Uniform ultimate boundedness and periodicity in functional differential equations (Q1124032)

We consider the existence of periodic solutions of functional differential equations with infinite delay, (1) \(x'=f(t,x_ t)\). Let \((X,\| \cdot \|)\) be the space of bounded continuous functions \(\phi\) : (-\(\infty,0]\to R^ n\) with the supremum norm and let G denote the set of continuous nonincreasing functions g: (-\(\infty,0]\to [1,\infty)\) such that g(r)\(\to \infty\) as \(r\to -\infty\) and \(g(0)=1\). For a given \(g\in G\), let \((X_ g,| \cdot |_ g)\) denote the Banach space of continuous functions \(\phi\) : (-\(\infty,0]\to R^ n\) for which \(| \phi |_ g=\sup_{t\leq 0}| \phi (t)/g(t)| <\infty.\) Our main result yields a T-periodic solution of (1) by asking that solutions be uniformly ultimately bounded. Uniform boundedness is not required. Theorem. Suppose that (i) solutions of (1) are uniformly ultimately bounded, (ii) f: \(R\times X\to R^ n\) takes bounded sets into bounded sets and \(f(t,\phi)=f(t+T,\phi)\) for some \(T>0\), (iii) for any bounded set \(\Omega\) \(\subset X\), solutions \(x(t)=x(t,0,\phi)\) of (1) depend continuously on \(\phi\in \Omega\) in the space \((X_ g,| \cdot |_ g)\). Then (1) has a T-periodic solution. The proof is based on the construction of a compact set in \((X_ g,| \cdot |_ g)\), say \[ S_ B=\{\phi \in X:\| \phi \| \leq 2B,\quad | \phi (u)- \phi (v)| \leq L_ B| u-v| \}, \] and application of Horn's fixed point theorem.