We study the bifurcation diagrams of positive solutions of the two point boundary value problem, and λ > 0 is a bifurcation parameter. We assume that functions g and h satisfy hypotheses (H1)-(H3). Under hypotheses (H1)-(H3), we give a complete classification of bifurcation diagrams, and we prove that, on the (λ, u ∞ )-plane, each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left. Hence the problem has at most two positive solutions for each λ > 0. More precisely, we prove the exact multiplicity of positive solutions.
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