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

W. A. Horn
1970 Transactions of the American Mathematical Society  
Let S0OE Sic S2 be convex subsets of the Banach space X, with S0 and S2 closed and Si open in S2. If / is a compact mapping of S2 into X such that U*=i/'(Si)c^ and/m(5i) u/m+1(5i)c50 for some »i>0, then/has a fixed point in S0. (This extends a result of F. E. Browder published in 1959.) Also, if {Tt : t s R*} is a continuous flow on the Banach space X, 50cSic=52 are convex subsets of X with So and S2 compact and Si open in S2, and 7*¡0(Si)c.So for some i0>0, where 7"((Si) <=S2 for all f¿ta,
more » ... 2 for all f¿ta, then there exists x0 e S0 such that Tt(x0) = x0 for all /ä0. Minor extensions of Browder's work on "nonejective" and "nonrepulsive" fixed points are also given, with similar results for flows. Interest in fixed point theorems in Banach and locally convex topological linear spaces has been increasing during the past few years as more applications are found in the field of differential and integral equations ([4], [7], [9], [10], [11], and [12]). An important class of fixed point theorems deals with compact or completely continuous mappings on convex sets, going back to the Schauder [13] and Tychonoff [14] theorems, wherein a convex set is assumed mapped into itself, and continuing to more complicated recent theorems such as those of F. E. Browder ([1], [2], and [3]) and others. The theorems of [1], [2], and [3] deal with the action of iterates of a given mapping, rather than the mapping itself, and it is the purpose of this paper to extend these theorems in the same vein and to prove analogous theorems related to flows, i.e., semigroups of mappings indexed by the nonnegative real half line. This latter extension was suggested to the author by A. J. Goldman of the National Bureau of Standards. A. Extensions of Browder's first result. This section will generalize the work of Browder in [1] . The result to be generalized is Theorem 2 of [1]. Let S and Sx be open convex subsets of the Banach space X, S0 a closed convex subset of X, S0CSX<^S, f a compact mapping of 5 into X. Suppose that, for a
doi:10.1090/s0002-9947-1970-0267432-1 fatcat:hhbt5nitqbdurnc6ukzpw7drt4