Stability of an Additive-Cubic-Quartic Functional Equation in Multi-Banach Spaces

Abstract

We prove the Hyers-Ulam stability of the additive-cubic-quartic functional equation in multi-Banach spaces by using the fixed point alternative method. The first results on the stability in the multi-Banach spaces were presented in (Dales and Moslehian 2007).

1. Introduction

Stability is investigated when one is asking whether a small error of parameters in one problem causes a large deviation of its solution. Given an approximate homomorphism, is it possible to approximate it by a true homomorphism? In other words, we are looking for situations when the homomorphisms are stable, that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it with small error as much as possible. This problem was posed by Ulam in 1940 (cf. [1]) and is called the stability of functional equations. For Banach spaces, the problem was solved by Hyers [2] in the case of approximately additive mappings. Later, Hyers' result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by allowing the Cauchy difference to be unbounded. During the last decade, stability of functional equations was studied by several mathematicians for mappings in various spaces including random normed spaces and fuzzy Banach spaces (cf. [5, 6]). For various other results on the stability of functional equations, one is referred to [7–26].

Most of the proofs of stability theorems in the Hyers-Ulam context have applied Hyers’ direct method. The exact solution of the functional equation is explicitly constructed as the limit of a sequence, which is originating from the given approximate solution. In 2003, Radu [27] proposed the fixed point alternative method for obtaining the existence of exact solutions and error estimations and noticed that a fixed point alternative method is essential for the solution of Ulam problem for approximate homomorphisms. Subsequently, some authors [28, 29] applied the fixed alternative method to investigate the stability problems of several functional equations.

The notion of multi-normed space was introduced by Dales and Polyakov [30] (or see [31, 32]). This concept is somewhat similar to operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples were given in [30, 31]. In 2007, stability of mappings on multi-normed spaces was first given in [31], and asymptotic aspect of the quadratic functional equation in multi-normed spaces was investigated in [33].

In this paper, we consider the following functional equation derived from additive, cubic, and quartic mappings: 𝑓𝑓𝑦(𝑥).11𝑥+2𝑦+𝑓𝑥−2𝑦=44𝑥+𝑦+𝑓𝑥−𝑦+12𝑓3𝑦−48𝑓2𝑦+60𝑓−66𝑓 (1.1) It is easy to see that the function 𝑓(𝑥)=𝑎𝑥+𝑏𝑥³+𝑐𝑥⁴ satisfies the functional equation (1.1). Eshaghi Gordji et al. [34] established the general solution and proved the generalized Hyers-Ulam stability for (1.1). The main purpose of this paper is to establish the Hyers-Ulam stability of (1.1) in multi-Banach spaces by using the fixed point alternative method.

2. Preliminaries

In this section, some useful results are pointed out. We begin with the alternative of a fixed point of Diaz and Margolis, which we will refer to as follows.

Lemma 2.1 (cf. [27, 35]). Let (𝑋,𝑑) be a complete generalized metric space and 𝐽∶𝑋→𝑋 be a strictly contractive mapping, that is, 𝑑𝐽𝑥,𝐽𝑦≤𝐿𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋, (2.1) for some 𝐿≤1. Then,  for each fixed element 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=+∞∀𝑛≥0, (2.2)or𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥<+∞∀𝑛≥𝑛₀, (2.3) for some natural number 𝑛₀. Moreover, if the second alternative holds, then(i)the sequence {𝐽^𝑛𝑥} is convergent to a fixed point 𝑦^∗ of 𝐽;(ii)𝑦^∗ is the unique fixed point of 𝐽 in the set 𝑌∶={𝑦∈𝑋∣𝑑(𝐽^𝑛0𝑥,𝑦)<+∞} and 𝑑(𝑦,𝑦^∗)≤1/(1−𝐿)𝑑(𝑦,𝐽𝑦),forall𝑥,𝑦∈𝑌.

Following [30, 31], we recall some basic facts concerning multi-normed spaces and some preliminary results.

Let (ℰ,‖⋅‖) be a complex normed space, and let 𝑘∈ℕ. We denote by ℰ^𝑘 the linear space ℰ⊕⋯⊕ℰ consisting of 𝑘-tuples (𝑥₁,…,𝑥_𝑘), where 𝑥₁,…,𝑥_𝑘∈ℰ. The linear operations ℰ^𝑘 are defined coordinatewise. The zero element of either ℰ or ℰ^𝑘 is denoted by 0. We denote by ℕ_𝑘 the set {1,2,…,𝑘} and by 𝔖_𝑘 the group of permutations on 𝑘 symbols.

