We are concerned with the following nonlinear second-order three-point boundary value problem on time scales −xΔΔ t f t, x t , t ∈ a, b T , x a 0, x σ2 b δx η , where a, b ∈ T with a < b, η ∈ a, b T and 0 < δ < σ2 b − a / η − a . A new representation of Green’s function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution ...