The existence of countably many positive solutions for nonlinear singular -point boundary value problems

In this paper, we study the existence of countably many positive solutions for nonlinear singular boundary value problem ( ϕ ( u ′ ) ) ′ + a ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , subject to the boundary value conditions: u ( 0 ) = ∑ i = 1 m - 2 α i u ( ξ i ) , ϕ ( u ′ ( 1 ) ) = ∑ i = 1 m - 2 β i ϕ ( u ′ ( ξ i ) ) , where ϕ : R → R is an increasing homeomorphism and positive homomorphism and ϕ ( 0 ) = 0 , ξ i ∈ ( 0 , 1 ) with 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 and α i , β i satisfy α i , β i ∈ [ 0 , + ∞ ) , 0 < ∑ i = 1 m - 2 α i < 1 , 0 < ∑ i = 1 m - 2 β i < 1 , f ∈ C ( [ 0 , + ∞ ) , [ 0 , + ∞ ) ) , a : [ 0 , 1 ] → [ 0 , + ∞ ) and has countably many singularities in [ 0 , 1 2 ) . We show that there exist countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones.