Positive solutions of singular three-point boundary value problems for second-order differential equations

In this paper, we study the existence of positive solutions of a singular three-point boundary value problem for the following second-order differential equation { y ″ + μ a ( t ) f ( t , y ( t ) ) = 0 , t ∈ ( 0 , 1 ) , y ( 0 ) − β y ′ ( 0 ) = 0 , y ( 1 ) = α y ( η ) , where μ > 0 is a parameter, β > 0 , 0 < η < 1 , 0 < α η < 1 , ( 1 − α η ) + β ( 1 − α ) > 0 . By constructing an available integral operator and combining fixed point index theory with properties of Green’s function under some conditions concerning the first eigenvalues corresponding to the relevant linear operator, the sufficient conditions of the existence of positive solutions for the boundary value problems are established. The interesting point of the results is that the term a ( t ) may be singular at t = 0 and/or t = 1 , moreover the nonlinear f ( t , x ) is also allowed to have singularity at x = 0 .