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January  2020, 13(1): 31-46. doi: 10.3934/dcdss.2020002

Long-time behavior of positive solutions of a differential equation with state-dependent delay

Josef Diblík

CEITEC - Central European Institute of Technology, Brno University of Technology, Brno, Czech Republic

Received  December 2016 Revised  April 2017 Published  January 2019

The long-time behavior of positive solutions of a differential equation with state-dependent delay $ \dot{y}(t) = -c(t)y(t-\tau(t,y(t))) $, where $ c $ is a positive coefficient, is considered. Sufficient conditions are given for the existence of positive solutions bounded from below and from above by functions of exponential type. As a consequence, criteria for the existence of positive solutions are derived and their lower bounds are given. Relationships are discussed with the existing results on the existence of positive solutions for delayed differential equations.

Keywords: Long-time behavior, time-dependent delay, exponential estimation of solutions, positive solution, monotone iterative method.
Mathematics Subject Classification: Primary: 34K25; Secondary: 34K12.
Citation: Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002
References:
[1]

R. P. Agarwal, L. Berezanski, E. Braverman and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-3455-9.  Google Scholar

[2]

R. P. Agarwal, M. Bohner and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, Inc., New York, 2004. doi: 10.1201/9780203025741.  Google Scholar

[3]

J. Baštinec, J. Diblík and Z. Šmarda, An explicit criterion for the existence of positive solutions of the linear delayed equation $ \dot x(t) = -c(t)x(t-\tau(t) $, Abstr. Appl. Anal., 2011 (2011), Article ID 561902, 12 pages. doi: 10.1155/2011/561902.  Google Scholar

[4]

L. Berezanski, J. Diblík and Z. Šmarda, Positive solutions of a second-order delay differential equations with a damping term, Comput. Math. Appl., 60 (2010), 1332-1342.  doi: 10.1016/j.camwa.2010.06.014.  Google Scholar

[5]

J. Baštinec, L. Berezansky, J. Diblík and Z. Šmarda, On the critical case in oscillation for differential equations with a single delay and with several delays, Abstr. Appl. Anal., 2010 (2010), Article ID 417869, 20 pages. doi: 10.1155/2010/417869.  Google Scholar

[6]

K. L. Cooke, Asymptotic theory for the delay-differential equation $ u'(t) = au(t-r(u(t)) $, J. Math. Anal. Appl., 19 (1967), 160-173.  doi: 10.1016/0022-247X(67)90029-7.  Google Scholar

[7]

J. Diblík, A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonl. Anal., TMA, 38 (1999), 327-339.  doi: 10.1016/S0362-546X(98)00199-0.  Google Scholar

[8]

J. Diblík, Criteria for the existence of positive solutions to delayed functional differential equations, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 68, 15 pp. doi: 10.14232/ejqtde.2016.1.68.  Google Scholar

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J. Diblík, Positive and oscillating solutions of differential equations with delay in critical case, J. Comput. Appl. Mathem., 88 (1998), 185-202.  doi: 10.1016/S0377-0427(97)00217-3.  Google Scholar

[10]

J. Diblík and N. Koksch, Positive solutions of the equation $ \dot x(t) = -c(t)x(t-\tau) $ in the critical case, J. Math. Anal. Appl., 250 (2000), 635-659.  doi: 10.1006/jmaa.2000.7008.  Google Scholar

[11]

J. Diblík and Z. Svoboda, An existence criterion of positive solutions of $ p $-type retarded functional differential equations, J. Comput. Appl. Math., 147 (2002), 315-331.  doi: 10.1016/S0377-0427(02)00439-9.  Google Scholar

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J. Diblík, Z. Svoboda and Z. Šmarda, Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case, Comput. Math. Appl., 56 (2008), 556-564.  doi: 10.1016/j.camwa.2008.01.015.  Google Scholar

[13]

A. Domoshnitsky and M. Drakhlin, Nonoscillation of first order differential equations with delay, J. Math. Anal. Appl., 206 (1997), 254-269.  doi: 10.1006/jmaa.1997.5231.  Google Scholar

[14]

A. Domoshnitsky, M. Drakhlin and E. Litsyn, Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments, J. Differential Equations, 228 (2006), 39-48.  doi: 10.1016/j.jde.2006.05.009.  Google Scholar

[15]

A. Domoshnitsky, M. Drakhlin and E. Litsyn, On equations with delay depending on solution, Nonlinear Anal., 49 (2002), 689-701.  doi: 10.1016/S0362-546X(01)00132-8.  Google Scholar

[16]

Y. Domshlak and I. P. Stavroulakis, Oscillation of first-order delay differential equations in a critical state, Appl. Anal., 61 (1996), 359-371.  doi: 10.1080/00036819608840464.  Google Scholar

[17]

Á. Elbert and I. P. Stavroulakis, Oscillation and non-oscillation criteria for delay differential equations, Proc. Amer. Math. Soc., 123 (1995), 1503-1510.  doi: 10.1090/S0002-9939-1995-1242082-1.  Google Scholar

