Contraction criterion on a closed norm ball

Statement

Let $X$ be a Banach space, $r>0$ , and $B=\{x:\|x-x_0\|\le r\}$ . Suppose $T:B\to X$ satisfies

\|T(x)-T(y)\|\le q\|x-y\|,\qquad q<1, \quad \|T(x_0)-x_0\|\le(1-q)r.

Then $T(B)\subseteq B$ , since $\|T(x)-x_0\|\le qr+(1-q)r$ . The ball is complete, so the contraction theorem gives a unique fixed point in it.

Small-parameter form

For $T(x)=x_0+\varepsilon N(x)$ , it suffices that $|\varepsilon|\sup_B\|N\|\le r$ and $|\varepsilon|\operatorname{Lip}_B N<1$ . A contraction estimate without the self-map condition is insufficient to apply the theorem on the ball.