In this paper we study the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state. A fixed point problem related to a Poincare operator is constructed and solved using an ejective fixed point theorem.1998 Academic Press

We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov-Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state.

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We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C 0, 1 . We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall-Liggett type for the semigroup is obtained in a dense subset of (C, || · || . ). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations.

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