Electron. J. Differential Equations,
Vol. 2017 (2017), No. 21, pp. 145.
HartmanWintner growth results for sublinear functional differential equations
John A. D. Appleby, Denis D. Patterson
Abstract:
This article determines the rate of growth to infinity of scalar
autonomous nonlinear functional and Volterra differential equations.
In these equations, the righthand side is a positive continuous
linear functional of f(x). We assume f grows sublinearly,
leading to subexponential growth in the solutions. The main results
show that the solution of the functional differential equations are
asymptotic to that of an auxiliary autonomous ordinary differential
equation with righthand side proportional to f. This happens provided f
grows more slowly than l(x)=x/log(x). The linearlogarithmic 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 nonunit factor;
if f grows more slowly than l, the ODE and FDE grow at exactly the same rate.
A partial converse of the last result is also proven. In the case when the
growth rate is slower than that of the ODE, sharp bounds on the growth rate
are determined. The Volterra and finite memory equations can have differing
asymptotic behaviour and we explore the source of these differences.
Submitted August 12, 2016. Published January 16, 2017.
Math Subject Classifications: 34K25, 34K28.
Key Words: Functional differential equations; Volterra equations; asymptotics;
subexponential growth; bounded delay; unbounded delay.
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John A. D. Appleby
School of Mathematical Sciences
Dublin City University,
Glasnevin, Dublin 9, Ireland
email: john.appleby@dcu.ie


Denis D. Patterson
School of Mathematical Sciences
Dublin City University
Glasnevin, Dublin 9, Ireland
email: denis.patterson2@mail.dcu.ie

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