Some fixed point theorems for compact maps and flows in Banach spaces.
Author:
W. A. Horn
Journal:
Trans. Amer. Math. Soc. 149 (1970), 391404
MSC:
Primary 47.85
MathSciNet review:
0267432
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Abstract: Let be convex subsets of the Banach space X, with and closed and open in . If f is a compact mapping of into X such that and for some , then f has a fixed point in . (This extends a result of F. E. Browder published in 1959.) Also, if is a continuous flow on the Banach space X, are convex subsets of X with and compact and open in , and for some , where for all , then there exists such that for all . Minor extensions of Browder's work on ``nonejective'' and ``nonrepulsive'' fixed points are also given, with similar results for flows.
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 F. E. Browder, On a generalization of the Schauder fixed point theorem, Duke Math. J. 26 (1959), 291303. MR 21 #4368. MR 0105629 (21:4368)
 [2]
 , Another generalization of the Schauder fixed point theorm, Duke Math. J. 32 (1965), 399406. MR 0203718 (34:3567)
 [3]
 , A further generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965), 575578. MR 0203719 (34:3568)
 [4]
 J. Cronin, Fixed points and topological degree in nonlinear analysis, Math. Surveys, no. 11, Amer. Math. Soc., Providence, R. I., 1964. MR 29 #1400. MR 0164101 (29:1400)
 [5]
 J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353367. MR 13, 373. MR 0044116 (13:373c)
 [6]
 A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962). MR 26 #6743. MR 0149253 (26:6743)
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 A. Halanaĭ, Asymptotic stability and small perturbations of periodic systems of differential equations with retarded arguments, Uspehi Mat. Nauk 17 (1962), no. 1 (103), 231233. (Russian) MR 25 #301. MR 0136840 (25:301)
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 W. A. Horn, A generalization of Browder's fixed point theorem, Abstract #61130, Notices Amer. Math. Soc. 11 (1964), 325.
 [9]
 G. S. Jones, The existence of periodic solutions of , J. Math. Anal. Appl. 5 (1962), 435450. MR 25 #5234. MR 0141837 (25:5234)
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 , Asymptotic fixed point theorems and periodic systems of functionaldifferential equations, Contributions to Differential Equations 2 (1963), 385405. MR 28 #1361. MR 0158135 (28:1361)
 [11]
 , Periodic motions in Banach space and applications to functionaldifferential equations, Contributions to Differential Equations 3 (1964), 75106. MR 29 #342. MR 0163039 (29:342)
 [12]
 , Hereditary dependence in the theory of differential equations. I, University of Maryland Tech. Note BN385, 1965.
 [13]
 J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171180.
 [14]
 A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (1935), 767776. MR 1513031
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197002674321
PII:
S 00029947(1970)02674321
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
