On the relation between interior critical points and parameters for a class of nonlinear problems with Neumann-Robin boundary conditions

We consider boundary value problem-u''(x)=λf(u(x)),x-bar (0,1),u'(0)=0,u'(1)+αu(1)=0,where α>=0, λ>0 are parameters and f-bar C2[0,∼) such that f(0)<0. In this paper we study for the cases p-bar (0,β) and p-bar (β,θ) (p is the value of the solution at x=0 and β, θ are such that f(β)=0, F(θ)=∫0θf(t)dt=0), the relation between λ and the number of interior critical points of the positive solutions of the above system