The existence of countably many positive solutions for nonlinear singular -point boundary value problems on the half-line

In this paper, by introducing a new operator, improving and generating a p -Laplace operator for some p > 1 , we study the existence of countably many positive solutions for nonlinear boundary value problems on the half-line ( φ ( u ′ ) ) ′ + a ( t ) f ( u ( t ) ) = 0 , 0 < t < + ∞ , u ( 0 ) = ∑ i = 1 m − 2 α i u ( ξ i ) , u ′ ( ∞ ) = 0 , where φ : R → R is the increasing homeomorphism and positive homomorphism and φ ( 0 ) = 0 . We show the sufficient conditions for the existence of countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones.