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November  2014, 34(11): 4577-4588. doi: 10.3934/dcds.2014.34.4577

## Delay-dependent stability criteria for neutral delay differential and difference equations

 1 Institute of Mathematics, Brno University of Technology, Technická 2, CZ-61669 Brno, Czech Republic, Czech Republic

Received  October 2013 Revised  January 2014 Published  May 2014

This paper discusses asymptotic stability properties of the neutral delay differential equation \begin{eqnarray*} y'(t) = a y (t) + b y ( t - \tau ) + c y'( t - \tau ),       t > 0, \\ \end{eqnarray*} where $a,\,b,\,c$ and $\tau >0$ are real scalars. We consider the exact as well as discretized delay-dependent asymptotic stability regions for this equation and describe them in terms of explicit necessary and sufficient conditions imposed on $a,\,b,\,c$ and $\tau$. Such descriptions enable us to observe some fundamental properties of these stability regions, especially with respect to stability of corresponding numerical formulae. As a consequence of our investigations, we extend existing results on this topic.
Citation: Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
##### References:
 [1] A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\Theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293. doi: 10.1016/S0168-9274(97)00026-3.  Google Scholar [2] A. Bellen, Z. Jackiewicz and M. Zennaro, Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52 (1988), 605-619. doi: 10.1007/BF01395814.  Google Scholar [3] A. Bellen and M. Zennaro, Numerical Methods For Delay Differential Equations, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [4] W. E. Brumley, On the asymptotic behavior of solutions of differential-difference equations of neutral type, J. Differential Equations, 7 (1970), 175-188. doi: 10.1016/0022-0396(70)90131-2.  Google Scholar [5] M. Calvo and T. Grande, On the asymptotic stability of $\Theta$-methods for delay differential equations, Numer. Math., 54 (1988), 257-269. doi: 10.1007/BF01396761.  Google Scholar [6] J. Čermák, The stability and asymptotic properties of the $\Theta$-methods for the pantograph equation, IMA J. Numer. Anal., 31 (2011), 1533-1551. doi: 10.1093/imanum/drq021.  Google Scholar [7] J. Čermák and J. Hrabalová, On stability regions for some delay differential equations and their discretizations,, Period. Math. Hung., ().   Google Scholar [8] J. Čermák, J. Jánský and P. Kundrát, On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations, J. Difference Equ. Appl., 18 (2012), 1781-1800. doi: 10.1080/10236198.2011.595406.  Google Scholar [9] S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005.  Google Scholar [10] H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcial. Ekvac., 34 (1991), 187-209.  Google Scholar [11] P. S. Gromova, Stability of solutions of nonlinear equations of the neutral type in the asymptotically critical case, Math. Notes, 1 (1967), 715-726.  Google Scholar [12] N. Guglielmi, Delay dependent stability regions of $\Theta$-methods for delay differential equations, IMA J. Numer. Anal., 18 (1998), 399-418. doi: 10.1093/imanum/18.3.399.  Google Scholar [13] N. Guglielmi, Asymptotic stability barriers for natural Runge-Kutta processes for delay equations, SIAM J. Numer. Anal., 39 (2001), 763-783. doi: 10.1137/S0036142900375396.  Google Scholar [14] N. Guglielmi, On the qualitative behaviour of numerical methods for delay differential equations of neutral type. A case study: $\Theta$-methods, Recent Trends in Numerical Analysis (L. Brugnano and D. Trigiante, eds.), 3 (2001), 175-184.  Google Scholar [15] N. D. Hayes, Roots of the transcendental equations associated with certain difference-differential equations, J. London Math. Soc., 25 (1950), 226-232.  Google Scholar [16] C. Huang, Delay-dependent stability of high order Runge-Kutta methods, Numer. Math., 111 (2009), 377-387. doi: 10.1007/s00211-008-0197-z.  Google Scholar [17] A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math., 24 (1997), 295-308. doi: 10.1016/S0168-9274(97)00027-5.  Google Scholar [18] Z. Jackiewicz, Asymptotic stability analysis of $\Theta$-methods for functional differential equations, Numer. Math., 43 (1984), 389-396. doi: 10.1007/BF01390181.  Google Scholar [19] S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation, J. Differential Equations, 256 (2014), 2368-2391. doi: 10.1016/j.jde.2014.01.004.  Google Scholar [20] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0.  Google Scholar [21] J. Kuang and Y. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005. Google Scholar [22] Y. Liu, On the $\Theta$-method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190. doi: 10.1016/0377-0427(95)00222-7.  Google Scholar [23] H. Matsunaga, Stability switches in a system of linear differential equations with diagonal delay, Appl. Math. Comput., 212 (2009), 145-152. doi: 10.1016/j.amc.2009.02.010.  Google Scholar [24] H. Matsunaga and H. Hashimoto, Asymptotic stability and stability switches in a linear integro-differential system, Differ. Equ. Appl., 3 (2011), 43-55. doi: 10.7153/dea-03-04.  Google Scholar [25] W. Snow, Existence, Uniqueness and Stability for Nonlinear Differential-Difference Equations in the Neutral Case, Thesis (Ph.D.)–New York University. 1964. 79 pp.  Google Scholar [26] V. V. Vlasov and D. A. Medvedev, Functional-differential equations and related problems in spectral theory, J. Math. Sci. (N. Y.), 164 (2010), 659-841. doi: 10.1007/s10958-010-9768-5.  Google Scholar

