Two positive solutions to three-point singular boundary value problems

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: −u″(t)=λ f(t,u(t)) for all t ∈ (0,1) subjecting to u(0)=0 and α u(η)=u(1), where η ∈ (0,1),α ∈[0,1), and λ is a positive parameter. The nonlinear term f(t,u) is nonnegative, and may be singular at t=0,t=1, and u=0. By the fixed point index theory and approximation method, we establish that there exists λ*∈(0,+∞], such that the above problem has at least two positive solutions for any λ ∈(0,λ*) under certain conditions on the nonlinear term f.