Let T, S : A ∪ B → A ∪ B be mappings such that T(A) ⊆ B, T(B) ⊆ A and S(A) ⊆ A, S(B) ⊆ B. Then the pair (T; S) of mappings defined on A ∪ B is called cyclic-noncyclic pair, where A and B are two nonempty subsets of a metric space (X, d). A coincidence best proximity point p ∈ A ∪ B for such a pair of mappings (T; S) is a point such that d(Sp, Tp) = dist(A, B). In this paper, we study the existence and convergence of coincidence best proximity points in the setting of convex metric spaces. We also present an application of one of our results to an integral equation.