[18]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, 1977.  Google Scholar

[19]

L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.  Google Scholar

[20]

J. Gallardo and M. Pinto, Asymptotic integration of nonautonomous delay-differential systems, J. Math. Anal. Appl., 199 (1996), 654-675.  doi: 10.1006/jmaa.1996.0168.  Google Scholar

[21] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992.  doi: 10.1007/978-94-015-7920-9.  Google Scholar
[22]

I. Györi and F. Hartung, On equi-stability with respect to parameters in functional differential equations, Nonlinear Funct. Anal. Appl., 7 (2002), 329-351.   Google Scholar

[23] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.   Google Scholar
[24] J. K. Hale and S. M. Verdun Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.  doi: 10.1007/978-1-4612-4342-7.  Google Scholar
[25]

F. Hartung, T. Kristin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in Handbook of Differential Equations, Ordinary Differential Equations, vol. 3, Edited by A. Cañada, P. Drábek and A. Fonda, Elsewier, 2006,435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

[26]

R. G. Koplatadze and T. A. Chanturia, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differentsialnyje Uravnenija, 18 (1982), 1463-1465.   Google Scholar

[27]

M. Pinto, Asymptotic integration of the functional-differential equation $ y'(t) = a(t)y(t-r(t,y)) $, J. Math. Anal. Appl., 175 (1993), 46-52.  doi: 10.1006/jmaa.1993.1150.  Google Scholar

[28]

M. Pituk and G. Röst, Large time behavior of a linear delay differential equation with asymptotically small coefficient, Bound. Value Probl., 2014 (2014), 1-9.  doi: 10.1186/1687-2770-2014-114.  Google Scholar

[29]

P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp.  Google Scholar

[30] V. Kolmanovski and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992.  doi: 10.1007/978-94-015-8084-7.  Google Scholar
[31]

V. E. Sljusarchuk, The necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with pulse influence in the Banach space, Ukrain. Mat. Zh., 51 (1999), 98-109.  doi: 10.1007/BF02591918.  Google Scholar

[32]

I. P. Stavroulakis, Oscillation criteria for first order delay difference equations, Mediterr. J. Math., 1 (2004), 231-240.  doi: 10.1007/s00009-004-0013-7.  Google Scholar

[33] E. Zeidler, Nonlinear Functional Analysis and its Application, part Ⅰ, Fixed-Point Theorems, Springer-Verlag, New York, 1986.  doi: 10.1007/978-1-4612-5020-3.  Google Scholar
[34]

D. Zhou, On a problem of I. Györi, J. Math. Anal. Appl., 183 (1994), 620-623.  doi: 10.1006/jmaa.1994.1168.  Google Scholar

[35]

D. Zhou, Negative answer to a problem of Győri, J. Shandong University, 24 (1989), 117-121. [In Chinese] Google Scholar

show all references

References:
[1]

R. P. Agarwal, L. Berezanski, E. Braverman and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012. doi: 10.1007/978-1-4614-3455-9.  Google Scholar

[2]

R. P. Agarwal, M. Bohner and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, Inc., New York, 2004. doi: 10.1201/9780203025741.  Google Scholar

[3]

J. Baštinec, J. Diblík and Z. Šmarda, An explicit criterion for the existence of positive solutions of the linear delayed equation $ \dot x(t) = -c(t)x(t-\tau(t) $, Abstr. Appl. Anal., 2011 (2011), Article ID 561902, 12 pages. doi: 10.1155/2011/561902.  Google Scholar

[4]

L. Berezanski, J. Diblík and Z. Šmarda, Positive solutions of a second-order delay differential equations with a damping term, Comput. Math. Appl., 60 (2010), 1332-1342.  doi: 10.1016/j.camwa.2010.06.014.  Google Scholar

[5]

J. Baštinec, L. Berezansky, J. Diblík and Z. Šmarda, On the critical case in oscillation for differential equations with a single delay and with several delays, Abstr. Appl. Anal., 2010 (2010), Article ID 417869, 20 pages. doi: 10.1155/2010/417869.  Google Scholar

[6]

K. L. Cooke, Asymptotic theory for the delay-differential equation $ u'(t) = au(t-r(u(t)) $, J. Math. Anal. Appl., 19 (1967), 160-173.  doi: 10.1016/0022-247X(67)90029-7.  Google Scholar

[7]

J. Diblík, A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonl. Anal., TMA, 38 (1999), 327-339.  doi: 10.1016/S0362-546X(98)00199-0.  Google Scholar

[8]

J. Diblík, Criteria for the existence of positive solutions to delayed functional differential equations, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), Paper No. 68, 15 pp. doi: 10.14232/ejqtde.2016.1.68.  Google Scholar

[9]