Definition 2.2 (cf. [30, 31]). A multi-norm on {ℰ^𝑘∶𝑘∈ℕ} is a sequence (‖⋅‖_𝑘)=(‖⋅‖_𝑘∶𝑘∈ℕ) such that ‖⋅‖_𝑘 is a norm on ℰ^𝑘 for each 𝑘∈ℕ, ‖𝑥‖₁=‖𝑥‖ for each 𝑥∈ℰ, and the following axioms are satisfied for each 𝑘∈ℕ with 𝑘≥2:
(N1)‖(𝑥_𝜎(1),…,𝑥_𝜎(𝑘))‖_𝑘=‖(𝑥₁,…,𝑥_𝑘)‖_𝑘, for 𝜎∈𝔖_𝑘, 𝑥₁,…,𝑥_𝑘∈ℰ;(N2)‖(𝛼₁𝑥₁,…,𝛼_𝑘𝑥_𝑘)‖_𝑘≤(max_{𝑖∈ℕ𝑘}|𝛼_𝑖|)‖(𝑥₁,…,𝑥_𝑘)‖_𝑘, for 𝛼₁,…,𝛼_𝑘∈ℂ,𝑥₁,…,𝑥_𝑘∈ℰ; (N3)‖(𝑥₁,…,𝑥_𝑘−1,0)‖_𝑘=‖(𝑥₁,…,𝑥_𝑘−1)‖_𝑘−1, for 𝑥₁,…,𝑥_𝑘−1∈ℰ;(N4)‖(𝑥₁,…,𝑥_𝑘−1,𝑥_𝑘−1)‖_𝑘=‖(𝑥₁,…,𝑥_𝑘−1)‖_𝑘−1, for 𝑥₁,…,𝑥_𝑘−1∈ℰ.In this case, we say that ((ℰ^𝑘,‖⋅‖_𝑘)∶𝑘∈ℕ) is a multi-normed space.

Suppose that ((ℰ^𝑘,‖⋅‖_𝑘)∶𝑘∈ℕ) is a multi-normed space, and take 𝑘∈ℕ. We need the following two properties of multi-norms. They can be found in [30](a)‖(𝑥,…,𝑥)‖_𝑘=‖𝑥‖, for 𝑥∈ℰ,(b)max_{𝑖∈ℕ𝑘}‖𝑥_𝑖‖≤‖(𝑥₁,…,𝑥_𝑘)‖_𝑘≤∑^𝑘_𝑖=1‖𝑥_𝑖‖≤𝑘max_{𝑖∈ℕ𝑘}‖𝑥_𝑖‖, for 𝑥₁,…,𝑥_𝑘∈ℰ.

It follows from (b) that if (ℰ,‖⋅‖) is a Banach space, then (ℰ^𝑘,‖⋅‖_𝑘) is a Banach space for each 𝑘∈ℕ; in this case, ((ℰ^𝑘,‖⋅‖_𝑘)∶𝑘∈ℕ) is a multi-Banach space.

Lemma 2.3 (cf. [30, 31]). Suppose that 𝑘∈ℕ and (𝑥₁,…,𝑥_𝑘)∈ℰ^𝑘. For each 𝑗∈{1,…,𝑘}, let (𝑥^𝑗_𝑛)_𝑛=1,2,… be a sequence in ℰ such that lim_𝑛→∞𝑥^𝑗_𝑛=𝑥_𝑗. Then lim_𝑛→∞𝑥¹_𝑛−𝑦₁,…,𝑥^𝑘_𝑛−𝑦_𝑘=𝑥₁−𝑦₁,…,𝑥_𝑘−𝑦_𝑘 (2.4) holds for all (𝑦₁,…,𝑦_𝑘)∈ℰ^𝑘.

Definition 2.4 (cf. [30, 31]). Let ((ℰ^𝑘,‖⋅‖_𝑘)∶𝑘∈ℕ) be a multi-normed space. A sequence (𝑥_𝑛) in ℰ is a multi-null sequence if for each 𝜀>0, there exists 𝑛₀∈ℕ such that sup_𝑘∈ℕ𝑥_𝑛,…,𝑥_{𝑛+𝑘−1𝑘}≤𝜀𝑛≥𝑛₀. (2.5)

Let 𝑥∈ℰ, we say that the sequence (𝑥_𝑛) is multi-convergent to 𝑥 in ℰ and write lim_𝑛→∞𝑥_𝑛=𝑥 if (𝑥_𝑛−𝑥) is a multi-null sequence.

