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We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Neumann–Robin boundary value problem { u′′(x) + λ f (u(x)) = 0, 0 < x < 1, u′(0) = 0 and u′(1) + αu(1) = 0, where λ > 0 is a bifurcation parameter, α > 0 is an evolution parameter, and nonlinearity f satisfies f (0) ≥ 0 and f (u) > 0 for u > 0. We obtain the multiplicity of positive solutions for α > 0 and λ > 0. Applications are given.

Classification and evolution of bifurcation curves for a one-dimensional Neumann–Robin problem and its applications