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

We study the existence of fixed points to a parameterized Hammertstain operator $\cH_β,$ $β\in (0,\infty],$ with sigmoid type of nonlinearity. The parameter $β<\infty$ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case $β=\infty$ 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 $\cH_\infty.$ 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 Hamiltionian 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.