Nearly Radical Quadratic Functional Equations in p-2-Normed Spaces

Abstract

We establish some stability results in 2-normed spaces for the radical quadratic functional equation 𝑓(∑^𝑛_𝑖=1(𝑥_𝑖+𝑦_𝑖)²)+𝑓(∑^𝑛_𝑖=1(𝑥_𝑖−𝑦_𝑖)²)=2∑^𝑛_𝑖=1(𝑓(𝑥_𝑖)+𝑓(𝑦_𝑖)) and then use subadditive functions to prove its stability in 𝑝-2-normed spaces.

1. Introduction and Preliminaries

The story of the stability of functional equations dates back to 1925 when a stability result appeared in the celebrated book by Póolya and Szeg [1]. In 1940, Ulam [2, 3] posed the famous Ulam stability problem which was partially solved by Hyers [4] in the framework of Banach spaces. Later Aoki [5] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [6] provided a generalization of Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. Găvruţa [7] obtained the generalized result of T. M. Rassias’ theorem which allows the Cauchy difference to be controlled by a general unbounded function. On the other hand, Rassias, Găvruţa, and several authors proved the Ulam-Gavruta-Rassias stability of several functional equations. For more details about the results concerning such problems, the reader is referred to [8–30].

Gajda and Ger [31] showed that one can get analogous stability results, for subadditive multifunctions. For further results see [32–42], among others.

The most famous functional equation is the Cauchy equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) any solution of which is called additive. It is easy to see that the function 𝑓∶ℝ→ℝ defined by 𝑓(𝑥)=𝑐𝑥² with 𝑐 an arbitrary constant is a solution of the functional equation 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦. (1.1) So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known [43, 44] that a function 𝑓∶𝑋→𝑌 between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function 𝐵₁∶𝑋×𝑋→𝑌 such that 𝑓(𝑥)=𝐵₁(𝑥,𝑥) for all 𝑥∈𝑋. The 𝐵₁(𝑥,𝑦)=(1/4)(𝑓(𝑥+𝑦)−𝑓(𝑥−𝑦)) for all 𝑥,𝑦∈𝑋.

We briefly recall some definitions and results used later on in the paper. For more details, the reader is referred to [45–49]. The theory of 2-normed spaces was first developed by Gähler [46] in the mid-1960s, while that of 2-Banach spaces was studied later by Gähler and White [47, 49].

Definition 1.1 (see [45]). Let  𝒳  be a real linear space over  ℝ  with dim 𝒳>1  and  ‖⋅,⋅‖∶𝒳×𝒳→ℝ  a function.
Then (𝒳,‖⋅,⋅‖)  is called a linear 2-normed space if(²𝑁₁)‖𝑥,𝑦‖>0 and ‖𝑥,𝑦‖=0 if and only if 𝑥 and 𝑦 are linearly dependent,(²𝑁₂)‖𝑥,𝑦‖=‖𝑦,𝑥‖,(²𝑁₃)‖𝛼𝑥,𝑦‖=|𝛼|‖𝑥,𝑦‖, for any 𝛼∈ℝ,(²𝑁₄)‖𝑥,𝑦+𝑧‖≤‖𝑥,𝑦‖+‖𝑥,𝑧‖,for all 𝑥,𝑦,𝑧∈𝒳. The function ‖⋅,⋅‖ is called the 2-norm on 𝒳.

Let (𝒳,‖⋅,⋅‖) be a linear 2-normed space. If 𝑥∈𝒳 and ‖𝑥,𝑦‖=0, for all 𝑦∈𝒳, then 𝑥=0. Moreover, for a linear 2-normed space (𝒳,‖⋅,⋅‖), the functions 𝑥→‖𝑥,𝑦‖ are continuous functions of 𝒳 into ℝ for each fixed 𝑦∈𝒳 (see [48]).