3. Main Results

Throughout this section, let 𝜀≥0, ℰ be a linear space, and let ((ℱ^𝑛,‖⋅‖_𝑛)∶𝑛∈ℕ) be a multi-Banach space. For convenience, we use the following abbreviation for a given mapping 𝑓∶ℰ→ℱ: 𝑓𝑓𝑦(𝑥).𝐷𝑓𝑥,𝑦=11𝑥+2𝑦+𝑓𝑥−2𝑦−44𝑥+𝑦+𝑓𝑥−𝑦−12𝑓3𝑦+48𝑓2𝑦−60𝑓+66𝑓 (3.1) Before proceeding to the proof of the main results in this section, we shall need the following two lemmas.

Lemma 3.1 (cf. [34]). If an even mapping 𝑓∶𝑋→𝑌 satisfies (1.1), then 𝑓 is quartic.

Lemma 3.2 (cf. [34]). If an odd mapping 𝑓∶𝑋→𝑌 satisfies (1.1), then 𝑓 is cubic-additive.

Theorem 3.3. Suppose that an even mapping 𝑓∶ℰ→ℱ satisfies 𝑓(0)=0 and sup_𝑘∈ℕ𝐷𝑓𝑥₁,𝑦₁,…,𝐷𝑓𝑥_𝑘,𝑦_𝑘𝑘≤𝜀 (3.2) for all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈ℰ. Then there exists a unique quartic mapping 𝑄∶ℰ→ℱ satisfying (1.1) and sup_𝑘∈ℕ𝑓𝑥₁−𝑄𝑥₁,…,𝑓𝑥_𝑘−𝑄𝑥_𝑘𝑘≤13𝜀330 (3.3) for all 𝑥₁,…,𝑥_𝑘∈ℰ.

