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Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity

Oleynik, A; Ponosov, A; Kostrykin, V; Sobolev, AV; (2016) Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity. Journal of Differential Equations , 261 (10) pp. 5844-5874. 10.1016/j.jde.2016.08.026. Green open access

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Abstract

We study the existence of fixed points to a parameterized Hammerstein operator Hβ, β∈(0,∞], with sigmoid type of nonlinearity. The parameter β<∞ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case β=∞ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large β exist and can be approximated by the fixed points of H∞. These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discontinuous unit step function. Moreover, in order to achieve even better approximation than a solution of the limit problem, we employ the iterative method that has several advantages compared to other existing methods. For example, this method can be used to construct non-isolated homoclinic orbit of a Hamiltonian system of equations. We illustrate the results and advantages of the numerical method for stationary versions of the FitzHugh–Nagumo reaction–diffusion equation and a neural field model.

Type: Article
Title: Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.jde.2016.08.026
Publisher version: http://dx.doi.org/10.1016/j.jde.2016.08.026
Language: English
Additional information: © 2016 Elsevier Inc. All rights reserved. This manuscript version is made available under a Creative Commons Attribution Non-commercial Non-derivative 4.0 International license (CC BY-NC-ND 4.0). This license allows you to share, copy, distribute and transmit the work for personal and non-commercial use providing author and publisher attribution is clearly stated. Further details about CC BY licenses are available at https://creativecommons.org/licenses/. Access may be initially restricted by the publisher.
Keywords: Nonlinear integral equations; Sigmoid type nonlinearities; Neural field model; FitzHugh–Nagumo model; Bumps
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1542761
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