July  2022, 27(7): 3605-3624. doi: 10.3934/dcdsb.2021198

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 Published  July 2022 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, 2022, 27 (7) : 3605-3624. 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]

Huaiyu Zhou, Jingbo Dou. Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022070

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Fen-Fen Yang. Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise. Probability, Uncertainty and Quantitative Risk, , () : -. doi: 10.3934/puqr.2022008

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875

[18]

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

[19]

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

[20]

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

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (454)
  • HTML views (340)
  • Cited by (0)

Other articles
by authors

[Back to Top]