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

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. 1.

    Barbarossa, M.V.: On a class of neutral equations with state-dependent delay in population dynamics. Doctoral Dissertation, Technical University Munich, Munich (2013)

  2. 2.

    Barbarossa, M.V., Hadeler, K.P., Kuttler, C.: State-dependent neutral delay equations from population dynamics, submitted for publication (2013)

  3. 3.

    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    Book  MATH  Google Scholar 

  4. 4.

    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)

    Book  MATH  Google Scholar 

  5. 5.

    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  6. 6.

    Mallet-Paret, J., Nussbaum, R.D., Paraskevopoulos, P.: Periodic solutions for functional differential equations with multiple state-dependent time lags. Topol. Methods Nonlinear Anal. 3(1), 101–162 (1994)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Walther, H.-O.: Smoothness properties of semiflows for differential equations with state dependent delay. In: Proceedings of the International Conference on Differential and Functional Differential Equations (Russian), Moscow, 2002. Moscow State Aviation Institute (MAI), Moscow (2003). English version. In: Journal of the Mathematical Sciences, 12, pp. 5193–5207 (2004)

  8. 8.

    Walther, H.-O.: Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays. J. Dyn. Differ. Equ. 22(3), 439–462 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. 9.

    Walther, H.-O.: More on linearized stability for neutral equations with state-dependent delay. Differ. Equ. Dyn. Syst. 19, 315–333 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. 10.

    Walther, H.-O.: Semiflows for neutral equations with state-dependent delays. Infinite Dymensional Dynamical Systems. In: Mallet-Paret, J., Wu, J., Yi, Y., Zhu, H. (eds.) Fields Institute Communications, vol. 64, pp. 211–267. Springer, New York (2013)

Download references

Acknowledgments

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. V. Barbarossa.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s12591-014-0204-z

Download citation

Keywords

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

Mathematics Subject Classification

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