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2015, 2015(special): 56-65. doi: 10.3934/proc.2015.0056

## Subexponential growth rates in functional differential equations

 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.
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. [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. [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. [4] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics, 34, Cambridge University Press, Cambridge, 1990. [5] P. Hartman and A. Wintner, Asymptotic integration of ordinary nonlinear differential equations, Amer. J. Math., 77 (1955), 692-724. [6] P. Hartman, Ordinary differential equations, $2^{nd}$ edition, SIAM, Philadelphia, 2002. [7] T. Kusano and H. Onose, Oscillatory and asymptotic behavior of sublinear retarded differential equations Hiroshima Math. J., 4 (1974), 343-355. [8] M. Pituk, The Hartman-Wintner Theorem for Functional Differential Equations, J. Differential Equations, 155 (1), (1999), 1-16.

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##### 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. [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. [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. [4] G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics, 34, Cambridge University Press, Cambridge, 1990. [5] P. Hartman and A. Wintner, Asymptotic integration of ordinary nonlinear differential equations, Amer. J. Math., 77 (1955), 692-724. [6] P. Hartman, Ordinary differential equations, $2^{nd}$ edition, SIAM, Philadelphia, 2002. [7] T. Kusano and H. Onose, Oscillatory and asymptotic behavior of sublinear retarded differential equations Hiroshima Math. J., 4 (1974), 343-355. [8] M. Pituk, The Hartman-Wintner Theorem for Functional Differential Equations, J. Differential Equations, 155 (1), (1999), 1-16.
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