A Pexider system of additive functional equations in Banach algebras

In this paper, we solve the system of functional equations


Introduction
Let B be a complex Banach algebra and let g : B → B be a C-linear mapping.Mirzavaziri and Moslehian [1] introduced the concept of g-derivation f : B → B as follows: f (xy) = f (x)g(y) + g(x)f (y) (1.1) for all x, y ∈ B. Park et al. [2] introduced the concept of hom-derivation on B, i.e., g : B → B is a homomorphism and f satisfies (1.1) for all x, y ∈ B.
The stability problem of functional equations originated from a question of Ulam [3] concerning the stability of group homomorphisms.Hyers [4] gave a first affirmative partial answer to the question of Ulam for Banach spaces.Hyers' theorem was generalized by Aoki [5] for additive mappings and by Th.M. Rassias [6] for linear mappings by considering an unbounded Cauchy difference.A generalization of the Th.M. Rassias theorem was obtained by Găvruta [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias' approach.Recently, Lee et al. [8,9] extended more general functional equations, which were mixed types of additive, quadratic and cubic functional equations in Banach spaces, and Park and Rassias [10] applied the functional equation theory to study partial multipliers in C * -algebras.Many mathematicians developed the Hyers results in various directions [11][12][13][14][15][16][17][18].
The method provided by Hyers [4] which produces the additive function will be called a direct method.This method is the most significant and strongest tool to study the stability of different functional equations.That is, the exact solution of the functional equation is explicitly constructed as a limit of a sequence, starting from the given approximate solution [19,20].The other significant method is the fixed point theorem, that is, the exact solution of the functional equation is explicitly created as a fixed point of some certain map (see [21,22]).
We consider a fixed point alternative theorem.
Theorem 1.1 [23] Assume that (B, d) is a complete generalized metric space and I : B → B is a strictly contractive mapping, that is, for all u, v ∈ B and a Lipschitz constant L < 1.Then for each given element u ∈ B, either for some positive integer n 0 .Furthermore, if the second alternative holds, then: (i) the sequence (I n u) is convergent to a fixed point p of I; (ii) p is the unique fixed point of I in the set In this paper, we consider the following system of functional equations: The aim of the present paper is to solve the system of functional equations (1.2) and prove the Hyers-Ulam stability of g-derivations in complex Banach algebras by using the fixed point method.
Throughout this paper, we assume that B is a complex Banach algebra.

Stability of the system of functional equations (1.2)
We solve and investigate the system of additive functional equations (1.2) in complex Banach algebras.Proof Letting x = y = 0 in (1.2), we get 2), we have for all x ∈ B. Setting y = 0 in (1.2), we obtain 2), we have for all x, y ∈ B. Hence the mapping f : B → B is additive and thus by (2.1) the mapping g : B → B is additive.
Using the fixed point technique, we prove the Hyers-Ulam stability of the system of the additive functional equations (1.2) in complex Banach algebras.
for all x, y ∈ B. Then there exist unique additive mappings F, G : B → B such that for all x ∈ B.
Proof Putting x = y = 0 in (2.3), we get 3), we obtain (2.4) and we consider inf ∅ = +∞.Then d is a complete generalized metric on (see [25]).Now, we define the mapping J : ( , Actually, let δ, γ ∈ ( , d) be given such that d(δ, γ ) = μ.Then for all x ∈ B. Hence for all x ∈ B and all δ, γ ∈ .Using (2.4), we obtain for all x ∈ B, which imply that d(f , J f ) ≤ L 2 and d(g, J g) ≤ L 2 .Using the fixed point alternative, we deduce the existence of unique fixed points of J , that is, the existence of mappings F, G : B → B, respectively, such that with the following property: there exist μ, η ∈ (0, ∞) satisfying for all x ∈ B. Since lim n→∞ d(J n f , F) = 0 and lim n→∞ d(J n g, G) = 0, for all x ∈ B. Using (2.2) and ( 2.3), we conclude that for all x, y ∈ B, since L < 1. Therefore by Lemma 2.2, the mappings F, G : B → B are additive.
Corollary 2.4 Let η, p be nonnegative real numbers with p ≥ 1 and let f , g : B → B be mappings satisfying for all x, y ∈ B. Then there exist unique additive mappings F, G : B → B such that Proof The proof follows from Theorem 2.3 by taking L = 2 1-p and (x, y) = η( x p + y p ) for all x, y ∈ B.

Stability of G-derivations in Banach algebras
In this section, by using the fixed point technique, we prove the Hyers-Ulam stability of g-derivations in complex Banach algebras.
for all x, y ∈ B and all λ ∈ T 1 .Then the mappings f , g : B → B are C-linear.
Proof If we put λ = 1 in (3.1), then f and g are additive by Lemma 2.2.
Letting y = x in (3.1), we have for all x ∈ B and all λ ∈ T 1 .Since the mappings f and g are additive, for all x ∈ B and all λ ∈ T 1 .So by Lemma 2.1 the mappings f and g are C-linear.
Theorem 3.2 Suppose that : B 2 → [0, ∞) is a function such that there exists an L < 1 with for all x, y ∈ B. Let f , g : B → B be mappings satisfying for all x, y ∈ B and all λ ∈ T 1 .Then there exist unique C-linear mappings F, G : B → B such that F is a G-derivation and for all x ∈ B.
Proof Let λ = 1 in (3.3).By Theorem 2.3, there are unique additive mappings F, G : B → B satisfying (3.5) and (3.6) given by for all x ∈ B. Using (3.2) and (3.3), we conclude that for all x, y ∈ B and all λ ∈ T 1 , since L < 1. Therefore by Lemma 3.1, the mappings F, G : Corollary 3.3 Let p, q, η be nonnegative real numbers with p + q > 2 and let f , g : B → B be mappings satisfying ⎧ ⎨ ⎩ f (λ(x + y)) + g(λ(yx)) -2λf (x) ≤ η x p y q , g(λ(x + y))f (λ(yx)) -2λg(y) η x p y q and f (xy)f (x)g(y)g(x)f (y) ≤ x p y q for all x, y ∈ B and all λ ∈ T 1 .Then there exist unique C-linear mappings F, G : B → B such that F is a G-derivation and x p+q , G(x)g(x) ≤ η 2 p+q -2 x p+q for all x ∈ B.
Proof The proof follows from Theorem 3.2 by taking (x, y) = η x p y q for all x, y ∈ B.
Choosing L = 2 2-p-q , we obtain the desired result.

Conclusion
We solved the system of functional equations (1.2) and we proved the Hyers-Ulam stability of g-derivations in Banach algebras.
x, y) = 0 for all x, y ∈ B. SoF(xy) = F(x)G(y) + G(x)F(y)for all x, y ∈ B. Thus the C-linear mapping F is a G-derivation.