Nearly General Septic Functional Equation

If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of Gă vruta, we examine the stability of general septic functional equation ∑i=08Cið−1Þ f ðx + ði − 4ÞyÞ = 0 which considered. The method of Găvruta as just mentioned was given in the reference Gavruta (1994).


Introduction
The concept of stability for a functional equation arising when replacing the functional equation by an inequality which acts as a perturbation of the equation. Ulam [1] posed the question concerning the stability of group homomorphisms. Hyers [2] gave the first partial affirmative answer to the question of Ulam, which states that if δ > 0 and f : X ⟶ Y is a mapping with X a normed space, Y a Banach space such that then there exists a unique additive mapping T : X ⟶ Y such that Thereafter, many authors have generalized the Hyers' result (refer to [3][4][5][6][7][8]). On the other hand, a generalization of the Hyers result was also obtained by Gȃvruta [9]. That is, let ðG, +Þ be an abelian group and let Y be a Banach space. Suppose that a function φ : G 2 ⟶ ½0,∞Þ satisfies the assumptionφ for all x, y ∈ G, and that f : G ⟶ Y is a mapping such that the inequality holds for all x, y ∈ G: Then, there exists a unique additive map- We, meanwhile, call the functional equation the n-monomial functional equation. The 1-monomial, the 2-monomial, the 3-monomial, the 4-monomial, the 5-monomial, the 6-monomial, and the 7-monomial functional equation are called additive, quadratic, cubic, quartic, quintic, sextic, and septic functional equation, respectively. In this case, we say that each solution of the previous equation is additive, quadratic, cubic, quartic, quintic, sextic, and septic mapping, respectively. The function f : R ⟶ R defined by f ðxÞ ≔ ax n is a particular solution of the n-monomial functional equation.
Quite recently, Lee [6] showed the stability of the n-monomial functional equation in the sense of Gȃvruta.
The following functional equation is called as Jensen, general quadratic, general cubic, general quartic, general quintic, general sextic, and general septic functional equation, respectively, for n = 2, 3, 4, 5, 6, 7, 8: The solution of the general septic functional equation is said to be a general septic mapping. The function f : R ⟶ R given by f ðxÞ ≔ ∑ 7 i=0 a i x i is a particular solution of the general septic mapping. More detailed term for the concept of a general septic mapping can be found in Baker's paper [10] by the term generalized polynomial mapping of degree at most 7.
Our principal purpose is to consider the following general septic functional equation and then we are going to obtain the stability theorems of the functional equation (8) in the spirit of Gȃvruta approach.

Stability of the General Septic Functional Equation (8)
In this section, we let Y and V be a real Banach space and a real vector space, respectively. For a given mapping f : V ⟶ Yand all x, y ∈ V, we use the following abbreviations On the other hand, iff is a mapping defined bỹ In addition, through tedious computation, we then get the following expressions Γf for all x ∈ V: for all x ∈ V and all integers n ≥ 0 and holds for all x ∈ V and all integers n ≥ 0 and is fulfilled for all x ∈ V and all integers n ≥ 0: Proof. By using (11) and the definitions of J n f and J ′ n f , we can obtain the result after tedious calculations. Therefore, the proof will be omitted here.

Lemma 2. Assume that
for all x, y ∈ V: Then, we have for all x ∈ V and all positive integers n: So, we figure out which implies J n f ðxÞ = f ðxÞ for all x ∈ V and all positive integers n: Similarly, we get the equality J′ n f ðxÞ = f ðxÞ for all x ∈ V and all positive integers n.
We are now in a position to prove the following theorem.
for all x, y ∈ V: Assume that f : V ⟶ Y is a mapping subject to the inequality for all x, y ∈ V: Then, there exists a unique general septic mapping F with Fð0Þ = 0 such that for all x ∈ V, where φ e : V 2 ⟶ ½0,∞Þ and Φ, Φ ′ : V ⟶ ½0,∞Þ are functions defined by Then, by (11) and the definitions of Γf and Δf , we obtain that Δf 3 Journal of Function Spaces hold for all x ∈ V: It follows from (15) and (26) for all x ∈ V. This gives that for all x ∈ V and all nonnegative integers n and m: By the definition of Φ and Φ′ together with (21) and (29), the sequence fJ nf ðxÞg is Cauchy in Y: And since Y is complete, the sequence fJ nf ðxÞg converges. Therefore, we can define a mapping F : Note that J 0f ðxÞ = f ðxÞ − f ð0Þ and Fð0Þ = 0 follow fromf ð0Þ = 0: Furthermore, by letting n = 0 and passing the limit as m ⟶ ∞ in (29), we arrive at the inequality (23). On the other hand, from (21) for all x, y ∈ V.
In order to prove the uniqueness of F, we suppose that F′ : V ⟶ Y is another general septic mapping satisfying (23) and F ′ð0Þ = 0: However, it is also possible to show uniqueness by replacing condition (23) with weaker condition. So, we want to prove that there is a unique mapping satisfying the weaker conditioñ for all x ∈ V. According to Lemma 2, we get F ′ ðxÞ = J n F ′ ðxÞ for all positive integers n: Thus, by using the condition (32) and the definition of J n , after tedious calculations, we have for all x ∈ V and all positive integer n: Taking the limit as n ⟶ ∞ in the last inequality, we then have This implies that F = F ′: for all x, y ∈ V: Assume that f : V ⟶ Y is a mapping subject to the inequality for all x, y ∈ V: Then, there exists a unique general septic mapping F with Fð0Þ = 0 such that for all x ∈ V, where φ e , Φ and Φ ′ are the functions in Theorem 3.