Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4609372 | Journal of Differential Equations | 2016 | 31 Pages |
We study the existence of fixed points to a parameterized Hammerstein operator HβHβ, β∈(0,∞]β∈(0,∞], with sigmoid type of nonlinearity. The parameter β<∞β<∞ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case β=∞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∞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.