We study the bifurcation curve of positive solutions of the combustion problem with nonlinear boundary conditions given by
-u′′(x)=λexp(((βu)/(β+u))), 0<x<1,
u(0)=0,
((u(1))/(u(1)+1))u′(1)+[1-((u(1))/(u(1)+1))]u(1)=0,
where λ>0 is called the Frank--Kamenetskii parameter or ignition parameter, β>0 is the activation energy parameter, u(x) is the dimensionless temperature, and the reaction term exp(((βu)/(β+u))) is the temperature dependence obeying the simple Arrhenius reaction-rate law in irreversible chemical reaction kinetics. We prove rigorously that, for β>β₁≈6.459 for some constant β₁, the bifurcation curve is double S-shaped on the (λ,∥u∥_{∞})-plane and the problem has at least six positive solutions for a certain range of positive λ. We give rigorous proofs of some computational results of Goddard II, Shivaji and Lee

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