State and prove the banach contraction mapping priciple?

Answer 1

The Banach Fixed-Point Theorem, or contraction mapping principle, states that if ( (X, d) ) is a complete metric space and ( T: X \to X ) is a contraction mapping (i.e., there exists a constant ( 0 < k < 1 ) such that ( d(T(x), T(y)) \leq k \cdot d(x, y) ) for all ( x, y \in X )), then ( T ) has a unique fixed point in ( X ).

Proof: Consider an arbitrary point ( x_0 \in X ) and construct the sequence ( x_{n+1} = T(x_n) ). By the contraction property, we have ( d(x_{n+1}, x_n) \leq k \cdot d(x_n, x_{n-1}) ), which implies ( d(x_n, x_m) \leq \frac{k^{m-n}}{1-k} d(x_0, x_1) ) for ( m > n ). Since ( k < 1 ), this shows that ( (x_n) ) is a Cauchy sequence, and because ( X ) is complete, it converges to a limit ( x^* ). Finally, ( x^* = T(x^) ) shows that ( x^ ) is a fixed point, and the uniqueness follows from the contraction property.