Stability and fixed points of point-dissipative systems

It is known that if T: X → X is completely continuous or if there exists an n₀ > 0 such that Tⁿ⁰ is completely continuous, then T point dissipative implies that there is a maximal compact invariant set which is uniformly asymptotically stable, attracts bounded sets, and has a fixed point (see Billotti and LaSalle [Bull. Amer. Math. Soc.6 1971]). The result is used, for example, in studying retarded functional differential equations, or parabolic partial differential equations. This result has been extended by Hale and Lopes [J. Differential Equations13 1973]. They get the result that if T is an α-contraction and compact dissipative then there is a maximal compact invariant set which is uniformly asymptotically stable, attracts neighborhoods of compact sets, and has a fixed point. The above result requires the stronger assumption of compact dissipative. The principal result of this paper is to get similar results under the weaker assumption of point dissipative. To do this we must make additional assumptions. We will show these assumptions are naturally satisfied by stable neutral functional differential equations and retarded functional differential equations with infinite delay. The result has applications to many other dynamical systems, of course.