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

Abstract

In this paper, we study the existence of countably many positive solutions for some nonlinear singular three-point boundary problems on the half-line ( ϕ ( u ′ ) ( t ) ) ′ + a ( t ) f ( t , u ( t ) ) = 0 , 0 < t < + ∞ , u ( 0 ) − B 0 ( u ′ ( η ) ) = 0 , u ′ ( ∞ ) = 0 , where ϕ ( s ) : R → R is an increasing homeomorphism and positive homomorphism and ϕ ( 0 ) = 0 and η ∈ ( 0 , + ∞ ) , a : [ 0 , + ∞ ) → [ 0 , + ∞ ) and has countably many singularities on [ 0 , + ∞ ) . By using the fixed-point index theorem and a new fixed-point theorem in cones, the existence of countably many solutions for singular three-point boundary value problem are obtained under conditions weaker than those used by Liu and Zhang [B.F. Liu, J.H. Zhang, The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions, J. Math. Anal. Appl. 309 (2005) 505–516].