American Institute of Mathematical Sciences

$\dot{x}(t)=-\mu x(t)+f(x(t-1))$

with continuous odd nonlinearities close to a step function $f_a(\xi)=-a$sign$\xi$, $a>0,$ we find sets of initial data to which solutions return. For Lipschitz nonlinearities the associated return maps become Lipschitz continuous. Monotonicity of $f$ close to 0 permits sharp estimates of the Lipschitz constants of the return map. If outside a neighbourhood of 0 the Lipschitz constant of $f$ is sufficiently small then the return map becomes a contraction, whose fixed point defines an attracting periodic orbit of the differential equation. Applications include the cases $f(x)=g(\gamma x)$ with $g'(x)=-(1+x^r)^{-1}$ for $x>0$, $r>\frac{3}{2}$ and $\gamma>0$ sufficiently large.

$(- \Delta )^{\frac{n}{2}} w = (n -1)! e^{n w} x \in R^n$

for all even dimensions n.
Let

$\alpha = \frac{1}{|S^n|} \int_{R^n} e^{n w} dx .$

We prove, for every $0 < \alpha < 1$, the existence of at least one solution. In particular, for $ n = 4$, we obtain the existence of radial solutions.