Common fixed point problem

<p>In mathematics, the common fixed point problem is the conjecture that, for any two <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> that map the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> into itself and commute under <a href="/facts/Function_composition/r5Pvnmth">functional composition</a>, there must be a point that is a <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> of both functions. In other words, if the functions 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and 
  
    
      
        g
      
    
    {\displaystyle g}
  
 are continuous, and 
  
    
      
        f
        (
        g
        (
        x
        )
        )
        =
        g
        (
        f
        (
        x
        )
        )
      
    
    {\displaystyle f(g(x))=g(f(x))}
  
 for all 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in the unit interval, then there must be some 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in the unit interval for which 
  
    
      
        f
        (
        x
        )
        =
        x
        =
        g
        (
        x
        )
      
    
    {\displaystyle f(x)=x=g(x)}
  
. 
</p><p>First posed in 1954, the problem remained unsolved for more than a decade, during which several mathematicians made incremental progress toward an affirmative answer. In 1967, William M. Boyce and John P. Huneke independently: 3  proved the conjecture to be false by providing examples of commuting functions on a closed interval that do not have a common fixed point.
</p>

Common fixed point problem open-in-new

Common fixed point problem