Functional equations on discrete sets

Abstract

Let Y (+) be a group, D ⊆ ℤ² where ℤ(+, ⩽) denotes the ordered group of all integers, and ℤ² := ℤ×ℤ. We shall use the notations D_x := {u ∈ ℤ | ∃v ∈ X : (u, v) ∈ D}, D_y := {v ∈ ℤ | ∃u ∈ ℤ : (u, v) ∈ D}, D _x+y := {z ∈ ℤ | ∃(u, v) ∈ D : z = u + v}. The main purpose of the article is to find sets D ⊆ ℤ² that the general solution of the functional equation f (x+y) = g(x)+h(y) for all (x, y) ∈ D with unknown functions f : D _x+y → Y, g : D_x → Y, h : D_y → Y is in the form of f (u) = a(u) + C ₁ + C ₂ for all u ∈ D_x + _y , g(v) = a(v) + C ₁ for all v ∈ D_x , h(z) = a(z) + C ₂ for all z ∈ D_y where a : ℤ → Y is an additive function, C ₁, C ₂ ∈ Y are constants.