The existence of multiple positive solutions to boundary value problems of nonlinear delay differential equations with countably many singularities on infinite interval

In this paper, we consider the existence of countably many positive solutions to a boundary value problem of a nonlinear delay differential equation with countably many singularities on infinite interval ( ϕ ( x ′ ( t ) ) ) ′ + a ( t ) f ( t , x ( t ) , x t ) = 0 , 0 < t < ∞ , x 0 = ξ , lim t → ∞ x ′ ( t ) = 0 , where ϕ : R → R is an increasing homeomorphism and a positive homomorphism with ϕ ( 0 ) = 0 , x t is a function in C ( [ − r , 0 ] , R ) defined by x t ( σ ) = x ( t + σ ) for − r ≤ σ ≤ 0 , and ξ ∈ C ( [ − r , 0 ] , R ) . By using the fixed-point index theory and a new fixed-point theorem in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem. The conclusions in this paper essentially extend and improve the known results.