June  2017, 37(6): 3211-3242. doi: 10.3934/dcds.2017137

Sharp estimation for the solutions of delay differential and Halanay type inequalities

Department of Mathematics, University of Pannonia, 8200 Veszprém, Egyetem u. 10., Hungary

Received  October 2016 Revised  January 2017 Published  February 2017

Fund Project: The research of the authors has been supported by Hungarian National Foundations for Scientific Research Grant No. K120186.

The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. The emphasis is put on the rate of convergence based on the theory of the generalized characteristic equation. The applicability and the sharpness of the results are illustrated by examples. This work aspires to serve as a remarkable step towards a unified theory of the nonautonomous Halanay inequality.

Citation: István Győri, László Horváth. Sharp estimation for the solutions of delay differential and Halanay type inequalities. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3211-3242. doi: 10.3934/dcds.2017137
References:
[1]

M. Adivar and E. A. Bohner, Halanay type inequalities on time scales with applications, Nonlinear Analysis, 74 (2011), 7519-7531.  doi: 10.1016/j.na.2011.08.007.

[2]

R. P. AgarwalY. H. Kim and S. K. Sen, New discrete Halanay inequalities: Stability of difference equations, Communications in Applied Analysis, 12 (2008), 83-90. 

[3]

R. P. Agarwal, Y. H. Kim and S. K. Sen, Advanced discrete Halanay type inequalities: Stability of difference equations Journal of Inequalities and Applications (2009), Art. ID 535849, 11 pp. doi: 10.1155/2009/535849.

[4]

E. AkınY. N. Raffoul and C. Tisdell, Exponential stability in functional dynamic equations on time scales, Commun. Math. Anal., 9 (2010), 93-108. 

[5]

C. T. H. Baker and A. Tang, Generalized Halanay inequalities for Volterra functional differential equations and discretized versions, in Volterra Equations and Applications (eds. C. Corduneanu and I Sandberg), Stability Control Theory Methods Appl., vol. 10, Gordon and Breach, Amsterdam, (2000), 39-55. 

[6]

C. T. H. Baker, Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag, J. Comput. Appl. Math., 234 (2010), 2663-2682.  doi: 10.1016/j.cam.2010.01.027.

[7]

L. Berezansky and E. Braverman, On oscillation of a second order impulsive linear delay differential equation, J. Math. Anal. Appl., 233 (1999), 276-300.  doi: 10.1006/jmaa.1999.6297.

[8]

S. N. Busenberg and K. L. Cooke, Stabilty conditions for linear non-autonomous delay differential equations, Quarterly of Applied Math., 42 (1984), 295-306.  doi: 10.1090/qam/757167.

[9]

C. Cuevas and M. V. S. Frasson, Asymptotic properties of solutions to linear nonautonomous delay differential equations through generalized characteristic equations, Electron. J. Qual. Theory Differ. Equ., No. 95 (2010), 1-5. 

[10]

J. G. DixC. G. Philos and I. K. Purnaras, An asymptotic property of solutions to linear nonautonomous delay differential equations, Electron.J. Differential Equations, (2005), 1-9. 

[11]

J. G. DixC. G. Philos and I. K. Purnaras, Asymptotic properties of solutions to linear non-autonomous neutral differential equations, J. Math. Anal. Appl., 318 (2006), 296-304.  doi: 10.1016/j.jmaa.2005.06.005.

[12]

R. D. Driver, Ordinary and Delay Differential Equations Applied Mathematics Series 20, Springer-Verlag, New York, 1977.

[13]

T. M. Flett, Differential Analysis Cambridge University Press, 1980.

[14]

I. Győri, Oscillation conditions in scalar linear delay differential equations, Bull. Austral. Math. Soc., 34 (1986), 1-9.  doi: 10.1017/S0004972700004457.

[15]

I. Győri and L. Horváth, Sharp Gronwall-Bellman type integral inequalities with delay, Electron. J. Qual. Theory Differ. Equ., (2016), 1-25. 

[16]

I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations Oxford Science Publications, Oxford, 1991.

[17]

A. Halanay, Differential Equations: Stability, Oscillations, Time Lags Academic Press, New York, 1966.

[18]

J. K. Hale, Theory of Functional Differential Equations Applied Math. Sciences, Vol. 3, Springer-Verlag, New York, 1977.

[19]

J. Hoffacker and C. Tisdell, Stability and instability for dynamic equations on time scales, Comput. Math. Appl., 49 (2005), 1327-1334.  doi: 10.1016/j.camwa.2005.01.016.

