Caristi fixed-point theorem

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the <a href="/facts/Banach_fixed-point_theorem/TRygeoCP">Banach fixed-point theorem</a> for maps of a <a href="/facts/Complete_metric_space/lsckmU8G">complete</a> <a href="/facts/Metric_space/gnBBRXtc">metric space</a> into itself. Caristi's <a href="/facts/Fixed-point_theorem/Jvkz4SUu">fixed-point theorem</a> modifies the 
  
    
      
        ε
      
    
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-<a href="/facts/Ekeland%27s_variational_principle/tnfEAbXk">variational principle of</a> <a href="/facts/Ivar_Ekeland/vLhPdySg">Ekeland</a> (1974, 1979). The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). 
The original result is due to the mathematicians James Caristi and <a href="/facts/William_Arthur_Kirk/HoblQ8h4">William Arthur Kirk</a>.
</p><p>Caristi fixed-point theorem can be applied to derive other classical fixed-point results, and also to prove the existence of bounded solutions of a <a href="/facts/Functional_equation/4ovU4CAt">functional equation</a>.
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Caristi fixed-point theorem open-in-new

Caristi fixed-point theorem