On a consequence of the Schauder theorem (Q2343951)

The authors prove the following easy consequence of the Schauder fixed point theorem: Let \(E\) be a Banach space and \(A\subset E\) a closed bounded convex set with nonempty interior. Let \(f:A\to E\) be continuous and assume that \((I-f)(A)\) is relatively compact and such that here is an interior point \(y_0\) of \(A\) such that \(\|f(x)-x\|\leq\|x-y_0\|\) whenever \(x\in \partial A\). Then there is an \(x_0\in A\) with \(f(x_0)=y_0\).