Positive solutions of fourth-order singular three point eigenvalue problems

In this paper, by establishing a new comparison theorem and constructing upper and lower solutions, some sufficient conditions of existence of positive solutions for the following singular fourth order three point eigenvalue problemu(4)(t)=λf(t,u),t∈(0,1),u(0)=αu(η),u(1)=0,u″(0)=βu″(η),u″(1)=0,are established due to Schauder's fixed point theorem for λ large enough, where α,β,η∈(0,1) are constants, f can be singular at t=0 and/or 1, u=0. Moreover, some peculiar cases are discussed and some further results are obtained.