A Fixed Point Theorem and Attractors

We investigate attractors for compact sets by considering a certain quotient space. The following theorem is included. Let /: G -> G, G a closed convex subset of a Banach space, / a mapping satisfying (i) there exists M c G which is an attractor for compact sets under/; (ii) the family {/"}Ü°_i is equicontinuous. Then/has a fixed point. 1. Since the appearance of the celebrated Schauder fixed point theorem, the following has been a longstanding open question. Conjecture 1. Let/: G -» G, G a closed, convex subset of a Banach space, /a continuous mapping with/^ compact, A > 1. Then/has a fixed point (?) In a paper which appeared in 1972, Roger Nussbaum [6] states Conjecture 1 along with several new conjectures concerning fixed points. In that paper he introduces the concept of an attractor for compact sets. Definition 1.1. Let A" be a topological space, /: X -+ X a map, and M a nonempty subset of A. M is an attractor for compact sets under/if (1) M is compact and f(M) c M, and (2) given any compact set C c X and any open neighborhood U of M, there exists an integer N = N(C, U) such that f(C) c U for n > A. M is an attractor for points under/ if (2) holds for C = (x) g X. For brevity, we shall use "a.c.s. under /" for the phrase "attractor for compact sets under/." In this setting of Conjecture 1, we see that c\(fN(G)) is an a.c.s. under/; hence, an affirmative answer to the following conjecture of Nussbaum would yield an affirmative answer to Conjecture 1. Conjecture 2. Let G be a closed, convex subset of a Banach space A and /: G -» G a continuous map. If there exists a set M G G which is an attractor for compact sets under/, then/has a fixed point (?) Nussbaum [6] states that he does not think that Conjecture 2 has an affirmative answer. If one attempts to provide a counterexample to Conjecture 2, a characterization of functions which have attractors seems desirable. In the direction of obtaining information concerning the structure of a.c.s., see Solomon [7]. In §2, we provide several results concerning those functions which have a.c.s. In §3, we provide a partial affirmative answer to Conjecture 2 and thus Presented to the Society, January 6, 1978; received by the editors April 20, 1977. AMS (MOS) subject classifications (1970). Primary 47H10; Secondary 54H25, 54B15.