American Institute of Mathematical Sciences

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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

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