On new stability results for composite functional equations in quasi - β - normed spaces

: In this article, we prove the generalized Hyers - Ulam - Rassias stability for the following composite functional equation: where f maps from a β p , ( ) - Banach space into itself, by using the fixed point method and the direct method. Also, the generalized Hyers - Ulam - Rassias stability for the composite s - functional inequality is discussed via our results.


Introduction and preliminaries
The stability problem of functional equations was first originated by Ulam [1] in 1940 as the following question concerning the stability of homomorphisms. Let G , 1 ( * ) be a group, G , • 2 ( )be a metric group with the metric d and f G G be a mapping such that such that The above problem was partially answered by Hyers [2] in 1941 under the assumption that the function f maps between two Banach spaces.
In 1978, Rassias [3] considered an unbounded Cauchy difference which generalizes Hyers's result as the following inequality: for all x y X , ∈ , where f maps from a Banach space X into a Banach space Y , ε 0 > and p 0 1 < < . Moreover, a generalization of this stability theorem was published by Gãvruta [4] in 1994 by replacing the un-bounded Cauchy difference with some condition involving a general control function in the spirit of Rassias approach.
In 2009, Fechner [14] discussed the following composite functional equation: f f x f y f x y f x y f x f y ( ( )  for all x y G , ∈ , where f is a self-mapping on an abelian group G, which is motivated from the following absolute value identity due to Tarski [15]: for all x y , ∈ . Fechner [14] also solved the solution of the composite functional equation (1.1) as follows.
Theorem 1.1. [14] Let G, ( +) be an abelian group uniquely divisible by 2. A function f G G : → is a solution of (1.1) and ( ) if and only if f is additive and f f f ∘ = .
Theorem 1.2. [14] Suppose that f : → is continuous at zero. Then f is a solution of (1.1) if and only if one of the following possibilities holds: In 2010, Kochanek [16] investigated the problem of determining general solutions f : → of the composite functional equation (1.1) under some assumptions upon f ( ). In the same year, Fechner [17] investigated three composite functional inequalities which are motivated from the composite functional equation (1.1) as follows: for all x y , ∈ , where f is in the class of mappings from into itself.
One year later, Fechner [18] studied the approximated solutions of the composite functional equation (1.1), where f is a continuous function from a Banach space E into itself, that is, the stability result under the following assumption: Kenary [19] proved the generalized Hyers-Ulam-Rassias stability of the composite functional equation (1.1) in random normed spaces and non-Archimedean normed spaces by using the direct method and the fixed point method. It follows from the different behavior on the tool for measuring distance between points of abstract spaces that the results of Kenary [19] differ from the main results of Fechner [18]. Nowadays, many results concerning the composite functional equation (1.1) in various abstract spaces are given. Moreover, many mathematicians also investigated functional equations related to the Ulam stability theory in various forms of functional equations such as quintic, sextic, septic, octic, nonic, decic functional equations etc. (see [20][21][22][23][24][25][26][27] and references therein).
To the best of our knowledge, there is no discussion so far concerning the generalized Hyers-Ulam-Rassias stability of the composite functional equation (1.1)

Preliminaries
In this section, we recall some basic concepts of quasi-β-normed spaces and many results which are needed in the main results.
Definition 2.1. [28] Let β be a real number with β 0 1 < ≤ , and X be a vector space over a field with = or . A function is called a quasi-β-norm on X if it satisfies the following conditions: Also, the pair X, is called a quasi-β-normed space. The smallest possible K is called the modulus of concavity of ∥⋅∥.
From the above definition, it is easy to see that a β p , ( )-norm is continuous. For more details of quasi-β-normed spaces, we refer to [28]. Next, we introduce a definition of b-metric spaces.
Definition 2.6. [29] Let X be a nonempty set, K 1 ≥ and d X X : 0 , × → [ ∞) be a function satisfying the following conditions for all x y z X , , ∈ : .
)is said to be complete if every Cauchy sequence is convergent.
To prove our main results, the following result is needed.
Let G be an abelian group and f G G : → be a mapping satisfying (1.1). Then the following assertions hold: Next, we recall the classical fixed point theorem in generalized metric spaces which is the important tool for investigating many stability results.
) be a complete generalized metric space and let J X X : → be a strictly contractive mapping with the Lipschitz constant L with L 0 1 < < . Then for each given element for all nonnegative integers n or there exists a positive integer n 0 such that the following assertions hold: A generalization of Theorem 2.11 was proved by Aydi and Czerwik [31] by considering the fixed point theorem in generalized b-metric spaces.
)be a complete generalized b-metric space and T X X : → satisfies the condition for all x y X , ∈ with D x y , Suppose that x X ∈ is arbitrarily fixed. Then either for every nonnegative integer n or there exists an k 0 ∈ such that the following assertions hold: } is a Cauchy sequence in X; 3. there exists a point u X ∈ such that Moreover, if D is continuous (with respect to one variable) and This implies that for all y X ∈ .

