Contracting return maps for monotone delayed feedback. (Q1864094)

The existence and stability of slowly oscillating periodic solutions to the delay differential equation \[ \dot x(t)=-\mu x(t)+f(x(t-1)) \] is derived for nonlinearities \(f\) which satisfy the standard negative feedback condition, \(xf(x)<0, x\neq0\), and are close to the step function \(f(x)=-\text{sign\,}(x)=-1\) for \(x>0\), \(f(x)=\text{sign\,} (x)=+1\) for \(x<0\), outside a small neighborhood \((-\beta,\beta)\) of \(0\). A crucial assumption is that \(f(x)\) is convex on an interval \((0,\varepsilon)\) for some \(0<\varepsilon<\beta\) [respectively, \(f(x)\) is concave in \((-\varepsilon,0)\)]. The proof involves the construction of a return map on a subset of initial functions \(C([-1,0], {\mathbb R})\), which is Lipschitz continuous and a contraction. Applications include \(f(x)=\text{arctan}(\gamma x)\) and \(f(x)=\text{tanh}(\gamma x)\) among others.