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

Author:
W. A. Horn

Journal:
Trans. Amer. Math. Soc. **149** (1970), 391-404

MSC:
Primary 47.85

DOI:
https://doi.org/10.1090/S0002-9947-1970-0267432-1

MathSciNet review:
0267432

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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.

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Keywords:
Banach space,
fixed points,
asymptotic fixed point theorems,
compact mappings,
flows,
nonejective fixed points,
nonrepulsive fixed points

Article copyright:
© Copyright 1970
American Mathematical Society