Generalized Hyers-Ulam Stability of the Pexider Functional Equation

In this paper, we investigate the generalized Hyers-Ulam stability of the Pexider functional equation f (x + y, z + w) = g(x, z) + h(y, w).


Introduction
Throughout this paper, let V and W be real vector spaces and Y be a real Banach space.In 1940, Ulam [1] raised the question about the stability of group homomorphisms.In 1941, for a given mapping f : V → Y, Hyers [2] solved the stability problem for the Cauchy additive functional equation f (x + y) − f (x) − f (y) = 0, for all x, y ∈ V.In 1978, Rassias [3] generalized Hyers' result and Gȃvruta [4] made Rassias' result more general.The concept of stability shown by Gȃvruta is called the 'generalized Hyers-Ulam stability'.
For given mappings f , g, h : V → Y, the stability of the Pexider functional equation f (x + y) − g(x) − h(y) = 0 ∀x, y ∈ V was investigated by Lee and Jun [5] (see also [6][7][8][9]).Now, for given mappings f , g, h : V × V → Y, we consider the Pexider functional equation for all x, y, z, w in the vector space V.One of typical examples of solutions of the functional Equation ( 1) are the mappings f , g, h : R × R → R given by f (x, z) = ax + bz + c + d, g(x, z) = ax + bz + c and h(x, z) = ax + bz + d with real constants a, b, c, d.
In this paper, we will investigate the generalized Hyers-Ulam stability of the Pexider functional Equation (1).

