# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021198
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## Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

 1 College of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610066, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Jun Shen, junshen85@163.com

Received  April 2021 Revised  June 2021 Early access August 2021

Fund Project: This work was supported by NSFC #11701400, #11831012, #12090013 and #12071317, and Sichuan Science and Technology Program #2020YJ0328

In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation $\dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)).$ As $n = 2$, this equation can be regarded as a mixed-type functional differential equation with state-dependence $\dot{x}(t) = f(t,x(t),x(T(t,x(t))))$ of a special form but, being a nonlinear operator, $n$-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.

Citation: Jun Zhou, Jun Shen. Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021198
##### References:
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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. [8] R. D. Driver, A two-body problem of classical electrodynamics: The one-dimensional case, Ann. Phys., 21 (1963), 122-142.  doi: 10.1016/0003-4916(63)90227-6. [9] G. M. Dunkel, On nested functional differential equations, SIAM J. Appl. Math., 18 (1970), 514-525.  doi: 10.1137/0118044. [10] E. Eder, The functional differential equation $x'(t) = x(x(t))$, J. Diff. Eqns., 54 (1984), 390-400.  doi: 10.1016/0022-0396(84)90150-5. [11] M. Fečkan, On a certain type of functional differential equations, Math. Slovaca, 43 (1993), 39-43. [12] C. G. Gal, Nonlinear abstract differential equations with deviated argument, J. Math. Anal. Appl., 333 (2007), 971-983.  doi: 10.1016/j.jmaa.2006.11.033. [13] P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, J. Diff. 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Kreher, A weak law of large numbers for a limit order book model with fully state dependent order dynamics, SIAM J. Financ. Math., 8 (2017), 314-343.  doi: 10.1137/15M1024226. [19] Q. Hu, A model of cold metal rolling processes with state-dependent delay, SIAM J. Appl. Math., 76 (2016), 1076-1100.  doi: 10.1137/141000257. [20] Q. Hu, W. Krawcewicz and J. Turi, Stabilization in a state-dependent model of turning processes, SIAM J. Appl. Math., 72 (2012), 1-24.  doi: 10.1137/110823468. [21] B. Kennedy, The Poincaré-Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback, J. Diff. Eqns., 266 (2019), 1865-1898.  doi: 10.1016/j.jde.2018.08.012. [22] M. Kloosterman, S. A. Campbell and F. J. Poulin, An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton, SIAM J. Appl. Math., 76 (2016), 551-577.  doi: 10.1137/15M1021271. [23] M. A. Krasnoselskii, Positive Solutions of Operator Equations, Translated from the Russian by Richard E. Flaherty; Edited by Leo F. Boron P. Noordhoff Ltd. Groningen, 1964. [24] Y. Kuang, $3/2$ stability results for nonautonomous state-dependent delay differential equations, Differential Equations and Applications to Biology and to Industry (Claremont, CA, 1994), World Sci. Publ., River Edge, NJ, (1996), 261–269. [25] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639. [26] K. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Diff. Eqns., 148 (1998), 407-421.  doi: 10.1006/jdeq.1998.3475. [27] Y. Liu, Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Appl. Math. Comput., 144 (2003), 543-556.  doi: 10.1016/S0096-3003(02)00431-9. [28] J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Diff. Eqns., 250 (2011), 4085-4103.  doi: 10.1016/j.jde.2010.10.023. [29] H. Müller-Krumbhaar and J. P. v. d. Eerden, Some properties of simple recursive differential equations, Z. Phys. B: Condensed Matter, 67 (1987), 239-242.  doi: 10.1007/BF01303988. [30] R. Oberg, On the local existence of solutions of certain functional-differential equations, Proc. Amer. Math. Soc., 20 (1969), 295-302.  doi: 10.1090/S0002-9939-1969-0234094-6. [31] J. Si and X. Wang, Smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients, J. Math. Anal. Appl., 226 (1998), 377-392.  doi: 10.1006/jmaa.1998.6086. [32] J. Si, X. Wang and S. Cheng, Nondecreasing and convex $C^2$-solutions of an iterative functional-differential equation, Aequat. Math., 60 (2000), 38-56.  doi: 10.1007/s000100050134. [33] J. Si and W. Zhang, Analytic solutions of a class of iterative functional differential equations, J. Comput. Appl. Math., 162 (2004), 467-481.  doi: 10.1016/j.cam.2003.08.049. [34] S. Staněk, On global properties of solutions of functional-differential equation $x'(t)=x(x(t))+x(t)$, Dyn. Syst. Appl., 4 (1995), 263-277. [35] E. Turdza, On a functional inequality with $n$-th iterate of the unknown function, Zeszyty Nauk. Uniw. Jagiello. Prace Mat., 16 (1974), 189-194. [36] E. Turdza, The solutions of an inequality for the $n$-th iterate of a function, Amer. Math. Month., 86 (1979), 281-283.  doi: 10.1080/00029890.1979.11994789. [37] H.-O. Walther, Merging homoclinic solutions due to state-dependent delay, J. Diff. Eqns., 259 (2015), 473-509.  doi: 10.1016/j.jde.2015.02.009. [38] K. Wang, On the equation $x'(t)=f(x(x(t)))$, Funk. Ekv., 33 (1990), 405-425. [39] B. Xu, W. Zhang and J. Si, Analytic solutions of an iterative functional differential equation which may violate the Diophantine condition, J. Difference Equ. Appl., 10 (2004), 201-211.  doi: 10.1080/1023-6190310001596571. [40] D. Yang and W. Zhang, Solutions of equivariance for iterative differential equations, Appl. Math. Lett., 17 (2004), 759-765.  doi: 10.1016/j.aml.2004.06.002. [41] Y. Zeng, P. Zhang, T.-T. Lu and W. Zhang, Existence of solutions for a mixed type differential equation with state-dependence, J. Math. Anal. Appl., 453 (2017), 629-644.  doi: 10.1016/j.jmaa.2017.04.020. [42] M. Zima, On positive solutions of boundary value problems on the half-Line, J. Math. Anal. Appl., 259 (2001), 127-136.  doi: 10.1006/jmaa.2000.7399.