A sequence {𝑥_𝑛} in a linear 2-normed space 𝒳 is called a Cauchy sequence if there are two points 𝑦,𝑧∈𝒳 such that 𝑦 and 𝑧 are linearly independent, lim_{𝑛,𝑚→∞}‖𝑥_𝑛−𝑥_𝑚,𝑦‖=0 and lim_{𝑛,𝑚→∞}‖𝑥_𝑛−𝑥_𝑚,𝑧‖=0.

A sequence {𝑥_𝑛} in a linear 2-normed space 𝒳 is called a convergent sequence if there is an 𝑥∈𝒳 such that lim_𝑛→∞‖𝑥_𝑛−𝑥,𝑦‖=0, for all 𝑦∈𝒳. If {𝑥_𝑛} converges to 𝑥, write 𝑥_𝑛→𝑥 as 𝑛→∞ and call 𝑥 the limit of {𝑥_𝑛}. In this case, we also write lim_𝑛→∞𝑥_𝑛=𝑥.

A linear 2-normed space in which every Cauchy sequence is a convergent sequence is called a 2-Banach space. For a convergent sequence {𝑥_𝑛} in a 2-normed space 𝒳, lim_𝑛→∞‖𝑥_𝑛,𝑦‖=‖lim_𝑛→∞𝑥_𝑛,𝑦‖, for all 𝑦∈𝒳 (see [48]).

We fix a real number 𝑝 with 0<𝑝≤1, and let 𝒴 be a linear space. A 𝑝-2-norm is a function on 𝒴×𝒴 satisfying Definition 1.1; (²𝑁₁), (²𝑁₂), and (²𝑁₄); the following: ‖𝛼𝑥,𝑦‖=|𝛼|^𝑝‖𝑥,𝑦‖, for all 𝑥,𝑦∈𝒴 and all 𝛼∈ℝ. The pair (𝒴,‖⋅,⋅‖) is called a 𝑝-2-normed space if ‖⋅,⋅‖ is a 𝑝-2-norm on 𝒴. A 𝑝-2-Banach space is a complete 𝑝-2-normed space.

We recall that a subadditive function is a function 𝜑_𝑎∶𝐴→𝐵, having a domain 𝐴 and a codomain (𝐵,≤) that are both closed under addition, with the following property: 𝜑_𝑎𝑥+𝑦≤𝜑_𝑎(𝑥)+𝜑_𝑎𝑦, (1.2) for all 𝑥,𝑦∈𝐴. Let ℓ∈{−1,1} be fixed. If there exists a constant 𝐿 with 0<𝐿<1 such that a function 𝜑_𝑎∶𝐴→𝐵 satisfies ℓ𝜑_𝑎𝑥+𝑦≤ℓ𝐿^ℓ𝜑_𝑎(𝑥)+𝜑_𝑎𝑦, (1.3) for all 𝑥,𝑦∈𝐴, then we say that 𝜑_𝑎 is contractively subadditive if ℓ=1, and 𝜑_𝑎 is expansively superadditive if ℓ=−1. It follows by the last inequality that 𝜑_𝑎 satisfies the following properties: 𝜑_𝑎2^ℓ𝑥≤2^ℓ𝐿𝜑_𝑎(𝑥),𝜑_𝑎2^ℓ𝑘𝑥≤2^ℓ𝐿^𝑘𝜑_𝑎(𝑥), (1.4) for all 𝑥∈𝐴 and integers 𝑘≥1.

Now, we consider the radical quadratic functional equation 𝑓^𝑛_𝑖=1𝑥_𝑖+𝑦_𝑖²+𝑓^𝑛_𝑖=1𝑥_𝑖−𝑦_𝑖²=2^𝑛_𝑖=1𝑓𝑥_𝑖+𝑓𝑦_𝑖, (1.5) where 𝑛∈ℕ is a fixed integer and prove generalized Ulam stability, in the spirit of Găvruta (see [7]), of this functional equation in 2-normed spaces. Moreover, in this paper, we investigate new results about the generalized Ulam stability by using subadditive functions in 𝑝-2-normed spaces for the radical quadratic functional equation (1.5).

