Stability of linear differential equations with a distributed delay

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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

Abstract
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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, (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 Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).
[8]	N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential 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 functional-differential equation,, Boundary value problems, (1987), 41.
[14]	S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations,, Functional-differential 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 Functional-Differential Equations,", Applied Mathematical Sciences, 99 (1993).
[8]	N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functional-differential 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 functional-differential equation,, Boundary value problems, (1987), 41.
[14]	S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functional-differential equations,, Functional-differential 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.

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