Positive solutions for a Hammerstein integral equation with a parameter

We discuss the existence of positive solutions for the Hammerstein integral equation u(x)=λ∫01K(x,y)f(y,u(y))dy. By calculation of the fixed point index in a cone, we obtain that there exists a critical value λ∗>0λ∗>0 such that the above equation has at least two, one positive solutions for λ∈(0,λ∗)λ∈(0,λ∗), λ=λ∗λ=λ∗, respectively, and has no positive solution for λ>λ∗λ>λ∗.