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A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback

## Existence of strictly decreasing positive solutions of linear differential equations of neutral type

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

* Corresponding author: Josef Diblík

Received  December 2016 Revised  April 2017 Published  January 2019

The paper is concerned with a linear neutral differential equation
 $\dot y(t) = -c(t)y(t-\tau(t))+d(t)\dot y(t-\delta(t))$
where
 $c\colon [t_0,\infty)\to (0,\infty)$
,
 $d\colon [t_0,\infty)\to [0,\infty)$
,
 $t_0\in {\Bbb{R}}$
and
 $\tau, \delta \colon [t_0,\infty)\to (0,r]$
,
 $r\in{\mathbb{R}}$
,
 $r>0$
are continuous functions. A new criterion is given for the existence of positive strictly decreasing solutions. The proof is based on the Rybakowski variant of a topological Ważewski principle suitable for differential equations of the delayed type. Unlike in the previous investigations known, this time the progress is achieved by using a special system of initial functions satisfying a so-called sewing condition. The result obtained is extended to more general equations. Comparisons with known results are given as well.
Citation: Josef Diblík, Zdeněk Svoboda. Existence of strictly decreasing positive solutions of linear differential equations of neutral type. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020004
##### 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., 2004. doi: 10.1201/9780203025741. Google Scholar [3] D. D. Bainov and D. P. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, 1991. Google Scholar [4] 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 [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] L. Berezanski and E. Braverman, On oscillation of a logistic equation with several delays, J. Comput. and Appl. Mathem., 113 (2000), 255-265. doi: 10.1016/S0377-0427(99)00260-5. Google Scholar [7] 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 [8] J. Čermák, A change of variables in the asymptotic theory of differential equations with unbounded delay, J. Comput. Appl. Mathem., 143 (2002), 81-93. doi: 10.1016/S0377-0427(01)00500-3. Google Scholar [9] 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 [10] J. Diblík, Existence of solutions with asymptotic behavior of a certain type of some systems of delay functional-differential equations, (Russian) Sibirsk. Mat. Zh., 32 (1991), 55-60; translation in Siberian Math. J., 32 (1991), 222-226. doi: 10.1007/BF00972768. Google Scholar [11] 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 [12] 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 [13] J. Diblík and M. Kúdelčíková, Existence and asymptotic behavior of positive solutions of functional differential equations of delayed type, Abstr. Appl. Anal., 2011 (2011), Article ID 754701, 16 pages. doi: 10.1155/2011/754701. Google Scholar [14] J. Diblík and M. Kúdelčíková, Two classes of asymptotically different positive solutions of the equation $\dot y(t) = -f(t,y(t))$, Nonlinear Anal., 70 (2009), 3702-3714. doi: 10.1016/j.na.2008.07.026. Google Scholar [15] J. Diblík and M. Kúdelčíková, Two classes of positive solutions of first order functional differential equations of delayed type, Nonlinear Anal., 75 (2012), 4807-4820. doi: 10.1016/j.na.2012.03.030. Google Scholar [16] 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 [17] 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 [18] J. Diblík, Z. Svoboda and Z. Šmarda, Retract principle for neutral functional differential equations, Nonlinear Anal., 71 (2009), e1393-e1400. doi: 10.1016/j.na.2009.01.164. Google Scholar [19] 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 [20] 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 [21] B. Dorociaková and R. Olach, Existence of positive solutions of delay differential equations, Tatra Mt. Math. Publ., 43 (2009), 63-70. doi: 10.2478/v10127-009-0025-6. Google Scholar [22] 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 [23] L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995. Google Scholar [24] 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 [25] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. Google Scholar [26] J. K. Hale and Sjoerd M. Verdun Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [27] P. Hartman, Ordinary Differential Equations, Second Edition, SIAM, 2002. doi: 10.1137/1.9780898719222. Google Scholar [28] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. Google Scholar [29] 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 [30] B. S. Razumikhin, A method of studying the stability of systems with an aftereffect, Sov. Math., Dokl., 7 (1966), 559-562, translation from Dokl. Akad. Nauk SSSR, 167 (1966), 1234-1237. Google Scholar [31] B. S. Razumikhin, Stability of Hereditary Systems, Nauka, 1988, (Russian, Ustoichivost ereditarnykh sistem). Google Scholar [32] B. S. Razumikhin, The application of Lyapunov’s method to problems in the stability of systems with delay, Autom. Remote Control, 21 (1960), 515-520, translation from Avtom. Telemekh., 21 (1960), 740-748. Google Scholar [33] K. P. Rybakowski, A topological principle for retarded functional differential equations of Carathéodory type, J. Diff. Equat., 39 (1981), 131-150. doi: 10.1016/0022-0396(81)90070-X. Google Scholar [34] K. P. Rybakowski, Ważewski's principle for retarded functional differential equations, J. Diff. Equat., 36 (1980), 117-138. doi: 10.1016/0022-0396(80)90080-7. Google Scholar [35] 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 [36] R. Srzednicki, Ważewski Method and Conley Index, Handbook of Differential Equations, Ordinary Differential Equations, Edited by A. Cañada, P. Drábek and A. Fonda, Elsevier Inc., 1 (2004), 591-684. Google Scholar [37] 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 [38] T. Ważewski, Sur un principle topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles, Ann. Soc. Polon. Math., 20 (1947), 279-313. Google Scholar [39] D. Zhou, On a problem of I. Györi, J. Math. Anal. Appl., 183 (1994), 620-623. doi: 10.1006/jmaa.1994.1168. 