Main results
In this section, we present the stability results of the composite functional equation (1.1) by using the fixed point method and the direct method. The first part is related to the fixed point method, and the results which are obtained by the direct method are given in the second part. Throughout this section, let X be a β p , ( )-Banach space and f X X : → be a mapping. For each x y X , ∈ , we will use the following symbol:

Stability of the composite functional equation by using the fixed point method
In this section, the stability of the composite functional equation )-Banach spaces by using a fixed point approach.
)-Banach space with the modulus of concavity K and ϕ X X : for all x y X , ∈ and f satisfies the following condition: , . Then there exists a unique composite mapping A X X : → such that for all x X ∈ .
Proof. Let g X X Ω : and so for all x X ∈ . Define a mapping J : Ω Ω → by for all g Ω ∈ . We want to show that ) = ∞ for all g h , Ω ∈ , then the above inequality is true. So we may assume that ). By Remark 2.13 and (3.6), we have  ( ) for all x y X , ∈ . Letting n → ∞ in the last inequality and using the condition (A), we obtain for all x y X , ∈ and so A is a composite mapping. From (3.8), we obtain for all x X ∈ . To show the uniqueness of A, suppose that A X X : ′ → is a composite mapping and for all x X ∈ . By Lemma 2.10, we have A x A x 2 2 n n ( ) = ( ) and A x A x 2 2 n n ′( ) = ′( ) for all x X ∈ and for all n ∈ . From (3.1) and (3.11), we get for all x X ∈ . Since the right-hand side of the inequality tends to zero as n → ∞, we get A A = ′. □ for all x y X , 0 ∈ ⧹{ } and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that for all x X 0 ∈ ⧹{ }.
Proof. Define a mapping ϕ X X : We will show that ϕ x y Lϕ x y 2 , 2 for all x y X , ∈ , where L 0 1 ≤ < . By Theorem 3.1, there exists a unique composite mapping A X X : → such that )-Banach space with the modulus of concavity K and f X X : → be a mapping. Suppose that there are a positive real number λ and real numbers p q , with p q 0 + < and K 2 2 p q β β < ( + ) such that for all x y X , 0 ∈ ⧹{ } and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that Proof. Define a mapping ϕ X X : We will show that ϕ x y Lϕ x y 2 , 2 , ( ) ≤ ( ) for all x y X , ∈ , where L 0 1 ≤ < . We can easily see that it holds for x 0 = or y 0 = . Suppose that x 0 ≠ and y 0 ≠ . Then we get ϕ x y λ x y ϕ x y Lϕ x y 2 , 2 2 2 , , , ( + ) where L 2 0 ,1.
p q β ≔ ∈( ) ( + ) So we have for all x y X , ∈ , where L 0 1 ≤ < . By Theorem 3.1, there exists a unique composite mapping A X X : → such that )-Banach space with the modulus of concavity K and f X X : → be a mapping. Suppose that there are a positive real number λ and a negative real number s with K 2 2 for all x y X , 0 ∈ ⧹{ } and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that for all x X 0 ∈ ⧹{ }.
Proof. Define a mapping ϕ X X : for all x y X , ∈ . We will show that ϕ x y Lϕ x y 2 , 2 , for all x y X , ∈ , where L 0 1 ≤ < . By Theorem 3.1, there exists a unique composite mapping A X X : → such that for all x X 0 ∈ ⧹{ }. □ Next, we present the following lemma in order to give the final stability result of the composite functional equation (1.1) in this part.
Lemma 3.5. Let X be a quasi-β-normed space and f X X : → be a mapping satisfying (1.1). If f is odd, then f is additive.
for any x X ∈ . Putting x 0 = in (3.12), we get f 0 0 ( ) = . Letting y 0 = in (1.1), we have f f x f x ( ( )) = ( ) (3.13) for any x X ∈ . Substituting y by y − into (1.1), by the oddness of f yields that for any x y X , ∈ . From (1.1) and (3.14), we obtain for any x y X , ∈ . By (3.13) and (3.15), we get (3.16) for any x y X , ∈ . From (1.1), (3.14) and (3.15), we obtain and so f  for all x y X , ∈ . From (3.18) and (3.19), we obtain f x y f x f y ( + ) = ( ) + ( ) (3.20) for any x y X , ∈ , and so f is additive. □ Theorem 3.6. Let X be a β p , ( )-Banach space with the modulus of concavity K and ϕ X X : 0 , × → [ ∞) be a function such that for all x y X , ∈ , where L 0 1 ≤ < and KL 2 β < . Suppose that f X X : → is a mapping satisfying for all x y X , ∈ and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that for all x X ∈ . Moreover, if A is odd, then A is additive.
Proof. The assertion immediately follows from Theorem 3.1 combined with Lemma 3.5. □

