American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stabilization of a hyperbolic/elliptic system modelling the viscoelastic-gravitational deformation in a multilayered Earth
  • PROC Home
  • This Issue
  • Next Article
    Hartman-type conditions for multivalued Dirichlet problem in abstract spaces
2015, 2015(special): 56-65. doi: 10.3934/proc.2015.0056

Subexponential growth rates in functional differential equations

John A. D. Appleby 1, and  Denis D. Patterson 2,

1. 

Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
2. 

School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9,, Ireland

Received  September 2014 Revised  January 2015 Published  November 2015

This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows sublinearly, and is such that solutions should exhibit growth faster than polynomial, but slower than exponential. Under some technical conditions on $f$, it is shown that the solution of the functional differential equation is asymptotic to that of an auxiliary autonomous ordinary differential equation with righthand side proportional to $f$ (with the constant of proportionality equal to the mass of the finite measure associated with the linear functional), provided $f$ grows more slowly than $l(x)=x/\log x$. This linear--logarithmic growth rate is also shown to be critical: if $f$ grows more rapidly than $l$, the ODE dominates the FDE; if $f$ is asymptotic to a constant multiple of $l$, the FDE and ODE grow at the same rate, modulo a constant non--unit factor.
Keywords: Functional differential equations, subexponential growth., asymptotics.
Mathematics Subject Classification: Primary: 34K25; Secondary: 34C1.
Citation: 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
References:
[1]

J. A. D. Appleby, M. J. McCarthy and A. Rodkina, Growth rates of delay-differential equations and uniform Euler schemes, Difference equations and applications, (eds. M. Bohner et al.), Uǧur-Bahçeşehir Univ. Publ. Co., Istanbul, (2009), 117-124.  Google Scholar

[2]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987.  Google Scholar

[3]

J. R. Graef, Oscillation, nonoscillation, and growth of solutions of nonlinear functional differential equations of arbitrary order, J. Math. Anal. Appl., 60 (1977), 398-409.  Google Scholar

[4]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics, 34, Cambridge University Press, Cambridge, 1990.  Google Scholar

[5]

P. Hartman and A. Wintner, Asymptotic integration of ordinary nonlinear differential equations, Amer. J. Math., 77 (1955), 692-724.  Google Scholar

[6]

P. Hartman, Ordinary differential equations, $2^{nd}$ edition, SIAM, Philadelphia, 2002.  Google Scholar

[7]

T. Kusano and H. Onose, Oscillatory and asymptotic behavior of sublinear retarded differential equations Hiroshima Math. J., 4 (1974), 343-355.  Google Scholar

[8]

M. Pituk, The Hartman-Wintner Theorem for Functional Differential Equations, J. Differential Equations, 155 (1), (1999), 1-16. Google Scholar

show all references

References:
[1]

J. A. D. Appleby, M. J. McCarthy and A. Rodkina, Growth rates of delay-differential equations and uniform Euler schemes, Difference equations and applications, (eds. M. Bohner et al.), Uǧur-Bahçeşehir Univ. Publ. Co., Istanbul, (2009), 117-124.  Google Scholar

[2]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987.  Google Scholar

[3]

J. R. Graef, Oscillation, nonoscillation, and growth of solutions of nonlinear functional differential equations of arbitrary order, J. Math. Anal. Appl., 60 (1977), 398-409.  Google Scholar

[4]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics, 34, Cambridge University Press, Cambridge, 1990.  Google Scholar

[5]

P. Hartman and A. Wintner, Asymptotic integration of ordinary nonlinear differential equations, Amer. J. Math., 77 (1955), 692-724.  Google Scholar

[6]

P. Hartman, Ordinary differential equations, $2^{nd}$ edition, SIAM, Philadelphia, 2002.  Google Scholar

[7]

T. Kusano and H. Onose, Oscillatory and asymptotic behavior of sublinear retarded differential equations Hiroshima Math. J., 4 (1974), 343-355.  Google Scholar

[8]

M. Pituk, The Hartman-Wintner Theorem for Functional Differential Equations, J. Differential Equations, 155 (1), (1999), 1-16. Google Scholar

[1]

Frank Blume. Realizing subexponential entropy growth rates by cutting and stacking. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3435-3459. doi: 10.3934/dcdsb.2015.20.3435
[2]

Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793
[3]

Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251
[4]

Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016
[5]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[6]

Olesya V. Solonukha. On nonlinear and quasiliniear elliptic functional differential equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 869-893. doi: 10.3934/dcdss.2016033
[7]

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
[8]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[9]

Olivier Hénot. On polynomial forms of nonlinear functional differential equations. Journal of Computational Dynamics, 2021, 8 (3) : 307-323. doi: 10.3934/jcd.2021013
[10]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[11]

Jun Zhou, Jun Shen. Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021198
[12]

Qingwu Gao, Zhongquan Huang, Houcai Shen, Juan Zheng. Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims. Journal of Industrial & Management Optimization, 2016, 12 (1) : 31-43. doi: 10.3934/jimo.2016.12.31
[13]

John A. D. Appleby, John A. Daniels. Exponential growth in the solution of an affine stochastic differential equation with an average functional and financial market bubbles. Conference Publications, 2011, 2011 (Special) : 91-101. doi: 10.3934/proc.2011.2011.91
[14]

Daniel Balagué, José A. Cañizo, Pierre Gabriel. Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. Kinetic & Related Models, 2013, 6 (2) : 219-243. doi: 10.3934/krm.2013.6.219
[15]

Burcu Gürbüz. A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021069
[16]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283
[17]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[18]

Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220
[19]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[20]

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

 Impact Factor: 

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences