American Institute of Mathematical Sciences

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

Stability of linear differential equations with a distributed delay

 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.

Citation: Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and 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. [2] V. Volterra, Fluctuations in the abundance of species considered mathematically, Nature, 118 (1926), 558-560. [3] G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. [4] A. D. Myshkis, "Linear Differential Equations with Retarded Argument," Nauka, Moscow, 1972. [5] L. Berezansky and E. Braverman, On oscillation of equations with distributed delay, Zeitschrift für Analysis und ihre Anwendungen, 20 (2001), 489-504. [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. [7] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [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. [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. [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. [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. [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. [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). [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). [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.

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References:
 [1] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191. Academic Press, Boston, MA, 1993. [2] V. Volterra, Fluctuations in the abundance of species considered mathematically, Nature, 118 (1926), 558-560. [3] G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. [4] A. D. Myshkis, "Linear Differential Equations with Retarded Argument," Nauka, Moscow, 1972. [5] L. Berezansky and E. Braverman, On oscillation of equations with distributed delay, Zeitschrift für Analysis und ihre Anwendungen, 20 (2001), 489-504. [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. [7] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [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. [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. [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. [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. [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. [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). [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). [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.
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