Convergence of Solutions for an Equation with State-Dependent Delay

Abstract A result of Smith and Thieme shows that if a semiflow is strongly order preserving, then a typical orbit converges to the set of equilibria. For the equation with state-dependent delay ẋ ( t ) = −μ x ( t ) +  f ( x ( t  −  r ( x ( t )))), where μ > 0 and f and r are smooth real functions with f (0) = 0 and f ′ > 0, we construct a semiflow which is monotone but not strongly order preserving. We prove a convergence result under a monotonicity condition different from the strong order preserving property, and apply it to the above equation to obtain generic convergence.

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