Contraction mapping theorem

• Definition
  contraction mapping
  • Let $(X,d)$ be a complete metric space. Then a map $T : X \to X$ is called a contraction mapping on $X$ if there exists $c \in [0,1)$ such that $$ d(T(x),T(y)) \leq c ~ d(x,y) $$ for all $x,y \in X.$
• Definition
  fixed point
  • If $T : X \to X$, then a point $x \in X$ such that $$ T(x) = x $$ is called a fixed point of $T.$
• Theorem
  Contraction mapping theorem (also known as the Banach fixed-point theorem)
  
  • If $T : X \to X$ is a contraction mapping on a complete metric space $(X, d)$, then $T$ admits a unique fixed point $x^* \in X$ (i.e. $T(x^*) = x^*$).