Schauder fixed-point theorem

The Schauder fixed-point theorem is an extension of the <a href="/facts/Brouwer_fixed-point_theorem/xP2p03lQ">Brouwer fixed-point theorem</a> to <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a>, which may be of infinite dimension. It asserts that if 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a nonempty <a href="/facts/Convex_set/vdAuJRJl">convex</a> closed subset of a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> topological vector space 
 
 
 
 V
 
 
 {\displaystyle V}
 
 and 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a continuous mapping of 
 
 
 
 K
 
 
 {\displaystyle K}
 
 into itself such that 
 
 
 
 f
 (
 K
 )
 
 
 {\displaystyle f(K)}
 
 is contained in a <a href="/facts/Compact_set/d0cgXJH7">compact</a> subset of 
 
 
 
 K
 
 
 {\displaystyle K}
 
, then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has a <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a>.
A consequence, called <a href="/facts/Helmut_Schaefer/xorHTDBE">Schaefer</a>'s fixed-point theorem, is particularly useful for proving existence of solutions to <a href="/facts/Nonlinear/4aYItALC">nonlinear</a> <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a>.
Schaefer's theorem is in fact a special case of the far reaching Leray–Schauder theorem which was proved earlier by <a href="/facts/Juliusz_Schauder/VSpCQfY1">Juliusz Schauder</a> and <a href="/facts/Jean_Leray/MyAEDyc7">Jean Leray</a>.
The statement is as follows:
Let 
 
 
 
 f
 
 
 {\displaystyle f}
 
 be a continuous and compact mapping of a Banach space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 into itself, such that the set

{
        x
        ∈
        X
        :
        x
        =
        λ
        f
        (
        x
        )
        
          
             for some 
          
        
        0
        ≤
        λ
        ≤
        1
        }
      
    
    {\displaystyle \{x\in X:x=\lambda f(x){\mbox{ for some }}0\leq \lambda \leq 1\}}

is bounded. Then 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has a fixed point. (A compact mapping in this context is one for which the image of every bounded set is <a href="/facts/Relatively_compact/z5uIqMDp">relatively compact</a>.)

Schauder fixed-point theorem open-in-new

Schauder fixed-point theorem