Main Results
For given mappings f , g, h : V × V → Y, we use the following abbreviations: D f gh (x, y, z, w) := f (x + y, z + w) − g(x, z) − h(y, w), for all x, y, z, w ∈ V. We need the following lemma to prove the main theorems.
Using the previous lemma, we obtain the following generalized Hyers-Ulam stability results for Equation (1).
Theorem 1. Suppose that f , g, h : V × V → Y are mappings for which there exists a function ϕ : and for all x, y, z, w ∈ V.Then, there exists a unique additive mapping F : for all x, z ∈ V, where the functions µ, ν, ξ : hold for all x, z ∈ V, the inequalities for all x, z ∈ V, follow from Inequality (3).Using the above inequalities, and the equalities we get the following inequalities for all x, z ∈ V and all n, m ∈ N ∪ {0}.So, the sequences { f (2 n x,2 n z) } n∈N , and } n∈N are Cauchy sequences for all x, z ∈ V\{0}.As Y is a real Banach space, we can define the mappings F, G, H : ) for all x, z ∈ V.By putting n = 0 and letting m → ∞ in Inequalities ( 10)-( 12), we obtain Inequalities ( 4)-( 6) for all x, z ∈ V.
From Inequality (3), we get for all x, y, z, w ∈ V. Since the right-hand side of the above equality tends to zero as n → ∞, we obtain that F, G, and H satisfy the functional Equation (1).Thus, by Lemma 1, F is an additive mapping.
If F : V × V → Y is another additive mapping satisfying ( 4)-( 6), we obtain Hence, the mapping is the unique additive mapping, as desired.
Corollary 1. Suppose that f : V × V → Y is a mapping for which there exists a function ϕ : for all x, y, z, w ∈ V.Then, there exists a unique additive mapping F : for all x, z ∈ V.
Theorem 2. Suppose that f : V × V → Y is a mapping for which there exists a function ϕ : and (3) for all x, y, z, w ∈ V.Then, there exists a unique additive mapping F : for all x, z ∈ V, where the functions µ, ν, ξ : V 2 → R are defined as in Theorem 1.
Proof.The inequalities for all x, z ∈ V follow from equalities ( 7)-( 9) for all x, z ∈ V and inequality (3).Using the above inequalities, we easily get the following inequalities for all x, z ∈ V and all n, m ∈ N ∪ {0}.Thus, the sequences for all x, z ∈ V, we have F(x, z) = G(x, z) = H(x, z) for all x, z ∈ V.By putting n = 0 and letting m → ∞ in Inequalities ( 18)-(20), we obtain Inequalities (15)-( 17) for all x, z ∈ V.
From Inequality (3), we get for all x, y, z, w ∈ V. Since the right-hand side of the above equality tends to zero as n → ∞, we obtain that F, G, and H satisfy the functional Equation (1).Hence, by Lemma 1, F is an additive mapping.
Corollary 2. Suppose that f : V × V → Y is a mapping for which there exists a function ϕ : V 2 → [0, ∞) satisfying inequality ( 14) and f satisfies inequality (13) for all x, y, z, w ∈ V.Then, there exists a unique additive mapping F : , for all x, z ∈ V.
Aoki ( [10]) and Gajda ([11]) proved the generalized Hyers-Ulam stability for an additive mapping in the cases where 0 ≤ p < 1 and 1 < p, respectively.It was also proved by Gajda ([11]) and Rassias and Semrl ( [12]) that the generalized Hyers-Ulam stability for an additive mapping does not holds for the case p = 1.
The above results for the cases p > 0 and p = 1 can be applied to the next results, due to Theorems 1 and 2.
Corollary 3. Let X be a normed space and p, θ be positive real numbers with p = 1.Suppose that f , g, h : X 2 → Y are mappings, such that f (x + y, z + w) − g(x, z) − h(y, w) ≤ θ( x p + y p + z p + w p ) for all x, y, z, w ∈ X.Then, there exists a unique additive mapping F : for all x, z ∈ X.
Corollary 4. Let X be a normed space and p, θ be positive real numbers with p = 1.If f : X 2 → Y is a mapping satisfying Inequality (13) for all x, y, z, w ∈ X.Then, there exists a unique additive mapping F : for all x, z ∈ X.
Since the equalities hold for all x, z ∈ V\{0}, we know that the limits of g(2 n x,2 n z) are given by lim n→∞ g (2 n x, 2 n z) for all x, z ∈ V\{0}.From inequality (25), we get for all x, y, z, w ∈ V\{0}.Since the right-hand side in the above equality equals zero, we obtain that the equality D FFF (x, y, z, w) = 0 holds for all x, y, z, w ∈ V\{0}.By Lemma 2, F satisfies the equality D FFF (x, y, z, w) = 0 for all x, y, z, w ∈ V.If F : V × V → Y is another mapping satisfying inequalities ( 26)-( 28) and the equality D F F F (x, y, z, w) = 0 for all x, y, z, w ∈ V, then we obtain the inequalities → 0 as k → ∞, we have F (x, z) = F(x, z) for all x, z ∈ V. Hence, the mapping F is the unique additive mapping, as desired.
Lee (Theorem 5 in [13] and Corollary 1 in [14]) proved the generalized Hyers-Ulam stability for an additive mapping in the case n = 1 and p < 0.
The next corollary, for the case p < 0, follows from Theorem 3.
Corollary 5. Let X be a normed space and p be a negative real number.Suppose that f , g, h : X 2 → Y are odd mappings, such that f (x + y, z + w) − g(x, z) − h(y, w) ≤ θ( x p + y p + z p + w p ) for all x, y, z, w ∈ X\{0}.Then f , g, h satisfy the equality D f f f (x, y, z, w) = 0 for all x, y, z, w ∈ X and the inequalities for all x, z ∈ X\{0}.
Proof.By Theorem 3, there exists a mapping F : X × X → Y, such that D FFF (x, y, z, w) = 0 for all x, y, z, w ∈ X and for all x, z, u ∈ X\{0}.Therefore, we obtain the inequalities  for all x, z ∈ X\{0} and k ∈ N. Since the right-hand side of the above equalities tends to zero as k → ∞ when p < 0, we know that f (x, z) = F(x, z) for all x, z ∈ X.So, f satisfies the equality D f f f (x, y, z, w) = 0 for all x, y, z, w ∈ X, which implies the equality f (x, z) = f (2x,2z) 2 for all x, z ∈ X, and the inequalities g(x, z) − f (x, z) = g(x, z) − f (2x, 2z) 2