Functional equations on discrete sets

Abstract Let Y (+) be a group, D ⊆ ℤ 2 where ℤ(+, ⩽) denotes the ordered group of all integers, and ℤ 2 := 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 ⊆ ℤ 2 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.