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

الأصل · EN

We study the existence of fixed points to a parameterized Hammertstain operator β, 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 ∞. 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.