Open Access
2017 Memory dependent growth in sublinear Volterra differential equations
John A.D. Appleby, Denis D. Patterson
J. Integral Equations Applications 29(4): 531-584 (2017). DOI: 10.1216/JIE-2017-29-4-531


We investigate memory dependent asymptotic growth in scalar Volterra equations with sublinear nonlinearity. In order to obtain precise results we extensively utilize the powerful theory of regular variation. By computing the growth rate in terms of a related ordinary differential equation we show that, when the memory effect is so strong that the kernel tends to infinity, the growth rate of solutions depends explicitly upon the memory of the system. Finally, we employ a fixed point argument for determining analogous results for a perturbed Volterra equation and show that, for a sufficiently large perturbation, the solution tracks the perturbation asymptotically, even when the forcing term is potentially highly non-monotone.


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John A.D. Appleby. Denis D. Patterson. "Memory dependent growth in sublinear Volterra differential equations." J. Integral Equations Applications 29 (4) 531 - 584, 2017.


Published: 2017
First available in Project Euclid: 10 November 2017

zbMATH: 1384.45002
MathSciNet: MR3722841
Digital Object Identifier: 10.1216/JIE-2017-29-4-531

Primary: 34K25
Secondary: 34K28

Keywords: asymptotics , regular variation , subexponential growth , unbounded delay , Volterra equations

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

Vol.29 • No. 4 • 2017
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