Complete guiding sets for a class of almost-periodic differential equations

A functional differential equation admitting a complete guiding set always has an a priori bounded solution. The introduction of this concept and the idea of reducibility allow the authors to apply a well-known method for the autonomous case to a non-autonomous problem. A previous result of Ortega and Tineo is extended to a non-hyperbolic semilinear functional differential equation with almost-periodic coefficients in its lineal part. The non-hyperbolicity condition is established in terms of the Sacker-Sell spectrum of the associated linear system.