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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, 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, 191. Academic Press, Boston, MA, 1993. 
[2] 
V. Volterra, Fluctuations in the abundance of species considered mathematically, Nature, 118 (1926), 558560. 
[3] 
G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221246. 
[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), 489504. 
[6] 
L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Mathematical and Computer Modelling, 48 (2008), 287304. 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, SpringerVerlag, New York, 1993. 
[8] 
N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functionaldifferential equation of evolution type, Differential Equations, 13 (1977), 19151925, 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), 745754, 914, II. Differential Equations, 27 (1991), 383388, III. Differential Equations, 27 (1991), 11651172, IV. Differential Equations, 29 (1993), 153160. 
[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), 391411. 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), 233256. doi: 10.3934/dcdsb.2001.1.233. 
[13] 
S. A. Gusarenko, Criteria for the stability of a linear functionaldifferential equation, Boundary value problems, 4145, Perm. Politekh. Inst., Perm, 1987 (Russian). 
[14] 
S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functionaldifferential equations, Functionaldifferential equations, 3040, 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), 290312. 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), 558560. 
[3] 
G. E. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221246. 
[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), 489504. 
[6] 
L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Mathematical and Computer Modelling, 48 (2008), 287304. 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, SpringerVerlag, New York, 1993. 
[8] 
N. V. Azbelev, L. Berezansky and L. F. Rahmatullina, A linear functionaldifferential equation of evolution type, Differential Equations, 13 (1977), 19151925, 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), 745754, 914, II. Differential Equations, 27 (1991), 383388, III. Differential Equations, 27 (1991), 11651172, IV. Differential Equations, 29 (1993), 153160. 
[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), 391411. 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), 233256. doi: 10.3934/dcdsb.2001.1.233. 
[13] 
S. A. Gusarenko, Criteria for the stability of a linear functionaldifferential equation, Boundary value problems, 4145, Perm. Politekh. Inst., Perm, 1987 (Russian). 
[14] 
S. A. Gusarenko, Conditions for the solvability of problems on the accumulation of perturbations for functionaldifferential equations, Functionaldifferential equations, 3040, 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), 290312. doi: 10.1006/jmaa.1997.5883. 
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