# American Institute of Mathematical Sciences

May  2014, 13(3): 1105-1117. doi: 10.3934/cpaa.2014.13.1105

## Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients

 1 Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo -- Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil, Brazil

Received  April 2013 Revised  October 2013 Published  December 2013

We study the asymptotic behaviour of the solutions of a class of linear neutral delay differential equations with discrete delay where the coefficients of the non neutral part are periodic functions which are rational multiples of all time delays. We show that this technique is applicable to a broader class where the coefficients of the neutral part are periodic functions as well.
Citation: Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105
##### References:
 [1] R. D. Driver, Some harmless delays, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972), New York: Academic Press, (1972), 103-119. [2] R. D. Driver, D. W. Sasser and M. L. Slater, The equation $x' (t)=ax(t)+bx(t-\tau )$ with "small'' delay, Amer. Math. Monthly, 80 (1973), 990-995. doi: 10.2307/2318773. [3] M. V. S. Frasson, On the dominance of roots of characteristic equations for neutral functional differential equations, Appl. Math. Comput., 214 (2009), no. 1, 66-72. doi: 10.1016/j.amc.2009.03.058. [4] M. V. S. Frasson and S. M. Verduyn Lunel, Large time behaviour of linear functional differential equations, Integral Equations Operator Theory, 47 (2003), 91-121. doi: 10.1007/s00020-003-1155-x. [5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99 New York: Springer-Verlag, 1993. [6] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, vol. 463 of Mathematics and its Applications, Dordrecht: Kluwer Academic Publishers, 1999. [7] I.-G. E. Kordonis, N. T. Niyianni and C. G. Philos, On the behavior of the solutions of scalar first order linear autonomous neutral delay differential equations, Arch. Math. (Basel), 71 (1998), 454-464. doi: 10.1007/s000130050290. [8] J. C. Lillo, Periodic differential difference equations, J. Math. Anal. Appl., 15 (1966), 434-441. [9] C. G. Philos, Asymptotic behaviour, nonoscillation and stability in periodic first-order linear delay differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1371-1387. doi: 10.1017/S0308210500027372. [10] C. G. Philos and I. K. Purnaras, Periodic first order linear neutral delay differential equations, Appl. Math. Comput., 117 (2001), 203-222. doi: 10.1016/S0096-3003(99)00174-5.

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##### References:
 [1] R. D. Driver, Some harmless delays, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972), New York: Academic Press, (1972), 103-119. [2] R. D. Driver, D. W. Sasser and M. L. Slater, The equation $x' (t)=ax(t)+bx(t-\tau )$ with "small'' delay, Amer. Math. Monthly, 80 (1973), 990-995. doi: 10.2307/2318773. [3] M. V. S. Frasson, On the dominance of roots of characteristic equations for neutral functional differential equations, Appl. Math. Comput., 214 (2009), no. 1, 66-72. doi: 10.1016/j.amc.2009.03.058. [4] M. V. S. Frasson and S. M. Verduyn Lunel, Large time behaviour of linear functional differential equations, Integral Equations Operator Theory, 47 (2003), 91-121. doi: 10.1007/s00020-003-1155-x. [5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99 New York: Springer-Verlag, 1993. [6] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, vol. 463 of Mathematics and its Applications, Dordrecht: Kluwer Academic Publishers, 1999. [7] I.-G. E. Kordonis, N. T. Niyianni and C. G. Philos, On the behavior of the solutions of scalar first order linear autonomous neutral delay differential equations, Arch. Math. (Basel), 71 (1998), 454-464. doi: 10.1007/s000130050290. [8] J. C. Lillo, Periodic differential difference equations, J. Math. Anal. Appl., 15 (1966), 434-441. [9] C. G. Philos, Asymptotic behaviour, nonoscillation and stability in periodic first-order linear delay differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1371-1387. doi: 10.1017/S0308210500027372. [10] C. G. Philos and I. K. Purnaras, Periodic first order linear neutral delay differential equations, Appl. Math. Comput., 117 (2001), 203-222. doi: 10.1016/S0096-3003(99)00174-5.
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