Hyperstability of the k-Cubic Functional Equation in Non-Archimedean Banach Spaces

)e starting point of studying the stability of functional equations seems to be the famous talk of Ulam [1] in 1940, in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms. Ulam’s problem: let G1 be a group and let G2 be a metric group with ametric d. Given ε> 0, there exists δ > 0 such that if a mapping h: G1⟶ G2 satisfies the inequality


Introduction
e starting point of studying the stability of functional equations seems to be the famous talk of Ulam [1] in 1940, in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms.
Ulam's problem: let G 1 be a group and let G 2 be a metric group with a metric d. Given ε > 0, there exists δ > 0 such that if a mapping h: G 1 ⟶ G 2 satisfies the inequality for all x, y ∈ G 1 , then there exists a homomorphism H: for all x ∈ G 1 . e first partial answer, in the case of Cauchy equation in Banach spaces, to Ulam question was given by Hyers [2]. Later, the result of Hyers was first generalized by Aoki [3], and only much later by Rassias [4] and Gȃvruţa [5]. Since then, the stability problems of several functional equations have been extensively investigated [6][7][8][9][10].
We say a functional equation is hyperstable if any function f satisfying the equation approximately (in some sense) must be actually a solution to it. It seems that the first hyperstability result was published in [11] and concerned the ring homomorphisms. However, the term hyperstability has been used for the first time in [12]. Quite often, hyperstability is confused with superstability, which also admits bounded functions. Numerous papers on this subject have been published, and we refer, for example, to [3,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. roughout this paper, N stands for the set of all positive integers and N m 0 , the set of integers greater than or equal m 0 , R + ≔ [0, ∞), and we use the notation X 0 for the set X∖ 0 { }. Let us recall (see, for instance, [30]) some basic definitions and facts concerning non-Archimedean normed spaces. Definition 1. By a non-Archimedean field, we mean a field K equipped with a function (valuation) | · |: K ⟶ [0, ∞) such that, for all r, s ∈ K, the following conditions hold: (1) |r| � 0 if and only if r � 0 (2) |rs| � |r||s| (3) |r + s| ≤ max |r|, |s| { } e pair (K, |.|) is called a valued field. In any non-Archimedean field, we have |1| � | − 1| � 1 and |n| ≤ 1, for n ∈ N 0 . In any field K, the function | · |: K ⟶ R + given by is a valuation which is called trivial, but the most important examples of non-Archimedean fields are p-adic numbers which have gained the interest of physicists for their research in some problems coming from quantum physics, p-adic strings, and superstrings.
Definition 2. Let X be a vector space over a scalar field K with a non-Archimedean nontrivial valuation | · |. A function ‖ · ‖ * : X ⟶ R is a non-Archimedean norm (valuation) if it satisfies the following conditions: e strong triangle inequality (ultrametric), namely, en, (X, ‖ · ‖ * ) is called a non-Archimedean normed space or an ultrametric normed space.
Definition 3. Let x n be a sequence in a non-Archimedean normed space X.
converges to zero.
(2) e sequence x n is said to be convergent if there exists x ∈ X such that, for any ε > 0, there is a positive integer N such that ‖x n − x‖ * ≤ ε, for all n ≥ N. en, the point x ∈ X is called the limit of the sequence x n , which is denoted by lim n⟶∞ x n � x. (3) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space or an ultrametric Banach space.
Let X and Y be normed spaces. A function f: X ⟶ Y is called a k−cubic function provided it satisfies the functional equation: for all x, y ∈ X, (5) and we can say that f: In 2013, Bahyrycz et al. [31] used the fixed point theorem from eorem 1 in [24] to prove the stability results for the generalization of p-Wright affine equation in ultrametric spaces. Recently, corresponding results for more general functional equations (in classical spaces) have been proved in [32][33][34][35].
In this paper, by using the fixed point method derived from [20,21,36], we present some hyperstability results for equation (5) in ultrametric Banach spaces. Before proceeding to the main results, we state eorem 1 which is useful for our purpose. To present it, we introduce the following three hypotheses: (H1): X is a nonempty set, Y is an ultrametric Banach space over a non-Archimedean field, f 1 , . . . , f k : X ⟶ X, and L 1 , . . . , L k : X ⟶ R + are given. (H2): T: Y X ⟶ Y X is an operator satisfying the inequality anks to a result due to Brzdçk and Ciepliński ( [25], Remark 2), we state a slightly modified version of the fixed point theorem ( [24], eorem 1) in ultrametric spaces. We use it to assert the existence of a unique fixed point of operator T: Y X ⟶ Y X . Theorem 1. Let hypotheses (H1)-(H3) be valid, and functions ε: X ⟶ R + and φ: X ⟶ Y fulfill the following two conditions: en, there exists a unique fixed point ψ ∈ Y X of T with Moreover,

