Periodic solutions of a differential-delay equation and the fixed point index

Abstract

For the simplest nonlinear differential-delay equation x(t)=g(x(t-l))i xg(x) < 0, x ≠0, periodic solutions of a certain natural type are in 1-1 correspondence with (nontrivial) fixed points of an abstract map (analogous to the Poincare map for O.D.E.s) which takes a cone in a Banach space into itself. In this paper we study the fixed point set corresponding to the equation above for g = -αf,α>0, where f is a special function. Thus, we are interested in existence and uniqueness of fixed points and the fixed point index. Part of the motivation for this paper comes from a bifurcation problem for a related differential-delay equation.

Keywords:

• Differential delay equation
• periodic solutions

AMS(MOS):

• 34K15
• 55M25