CAL1_L10

The Banach contraction principle

Definition 1(Definition).

A point $x\in X$ is called a fixed point of the mapping $T:X\to X$ if $T(x)=x$.

Definition 2(Definition).

A mapping $T$ from a metric space $(X,\rho )$ into itself is said to be Lipschitz if there exists $c\ge 0$, called a Lipschitz constant for the mapping, for which

$$\rho (T(u),T(v))\le c\rho (u,v)\forall u,v\in X.$$

If $c<1$, the Lipschitz mapping is called a contraction.

Theorem 3(Banach contraction principle).

Let $X$ be a complete metric space and the mapping $T:X\to X$ be a contraction. Then $T$ has exactly one fixed point.