Periodic solutions to functional differential equations

Abstract

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

Periodic solutions of abstract differential equations with infinite delay

“…THEOREM (Horn). Let S Q c: S 1 is both compact and convex. Note that 5 X and 5 2 are also convex sets (Figure 1).…”

“…In [3] it was shown that if unbounded initial functions are allowed, then for certain integrodifferential equations with infinite delay, it is the case that UB and UUB solutions imply the existence of an mT-periodic solution. That result was extended in [1] to Aperiodic solutions; unbounded initial functions were still required.…”

An asymptotic fixed point theorem for a locally convex space

“…In a series of papers (cf., [1,4,11]) investigators have searched for a natural space in which to discuss the existence of periodic solutions for functional differential equations with infinite delay. From the very start of such investigations it seemed that the proper setting was the metric space (Y, The Schauder-Tychonov theorem [13] states that if S is a compact convex subset of (Y, p) and if P: S -> S is continuous, then P has a fixed point.…”

“…Detailed discussions of applications for finite delay equations are found in [7,8] and for infinite delay in [1,4,5,6]. Continuity of P is extensively discussed in [4 and 10].…”

Uniqueness without continuous dependence

“…Introduction. In the classical theory of ordinary differential equations if solutions of a system (1) x' = h(t,x)…”