doi: 10.3934/dcdsb.2021198
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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:
[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. BoulluL. 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. ChengJ. 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. HaoJ. 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. HartungT. KrisztinH.-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. HernandezJ. 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. HuW. 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. KloostermanS. 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. KuczmaB. 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. SiX. 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. XuW. 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. ZengP. ZhangT.-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. BoulluL. 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. ChengJ. 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. HaoJ. 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. HartungT. KrisztinH.-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. HernandezJ. 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. HuW. 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. KloostermanS. 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. KuczmaB. 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. SiX. 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. XuW. 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. ZengP. ZhangT.-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.

[1]

Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure and Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929

[2]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[3]

Gennaro Infante. Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 691-699. doi: 10.3934/dcdsb.2019261

[4]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

[5]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167

[6]

Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416

[7]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

[8]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial and Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621

[9]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793

[10]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

[11]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[12]

Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134

[13]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038

[14]

Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155

[15]

Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure and Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016

[16]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103

[17]

Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033

[18]

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27

[19]

John A. D. Appleby, Denis D. Patterson. Subexponential growth rates in functional differential equations. Conference Publications, 2015, 2015 (special) : 56-65. doi: 10.3934/proc.2015.0056

[20]

Olivier Hénot. On polynomial forms of nonlinear functional differential equations. Journal of Computational Dynamics, 2021, 8 (3) : 309-323. doi: 10.3934/jcd.2021013

2020 Impact Factor: 1.327

Article outline

[Back to Top]