In this paper, we solve the additive ρ -functional equations where ρ is a fixed number with ρ ≠ 1, 2, and where ρ is a fixed number with ρ ≠ 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ -functional equations (0.1) and (0.2) in Banach spaces. The stability problem of functional equations originated from a question of Ulam [5] concerning the stability of group homomorphisms. The functional equation is called the Cauchy equation . In particular, every solution of the Cauchy equation is said to be an additive mapping . Hyers [3] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gǎvruta [2] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. In Section 2, we solve the additive functional equation (0.1) and prove the Hyers-Ulam stability of the additive functional equation (0.1) in Banach spaces. In Section 3, we solve the additive functional equation (0.2) and prove the Hyers-Ulam stability of the additive functional equation (0.2) in Banach spaces. Throughout this paper, assume that X is a normed space and that Y is a Banach space. Let ρ be a number with ρ ≠ 1, 2. We solve and investigate the additive ρ -functional equation (0.1) in normed spaces. Lemma 2.1. If a mapping f : X → Y satisfies for all x , y , z ∈ X , then f : X → Y is additive . Proof . Assume that f : X → Y satisfies (2.1). Letting x = y = z = 0 in (2.1), we get −2 f (0) = − ρf (0). So f (0) = 0. Letting y = x and z = 0 in (2.1), we get f (2 x )−2 f ( x ) = 0 and so f (2 x ) = 2 f ( x ) for all x ∈ X . Thus for all x ∈ X . It follows from (2.1) and (2.2) that and so f ( x + y + z ) = f ( x ) + f ( y ) + f ( z ) for all x, y, z ∈ X . Since f (0) = 0, for all x , y ∈ X .   ☐ We prove the Hyers-Ulam stability of the additive ρ -functional equation (2.1) in Banach spaces. Theorem 2.2. Let φ : X ³ → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and for all x , y , z ∈ X . Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . Letting y = x and z = 0 in (2.4), we get for all x ∈ X . So for all x ∈ X . Hence for all nonnegative integers m and l with m > l and all x ∈ X . It follows from (2.7) that the sequence is Cauchy for all x ∈ X . Since Y is a Banach space, the sequence converges. So one can define the mapping A : X → Y by for all x ∈ X . Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5). Now, let T : X → Y be another additive mapping satisfying (2.5). Then we have which tends to zero as q → ∞ for all x ∈ X . So we can conclude that A ( x ) = T ( x ) for all x ∈ X . This proves the uniqueness of A . It follows from (2.3) and (2.4) that for all x , y , z ∈ X . So for all x , y , z ∈ X . By Lemma 2.1, the mapping A : X → Y is additive.   ☐ Corollary 2.3. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and for all x , y , z ∈ X . Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . Letting φ ( x , y , z ) := in Theorem 2.2, we get the desired result.   ☐ Theorem 2.4. Let φ : X ³ → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (2.4) and for all x , y , z ∈ X . Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . It follows from (2.6) that for all x ∈ X . Hence for all nonnegative integers m and l with m > l and all x ∈ X . It follows from (2.10) that the sequence is a Cauchy sequence for all x ∈ X . Since Y is complete, the sequence converges. So one can define the mapping A : X → Y by for all x ∈ X . Moreover, letting l = 0 and passing the limit m → ∞ in (2.10), we get (2.9). The rest of the proof is similar to the proof of Theorem 2.2.   ☐ Corollary 2.5. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (2.8). Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . Letting φ ( x , y , z ) := in Theorem 2.4, we get the desired result.   ☐ Let ρ be a number with ρ ≠ 1. We solve and investigate the additive ρ -functional equation (0.2) in normed spaces. Lemma 3.1. If a mapping f : X → Y satisfies for all x , y , z ∈ X , then f : X → Y is additive . Proof . Assume that f : X → Y satisfies (3.1). Letting x = y = z = 0 in (2.1), we get -2 f (0) = -2 ρf (0). So f (0) = 0. Letting y = x and z = 0 in (2.1), we get f (2 x )-2 f ( x ) = 0 and so f (2 x ) = 2 f ( x ) for all x ∈ X . Thus for all x ∈ X . It follows from (3.1) and (3.2) that and so for all x , y ∈ X .   ☐ We prove the Hyers-Ulam stability of the additive ρ -functional equation (3.1) in Banach spaces. Theorem 3.2. Let φ : X ³ → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0 and for all x , y , z ∈ X . Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . Letting y = x and z = 0 in (3.3), we get for all x ∈ X . The rest of the proof is similar to the proof of Theorem 2.2.   ☐ Corollary 3.3. Let r > 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and for all x , y , z ∈ X . Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . Letting φ ( x , y , z ) := in Theorem 3.2, we get the desired result.   ☐ Theorem 3.4. Let φ : X ³ → [0, ∞) be a function and let f : X → Y be a mapping satisfying f (0) = 0, (3.3) and for all x , y , z ∈ X . Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . It follows from (3.4) that for all x ∈ X . The rest of the proof is similar to the proofs of Theorems 2.2 and 2.4.   ☐ Corollary 3.5. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying f (0) = 0 and (3.5). Then there exists a unique additive mapping A : X → Y such that for all x ∈ X . Proof . Letting φ ( x , y , z ) := in Theorem 3.4, we get the desired result.   ☐