American Institute of Mathematical Sciences

July  2005, 12(4): 687-722. doi: 10.3934/dcds.2005.12.687

A saddle point theorem for functional state-dependent delay differential equations

 1 Depto. de Matemática Aplicada, E.T.S.I. Industriales, c. José Gutiérrez Abascal, 2, 28006 Madrid, Spain, Spain

Received  December 2003 Revised  November 2004 Published  January 2005

The aim of the work is to study the stability of equilibrium points for some functional differential equations with state-dependent delay. As a preliminary step, existence, continuation, uniqueness and smoothness results have been shown for solutions. From a given functional differential equation with state-dependent delay, a linearized equation is constructed which gives sufficient conditions for asymptotic stability of equilibrium solutions. Also, a saddle point theorem is shown in the case where the equilibrium point is a hyperbolic equilibrium point for the linearized equation.
Citation: Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
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