Proof. Letting 𝑥₁=⋯=𝑥_𝑘=0 in (3.2), we get sup_𝑘∈ℕ−12𝑓3𝑦₁+70𝑓2𝑦₁−148𝑓𝑦₁,…,−12𝑓3𝑦_𝑘+70𝑓2𝑦_𝑘−148𝑓𝑦_𝑘𝑘≤𝜀 (3.4) for all 𝑦₁,…,𝑦_𝑘∈ℰ. Replacing 𝑥₁,…,𝑥_𝑘 with 𝑦₁,…,𝑦_𝑘 in (3.2), we obtain sup_𝑘∈ℕ−𝑓3𝑦₁+4𝑓2𝑦₁+17𝑓𝑦₁,…,−𝑓3𝑦_𝑘+4𝑓2𝑦_𝑘+17𝑓𝑦_𝑘𝑘≤𝜀 (3.5) for all 𝑦₁,…,𝑦_𝑘∈ℰ.
It follows from (3.4) and (3.5) that sup_𝑘∈ℕ𝑓2𝑥₁−16𝑓𝑥₁,…,𝑓2𝑥_𝑘−16𝑓𝑥_𝑘𝑘≤13𝜀22 (3.6) for all 𝑥₁,…,𝑥_𝑘∈ℰ.
Let 𝐸∶={𝑔𝑔∶ℰ→ℱ,𝑔(0)=0}, and introduce the generalized metric 𝑑 defined on 𝐸 by 𝑑𝑔,ℎ=inf𝑐∈(0,∞)∣sup_𝑘∈ℕ𝑔𝑥₁−ℎ𝑥₁,…,𝑔𝑥_𝑘−ℎ𝑥_𝑘𝑘≤𝑐,∀𝑥₁,…,𝑥_𝑘∈ℰ. (3.7) Then, it is easy to show that 𝑑 is a complete generalized metric on 𝐸 (see the proof in [36] or [5]). We now define a function 𝐽₁∶𝐸→𝐸 by 𝐽₁𝑔(𝑥)=1𝑔()162𝑥,∀𝑥∈ℰ. (3.8) We assert that 𝐽₁ is a strictly contractive mapping. Given 𝑔,ℎ∈𝐸, let 𝑐∈(0,∞) be an arbitrary constant with 𝑑(𝑔,ℎ)≤𝑐. From the definition of 𝑑, it follows that sup_𝑘∈ℕ𝑔𝑥₁−ℎ𝑥₁,…,𝑔𝑥_𝑘−ℎ𝑥_𝑘𝑘≤𝑐 (3.9) for all 𝑥₁,…,𝑥_𝑘∈ℰ. Therefore, sup_𝑘∈ℕ𝐽₁𝑔𝑥₁−𝐽₁ℎ𝑥₁,…,𝐽₁𝑔𝑥_𝑘−𝐽₁ℎ𝑥_𝑘𝑘=sup_𝑘∈ℕ1𝑔162𝑥₁−1ℎ162𝑥₁1,…,𝑔162𝑥_𝑘−1ℎ162𝑥_𝑘𝑘≤1𝑐16 (3.10) for all 𝑥₁,…,𝑥_𝑘∈ℰ. Hence, it holds that 𝑑(𝐽₁𝑔,𝐽₁ℎ)≤(1/16)𝑐, that is, 𝑑(𝐽₁𝑔,𝐽₁ℎ)≤(1/16)𝑑(𝑔,ℎ) for all 𝑔,ℎ∈𝐸.
By using (3.6), we have 𝑑(𝐽₁𝑓,𝑓)≤(13/352)𝜀. According to Lemma 2.1, we deduce the existence of a fixed point of 𝐽₁, that is, the existence of a mapping 𝑄∶𝑋→𝑌 such that 𝑄(2𝑥)=16𝑄(𝑥) for all 𝑥∈ℰ. Moreover, we have 𝑑(𝐽^𝑛₁𝑓,𝑄)→0, which implies that 𝑄(𝑥)=lim_𝑛→∞𝐽^𝑛₁𝑓(𝑥)=lim_𝑛→∞𝑓2^𝑛𝑥16^𝑛,∀𝑥∈ℰ. (3.11) Also, 𝑑(𝑓,𝑄)≤1/(1−𝐿)𝑑(𝐽₁𝑓,𝑓) implies the inequality 𝑑≤1𝑓,𝑄()𝑑𝐽1−1/16¹𝑓,𝑓≤13330𝜀. (3.12)
Set 𝑥₁=⋯=𝑥_𝑘=2^𝑛𝑥,𝑦₁=⋯=𝑦_𝑘=2^𝑛𝑦 in (3.2), and divide both sides by 16^𝑛. Then, using property (𝑎), we obtain 𝐷𝑄𝑥,𝑦=lim_𝑛→∞116^𝑛𝐷𝑓2^𝑛𝑥,2^𝑛𝑦≤lim_𝑛→∞𝜀16^𝑛=0 (3.13) for all 𝑥,𝑦∈ℰ. Hence, by Lemma 3.1, 𝑄 is quartic.
The uniqueness of 𝑄 follows from the fact that Q is the unique fixed point of 𝐽₁ with the property that there exists 𝑙∈(0,∞) such that sup_𝑘∈ℕ𝑓𝑥₁−𝑄𝑥₁,…,𝑓𝑥_𝑘−𝑄𝑥_𝑘𝑘≤𝑙 (3.14) for all 𝑥₁,…,𝑥_𝑘∈ℰ. This completes the proof of the theorem.

Theorem 3.4. Suppose that an odd mapping 𝑓∶ℰ→ℱ satisfies sup_𝑘∈ℕ𝐷𝑓𝑥₁,𝑦₁,…,𝐷𝑓𝑥_𝑘,𝑦_𝑘𝑘≤𝜀 (3.15) for all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈ℰ. Then there exists a unique additive mapping 𝐴∶ℰ→ℱ and a unique cubic mapping 𝐶∶ℰ→ℱ satisfying (1.1) and sup_𝑘∈ℕ𝑓2𝑥₁−8𝑓𝑥₁−𝐴𝑥₁,…,𝑓2𝑥_𝑘−8𝑓𝑥_𝑘−𝐴𝑥_𝑘𝑘≤1733𝜀,sup_𝑘∈ℕ𝑓2𝑥₁−2𝑓𝑥₁−𝐶𝑥₁,…,𝑓2𝑥_𝑘−2𝑓𝑥_𝑘−𝐶𝑥_𝑘𝑘≤17𝜀231 (3.16) for all 𝑥₁,…,𝑥_𝑘∈ℰ.

