Some fixed point theorems for compact maps and flows in Banach spaces.
HTML articles powered by AMS MathViewer
- by W. A. Horn PDF
- Trans. Amer. Math. Soc. 149 (1970), 391-404 Request permission
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.References
- Felix E. Browder, On a generalization of the Schauder fixed point theorem, Duke Math. J. 26 (1959), 291–303. MR 105629
- Felix E. Browder, Another generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965), 399–406. MR 203718
- Felix E. Browder, A further generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965), 575–578. MR 203719
- Jane Cronin, Fixed points and topological degree in nonlinear analysis, Mathematical Surveys, No. 11, American Mathematical Society, Providence, R.I., 1964. MR 0164101
- J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116
- A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962), 93. MR 149253
- A. Halanaĭ, Asymptotic stability and small perturbations of periodic systems of differential equations with retarded argument, Uspehi Mat. Nauk 17 (1962), no. 1 (103), 231–233 (Russian). MR 0136840 W. A. Horn, A generalization of Browder’s fixed point theorem, Abstract #611-30, Notices Amer. Math. Soc. 11 (1964), 325.
- G. Stephen Jones, The existence of periodic solutions of $f^{\prime } (x)=-\alpha f(x-1)\{1+f(x)\}$, J. Math. Anal. Appl. 5 (1962), 435–450. MR 141837, DOI 10.1016/0022-247X(62)90017-3
- G. Stephen Jones, Asymptotic fixed point theorems and periodic systems of functional-differential equations, Contributions to Differential Equations 2 (1963), 385–405. MR 158135
- G. Stephen Jones, Periodic motions in Banach space and applications to functional-differential equations, Contributions to Differential Equations 3 (1964), 75–106. MR 163039 —, Hereditary dependence in the theory of differential equations. I, University of Maryland Tech. Note BN-385, 1965. J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171-180.
- A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (1935), no. 1, 767–776 (German). MR 1513031, DOI 10.1007/BF01472256
Additional Information
- © Copyright 1970 American Mathematical Society
- 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