Main Results
In this section, we use eorem 1 as a basic tool to prove the hyperstability results of the k-cubic functional equation (5) in ultrametric Banach spaces.

Theorem 2.
Let (X, ‖ · ‖) and (Y, ‖ · ‖ * ) be normed space and ultrametric Banach space, respectively, c ≥ 0, p, q ∈ R, and p + q < 0, and let f: for all x, y ∈ X 0 such that kx Since p + q < 0, one of p, q must be negative. Assume that q < 0 and replacing y by mx and x by ((m + 1)/k)x in (11), we obtain and write It is easily seen that Λ m has the form described in (H3) with k � 4, (11) can be written in the following way: Moreover, for every ξ, μ ∈ Y X 0 , x ∈ X 0 , So, (H2) is valid. By using mathematical induction, we will show that, for each x ∈ X 0 , we have where α m � ((m − 1)/k) p+q . We obtain that (18) holds for n � 0. Next, we will assume that (18) holds for n � r, where r ∈ N. en, we have

Journal of Mathematics
is shows that (18) holds for n � r + 1. Now, we can conclude that inequality (18) holds for all n ∈ N 0 . From (18), we obtain for all x ∈ X 0 . Hence, according to eorem 1, there exists a unique solution C m : X 0 ⟶ Y of the equation: such that Moreover, for all x ∈ X 0 . Now, we show that for every x, y ∈ X 0 such that x + y ≠ 0, x − y ≠ 0. Since the case n � 0 is just (11), take r ∈ N, and suppose the last inequality holds for n � r and every x, y ∈ X 0 such that Journal of Mathematics for all x, y ∈ X 0 such that x + y ≠ 0 and x − y ≠ 0. us, by induction, we have shown that suppose the last inequality holds for every n ∈ N 0 . Letting n ⟶ ∞, we obtain that C m (kx + y) + C m (kx − y) � kC m (x + y) + kC m (x − y) for all x, y ∈ X 0 such that x + y ≠ 0, x − y ≠ 0. In this way, we obtain a sequence C m m ≥ m 0 of k−cubic functions on X 0 such that is implies that It follows, with m ⟶ ∞, that f is k−cubic on X 0 . In a similar way, we can prove the following theorem. □ Theorem 3. Let (X, ‖ · ‖) and (Y, ‖ · ‖ * ) be normed space and ultrametric Banach space, respectively, c ≥ 0, p, q ∈ R, and p + q > 0, and let f: X ⟶ Y satisfies for all x, y ∈ X 0 such that x + y ≠ 0 and Since p + q > 0, one of p, q must be positive, and let q > 0 and replace y by (−2/m)y and x by ((m − 2)/km)y in (29). us, Writing e rest of the proof is similar to the proof of the last theorem. It easy to show the hyperstability of cubic equation on the set containing 0. e above theorems imply, in particular, the following corollary, which shows their simple application. □ Corollary 1. Let (X, ‖ · ‖) and (Y, ‖ · ‖ * ) be normed space and ultrametric Banach space, respectively, G: X 2 ⟶ Y and G(x 0 , y 0 ) ≠ 0 for some x 0 , y 0 ∈ X, and ‖G(x, y)‖ * ≤ c‖x‖ p ‖y‖ q , x, y ∈ X, where c ≥ 0, p, q ∈ R. Assume that the numbers p and q satisfy one of the following conditions: (1) p + q < 0, and (11) holds for all x, y ∈ X 0  G(x, y), x, y ∈ X, (36) has no solution in the class of functions g: X ⟶ Y.
Proof. Replacing x by mx and y by (−km + 1)x for m ∈ N in (39), we get for all x ∈ X 0 . For each m ∈ U, we define the operator T m : Y X 0 ⟶ Y X 0 by