Solvability for second-order three-point boundary value problems at resonance on a half-line

Abstract

This paper deals with the solvability and uniqueness of the second-order three-point boundary value problems at resonance on a half-line x ″ ( t ) = f ( t , x ( t ) , x ′ ( t ) ) , 0 < t < + ∞ , x ( 0 ) = x ( η ) , lim t → + ∞ x ′ ( t ) = 0 , and x ″ ( t ) = f ( t , x ( t ) , x ′ ( t ) ) + e ( t ) , 0 < t < + ∞ , x ( 0 ) = x ( η ) , lim t → + ∞ x ′ ( t ) = 0 , where f : [ 0 , + ∞ ) × R 2 → R , e : [ 0 , + ∞ ) → R are continuous and η ∈ ( 0 , + ∞ ) . By using the coincidence degree theory, we establish some existence and uniqueness criteria.