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Linearized Stability for a New Class of Neutral Equations with State-Dependent Delay


For neutral delay differential equations of the form

$$\begin{aligned} \dot{x}(t)=g(\partial x_t,x_t), \end{aligned}$$

with \(g\) defined on an open subset of the space \(C([-h,0],\mathbb {R}^n)\times C^1([-h,0],\mathbb {R}^n)\), we extend an earlier principle of linearized stability. The present result applies to a wider class of neutral differential equations

$$\begin{aligned} \dot{x}(t) = f(x(t),\dot{x}(t-\tau (x(t))), x(t-\sigma (x(t)))) \end{aligned}$$

with state-dependent delays which includes models for population dynamics with maturation delay.

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MVB was supported by the ERC Starting Grant No. 259559 as well as by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP-4.2.4. A/2-11-1-2012-0001 National Excellence Program.

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Correspondence to M. V. Barbarossa.

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Barbarossa, M.V., Walther, H.O. Linearized Stability for a New Class of Neutral Equations with State-Dependent Delay. Differ Equ Dyn Syst 24, 63–79 (2016).

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  • State-dependent delay
  • Neutral functional differential equation
  • Linearized stability
  • Population dynamics

Mathematics Subject Classification

  • 34K40
  • 34K20
  • 37L15
  • 92D25