show all references

##### References:
 [1] P. Andrzej, On some iterative-differential equations. I, Zeszyty Nauk. Uniw. Jagiello. Prace Mat., 12 (1968), 53-56. [2] I. Balázs and T. Krisztin, A differential equation with a state-dependent queueing delay, SIAM J. Math. Anal., 52 (2020), 3697-3737.  doi: 10.1137/19M1257585. [3] L. Boullu, L. Pujo-Menjouet and J. Wu, A model for megakaryopoiesis with state-dependent delay, SIAM. J. Appl. Math., 79 (2019), 1218-1243.  doi: 10.1137/18M1201020. [4] G. Brauer, Functional inequalities, Amer. Math. Month., 71 (1964), 1014-1017.  doi: 10.2307/2311919. [5] C. E. Carr and M. Konishi, A circuit for detection of interaural time differences in the brain stem of the barn owl, J. Neurosci., 10 (1990), 3227-3246.  doi: 10.1523/JNEUROSCI.10-10-03227.1990. [6] S. Cheng, J. Si and X. Wang, An existence theorem for iterative functional differential equations, Acta Math. Hungar., 94 (2002), 1-17.  doi: 10.1023/A:1015609518664. [7] 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. [8] R. D. Driver, A two-body problem of classical electrodynamics: The one-dimensional case, Ann. Phys., 21 (1963), 122-142.  doi: 10.1016/0003-4916(63)90227-6. [9] G. M. Dunkel, On nested functional differential equations, SIAM J. Appl. Math., 18 (1970), 514-525.  doi: 10.1137/0118044. [10] E. Eder, The functional differential equation $x'(t) = x(x(t))$, J. Diff. Eqns., 54 (1984), 390-400.  doi: 10.1016/0022-0396(84)90150-5. [11] M. Fečkan, On a certain type of functional differential equations, Math. Slovaca, 43 (1993), 39-43. [12] C. G. Gal, Nonlinear abstract differential equations with deviated argument, J. Math. Anal. Appl., 333 (2007), 971-983.  doi: 10.1016/j.jmaa.2006.11.033. [13] P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, J. Diff. Eqns., 260 (2016), 6176-6200.  doi: 10.1016/j.jde.2015.12.038. [14] L. J. Grimm, Existence and continuous dependence for a class of nonlinear neutral-differential equations, Proc. Amer. Math. Soc., 29 (1971), 467-473.  doi: 10.1090/S0002-9939-1971-0287117-1. [15] Z. Hao, J. Liang and T. Xiao, Positive solutions of operator equations on half-line, J. Math. Anal. Appl., 314 (2006), 423-435.  doi: 10.1016/j.jmaa.2005.04.004. [16] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delay: Theory and applications, Handbook of Differential Equations: Ordinary Differential Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 3 (2006), 435-545.  doi: 10.1016/S1874-5725(06)80009-X. [17] E. Hernandez, J. Wu and A. Chadha, Existence, uniqueness and approximate controllability of abstract differential equations with state-dependent delay, J. Diff. Eqns., 269 (2020), 8701-8735.  doi: 10.1016/j.jde.2020.06.030. [18] U. Horst and D. Kreher, A weak law of large numbers for a limit order book model with fully state dependent order dynamics, SIAM J. Financ. Math., 8 (2017), 314-343.  doi: 10.1137/15M1024226. [19] Q. Hu, A model of cold metal rolling processes with state-dependent delay, SIAM J. Appl. Math., 76 (2016), 1076-1100.  doi: 10.1137/141000257. [20] Q. Hu, W. Krawcewicz and J. Turi, Stabilization in a state-dependent model of turning processes, SIAM J. Appl. Math., 72 (2012), 1-24.  doi: 10.1137/110823468. [21] B. Kennedy, The Poincaré-Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback, J. Diff. Eqns., 266 (2019), 1865-1898.  