Ngô Quốc Anh

December 29, 2008

A fixed point theorem

Filed under: Các Bài Tập Nhỏ, Nghiên Cứu Khoa Học — Ngô Quốc Anh @ 17:06

Let (X,d) be a complete metric space and let F: X\to X be such that F^N : X \to X is a contraction for some positive integer N. Show that F has a unique fixed point u \in X and that for each x \in X, \mathop {\lim }\limits_{n \to \infty } {F^n}\left( x \right) = u.

Proof.
Since F^N : X \to X is a contraction, then F^N has a fixed point, say u_0, i.e., F^N u_0=u_0. Note that

\displaystyle d\left( {F{u_0},{u_0}} \right) = d\left( {{{\left( {{F^N}} \right)}^n}F{u_0},{{\left( {{F^N}} \right)}^n}{u_0}} \right) \leq {k^n}d\left( {F{u_0},{u_0}} \right).

Since k<1 then d\left( {F{u_0},{u_0}} \right)=0. In other word, u_0 is a fixed point of F.

To prove the uniqueness, assume $u’_0$ is also a fixed point of F. Then both $u_0$ and $u’_0$ are fixed points of F^N which implies that u_0 \equiv u'_0 due to the uniqueness of fixed point of a contractive mapping.

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