The existence of multiple positive solutions for singular functional differential equations with sign-changing nonlinearity

In this paper, we study the existence of multiple positive solutions for boundary value problems based on second-order functional differential equations with the form { y ″ ( t ) + f ( t , y ( t − τ ) ) = 0 , ∀ t ∈ ( 0 , 1 ) ∖ { τ } , y ( t ) = η ( t ) , ∀ t ∈ [ − τ , 0 ] , y ( 1 ) = 0 where 0 < τ < 1 and f : ( 0 , 1 ) × ( 0 , + ∞ ) → ( − ∞ , + ∞ ) is continuous, may be singular at t = 0 , 1 , y = 0 and takes negative values. By applying the fixed point index theorem, we obtain the conditions for the existence of at least two and of three positive solutions. An example to illustrate our results is given.