The <Leray-Schauder fixed point theorem> says that if $X$ is a Banach space and $T:X\to X$ is continuous and compact, and the set
$$
\{u\in X:u=tT(u)\text{ for some }0\leq t\leq1\}
$$
is bounded, then $T$ has a fixed point.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 5 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-107/5/b/ii/solution