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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

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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

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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

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

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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

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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

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