Abstract
We study the bifurcation curves of positive solutions of the boundary value problem {(u <sup>″</sup> (x) + f <sub>ε</sub> (u (x)) = 0, - 1 < x < 1,; u (- 1) = u (1) = 0,) where f <sub>ε</sub> (u) = g (u) - ε h (u), ε ∈ R is a bifurcation parameter, and functions g, h ∈ C [0, ∞) ∩ C <sup>2</sup> (0, ∞) satisfy five hypotheses presented herein. Assuming these hypotheses on fixed g and h, we prove that the bifurcation curve is reverse S-shaped on the (ε, {norm of matrix} u {norm of matrix} <sub>∞</sub> )-plane; that is, the bifurcation curve has exactly two turning points at some points (over(ε, ̃), {norm of matrix} u <sub>over(ε, ̃)</sub> {norm of matrix} <sub>∞</sub> ) and (ε <sup>*</sup> , {norm of matrix} u <sub>ε*</sub> {norm of matrix} <sub>∞</sub> ) such that over(ε, ̃) < ε <sup>*</sup> and {norm of matrix} u <sub>over(ε, ̃)</sub> {norm of matrix} <sub>∞</sub> < {norm of matrix} u <sub>ε*</sub> {norm of matrix} <sub>∞</sub> . In addition, we prove that ε <sup>*</sup> > 0. Thus the exact number of positive solutions can be precisely determined by the values of over(ε, ̃) and ε <sup>*</sup> . We give an application to the two-parameter bifurcation problem {(u <sup>″</sup> (x) + λ (1 + u <sup>2</sup> - ε u <sup>3</sup> ) = 0, - 1 < x < 1,; u (- 1) = u (1) = 0,) where λ, ε are two positive bifurcation parameters. Some new results are obtained. © 2008 Elsevier Ltd. All rights reserved.