Stability of the composite functional equation by using the direct method
In this section, we consider the generalized Hyers-Ulam-Rassias stability of the composite functional equation (1.1) in β p , ( )-Banach spaces by using the direct method.
Theorem 3.7. Let X be a β p , ( )-Banach space with the modulus of concavity K and ϕ X X : 0 , × → [ ∞) be a function such that for all x y X , ∈ . Suppose that f X X : → is a mapping satisfying for all x y X , ∈ and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that for all x X ∈ . Moreover, if A is odd, then A is additive.
Proof. Replacing y by x into (3.25), we get for all x X ∈ . From (3.24), the right-hand side of the above inequality tends to zero as m → ∞.  (3.33) for all x X ∈ . From (3.31), we arrive to the inequality (3.34) and so for all x X ∈ . To show that A satisfies (1.1), by using (3.24) and (3.25), we get for all x y X , ∈ and so A is a composite mapping. To show the uniqueness of A, suppose that A X X : ′ → is a composite mapping and it satisfies (3.26  for all x X ∈ . Since the right-hand side of the inequality tends to zero as n → ∞, we have A A = ′. If A is odd, by Lemma 3.5, then A is additive. □ for all x y X , 0 ∈ ⧹{ } and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that for all x X 0 ∈ ⧹{ }.
Proof. Define a mapping ϕ X X : By Theorem 3.7, there exists a unique composite mapping A X X : → such that for all x X 0 ∈ ⧹{ }. □ Corollary 3.9. Let X be a β p , ( )-Banach space with the modulus of concavity K and f X X : → be a mapping. Suppose that there are a positive real number λ and real numbers r s , with 2 1 for all x y X , 0 ∈ ⧹{ } and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that for all x X 0 ∈ ⧹{ }.
Proof. Define a mapping ϕ X X : for all x y X , 0 ∈ ⧹{ } and f satisfies the condition (A). Then there exists a unique composite mapping A X X : → such that this article, it is also the disadvantage at the same time. In the case of the upper bound of the norm of the difference between both sides of the composite functional equation (1.1) is controlled by some positive constant, the first main result in Section 3 (Theorem 3.1) cannot be applied in this case. It is still open for the researcher to seek the suitable or the generalized control function covering this case and all results in this article.

Conclusions and recommendations
Based on the fixed point method and the direct method, we proved the generalized Hyers-Ulam stability results of the composite functional equation (1.1) in β p , ( )-Banach spaces. Also, we gave many results which are obtained by choosing the suitable mapping ϕ like the sum or the multiplication of the power of norms. Moreover, the generalized Hyers-Ulam stability results of the composite s-functional inequality is solved and the generalized Hyers-Ulam stability result of this inequality is discussed. To recommend the way for making the research, the reader can use the main results in this article to provide the stability results for the pexiderized composite functional equation and the pexiderized composite s-functional inequality. Furthermore, our results can be extended to the generalized hyperstability results of the composite functional equation and the composite s-functional inequality by using the fixed point theorem in quasi-β-Banach spaces (see [33,34]).

Funding information: This work was supported by Thammasat University Research Unit in Fixed Points and Optimization.
Conflict of interest: Authors state no conflict of interest.