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.
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Masaharu Taniguchi, A Remark on Singular Perturbation Methods via the Lyapunov-Schmidt Reduction. Publ. Res. Inst. Math. Sci. 31 (1995), no. 6, pp. 1001–1010DOI 10.2977/PRIMS/1195163593