An Evolutionary Property of the Bifurcation Curves for a Positone Problem with Cubic Nonlinearity

We study an evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity { u ″ ( x ) + λ f ( u ) = 0 , − 1 < x < 1 , u ( − 1 ) = u ( 1 ) = 0 , f ( u ) = − ε u 3 + σ u 2 + τ u + ρ , where λ > 0 is a bifurcation parameters,ε> 0 is an evolution parameter, andσ, ρ> 0,τ≥ 0 are constants. In addition, we improve lower and upper bounds of the critical bifurcation valueε̃ of the problem. 2010Mathematics Subject Classification. 34B18, 74G35. Key words and phrases. Positive solution, Exact multiplicity, Turning point, S-shaped bifurcation curve.