Proof. Put 𝑥₁=⋯=𝑥_𝑘=0 in (3.15). Then, by the oddness of 𝑓, we have sup_𝑘∈ℕ12𝑓3𝑦₁−48𝑓2𝑦₁+60𝑓𝑦₁,…12𝑓3𝑦_𝑘−48𝑓2𝑦_𝑘+60𝑓𝑦_𝑘𝑘≤𝜀 (3.17) for all 𝑦₁,…,𝑦_𝑘∈ℰ. Replacing 𝑥₁,…,𝑥_𝑘 with 2𝑦₁,…,2𝑦_𝑘 in (3.15), we obtain sup_𝑘∈ℕ11𝑓4𝑦₁−56𝑓3𝑦₁+114𝑓2y₁−104𝑓𝑦₁,…,11𝑓4𝑦_𝑘−56𝑓3𝑦_𝑘+114𝑓2𝑦_𝑘−104𝑓𝑦_𝑘𝑘≤𝜀 (3.18) for all 𝑦₁,…,𝑦_𝑘∈ℰ. By (3.17) and (3.18), we have sup_𝑘∈ℕ𝑓4𝑥₁−10𝑓2𝑥₁+16𝑓𝑥₁,…,𝑓4𝑥_𝑘−10𝑓2𝑥_𝑘+16𝑓𝑥_𝑘𝑘≤17𝜀33 (3.19) for all 𝑥₁,…,𝑥_𝑘∈ℰ. Putting 𝛼(𝑥)∶=𝑓(2𝑥)−8𝑓(𝑥) for all 𝑥∈ℰ, we get sup_𝑘∈ℕ𝛼2𝑥₁−2𝛼𝑥₁,…,𝛼2𝑥_𝑘−2𝛼𝑥_𝑘𝑘≤1733𝜀 (3.20) for all 𝑥₁,…,𝑥_𝑘∈ℰ.
Let the same definitions for 𝐸 and 𝑑 as in the proof of Theorem 3.3 such that (𝐸,𝑑) becomes a complete generalized metric space. We now define a function 𝐽₂∶𝐸→𝐸 by 𝐽₂𝑔(𝑥)=12𝑔(2𝑥),∀𝑥∈ℰ. (3.21) Applying a similar technique as in the proof of Theorem 3.3, we obtain 𝑑(𝐽₂𝑔,𝐽₂ℎ)≤(1/2)𝑐, that is, 𝑑(𝐽₂𝑔,𝐽₂ℎ)≤(1/2)𝑑(𝑔,ℎ) for all 𝑔,ℎ∈𝐸.
By (3.20), we have 𝑑(𝐽₂𝛼,𝛼)≤(17/66)𝜀. According to Lemma 2.1, we deduce the existence of a fixed point of 𝐽₂, that is, the existence of a mapping 𝐴∶𝑋→𝑌 such that 𝐴(2𝑥)=2𝐴(𝑥) for all 𝑥∈ℰ. Moreover, we have 𝑑(𝐽^𝑛₂𝛼,𝐴)→0, which implies that 𝐴(𝑥)=lim_𝑛→∞𝐽^𝑛₂𝛼(𝑥)=lim_𝑛→∞𝛼2^𝑛𝑥2^𝑛,∀𝑥∈ℰ. (3.22) Also, 𝑑(𝛼,𝐴)≤1/(1−𝐿)𝑑(𝐽₂𝛼,𝛼) implies the inequality 𝑑()≤1𝛼,𝐴()𝑑𝐽1−1/2²𝛼,𝛼≤1733𝜀. (3.23)
