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Exterior differential systems and prolongations for three important nonlinear partial differential equations
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 |
$\dot{x}(t) + \sum_{k=1}^m \int_{h_k(t)}^t x(s) d_s R_k(t,s) =0, h_k(t)\leq t,$ 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.
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. |
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. |
[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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