Combined system of additive functional equations in Banach algebras

: In this study, we solve the system of additive functional equations:

https://doi.org/10.1515/math-2023-0177 received July 15, 2023; accepted January 9, 2024 Abstract: In this study, we solve the system of additive functional equations: 1 Introduction Let be a complex Banach algebra and → f : be a -linear mapping.Mirzavaziri and Moslehian [1] introduced the concept of f -derivation → g : as follows: for all ∈ x y , .Park et al. [2] introduced the concept of hom-derivation on , i.e., → f : is a homomorphism and → g : satisfies (1.1) for all ∈ x y , .Dehghanian et al. [3] introduced the concept of hom-der → g : as follows: for all ∈ x y , .Kheawborisuk et al. [4] defined and studied hom-ders in fuzzy Banach algebras. .Then, h is a homomorphism, f is a -linear mapping, and g , which is defined by We say that an equation is stable if any function satisfying the equation approximately is near to an exact solution of the equation.
The method provided by Hyers [6] that produces the additive function is called a direct method.This method is the most significant and strong tool to concerning the stability of different functional equations, i.e., the exact solution of the functional equation is explicitly constructed as a limit of a sequence, starting from the given approximate solution [21,22].The other significant method is a fixed point method, i.e., the exact solution of the functional equation is explicitly created as a fixed point of some certain mapping [23][24][25][26].
We remember a fixed point alternative theorem.

I I ( )
for some positive integer n 0 .Furthermore, if the second alternative holds, then (i) the sequence In this article, we consider the following system of additive functional equations: for all ∈ x y , .The aim of this article is to investigate the system of additive functional equations and prove the Hyers-Ulam stability of (homomorphism, derivation)-systems in complex Banach algebras by using the fixed point method.
Throughout this study, assume that is a complex Banach algebra.

Stability of system of additive functional equations
In this section, we investigate the system of additive functional equations (1.2) in complex Banach algebras.
[28] Let be a complex Banach algebra and → : be an additive mapping such that for all ∈ x y , and all ∈ λ 1 .Then, the mappings → f g h , , : are -linear.
Proof.Let = λ 1 in (2.1).Then, h is additive and for all ∈ x y , .Hence, the mapping → g : is additive, and thus, the mapping and so ( ) for all ∈ x .So by Lemma 2.1, the mappings h, g, and f are -linear.□ Using the fixed point technique, we prove the Hyers-Ulam stability of the system of additive functional equations (1.2) in complex Banach algebras.
) is a function such that there exists an < L 1 with: ) , for all ∈ x y , .Then, there exist unique -linear mappings , , and so , and so We define a generalized metric on Γ as follows: and we consider ∅ = +∞ inf .Then, d is a complete generalized metric on Γ [29].Now, we define the mapping ) be given such that for all ∈ x and all ∈ δ γ , Γ.
) , and Using the fixed point alternative, we deduce the existence of unique fixed points of , i.e., the existence of mappings → H G F , , : , respectively, such that with the following property: there exist , , for all ∈ x .Using (2.3) and (2.4), we conclude that for all ∈ x y , and all ∈ λ 1 .
Therefore, by Lemma 2.2, the mappings → H G F , , : are -linear.□ Corollary 2.4.Let p and q be nonnegative real numbers with + > p q 1 and → h g f , , : be mappings satisfying System of additive functional equations  5 for all ∈ x y , and all ∈ λ 1 .Then, there exist unique -linear mappings → H G F , , : for all ∈ x .
Proof.The proof follows from Theorem 2.3 by taking = Δ x y x y , , we obtain the desired result.□ Corollary 2.5.Let p and θ be nonnegative real numbers with > p 1 and → h g f , , : be mappings satisfying for all ∈ x y , and all ∈ λ 1 .Then, there exist unique -linear mappings → H G F , , : Proof.The proof follows from Theorem 2.3 taking 1 , we obtain the desired result.□

Stability of (homomorphism, derivation)-systems in Banach algebras
In this section, by using the fixed point technique, we prove the Hyers-Ulam stability of (homomorphism, derivation)-systems in complex Banach algebras.
) is a function such that there exists an < L 1 with:   Proof.The proof follows from Theorem 3.1 and Corollary 2.5.□

Conclusion
We solved the system of additive functional equation (1.2), and we investigated the Hyers-Ulam stability of (homomorphism, derivation)-systems in Banach algebras.
Proof.The proof follows from Theorem 3.1 and Corollary 2.4.satisfying (2.10) such that F is a homomorphism and G is a (homomorphism, derivation)-system.
□ Corollary 3.3.Let p and θ be nonnegative real numbers with >