Linearized Stability for a New Class of Neutral Equations with State-Dependent Delay
Tóm tắt
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.
Tài liệu tham khảo
Barbarossa, M.V.: On a class of neutral equations with state-dependent delay in population dynamics. Doctoral Dissertation, Technical University Munich, Munich (2013)
Barbarossa, M.V., Hadeler, K.P., Kuttler, C.: State-dependent neutral delay equations from population dynamics, submitted for publication (2013)
Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)
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)
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)
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)
Walther, H.-O.: More on linearized stability for neutral equations with state-dependent delay. Differ. Equ. Dyn. Syst. 19, 315–333 (2011)
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)