Hence, it follows that 𝐷𝐴𝑥,𝑦=lim_𝑛→∞12^𝑛𝐷𝛼2^𝑛𝑥,2^𝑛𝑦=lim_𝑛→∞12^𝑛𝐷𝑓2^𝑛+1𝑥,2^𝑛+1𝑦−8𝐷𝑓2^𝑛𝑥,2^𝑛𝑦≤lim_𝑛→∞𝜀2^𝑛=0 (3.24) for all 𝑥,𝑦∈ℰ. This means that 𝐴 satisfies (1.1). Then, by Lemma 3.2, 𝑥↦𝐴(2𝑥)−8𝐴(𝑥) is additive. Thus, by 𝐴(2𝑥)=2𝐴(𝑥), we conclude that 𝐴 is additive.
Putting 𝛽(𝑥)∶=𝑓(2𝑥)−2𝑓(𝑥) for all 𝑥∈ℰ, we get sup_𝑘∈ℕ𝛽2𝑥₁−8𝛽𝑥₁,…,𝛼2𝑥_𝑘−8𝛽𝑥_𝑘𝑘≤17𝜀33 (3.25) for all 𝑥₁,…,𝑥_𝑘∈ℰ. We now define a function 𝐽₃∶𝐸→𝐸 by 𝐽₃𝑔(𝑥)=18𝑔(2𝑥),∀𝑥∈ℰ. (3.26) Applying a similar technique as in the proof of Theorem 3.3, we obtain 𝑑(𝐽₃𝑔,𝐽₃ℎ)≤(1/8)𝑐, that is, 𝑑(𝐽₃𝑔,𝐽₃ℎ)≤(1/8)𝑑(𝑔,ℎ) for all 𝑔,ℎ∈𝐸.
By (3.25), we have 𝑑(𝐽₃𝛽,𝛽)≤(17/264)𝜀. According to Lemma 2.1, we deduce the existence of a fixed point of 𝐽₃, that is, the existence of a mapping 𝐶∶𝑋→𝑌 such that 𝐶(2𝑥)=8𝐶(𝑥) for all 𝑥∈ℰ. Moreover, we have 𝑑(𝐽^𝑛₃𝛽,𝐶)→0, which implies that 𝐶(𝑥)=lim_𝑛→∞𝐽^𝑛₃𝛽(𝑥)=lim_𝑛→∞𝛽2^𝑛𝑥8^𝑛,∀𝑥∈ℰ. (3.27) Also, 𝑑(𝛽,𝐶)≤1/(1−𝐿)𝑑(𝐽₃𝛽,𝛽) implies the inequality 𝑑≤1𝛽,𝐶()𝑑𝐽1−1/8³𝛽,𝛽≤17231𝜀. (3.28)
Then we have 𝐷𝐶𝑥,𝑦=lim_𝑛→∞18^𝑛𝐷𝛽2^𝑛𝑥,2^𝑛𝑦=lim_𝑛→∞18^𝑛𝐷𝑓2^𝑛+1𝑥,2^𝑛+1𝑦−2𝐷𝑓2^𝑛𝑥,2^𝑛𝑦≤lim_𝑛→∞𝜀8^𝑛=0 (3.29) for all 𝑥,𝑦∈ℰ. Hence, the mapping 𝐶 satisfies (1.1). Therefore, by Lemma 3.2, 𝑥↦𝐶(2𝑥)−2𝐶(𝑥) is cubic. Thus, 𝐶(2𝑥)=8𝐶(𝑥) implies that the mapping 𝐶 is cubic.
The uniqueness of 𝐴 and 𝐶 can be proved in the same reasoning as in the proof of Theorem 3.3. This completes the proof of the theorem.

