Continuity, compactness, fixed points, and integral equations

T. A. Burton, Northwest Research Institute, Port Angeles, WA, U.S.A.
G. Makay, Bolyai Institute, University of Szeged, Hungary

E. J. Qualitative Theory of Diff. Equ., No. 14. (2002), pp. 1-13.

Abstract: An integral equation, $x(t)=a(t)-\int^t_{-\infty} D(t,s)g(x(s))ds$ with $a(t)$ bounded, is studied by means of a Liapunov functional. There results an {\it{a}} {\it{priori}} bound on solutions. This gives rise to an interplay between continuity and compactness and leads us to a fixed point theorem of Schaefer type. It is a very flexible fixed point theorem which enables us to show that the solution inherits properties of $a(t)$, including periodic or almost periodic solutions in a Banach space.

You can download the full text of this paper in DVI, PostScript or PDF format.