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##### References:
 [1] A. Bellen, N. Guglielmi and L. Torelli, Asymptotic stability properties of $\Theta$-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 279-293. doi: 10.1016/S0168-9274(97)00026-3.  Google Scholar [2] A. Bellen, Z. Jackiewicz and M. Zennaro, Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52 (1988), 605-619. doi: 10.1007/BF01395814.  Google Scholar [3] A. Bellen and M. Zennaro, Numerical Methods For Delay Differential Equations, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [4] W. E. Brumley, On the asymptotic behavior of solutions of differential-difference equations of neutral type, J. Differential Equations, 7 (1970), 175-188. doi: 10.1016/0022-0396(70)90131-2.  Google Scholar [5] M. Calvo and T. Grande, On the asymptotic stability of $\Theta$-methods for delay differential equations, Numer. Math., 54 (1988), 257-269. doi: 10.1007/BF01396761.  Google Scholar [6] J. Čermák, The stability and asymptotic properties of the $\Theta$-methods for the pantograph equation, IMA J. Numer. Anal., 31 (2011), 1533-1551. doi: 10.1093/imanum/drq021.  Google Scholar [7] J. Čermák and J. Hrabalová, On stability regions for some delay differential equations and their discretizations,, Period. Math. Hung., ().   Google Scholar [8] J. Čermák, J. Jánský and P. Kundrát, On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations, J. Difference Equ. Appl., 18 (2012), 1781-1800. doi: 10.1080/10236198.2011.595406.  Google Scholar [9] S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005.  Google Scholar [10] H. I. Freedman and Y. Kuang, Stability switches in linear scalar neutral delay equations, Funkcial. Ekvac., 34 (1991), 187-209.  Google Scholar [11] P. S. Gromova, Stability of solutions of nonlinear equations of the neutral type in the asymptotically critical case, Math. Notes, 1 (1967), 715-726.  Google Scholar [12] N. Guglielmi, Delay dependent stability regions of $\Theta$-methods for delay differential equations, IMA J. Numer. Anal., 18 (1998), 399-418. doi: 10.1093/imanum/18.3.399.  Google Scholar [13] N. Guglielmi, Asymptotic stability barriers for natural Runge-Kutta processes for delay equations, SIAM J. Numer. Anal., 39 (2001), 763-783. doi: 10.1137/S0036142900375396.  Google Scholar [14] N. Guglielmi, On the qualitative behaviour of numerical methods for delay differential equations of neutral type. A case study: $\Theta$-methods, Recent Trends in Numerical Analysis (L. Brugnano and D. Trigiante, eds.), 3 (2001), 175-184.  Google Scholar [15] N. D. Hayes, Roots of the transcendental equations associated with certain difference-differential equations, J. London Math. Soc., 25 (1950), 226-232.  Google Scholar [16] C. Huang, Delay-dependent stability of high order Runge-Kutta methods, Numer. Math., 111 (2009), 377-387. doi: 10.1007/s00211-008-0197-z.  Google Scholar [17] A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math., 24 (1997), 295-308. doi: 10.1016/S0168-9274(97)00027-5.  Google Scholar [18] Z. Jackiewicz, Asymptotic stability analysis of $\Theta$-methods for functional differential equations, Numer. Math., 43 (1984), 389-396. doi: 10.1007/BF01390181.  Google Scholar [19] S. Junca and B. Lombard, Stability of a critical nonlinear neutral delay differential equation, J. Differential Equations, 256 (2014), 2368-2391. doi: 10.1016/j.jde.2014.01.004.  Google Scholar [20] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0.  Google Scholar [21] J. Kuang and Y. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005. Google Scholar [22] Y. Liu, On the $\Theta$-method for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190. doi: 10.1016/0377-0427(95)00222-7.  Google Scholar [23] H. Matsunaga, Stability switches in a system of linear differential equations with diagonal delay, Appl. Math. Comput., 212 (2009), 145-152. doi: 10.1016/j.amc.2009.02.010.  Google Scholar [24] H. Matsunaga and H. Hashimoto, Asymptotic stability and stability switches in a linear integro-differential system, Differ. Equ. Appl., 3 (2011), 43-55. doi: 10.7153/dea-03-04.  Google Scholar [25] W. Snow, Existence, Uniqueness and Stability for Nonlinear Differential-Difference Equations in the Neutral Case, Thesis (Ph.D.)–New York University. 1964. 79 pp.  Google Scholar [26] V. V. Vlasov and D. A. Medvedev, Functional-differential equations and related problems in spectral theory, J. Math. Sci. (N. Y.), 164 (2010), 659-841. doi: 10.1007/s10958-010-9768-5.  Google Scholar
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