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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0267432-1
PII: S 0002-9947(1970)0267432-1
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