doi: 10.1016/j.jde.2018.08.012. [22] M. Kloosterman, S. A. Campbell and F. J. Poulin, An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton, SIAM J. Appl. Math., 76 (2016), 551-577.  doi: 10.1137/15M1021271. [23] M. A. Krasnoselskii, Positive Solutions of Operator Equations, Translated from the Russian by Richard E. Flaherty; Edited by Leo F. Boron P. Noordhoff Ltd. Groningen, 1964. [24] Y. Kuang, $3/2$ stability results for nonautonomous state-dependent delay differential equations, Differential Equations and Applications to Biology and to Industry (Claremont, CA, 1994), World Sci. Publ., River Edge, NJ, (1996), 261–269. [25] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139086639. [26] K. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Diff. Eqns., 148 (1998), 407-421.  doi: 10.1006/jdeq.1998.3475. [27] Y. Liu, Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Appl. Math. Comput., 144 (2003), 543-556.  doi: 10.1016/S0096-3003(02)00431-9. [28] J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Diff. Eqns., 250 (2011), 4085-4103.  doi: 10.1016/j.jde.2010.10.023. [29] H. Müller-Krumbhaar and J. P. v. d. Eerden, Some properties of simple recursive differential equations, Z. Phys. B: Condensed Matter, 67 (1987), 239-242.  doi: 10.1007/BF01303988. [30] R. Oberg, On the local existence of solutions of certain functional-differential equations, Proc. Amer. Math. Soc., 20 (1969), 295-302.  doi: 10.1090/S0002-9939-1969-0234094-6. [31] J. Si and X. Wang, Smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients, J. Math. Anal. Appl., 226 (1998), 377-392.  doi: 10.1006/jmaa.1998.6086. [32] J. Si, X. Wang and S. Cheng, Nondecreasing and convex $C^2$-solutions of an iterative functional-differential equation, Aequat. Math., 60 (2000), 38-56.  doi: 10.1007/s000100050134. [33] J. Si and W. Zhang, Analytic solutions of a class of iterative functional differential equations, J. Comput. Appl. Math., 162 (2004), 467-481.  doi: 10.1016/j.cam.2003.08.049. [34] S. Staněk, On global properties of solutions of functional-differential equation $x'(t)=x(x(t))+x(t)$, Dyn. Syst. Appl., 4 (1995), 263-277. [35] E. Turdza, On a functional inequality with $n$-th iterate of the unknown function, Zeszyty Nauk. Uniw. Jagiello. Prace Mat., 16 (1974), 189-194. [36] E. Turdza, The solutions of an inequality for the $n$-th iterate of a function, Amer. Math. Month., 86 (1979), 281-283.  doi: 10.1080/00029890.1979.11994789. [37] H.-O. Walther, Merging homoclinic solutions due to state-dependent delay, J. Diff. Eqns., 259 (2015), 473-509.  doi: 10.1016/j.jde.2015.02.009. [38] K. Wang, On the equation $x'(t)=f(x(x(t)))$, Funk. Ekv., 33 (1990), 405-425. [39] B. Xu, W. Zhang and J. Si, Analytic solutions of an iterative functional differential equation which may violate the Diophantine condition, J. Difference Equ. Appl., 10 (2004), 201-211.  doi: 10.1080/1023-6190310001596571. [40] D. Yang and W. Zhang, Solutions of equivariance for iterative differential equations, Appl. Math. Lett., 17 (2004), 759-765.  doi: 10.1016/j.aml.2004.06.002. [41] Y. Zeng, P. Zhang, T.-T. Lu and W. Zhang, Existence of solutions for a mixed type differential equation with state-dependence, J. Math. Anal. Appl., 453 (2017), 629-644.  doi: 10.1016/j.jmaa.2017.04.020. [42] M. Zima, On positive solutions of boundary value problems on the half-Line, J. Math. Anal. Appl., 259 (2001), 127-136.  doi: 10.1006/jmaa.2000.7399.
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