Green’s function and existence of a unique solution for a third-order three-point boundary value problem

Abstract

The solutions of third-order three-point boundary value problem x′′′ + f(t, x) = 0, t ∈ [a, b], x(a) = x′ (a) = 0, x(b) = kx(η), where η ∈ (a, b), k ∈ ℝ, f ∈ C([a, b] × ℝ, ℝ) and f(t, 0) ≠ 0, are the subject of this investigation. In order to establish existence and uniqueness results for the solutions, attention is focused on applications of the corresponding Green’s function. As an application, also one example is given to illustrate the result.