[20]

A. IvanovE. Liz and S. Trofimchuk, Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima, Tohoku Math. J., 54 (2002), 277-295.  doi: 10.2748/tmj/1113247567.

[21]

E. Kaufmann and Y.N. Raffoul, Periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale, Electron. J. Differential Equations, (2007), 1-12. 

[22]

B. Li, Oscillation of first order delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 3729-3737.  doi: 10.1090/S0002-9939-96-03674-X.

[23]

E. Liz and J. B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett., 15 (2002), 655-659.  doi: 10.1016/S0893-9659(02)00024-1.

[24]

E. Liz and S. Trofimchuk, Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality, J. Math. Anal. Appl., 248 (2000), 625-644.  doi: 10.1006/jmaa.2000.6947.

[25]

S. Mohamad and K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Austral Math. Soc., 61 (2000), 371-385.  doi: 10.1017/S0004972700022413.

[26]

B. OuB. Jia and L. Erbe, An extended Halanay inequality of integral type on time scales, Electron. J. Qual. Theory Differ. Equ., (2015), 1-11. 

[27]

C. G. Philos and I. K. Purnaras, An asymptotic result for second order linear nonautonomous neutral delay differential equations, Hiroshima Math. J., 40 (2010), 47-63. 

[28]

H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl., 270 (2002), 143-149.  doi: 10.1016/S0022-247X(02)00056-2.

[29]

S. Udpin and P. Niamsup, New discrete type inequalities and global stability of nonlinear difference equations, Appl. Math. Lett., 22 (2009), 856-859.  doi: 10.1016/j.aml.2008.07.011.

[30]

W. Wang, A generalized Halanay inequality for stability of nonlinear neutral functional differential equations J. Inequal. Appl. (2010), Art. ID 475019, 16 pp. doi: 10.1155/2010/475019.

[31]

L. WenW. Wang and Y. Yu, Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 1746-1754.  doi: 10.1016/j.na.2009.09.016.

[32]

L. WenY. Yu and W. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169-178.  doi: 10.1016/j.jmaa.2008.05.007.

[33]

D. Xu and Z. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl., 305 (2005), 107-120.  doi: 10.1016/j.jmaa.2004.10.040.

[34]

T. Yoneyama and J. Sugie, Perturbing uniformly stable nonlinear scalar delay-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 303-311.  doi: 10.1016/0362-546X(88)90116-2.

show all references

References:
[1]

M. Adivar and E. A. Bohner, Halanay type inequalities on time scales with applications, Nonlinear Analysis, 74 (2011), 7519-7531.  doi: 10.1016/j.na.2011.08.007.

[2]

R. P. AgarwalY. H. Kim and S. K. Sen, New discrete Halanay inequalities: Stability of difference equations, Communications in Applied Analysis, 12 (2008), 83-90. 

[3]

R. P. Agarwal, Y. H. Kim and S. K. Sen, Advanced discrete Halanay type inequalities: Stability of difference equations Journal of Inequalities and Applications (2009), Art. ID 535849, 11 pp. doi: 10.1155/2009/535849.

[4]

E. AkınY. N. Raffoul and C. Tisdell, Exponential stability in functional dynamic equations on time scales, Commun. Math. Anal., 9 (2010), 93-108. 

[5]

C. T. H. Baker and A. Tang, Generalized Halanay inequalities for Volterra functional differential equations and discretized versions, in Volterra Equations and Applications (eds. C. Corduneanu and I Sandberg), Stability Control Theory Methods Appl., vol. 10, Gordon and Breach, Amsterdam, (2000), 39-55. 

[6]

C. T. H. Baker, Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag, J. Comput. Appl. Math., 234 (2010), 2663-2682.  doi: 10.1016/j.cam.2010.01.027.

[7]

L. Berezansky and E. Braverman, On oscillation of a second order impulsive linear delay differential equation, J. Math. Anal. Appl., 233 (1999), 276-300.  doi: 10.1006/jmaa.1999.6297.

[8]

S. N. Busenberg and K. L. Cooke, Stabilty conditions for linear non-autonomous delay differential equations, Quarterly of Applied Math., 42 (1984), 295-306.  doi: 10.1090/qam/757167.

