A Remark on Singular Perturbation Methods via the Lyapunov-Schmidt Reduction

Abstract

For some reaction-diffusion equations, Lyapunov-Schmidt reduction was shown to be applicable to construct singularly perturbed equilibrium solutions. For this application, it is indispensable to show that some inverse operator are uniformly bounded. In this paper, we give an elementary proof of this fact. § 1. IntroductionFor differential equations containing a small parameter in the spatial derivatives, there often exist solutions with internal transition layers. Hale and Sakamoto [1] applied Lyapunov-Schmidt reduction to construct singularly perturbed equilibrium solutions to Equation (1) below. This method also gave the stability condition for the solutions simultaneously.In the following, we briefly sketch the method of [1], with special attention to the part where our present contribution appears. We consider the following equation

Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II

“…The computation of the Taylor coefficients is intimately related to some singularly perturbed linearized operator L ǫ U which depends not only on the singular parameter ǫ, but also on the spatial inhomogeneity a(x; β) as well as some initial choice of approximate solution U . These dependences have not been carefully studied in [12,18,19,15,13,10,11,16,17]. There works are primarily concerned with the uniform boundedness and invertibility of L ǫ U .…”