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  November 2019 Early access  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 and Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134
References:
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M. BelmekkiM. 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.

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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.

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M. Eichmann, A local Hopf Bifurcation Theorem for Differential Equations with State Dependent Delay, Ph.D. thesis, Universitat Giessen, Giessen, 2006.

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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.

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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.

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J. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, New York, 1977.

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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.

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F. HartungT. KrisztinH. 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.

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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.

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, New York, 1981.

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E. HernandezM. 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.

[13]

E. HernandezA. 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.

[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.

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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.

[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. 

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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.

[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.

[19]

Y. LvR. 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.

[20]

Y. LvR. YuanY. 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.

[21]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326.

[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.

[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.

[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.

[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.

[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.

[27]

E. Stumpf, A note on local center manifolds for differential equations with state-dependent delay, Differ. Integral Equ., 29 (2016), 1093-1106. 

[28]

Y. SuJ. 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.

[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.

[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.

[31]

J. WangJ. 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.

[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.

[33]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[34]

H. ZhangM. 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.

[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.

[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.

show all references

References:
[1]

Z. BalanovQ. 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.

[2]

M. BelmekkiM. 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.

[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.

[4]

M. Eichmann, A local Hopf Bifurcation Theorem for Differential Equations with State Dependent Delay, Ph.D. thesis, Universitat Giessen, Giessen, 2006.

[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.

[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.

[7]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, New York, 1977.

[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.

[9]

F. HartungT. KrisztinH. 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.

[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.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, New York, 1981.

[12]

E. HernandezM. 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.

[13]

E. HernandezA. 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.

[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.

[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.

[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. 

[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.

[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.

[19]

Y. LvR. 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.

[20]

Y. LvR. YuanY. 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.

[21]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326.

[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.

[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.

[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.

[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.

[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.

[27]

E. Stumpf, A note on local center manifolds for differential equations with state-dependent delay, Differ. Integral Equ., 29 (2016), 1093-1106. 

[28]

Y. SuJ. 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.

[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.

[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.

[31]

J. WangJ. 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.

[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.

[33]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[34]

H. ZhangM. 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.

[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.

[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.

Figure 1.  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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