American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178
[2]

Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123
[3]

Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897
[4]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[5]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
[6]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[7]

Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245
[8]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[9]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509
[10]

Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3429-3440. doi: 10.3934/dcdsb.2016105
[11]

Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041
[12]

Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873
[13]

Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112
[14]

Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098
[15]

María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201
[16]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675
[17]

Jie Tang, Ziqing Xie, Zhimin Zhang. The long time behavior of a spectral collocation method for delay differential equations of pantograph type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 797-819. doi: 10.3934/dcdsb.2013.18.797
[18]

C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139
[19]

H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119
[20]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations & Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (140)
  • HTML views (657)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences