# American Institute of Mathematical Sciences

November  2019, 24(11): 6167-6188. doi: 10.3934/dcdsb.2019134

## Linearized stability for abstract functional differential equations subject to state-dependent delays with applications

 a. Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China b. School of Mathematics and Big Data, Foshan University, Foshan, Guangdong, 528000, China

* Corresponding author: Junjie Wei

Received  April 2018 Revised  December 2018 Published  July 2019

Fund Project: This work is supported by National Natural Science Foundation of China (No. 11771109)

In this paper, the linearized stability for a class of abstract functional differential equations (FDE) with state-dependent delays (SD) is investigated. In particular, such equations contain more general delay terms which not only cover the discrete delay and distributed delay as special cases, but also extend the SD to abstract integro-differential equation that the states belong to some infinite-dimensional space. The principle of linearized stability for such equations is established, which is nontrivial compared with ordinary differential equations with SD. Moreover, it should be stressed that such topic is untreated in the literatures up to date. Finally, we present an example to show the effectiveness of the proposed results.

Citation: Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134
##### References:
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O. Walther, The solution manifold and ${C }_{1}$-smoothness of solution operators for differential equations with state dependent delay, J. Differ. Equ., 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001. Google Scholar [31] J. Wang, J. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differ. Equ., 260 (2016), 3495-3523. doi: 10.1016/j.jde.2015.10.036. Google Scholar [32] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model, Nonlinear Anal. Real, 67 (2007), 2699-2711. doi: 10.1016/j.na.2006.09.034. Google Scholar [33] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar [34] H. Zhang, M. Yang and L. Wang, Existence and exponential convergence of the positive periodic solution for a model of hematopoiesis, Appl. Math. Lett., 26 (2013), 38-42. doi: 10.1016/j.aml.2012.02.034. Google Scholar [35] J. Zhao and J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Anal. Real, 22 (2015), 66-83. doi: 10.1016/j.nonrwa.2014.07.010. Google Scholar [36] W. Zuo and J. Wei, Stability and hopf bifurcation in a diffusive predator-prey system with delay effect, Nonlinear Anal. Real, 12 (2011), 1998-2011. doi: 10.1016/j.nonrwa.2010.12.016. Google Scholar