2. Main Results

In this section, let 𝑋 be a linear space, and let ℝ and ℝ⁺ denote the sets of real and positive real numbers, respectively. If a mapping 𝑓∶ℝ→𝑋 satisfies the functional equation (1.5), by letting 𝑥_𝑖=𝑦_𝑖=0(1≤𝑖≤𝑛) in (1.5), we get 𝑓(0)=0. Setting 𝑥_𝑖=𝑦_𝑖=𝑥(1≤𝑖≤𝑛) in (1.5) and using 𝑓(0)=0, we get 𝑓√4𝑛𝑥²=4𝑛𝑓(𝑥), (2.1) for all 𝑥∈ℝ. Putting 𝑥_𝑖=2𝑥,𝑦_𝑖=0(1≤𝑖≤𝑛) in (1.5) and using 𝑓(0)=0, we get 2𝑓√4𝑛𝑥²=2𝑛𝑓(2𝑥), (2.2) for all 𝑥∈ℝ. It follows from (2.1) and (2.2) that 𝑓2^𝑚𝑥=4^𝑚𝑓(𝑥), (2.3) for all 𝑥∈ℝ and integers 𝑚≥1. Setting 𝑦_𝑛=−𝑦_𝑛 in (1.5) and then comparing it with (1.5), we obtain 𝑓(−𝑦_𝑛)=𝑓(𝑦_𝑛), for all 𝑦_𝑛∈ℝ. Letting 𝑥_𝑖=𝑦_𝑖=0(2≤𝑖≤𝑛) in (1.5), we get 𝑓𝑥₁+𝑦₁²+𝑓𝑥₁−𝑦₁²=2𝑓𝑥₁+2𝑓𝑦₁, (2.4) for all 𝑥₁,𝑦₁∈ℝ. It follows from (2.4) and the evenness of 𝑓 that 𝑓 satisfies (1.1). So we have the following lemma.

Lemma 2.1. If a mapping 𝑓∶ℝ→𝑋 satisfies the functional equation (1.5), then 𝑓 is quadratic.

Corollary 2.2. If a mapping 𝑓∶ℝ→𝑋 satisfies the functional equation (1.5), then there exists a symmetric biadditive mapping 𝐵₁∶ℝ×ℝ→𝑋 such that 𝑓(𝑥)=𝐵₁(𝑥,𝑥), for all 𝑥∈ℝ.

Hereafter, we will assume that 𝒳 is a 2-Banach space. First, using an idea of Găvruţa [7], we prove the stability of (1.5) in the spirit of Ulam, Hyers, and Rassias.

Let 𝜙 be a function from ℝ^2𝑛+1 to ℝ⁺∪{0}. A mapping 𝑓∶ℝ→𝒳 is called a 𝜙-approximatively radical quadratic function if 𝑓^𝑛_𝑖=1𝑥_𝑖+𝑦_𝑖²+𝑓^𝑛_𝑖=1𝑥_𝑖−𝑦_𝑖²−2^𝑛_𝑖=1𝑓(𝑥_𝑖)+𝑓(𝑦_𝑖),𝑧_𝒳≤𝜙𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧, (2.5) for all 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧∈ℝ, where 𝑛∈ℕ is a fixed integer.

Theorem 2.3. Let ℓ∈{−1,1} be fixed, and let 𝑓∶ℝ→𝒳 be a 𝜙- approximatively radical quadratic function with 𝑓(0)=0. If the function 𝜙∶ℝ^2𝑛+1→ℝ⁺∪{0} satisfies Φ(𝑥,𝑧)∶=^∞_{()𝑗=1−ℓ/2}14^ℓ𝑗𝜙2^ℓ𝑗𝑥,…,^{2𝑛0𝑥0𝑓𝑒37}2^ℓ𝑗+1𝑥,𝑧2𝜙2^1+ℓ𝑗𝑥,…,^{𝑛0𝑥0𝑓𝑒37}2^1+ℓ𝑗𝑥,0,…,^{𝑛0𝑥0𝑓𝑒37}0,𝑧<∞, (2.6) and lim_𝑚→∞(1/4^ℓ𝑚)𝜙(2^ℓ𝑚𝑥₁,…,2^ℓ𝑚𝑥_𝑛,2^ℓ𝑚𝑦₁,…,2^ℓ𝑚𝑦_𝑛,𝑧)=0, for all 𝑥,𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧∈ℝ, then there exists a unique quadratic mapping ℱ∶ℝ→𝒳, satisfies (1.5) and the inequality 𝑓(𝑥)−ℱ(𝑥),𝑦_𝒳≤1Φ4𝑛𝑥,𝑦, (2.7) for all 𝑥,𝑦∈ℝ.

