The existence of multiple positive solutions for multi-point boundary value problems on the half-line

In this paper, we consider the existence of multiple positive solutions for some nonlinear m -point boundary value problems on the half-line ( φ ( u ′ ) ) ′ + a ( t ) f ( t , u ( t ) ) = 0 , 0 < t < + ∞ , u ( 0 ) = ∑ i = 1 m − 2 α i u ( ξ i ) , u ′ ( ∞ ) = 0 , where φ : R → R is an increasing homeomorphism and positive homomorphism and φ ( 0 ) = 0 . Using a fixed-point theorem for operator on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.