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., 2004. doi: 10.1201/9780203025741. Google Scholar [3] D. D. Bainov and D. P. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, 1991. Google Scholar [4] 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 [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] L. Berezanski and E. Braverman, On oscillation of a logistic equation with several delays, J. Comput. and Appl. Mathem., 113 (2000), 255-265. doi: 10.1016/S0377-0427(99)00260-5. Google Scholar [7] 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 [8] J. Čermák, A change of variables in the asymptotic theory of differential equations with unbounded delay, J. Comput. Appl. Mathem., 143 (2002), 81-93. doi: 10.1016/S0377-0427(01)00500-3. Google Scholar [9] 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 [10] J. Diblík, Existence of solutions with asymptotic behavior of a certain type of some systems of delay functional-differential equations, (Russian) Sibirsk. Mat. Zh., 32 (1991), 55-60; translation in Siberian Math. J., 32 (1991), 222-226. doi: 10.1007/BF00972768. Google Scholar [11] 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 [12] 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 [13] J. Diblík and M. Kúdelčíková, Existence and asymptotic behavior of positive solutions of functional differential equations of delayed type, Abstr. Appl. Anal., 2011 (2011), Article ID 754701, 16 pages. doi: 10.1155/2011/754701. Google Scholar [14] J. Diblík and M. Kúdelčíková, Two classes of asymptotically different positive solutions of the equation $\dot y(t) = -f(t,y(t))$, Nonlinear Anal., 70 (2009), 3702-3714. doi: 10.1016/j.na.2008.07.026. Google Scholar [15] J. Diblík and M. Kúdelčíková, Two classes of positive solutions of first order functional differential equations of delayed type, Nonlinear Anal., 75 (2012), 4807-4820. doi: 10.1016/j.na.2012.03.030. Google Scholar [16] 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 [17] 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 [18] J. Diblík, Z. Svoboda and Z. Šmarda, Retract principle for neutral functional differential equations, Nonlinear Anal., 71 (2009), e1393-e1400. doi: 10.1016/j.na.2009.01.164. Google Scholar [19] 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 [20] 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 [21] B. Dorociaková and R. Olach, Existence of positive solutions of delay differential equations, Tatra Mt. Math. Publ., 43 (2009), 63-70. doi: 10.2478/v10127-009-0025-6. Google Scholar [22] 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 [23] L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995. Google Scholar [24] 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 [25] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. Google Scholar [26] J. K. Hale and Sjoerd M. Verdun Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [27] P. Hartman, Ordinary Differential Equations, Second Edition, SIAM, 2002. doi: 10.1137/1.9780898719222. Google Scholar [28] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. Google Scholar [29] 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 [30] B. S. Razumikhin, A method of studying the stability of systems with an aftereffect, Sov. Math., Dokl., 7 (1966), 559-562, translation from Dokl. Akad. Nauk SSSR, 167 (1966), 1234-1237. Google Scholar [31] B. S. Razumikhin, Stability of Hereditary Systems, Nauka, 1988, (Russian, Ustoichivost ereditarnykh sistem). Google Scholar [32] B. S. Razumikhin, The application of Lyapunov’s method to problems in the stability of systems with delay, Autom. Remote Control, 21 (1960), 515-520, translation from Avtom. Telemekh., 21 (1960), 740-748. Google Scholar [33] K. P. Rybakowski, A topological principle for retarded functional differential equations of Carathéodory type, J. Diff. Equat., 39 (1981), 131-150. doi: 10.1016/0022-0396(81)90070-X. Google Scholar [34] K. P. Rybakowski, Ważewski's principle for retarded functional differential equations, J. Diff. Equat., 36 (1980), 117-138. doi: 10.1016/0022-0396(80)90080-7. Google Scholar [35] 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 [36] R. Srzednicki, Ważewski Method and Conley Index, Handbook of Differential Equations, Ordinary Differential Equations, Edited by A. Cañada, P. Drábek and A. Fonda, Elsevier Inc., 1 (2004), 591-684. Google Scholar [37] 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 [38] T. Ważewski, Sur un principle topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles, Ann. Soc. Polon. Math., 20 (1947), 279-313. Google Scholar [39] D. Zhou, On a problem of I. Györi, J. Math. Anal. Appl., 183 (1994), 620-623. doi: 10.1006/jmaa.1994.1168. Google Scholar
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