Some fixed point theorems for compact maps and flows in Banach spaces.

Abstract:

Let ${S_0} \subset {S_1} \subset {S_2}$ be convex subsets of the Banach space X, with ${S_0}$ and ${S_2}$ closed and ${S_1}$ open in ${S_2}$. If f is a compact mapping of ${S_2}$ into X such that $\cup _{j = 1}^m{f^j}({S_1}) \subset {S_2}$ and ${f^m}({S_1}) \cup {f^{m + 1}}({S_1}) \subset {S_0}$ for some $m > 0$, then f has a fixed point in ${S_0}$. (This extends a result of F. E. Browder published in 1959.) Also, if $\{ {T_t}:t \in {R^ + }\}$ is a continuous flow on the Banach space X, ${S_0} \subset {S_1} \subset {S_2}$ are convex subsets of X with ${S_0}$ and ${S_2}$ compact and ${S_1}$ open in ${S_2}$, and ${T_{{t_0}}}({S_1}) \subset {S_0}$ for some ${t_0} > 0$, where ${T_t}({S_1}) \subset {S_2}$ for all $t \leqq {t_0}$, then there exists ${x_0} \in {S_0}$ such that ${T_t}({x_0}) = {x_0}$ for all $t \geqq 0$. Minor extensions of Browder’s work on “nonejective” and “nonrepulsive” fixed points are also given, with similar results for flows.