[9]

C. Cuevas and M. V. S. Frasson, Asymptotic properties of solutions to linear nonautonomous delay differential equations through generalized characteristic equations, Electron. J. Qual. Theory Differ. Equ., No. 95 (2010), 1-5. 

[10]

J. G. DixC. G. Philos and I. K. Purnaras, An asymptotic property of solutions to linear nonautonomous delay differential equations, Electron.J. Differential Equations, (2005), 1-9. 

[11]

J. G. DixC. G. Philos and I. K. Purnaras, Asymptotic properties of solutions to linear non-autonomous neutral differential equations, J. Math. Anal. Appl., 318 (2006), 296-304.  doi: 10.1016/j.jmaa.2005.06.005.

[12]

R. D. Driver, Ordinary and Delay Differential Equations Applied Mathematics Series 20, Springer-Verlag, New York, 1977.

[13]

T. M. Flett, Differential Analysis Cambridge University Press, 1980.

[14]

I. Győri, Oscillation conditions in scalar linear delay differential equations, Bull. Austral. Math. Soc., 34 (1986), 1-9.  doi: 10.1017/S0004972700004457.

[15]

I. Győri and L. Horváth, Sharp Gronwall-Bellman type integral inequalities with delay, Electron. J. Qual. Theory Differ. Equ., (2016), 1-25. 

[16]

I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations Oxford Science Publications, Oxford, 1991.

[17]

A. Halanay, Differential Equations: Stability, Oscillations, Time Lags Academic Press, New York, 1966.

[18]

J. K. Hale, Theory of Functional Differential Equations Applied Math. Sciences, Vol. 3, Springer-Verlag, New York, 1977.

[19]

J. Hoffacker and C. Tisdell, Stability and instability for dynamic equations on time scales, Comput. Math. Appl., 49 (2005), 1327-1334.  doi: 10.1016/j.camwa.2005.01.016.

[20]

A. IvanovE. Liz and S. Trofimchuk, Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima, Tohoku Math. J., 54 (2002), 277-295.  doi: 10.2748/tmj/1113247567.

[21]

E. Kaufmann and Y.N. Raffoul, Periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale, Electron. J. Differential Equations, (2007), 1-12. 

[22]

B. Li, Oscillation of first order delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 3729-3737.  doi: 10.1090/S0002-9939-96-03674-X.

[23]

E. Liz and J. B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett., 15 (2002), 655-659.  doi: 10.1016/S0893-9659(02)00024-1.

[24]

E. Liz and S. Trofimchuk, Existence and stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality, J. Math. Anal. Appl., 248 (2000), 625-644.  doi: 10.1006/jmaa.2000.6947.

[25]

S. Mohamad and K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Austral Math. Soc., 61 (2000), 371-385.  doi: 10.1017/S0004972700022413.

[26]

B. OuB. Jia and L. Erbe, An extended Halanay inequality of integral type on time scales, Electron. J. Qual. Theory Differ. Equ., (2015), 1-11. 

[27]

C. G. Philos and I. K. Purnaras, An asymptotic result for second order linear nonautonomous neutral delay differential equations, Hiroshima Math. J., 40 (2010), 47-63. 

[28]

H. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl., 270 (2002), 143-149.  doi: 10.1016/S0022-247X(02)00056-2.

[29]

S. Udpin and P. Niamsup, New discrete type inequalities and global stability of nonlinear difference equations, Appl. Math. Lett., 22 (2009), 856-859.  doi: 10.1016/j.aml.2008.07.011.

[30]

W. Wang, A generalized Halanay inequality for stability of nonlinear neutral functional differential equations J. Inequal. Appl. (2010), Art. ID 475019, 16 pp. doi: 10.1155/2010/475019.

[31]

L. WenW. Wang and Y. Yu, Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 1746-1754.  doi: 10.1016/j.na.2009.09.016.

[32]

L. WenY. Yu and W. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169-178.  doi: 10.1016/j.jmaa.2008.05.007.

[33]

D. Xu and Z. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl., 305 (2005), 107-120.  doi: 10.1016/j.jmaa.2004.10.040.

[34]

T. Yoneyama and J. Sugie, Perturbing uniformly stable nonlinear scalar delay-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 303-311.  doi: 10.1016/0362-546X(88)90116-2.

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