# American Institute of Mathematical Sciences

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

 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.

Citation: Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, 1977. [19] L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995. [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. [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. [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. [23] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. [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. [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. [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. [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. [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. [29] P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp. [30] V. Kolmanovski and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992.  doi: 10.1007/978-94-015-8084-7. [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. [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. [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. [34] D. Zhou, On a problem of I. Györi, J. Math. Anal. Appl., 183 (1994), 620-623.  doi: 10.1006/jmaa.1994.1168. [35] D. Zhou, Negative answer to a problem of Győri, J. Shandong University, 24 (1989), 117-121. [In Chinese]

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [18] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, 1977. [19] L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995. [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. [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. [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. [23] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. [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. [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. [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. [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. [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. [29] P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp. [30] V. Kolmanovski and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992.  doi: 10.1007/978-94-015-8084-7. [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. [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. [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. [34] D. Zhou, On a problem of I. Györi, J. Math. Anal. Appl., 183 (1994), 620-623.  doi: 10.1006/jmaa.1994.1168. [35] D. Zhou, Negative answer to a problem of Győri, J. Shandong University, 24 (1989), 117-121. [In Chinese]
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