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 ∫ 0 1 K ( x , y ) f ( y , u ( y ) ) d y . By calculation of the fixed point index in a cone, we obtain that there exists a critical value λ ∗ > 0 such that the above equation has at least two, one positive solutions for λ ∈ ( 0 , λ ∗ ) , λ = λ ∗ , respectively, and has no positive solution for λ > λ ∗ .