The stability of a general sextic functional equation by fixed point theory

The stability of a general sextic functional equation by fixed point theory Yang-Hi Lee 1,† Soon-Mo Jung 2,† and Jaiok Roh 3,*,† 1 Department of Mathematics Education, Gongju National University of Education, Gongju 32553, Korea 2 Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Korea 3 Ilsong College of Liberal Arts, Hallym University, Chuncheon, Kangwon-Do 200-702, Korea * Correspondence: joroh@hallym.ac.kr; Tel.: +82-33-248-2015 † These authors contributed equally to this work.


Introduction
In 1940, Ulam [1] remarked the problem concerning the stability of group homomorphisms. In 1941, Hyers [2] gave an answer to this question for additive mappings between Banach spaces. Subsequently, many mathematicians came to deal with this question (cf. [3][4][5][6][7][8][9][10]). Let V and W be real vector spaces, X be a real normed space, Y be a real Banach space, n ∈ ℕ (the set of natural numbers), and f : V ⟶ W be a given mapping. Consider the functional equation for every x, y ∈ V, where n C i = n!/i!ðn − iÞ!. The functional equation (1) is called an n-monomial functional equation, and every solution of the functional equation (1) is called to be a monomial mapping of degree n. The function f : ℝ ⟶ ℝ given by is a particular solution of the functional equation (1). In particular, the functional equation (1) is called a sextic functional equation for the case n = 6, and every solution of the functional equation (1) is called to be a sextic mapping for the case n = 6. Many mathematicians [11][12][13][14][15][16][17] have previously investigated the stability of the sextic functional equation, and many authors [18][19][20][21][22][23][24][25][26] have studied the stability of the n-monomial functional equation in various spaces. The solution of the functional equation is called a generalized polynomial mapping of degree n ∈ ℕ (See Baker [27]). The function f : ℝ ⟶ ℝ given by is a particular solution of the functional equation (3). Some mathematicians [28][29][30][31] have previously investigated the stability of the functional equation (3) for the cases n = 4, 5, 6, 7. In particular, the functional equation is called a general sextic functional equation, and every solution of the functional equation (4) is said to be the general sextic mapping.
In this paper, we will partially generalize the results in [31] for the stability of the general sextic functional equation. For the details, one can refer Corollary 4 and Corollary 7 which are special cases of main theorems. Specifically, in this paper, we will show that there is only one solution F of the general sextic functional equation (4) near the function f , which approximates the functional equation (4) by using fixed point theorem [32][33][34][35]. Moreover, the solution mapping F of the functional equation (4) can be explicitly constructed by the formula which approximates the mapping f .

Main Results
We first recall the following Margolis and Diaz fixed point theorem, which is necessary to obtain the main results of this paper.
Proposition 1 (see [36]). Suppose ðX, dÞ is a complete generalized metric space, which means that the metric d may assume infinite values, and J : X ⟶ X is a strictly contractive mapping with the Lipschitz constant 0 < L < 1. Then, for each given element x ∈ X, either or there exists an integer k ≥ 0 such that: (i) dðJ n x, J n+1 xÞ < +∞ for all n ≥ k (ii) The sequence fJ n xg converge to a fixed point y * of J (iii) y * is a unique fixed point of J in Y = fy ∈ X : dðJ k x, yÞ<+∞g (iv) dðy, y * Þ ≤ 1/ð1 − LÞdðy, JyÞ for every y ∈ Y In this paper, we let V and W be real vector spaces, X be a real normed space, and Y be a real Banach space. For a mapping g : V ⟶ W, we use the following abbreviations for every x, y ∈ V.
Now, we will see useful lemma for the proof of main theorem.

Journal of Function Spaces
And, to obtain the equality (8), by (7), we obtain the following calculation: In the following main theorem, we will prove the generalized Hyers-Ulam stability of the functional equation (4) by using the direct method.
then there exists the unique solution mapping F : V ⟶ Y of (4) such that for all In particular, F is represented by for all x ∈ V.
Proof. We let the set S be the set of the functions g : V ⟶ Y with gð0Þ = 0. And we define a generalized metric on Sby Then, it is not so difficult to show that ðS, dÞ is a complete generalized metric space (see ([34], Theorem 2.5) or the proof of ( [37], Theorem 3.1)). Next, we see the mapping J : S ⟶ S, which is defined by for all x ∈ V: And, by using the oddness and the evenness of g o and g e and n C i−1 + n C i = n+1 C i , due to mathematical induction, we can get holds for all n ∈ ℕ and x ∈ V. Let g, h ∈ S and we choose K ∈ ½0,∞ as an arbitrary constant with dðg, hÞ ≤ K. Due to the definition of d and (7) in And, by (11) we obtain for every x ∈ V. It implies that dð f , J f Þ ≤ 1 < ∞ from the definition of d and due to Proposition 1, the sequence fJ n f g converges to only one fixed point F : V ⟶ Y of J in the set T = fg ∈ S : dðf , gÞ<∞g which implies (13). Moreover, by Proposition 1, we have which implies (12). Also, by the equality (8) in Lemma 2, since one has for all x, y ∈ V, we obtain for every x, y ∈ V. Therefore, F is the unique solution of the functional equation (4) with (12). Finally, we see that if F is a solution of the sextic functional equation (4) with Fð0Þ = 0, then we can derive that F is a fixed point of J from the equality In next corollary, we will consider special function φðx, yÞ = ∥x∥ p + ∥y∥ p in Theorem 3 to compare with the results in [31]. 4 Journal of Function Spaces Corollary 4. Let X be a real normed space, θ be as in Lemma 2, and p be a fixed real number such that 0 < p < log 2 ð4 ffiffiffiffiffi 21 p − 14Þ. If f : X ⟶ Y satisfies the equality f ð0Þ = 0 and the inequality for all x, y ∈ X, then there exists a unique solution mapping F : V ⟶ Y of (4) satisfying the inequality for all x ∈ X.
Next, we will try to prove the stability of the sextic functional equation (4) from another point of view. For that, we first will introduce useful facts in the following lemma.
holds for all x, y ∈ V.
Proof. When 0 < θ < π/4 and cos And by (17) and (18), we have for all x, y ∈ V. Therefore, by taking the limit, we complete the proof of (19).
In the following theorem, we will prove the stability of the solution for the sextic functional equation (4) with different types of functions compared to Theorem 3. Theorem 6. Let θ, L, and φ be as in Lemma 5. If f : V ⟶ Y is a mapping such that the inequality (11) in Theorem 3 holds for every x, y ∈ V, then there exists only one solution F : V ⟶ Y of (4) satisfying the following inequality !
And, by (11) in assumption, we obtain , for every x ∈ V.
It implies that dð f , J f Þ ≤ 1 < ∞ by the definition of d. Therefore, according to Proposition 1, the sequence fJ n f g converges to only one fixed point F : V ⟶ Y of J in the set T = fg ∈ S | dð f , gÞ<∞g, which is represented by (23) for every x ∈ V.
We also due to Proposition 1 obtain that which implies (22). Now, by (19) in Lemma 5, since we have which conclude that F is a solution of the sextic functional equation (4).
Finally, we see that if F is a solution of the sextic functional equation (4) implies that F is a fixed point of J.
In next corollary, we will consider special function φðx, yÞ = kxk p + kyk p in Theorem 6 to compare with the results in [31].