American Institute of Mathematical Sciences

Advanced
September  2011, 10(5): 1361-1375. doi: 10.3934/cpaa.2011.10.1361

Stability of linear differential equations with a distributed delay

Leonid Berezansky 1, and  Elena Braverman 2,

1. 

Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
2. 

Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4

Received  August 2009 Revised  August 2010 Published  April 2011

We present some new stability results for the scalar linear equation with a distributed delay

$\dot{x}(t) + \sum_{k=1}^m \int_{h_k(t)}^t x(s) d_s R_k(t,s) =0, h_k(t)\leq t,$ su$p_{t\geq 0}(t-h_k(t))<\infty,$

where the functions involved in the equation are not required to be continuous.
The results are applied to integro-differential equations, equations with several concentrated delays and equations of a mixed type.

Keywords: exponential stability, explicit stability conditions., Distributed delay.
Mathematics Subject Classification: Primary: 34K20; Secondary: 34K2.
Citation: Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
References:
[1]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191. Academic Press, Boston, MA, 1993.  Google Scholar

[2]

V. Volterra, Fluctuations in the abundance of species considered mathematically, Nature, 118 (1926), 558-560. Google Scholar

[3]

G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. Google Scholar

[4]

A. D. Myshkis, "Linear Differential Equations with Retarded Argument," Nauka, Moscow, 1972.  Google Scholar

[5]

L. Berezansky and E. Braverman, On oscillation of equations with distributed delay, Zeitschrift für Analysis und ihre Anwendungen, 20 (2001), 489-504.  Google Scholar

[6]

L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Mathematical and Computer Modelling, 48 (2008), 287-304. doi: 10.1016/j.mcm.2007.10.003.  Google Scholar

[7]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[8]

N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential equation of evolution type, Differential Equations, 13 (1977), 1915--1925, 2106.  Google Scholar

[9]

N. V. Azbelev, L. Berezansky, P. M. Simonov and A. V. Chistyakov, The stability of linear systems with aftereffect. I, Differential Equations, 23 (1987), 745-754, 914, II. Differential Equations, 27 (1991), 383-388, III. Differential Equations, 27 (1991), 1165-1172, IV. Differential Equations, 29 (1993), 153-160.  Google Scholar

[10]

N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect," Stability and Control: Theory, Methods and Applications, 20. Taylor $&$ Francis, London, 2003.  Google Scholar

[11]

L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl., 314 (2006), 391-411. doi: 10.1016/j.jmaa.2005.03.103.  Google Scholar

[12]

S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 1 (2001), 233-256. doi: 10.3934/dcdsb.2001.1.233.  Google Scholar

[13]

S. A. Gusarenko, Criteria for the stability of a linear functional-differential equation, Boundary value problems, 41-45, Perm. Politekh. Inst., Perm, 1987 (Russian).  Google Scholar

[14]

S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations, Functional-differential equations, 30-40, Perm. Politekh. Inst., Perm, 1987 (Russian).  Google Scholar

[15]

I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl., 220 (1998), 290-312. doi: 10.1006/jmaa.1997.5883.  Google Scholar

show all references

References:
[1]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191. Academic Press, Boston, MA, 1993.  Google Scholar

[2]

V. Volterra, Fluctuations in the abundance of species considered mathematically, Nature, 118 (1926), 558-560. Google Scholar

[3]

G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. Google Scholar

[4]

A. D. Myshkis, "Linear Differential Equations with Retarded Argument," Nauka, Moscow, 1972.  Google Scholar

[5]

L. Berezansky and E. Braverman, On oscillation of equations with distributed delay, Zeitschrift für Analysis und ihre Anwendungen, 20 (2001), 489-504.  Google Scholar

[6]

L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Mathematical and Computer Modelling, 48 (2008), 287-304. doi: 10.1016/j.mcm.2007.10.003.  Google Scholar

[7]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[8]

N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential equation of evolution type, Differential Equations, 13 (1977), 1915--1925, 2106.  Google Scholar

[9]

N. V. Azbelev, L. Berezansky, P. M. Simonov and A. V. Chistyakov, The stability of linear systems with aftereffect. I, Differential Equations, 23 (1987), 745-754, 914, II. Differential Equations, 27 (1991), 383-388, III. Differential Equations, 27 (1991), 1165-1172, IV. Differential Equations, 29 (1993), 153-160.  Google Scholar

[10]

N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect," Stability and Control: Theory, Methods and Applications, 20. Taylor $&$ Francis, London, 2003.  Google Scholar

[11]

L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl., 314 (2006), 391-411. doi: 10.1016/j.jmaa.2005.03.103.  Google Scholar

[12]

S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 1 (2001), 233-256. doi: 10.3934/dcdsb.2001.1.233.  Google Scholar

[13]

S. A. Gusarenko, Criteria for the stability of a linear functional-differential equation, Boundary value problems, 41-45, Perm. Politekh. Inst., Perm, 1987 (Russian).  Google Scholar

[14]

S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations, Functional-differential equations, 30-40, Perm. Politekh. Inst., Perm, 1987 (Russian).  Google Scholar

[15]

I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl., 220 (1998), 290-312. doi: 10.1006/jmaa.1997.5883.  Google Scholar

[1]

Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[2]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[3]

Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19
[4]

Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[5]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[6]

Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[7]

Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[8]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[9]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[10]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[11]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010
[12]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[13]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[14]

Adriana Flores de Almeida, Marcelo Moreira Cavalcanti, Janaina Pedroso Zanchetta. Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Evolution Equations & Control Theory, 2019, 8 (4) : 847-865. doi: 10.3934/eect.2019041
[15]

Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19
[16]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
[17]

Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457
[18]

Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048
[19]

Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028
[20]

Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences