On the relation between interior critical points of positive solutions and parameters for a class of nonlinear boundary value problems

We consider the boundary value problem u(x)=f(u(x)), x(0,1); u(0)=0; u(1)+u(1)=0, where > 0, > 0 are parameters and fc2[0,) such that f(0) < 0. In this paper, we study for the two cases =0 and = ( is the value of the solution at x=0 and is such that F()=0 where F(s)=0sf(t)dt) the relation between and the number of interior critical points of the nonnegative solutions of the above system.