
Previous Article
Exterior differential systems and prolongations for three important nonlinear partial differential equations
 CPAA Home
 This Issue

Next Article
Duffingvan der Poltype oscillator system and its first integrals
Stability of linear differential equations with a distributed delay
1.  Department of Mathematics, BenGurion University of the Negev, BeerSheva 84105, Israel 
2.  Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4 
$\dot{x}(t) + \sum_{k=1}^m \int_{h_k(t)}^t x(s) d_s R_k(t,s) =0, h_k(t)\leq t,$ su$p_{t\geq 0}(th_k(t))<\infty,$
where the functions involved in the equation are not required to be continuous.
The results are applied to integrodifferential equations,
equations with several concentrated delays and equations of a mixed type.
References:
[1] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, (1993). 
[2] 
V. Volterra, Fluctuations in the abundance of species considered mathematically,, Nature, 118 (1926), 558. 
[3] 
G. E. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221. 
[4] 
A. D. Myshkis, "Linear Differential Equations with Retarded Argument,", Nauka, (1972). 
[5] 
L. Berezansky and E. Braverman, On oscillation of equations with distributed delay,, Zeitschrift f\, 20 (2001), 489. 
[6] 
L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay,, Mathematical and Computer Modelling, 48 (2008), 287. doi: 10.1016/j.mcm.2007.10.003. 
[7] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations,", Applied Mathematical Sciences, 99 (1993). 
[8] 
N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functionaldifferential equation of evolution type,, Differential Equations, 13 (1977). 
[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. 
[10] 
N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect,", Stability and Control: Theory, (2003). 
[11] 
L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations,, J. Math. Anal. Appl., 314 (2006), 391. doi: 10.1016/j.jmaa.2005.03.103. 
[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. doi: 10.3934/dcdsb.2001.1.233. 
[13] 
S. A. Gusarenko, Criteria for the stability of a linear functionaldifferential equation,, Boundary value problems, (1987), 41. 
[14] 
S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functionaldifferential equations,, Functionaldifferential equations, (1987), 30. 
[15] 
I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations,, J. Math. Anal. Appl., 220 (1998), 290. doi: 10.1006/jmaa.1997.5883. 
show all references
References:
[1] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, (1993). 
[2] 
V. Volterra, Fluctuations in the abundance of species considered mathematically,, Nature, 118 (1926), 558. 
[3] 
G. E. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221. 
[4] 
A. D. Myshkis, "Linear Differential Equations with Retarded Argument,", Nauka, (1972). 
[5] 
L. Berezansky and E. Braverman, On oscillation of equations with distributed delay,, Zeitschrift f\, 20 (2001), 489. 
[6] 
L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay,, Mathematical and Computer Modelling, 48 (2008), 287. doi: 10.1016/j.mcm.2007.10.003. 
[7] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations,", Applied Mathematical Sciences, 99 (1993). 
[8] 
N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functionaldifferential equation of evolution type,, Differential Equations, 13 (1977). 
[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. 
[10] 
N. V. Azbelev and P. M. Simonov, "Stability of Differential Equations with Aftereffect,", Stability and Control: Theory, (2003). 
[11] 
L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations,, J. Math. Anal. Appl., 314 (2006), 391. doi: 10.1016/j.jmaa.2005.03.103. 
[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. doi: 10.3934/dcdsb.2001.1.233. 
[13] 
S. A. Gusarenko, Criteria for the stability of a linear functionaldifferential equation,, Boundary value problems, (1987), 41. 
[14] 
S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functionaldifferential equations,, Functionaldifferential equations, (1987), 30. 
[15] 
I. Györi, F. Hartung and J. Turi, Preservation of stability in delay equations under delay perturbations,, J. Math. Anal. Appl., 220 (1998), 290. doi: 10.1006/jmaa.1997.5883. 
[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) : 233256. 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) : 219223. 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) : 1941. doi: 10.3934/mbe.2016.13.19 
[4] 
Elena Braverman, Sergey Zhukovskiy. Absolute and delaydependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems  A, 2012, 32 (6) : 20412061. doi: 10.3934/dcds.2012.32.2041 
[5] 
István Györi, Ferenc Hartung. Exponential stability of a statedependent delay system. Discrete & Continuous Dynamical Systems  A, 2007, 18 (4) : 773791. 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) : 235250. 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) : 18551876. 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  A, 2007, 18 (2&3) : 295313. doi: 10.3934/dcds.2007.18.295 
[9] 
Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary timevarying delay. Discrete & Continuous Dynamical Systems  S, 2011, 4 (3) : 693722. doi: 10.3934/dcdss.2011.4.693 
[10] 
Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discretetime switched delay systems. Discrete & Continuous Dynamical Systems  B, 2017, 22 (1) : 199208. doi: 10.3934/dcdsb.2017010 
[11] 
Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with statedependent delay. Discrete & Continuous Dynamical Systems  B, 2017, 22 (8) : 31673197. doi: 10.3934/dcdsb.2017169 
[12] 
Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible nonNewtonian fluid with delay. Discrete & Continuous Dynamical Systems  B, 2018, 23 (10) : 42854303. doi: 10.3934/dcdsb.2018138 
[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) : 24512464. doi: 10.3934/dcdsb.2012.17.2451 
[14] 
Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factordependent stem cell dynamics model with distributed delay. Discrete & Continuous Dynamical Systems  B, 2007, 8 (1) : 1938. doi: 10.3934/dcdsb.2007.8.19 
[15] 
Jinhu Xu, Yicang Zhou. Global stability of a multigroup model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 10831106. doi: 10.3934/mbe.2015.12.1083 
[16] 
Aissa Guesmia, Nassereddine Tatar. Some wellposedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457491. doi: 10.3934/cpaa.2015.14.457 
[17] 
Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems  A, 2016, 36 (10) : 56575679. doi: 10.3934/dcds.2016048 
[18] 
Yaru Xie, Genqi Xu. Exponential stability of 1d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems  S, 2017, 10 (3) : 557579. doi: 10.3934/dcdss.2017028 
[19] 
ShiLiang Wu, WanTong Li, SanYang Liu. Exponential stability of traveling fronts in monostable reactionadvectiondiffusion equations with nonlocal delay. Discrete & Continuous Dynamical Systems  B, 2012, 17 (1) : 347366. doi: 10.3934/dcdsb.2012.17.347 
[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) : 33793393. doi: 10.3934/dcdsb.2018325 
2018 Impact Factor: 0.925
Tools
Metrics
Other articles
by authors
[Back to Top]