J. Diblík, Positive and oscillating solutions of differential equations with delay in critical case, J. Comput. Appl. Mathem., 88 (1998), 185-202.  doi: 10.1016/S0377-0427(97)00217-3.  Google Scholar

[10]

J. Diblík and N. Koksch, Positive solutions of the equation $ \dot x(t) = -c(t)x(t-\tau) $ in the critical case, J. Math. Anal. Appl., 250 (2000), 635-659.  doi: 10.1006/jmaa.2000.7008.  Google Scholar

[11]

J. Diblík and Z. Svoboda, An existence criterion of positive solutions of $ p $-type retarded functional differential equations, J. Comput. Appl. Math., 147 (2002), 315-331.  doi: 10.1016/S0377-0427(02)00439-9.  Google Scholar

[12]

J. Diblík, Z. Svoboda and Z. Šmarda, Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case, Comput. Math. Appl., 56 (2008), 556-564.  doi: 10.1016/j.camwa.2008.01.015.  Google Scholar

[13]

A. Domoshnitsky and M. Drakhlin, Nonoscillation of first order differential equations with delay, J. Math. Anal. Appl., 206 (1997), 254-269.  doi: 10.1006/jmaa.1997.5231.  Google Scholar

[14]

A. Domoshnitsky, M. Drakhlin and E. Litsyn, Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments, J. Differential Equations, 228 (2006), 39-48.  doi: 10.1016/j.jde.2006.05.009.  Google Scholar

[15]

A. Domoshnitsky, M. Drakhlin and E. Litsyn, On equations with delay depending on solution, Nonlinear Anal., 49 (2002), 689-701.  doi: 10.1016/S0362-546X(01)00132-8.  Google Scholar

[16]

Y. Domshlak and I. P. Stavroulakis, Oscillation of first-order delay differential equations in a critical state, Appl. Anal., 61 (1996), 359-371.  doi: 10.1080/00036819608840464.  Google Scholar

[17]

Á. Elbert and I. P. Stavroulakis, Oscillation and non-oscillation criteria for delay differential equations, Proc. Amer. Math. Soc., 123 (1995), 1503-1510.  doi: 10.1090/S0002-9939-1995-1242082-1.  Google Scholar

[18]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, 1977.  Google Scholar

[19]

L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.  Google Scholar

[20]

J. Gallardo and M. Pinto, Asymptotic integration of nonautonomous delay-differential systems, J. Math. Anal. Appl., 199 (1996), 654-675.  doi: 10.1006/jmaa.1996.0168.  Google Scholar

[21] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992.  doi: 10.1007/978-94-015-7920-9.  Google Scholar
[22]

I. Györi and F. Hartung, On equi-stability with respect to parameters in functional differential equations, Nonlinear Funct. Anal. Appl., 7 (2002), 329-351.   Google Scholar

[23] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991.   Google Scholar
[24] J. K. Hale and S. M. Verdun Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.  doi: 10.1007/978-1-4612-4342-7.  Google Scholar
[25]

F. Hartung, T. Kristin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in Handbook of Differential Equations, Ordinary Differential Equations, vol. 3, Edited by A. Cañada, P. Drábek and A. Fonda, Elsewier, 2006,435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

[26]

R. G. Koplatadze and T. A. Chanturia, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differentsialnyje Uravnenija, 18 (1982), 1463-1465.   Google Scholar

[27]

M. Pinto, Asymptotic integration of the functional-differential equation $ y'(t) = a(t)y(t-r(t,y)) $, J. Math. Anal. Appl., 175 (1993), 46-52.  doi: 10.1006/jmaa.1993.1150.  Google Scholar

[28]

M. Pituk and G. Röst, Large time behavior of a linear delay differential equation with asymptotically small coefficient, Bound. Value Probl., 2014 (2014), 1-9.  doi: 10.1186/1687-2770-2014-114.  Google Scholar

[29]

P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp.  Google Scholar

[30] V. Kolmanovski and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992.  doi: 10.1007/978-94-015-8084-7.  Google Scholar
[31]

V. E. Sljusarchuk, The necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with pulse influence in the Banach space, Ukrain. Mat. Zh., 51 (1999), 98-109.  doi: 10.1007/BF02591918.  Google Scholar

[32]

I. P. Stavroulakis, Oscillation criteria for first order delay difference equations, Mediterr. J. Math., 1 (2004), 231-240.  doi: 10.1007/s00009-004-0013-7.  Google Scholar

[33] E. Zeidler, Nonlinear Functional Analysis and its Application, part Ⅰ, Fixed-Point Theorems, Springer-Verlag, New York, 1986.  doi: 10.1007/978-1-4612-5020-3.  Google Scholar
[34]

D. Zhou, On a problem of I. Györi, J. Math. Anal. Appl., 183 (1994), 620-623.  doi: 10.1006/jmaa.1994.1168.  Google Scholar

[35]

D. Zhou, Negative answer to a problem of Győri, J. Shandong University, 24 (1989), 117-121. [In Chinese] Google Scholar

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