Theorem 3.5. Suppose that an odd mapping 𝑓∶ℰ→ℱ satisfies sup_𝑘∈ℕ𝐷𝑓𝑥₁,𝑦₁,…,𝐷𝑓𝑥_𝑘,𝑦_𝑘𝑘≤𝜀 (3.30) for all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈ℰ. Then there exists a unique additive mapping 𝐴∶ℰ→ℱ and a unique cubic mapping 𝐶∶ℰ→ℱ such that sup_𝑘∈ℕ𝑓𝑥₁−𝐴𝑥₁−𝐶𝑥₁,…,𝑓𝑥_𝑘−𝐴𝑥_𝑘−𝐶𝑥_𝑘𝑘≤68𝜀693 (3.31) for all 𝑥₁,…,𝑥_𝑘∈ℰ.

Proof. By Theorem 3.4, there exist an additive mapping 𝐴₀∶ℰ→ℱ and a cubic mapping 𝐶₀∶ℰ→ℱ such that sup_𝑘∈ℕ𝑓2𝑥₁−8𝑓𝑥₁−𝐴₀𝑥₁,…,𝑓2𝑥_𝑘−8𝑓𝑥_𝑘−𝐴₀𝑥_𝑘𝑘≤1733𝜀,sup_𝑘∈ℕ𝑓2𝑥₁−2𝑓𝑥₁−𝐶₀𝑥₁,…,𝑓2𝑥_𝑘−2𝑓𝑥_𝑘−𝐶₀𝑥_𝑘𝑘≤17𝜀231 (3.32) for all 𝑥₁,…,𝑥_𝑘∈ℰ. Combining the above two equations in (3.32) yields that sup_𝑘∈ℕ6𝑓𝑥₁+𝐴₀𝑥₁−𝐶₀𝑥₁,…,𝑓6𝑥_𝑘+𝐴₀𝑥_𝑘−𝐶₀𝑥_𝑘𝑘≤136𝜀231 (3.33) for all 𝑥₁,…,𝑥_𝑘∈ℰ. So we obtain (3.31) by letting 𝐴=−𝐴₀/6 and 𝐶=𝐶₀/6.
To prove the uniqueness of 𝐴 and 𝐶, let 𝐴₁,𝐶₁∶ℰ→ℱ be other additive and cubic mappings satisfying (3.31). Let 𝐴^′=𝐴−𝐴₁ and 𝐶^′=𝐶−𝐶₁. Then, using property (𝑎), we obtain 𝐴^′(𝑥)+𝐶^′(𝑥)≤𝑓(𝑥)−𝐴(𝑥)−𝐶(𝑥)+𝑓(𝑥)−𝐴₁(𝑥)−𝐶₁(𝑥)≤136𝜀693 (3.34) for all 𝑥∈ℰ, then (3.34) implies that lim_𝑛→∞18^𝑛𝐴^′2^𝑛𝑥+𝐶^′2^𝑛𝑥=0 (3.35) for all 𝑥∈ℰ. Therefore, 𝐶^′(𝑥)=0 for all 𝑥∈ℰ. By (3.34), we have 𝐴^′(𝑥)=0 for all 𝑥∈ℰ. This completes the proof of the theorem.

Theorem 3.6. Suppose that a mapping 𝑓∶ℰ→ℱ satisfies 𝑓(0)=0 and sup_𝑘∈ℕ𝐷𝑓𝑥₁,𝑦₁,…,𝐷𝑓𝑥_𝑘,𝑦_𝑘𝑘≤𝜀 (3.36) for all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈ℰ. Then there exists a unique additive mapping 𝐴∶ℰ→ℱ, a unique cubic mapping 𝐶∶ℰ→ℱ, and a unique quartic mapping 𝑄∶ℰ→ℱ such that sup_𝑘∈ℕ(𝑓𝑥₁−𝐴𝑥₁−𝐶𝑥₁−𝑄𝑥₁,…,𝑓𝑥_𝑘−𝐴𝑥_𝑘−𝐶𝑥_𝑘−𝑄𝑥_𝑘𝑘≤953𝜀6930 (3.37) for all 𝑥₁,…,𝑥_𝑘∈ℰ.

Proof. Let 𝑓_𝑜(𝑥)=(1/2)(𝑓(𝑥)−𝑓(−𝑥)) for all 𝑥∈ℰ, then 𝑓_𝑜(0)=0,𝑓_𝑜(−𝑥)=−𝑓_𝑜(𝑥) and sup_𝑘∈ℕ𝐷𝑓_𝑜𝑥₁,𝑦₁,…,𝐷𝑓_𝑜𝑥_𝑘,𝑦_𝑘𝑘≤𝜀 (3.38) for all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈ℰ. From Theorem 3.5, it follows that there exists a unique additive mapping 𝐴∶ℰ→ℱ and a unique cubic mapping 𝐶∶ℰ→ℱ satisfying (3.31).
Let 𝑓_𝑒(𝑥)=(1/2)(𝑓(𝑥)+𝑓(−𝑥)) for all 𝑥∈ℰ, then 𝑓_𝑒(0)=0,𝑓_𝑒(−𝑥)=𝑓_𝑒(𝑥) and sup_𝑘∈ℕ𝐷𝑓_𝑒𝑥₁,𝑦₁,…,𝐷𝑓_𝑒𝑥_𝑘,𝑦_𝑘𝑘≤𝜀 (3.39) for all 𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘∈ℰ. By Theorem 3.3, there exists a unique quartic mapping 𝑄∶ℰ→ℱ satisfying (3.3). Now it is obvious that (3.37) holds for all 𝑥₁,…,𝑥_𝑘∈ℰ. This completes the proof of the theorem.

Acknowledgments

The authors are very grateful to the referees for their helpful comments and suggestions. This research is supported by Guangdong Provincial Natural Science Foundation (07301595), China.