the Kakutani and Markov fixed point theorems, now subsumed in the Ryll-Nardzewski fixed point theorem (1967).

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>

==See also==

* [[Topological degree theory]]

==References==

* Vasile I. Istratescu, ''Fixed Point Theory, An Introduction'', D.Reidel, Holland (1981). {{isbn|90-277-1224-7}}.

* Andrzej Granas and [[James Dugundji]], ''Fixed Point Theory'' (2003) Springer-Verlag, New York, {{isbn|0-387-00173-5}}.

* William A. Kirk and [[Brailey Sims]], ''Handbook of Metric Fixed Point Theory'' (2001), Kluwer Academic, London {{isbn|0-7923-7073-2}}.

* Samuel A. Aniki and Kamilu Rauf, ''Some stability results in partially ordered metric spaces for coupled fixed point iteration of procedures for mixed monotone mappings'' (2019), Islamic University Multidisciplinary Journal, 6(3), 175-186 https://www.iuiu.ac.ug/journaladmin/iumj/ArticleFiles/49305.pdf

==External links==

* {{PlanetMath|TychonoffFixedPointTheorem|Tychonoff Fixed Point Theorem}}

[[Category:Fixed-point theorems]]

[[fr:Théorème du point fixe de Schauder]]