show all references

##### References:
 [1] Z. Balanov, Q. Hu and W. Krawcewicz, Global hopf bifurcation of differential equations with threshold type state-dependent delay, J. Differ. Equ., 257 (2014), 2622-2670. doi: 10.1016/j.jde.2014.05.053. Google Scholar [2] M. Belmekki, M. Benchohra and K. Ezzinbi, Existence results for some partial functional differential equations with state-dependent delay, Appl. Math. Lett., 24 (2011), 1810-1816. doi: 10.1016/j.aml.2011.04.039. Google Scholar [3] K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, P. Am. Math. Soc., 124 (1996), 1417-1426. doi: 10.1090/S0002-9939-96-03437-5. Google Scholar [4] M. Eichmann, A local Hopf Bifurcation Theorem for Differential Equations with State Dependent Delay, Ph.D. thesis, Universitat Giessen, Giessen, 2006.Google Scholar [5] P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, J. Differ. Equ., 260 (2016), 6176-6200. doi: 10.1016/j.jde.2015.12.038. Google Scholar [6] S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Springer-Verlag, Berlin, New York, 2013. doi: 10.1007/978-1-4614-6992-6. Google Scholar [7] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, New York, 1977. Google Scholar [8] F. Hartung, Linearized stability for a class of neutral functional differential equations with state-dependent delays, Nonlinear Anal. Real, 69 (2008), 1629-1643. doi: 10.1016/j.na.2007.07.004. Google Scholar [9] F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, Handbook of Differential Equations: Ordinary Differential Equations, 3 (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X. Google Scholar [10] F. Hartung, Nonlinear variation of constants formula for differential equations with state-dependent delays, J. Dyn. Differ. Equ., 28 (2016), 1187-1213. doi: 10.1007/s10884-015-9445-y. Google Scholar [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, New York, 1981. Google Scholar [12] E. Hernandez, M. Pierri and J. Wu, ${C} ^{1+\alpha}$-strict solutions and wellposedness of abstract differential equations with state dependent delay, J. Differ. Equ., 261 (2016), 6856-6882. doi: 10.1016/j.jde.2016.09.008. Google Scholar [13] E. Hernandez, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. Real, 7 (2006), 510-519. doi: 10.1016/j.nonrwa.2005.03.014. Google Scholar [14] A. Hou and S. Guo, Stability and bifurcation in a state-dependent delayed predator-prey system, Int. J. Bifurcat. Chaos, 26 (2016), 1650060, 15 pp. doi: 10.1142/S0218127416500607. Google Scholar [15] Q. Hu and J. Wu, Global hopf bifurcation for differential equations with state-dependent delay, J. Differ. Equ., 248 (2010), 2801-2840. doi: 10.1016/j.jde.2010.03.020. Google Scholar [16] T. Krisztin, ${C}_{1}$-smoothness of center manifolds for differential equations with state-dependent delay, Nonlinear Dynamics and Evolution Equations, Fields Institute Communications, 48 (2006), 213-226. Google Scholar [17] T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold, J. Differ. Equ., 260 (2016), 4454-4472. doi: 10.1016/j.jde.2015.11.018. Google Scholar [18] B. Liu, New results on the positive almost periodic solutions for a model of hematopoiesis, Nonlinear Anal. Real, 17 (2014), 252-264. doi: 10.1016/j.nonrwa.2013.12.003. Google Scholar [19] Y. Lv, R. Yuan and Y. Pei, Smoothness of semiflows for parabolic partial differential equations with state-dependent delay, J. Differ. Equ., 260 (2016), 6201-6231. doi: 10.1016/j.jde.2015.12.037. Google Scholar [20] Y. Lv, R. Yuan, Y. Pei and T. Li, Global stability of a competitive model with state-dependent delay, J. Dyn. Differ. Equ., 29 (2017), 501-521. doi: 10.1007/s10884-015-9475-5. Google Scholar [21] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar [22] B. Niu and W. Jiang, Multiple bifurcation analysis in a NDDE arising from van der pol's equation with extended delay feedback, Nonlinear Anal. Real, 14 (2013), 699-717. doi: 10.1016/j.nonrwa.2012.07.028. Google Scholar [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [24] A. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays, J. Math. Anal. Appl., 326 (2007), 1031-1045. doi: 10.1016/j.jmaa.2006.03.049. Google Scholar [25] A. Rezounenko, A condition on delay for differential equations with discrete state-dependent delay, J. Math. Anal. Appl., 385 (2012), 506-516. doi: 10.1016/j.jmaa.2011.06.070. Google Scholar [26] A. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space, Discrete Cont. Dyn. S., 33 (2013), 819-835. doi: 10.3934/dcds.2013.33.819. Google Scholar [27] E. Stumpf, A note on local center manifolds for differential equations with state-dependent delay, Differ. Integral Equ., 29 (2016), 1093-1106. Google Scholar [28] Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusionpopulation model with delay effect, J. Differ. Equ., 247 (2009), 1156-1184. doi: 10.1016/j.jde.2009.04.017. Google Scholar [29] H. O. Walther, Linearized stability for semiflows generated by a class of neutral equations, with applications to state-dependent delays, J. Dyn. Differ. Equ., 22 (2010), 439-462. doi: 10.1007/s10884-010-9168-z. Google Scholar [30] H. O. Walther, The solution manifold and ${C }_{1}$-smoothness of solution operators for differential equations with state dependent delay, J. Differ. Equ., 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001. Google Scholar [31] J. Wang, J. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differ. Equ., 260 (2016), 3495-3523. doi: 10.1016/j.jde.2015.10.036. Google Scholar [32] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model, Nonlinear Anal. Real, 67 (2007), 2699-2711. doi: 10.1016/j.na.2006.09.034. Google Scholar [33] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar [34] H. Zhang, M. Yang and L. Wang, Existence and exponential convergence of the positive periodic solution for a model of hematopoiesis, Appl. Math. Lett., 26 (2013), 38-42. doi: 10.1016/j.aml.2012.02.034. Google Scholar [35] J. Zhao and J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Anal. Real, 22 (2015), 66-83. doi: 10.1016/j.nonrwa.2014.07.010. Google Scholar [36] W. Zuo and J. Wei, Stability and hopf bifurcation in a diffusive predator-prey system with delay effect, Nonlinear Anal. Real, 12 (2011), 1998-2011. doi: 10.1016/j.nonrwa.2010.12.016. Google Scholar
Numerical simulations of system (37) with the initial value $\phi ( \zeta ,y) = 0.1sin^{2}(y)$. (a) The equilibrium point $u^{\ast } = 0$ is asymptotically stable with the system parameters $m = 3,$ $D = 0.01,$ $B = 0.1,$ $d = 0.2.$ (b) The equilibrium point $u^{\ast } = 0$ is unstable with the system parameters $m = 3,$ $D = 0.01,$ $B = 0.6,$ $d = 0.2.$
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