Proof. Letting 𝑥_𝑖=𝑥+𝑦,𝑦_𝑖=𝑥−𝑦(1≤𝑖≤𝑛) in (2.5), we get 𝑓√4𝑛𝑥²+𝑓4𝑛𝑦²−2𝑛𝑓(𝑥+𝑦)−2𝑛𝑓(𝑥−𝑦),𝑧_𝒳≤𝜙𝑥+𝑦,…,^{𝑛0𝑥0𝑓𝑒37}𝑥+𝑦,𝑥−𝑦,…,^{𝑛0𝑥0𝑓𝑒37}𝑥−𝑦,𝑧, (2.8) for all 𝑥,𝑦,𝑧∈ℝ. Setting 𝑥_𝑖=𝑦_𝑖=𝑥(1≤𝑖≤𝑛) in (2.5), we get 𝑓√4𝑛𝑥²−4𝑛𝑓(𝑥),𝑧_𝒳≤𝜙𝑥,…,^{2𝑛0𝑥0𝑓𝑒37}𝑥,𝑧, (2.9) for all 𝑥,𝑧∈ℝ. Replacing 𝑦 by 𝑥 in (2.8), we obtain 𝑓√4𝑛𝑥²−𝑛𝑓(2𝑥),𝑧_𝒳≤12𝜙2𝑥,…,^{𝑛0𝑥0𝑓𝑒37}2𝑥,0,…,^{𝑛0𝑥0𝑓𝑒37}0,𝑧, (2.10) for all 𝑥,𝑧∈ℝ. It follows from (2.9) and (2.10) that 4𝑓(𝑥)−𝑓(2𝑥),𝑦_𝒳≤1𝑛𝜙^{2𝑛0𝑥0𝑓𝑒37}+1𝑥,…,𝑥,𝑦𝜙2𝑛^{𝑛0𝑥0𝑓𝑒37}2𝑥,…,2𝑥,0,…,^{𝑛0𝑥0𝑓𝑒37}0,𝑦, (2.11) for all 𝑥,𝑦∈ℝ. Thus, 𝑓(𝑥)−14𝑓(2𝑥),𝑦_𝒳≤1𝜙4𝑛^{2𝑛0𝑥0𝑓𝑒37}+1𝑥,…,𝑥,𝑦𝜙8𝑛^{𝑛0𝑥0𝑓𝑒37}2𝑥,…,2𝑥,^{𝑛0𝑥0𝑓𝑒37},𝑓(𝑥)𝑥0,…,0,𝑦−4𝑓2,𝑦_𝒳≤1𝑛𝜙^{2𝑛0𝑥0𝑓𝑒37}𝑥2𝑥,…,2+1,𝑦𝜙2𝑛^{𝑛0𝑥0𝑓𝑒37}𝑥,…,𝑥,0,…,^{𝑛0𝑥0𝑓𝑒37}0,𝑦, (2.12) for all 𝑥,𝑦∈ℝ. Hence, 14^ℓ𝑘𝑓2^ℓ𝑘𝑥−14^ℓ𝑟𝑓2^ℓ𝑟𝑥,𝑦_𝒳≤14𝑛^{𝑟−(1+ℓ)/2}_{()𝑗=𝑘+1−ℓ/2}14^ℓ𝑗𝜙2^ℓ𝑗𝑥,…,^{2𝑛0𝑥0𝑓𝑒37}2^ℓ𝑗+1𝑥,𝑦2𝜙2^1+ℓ𝑗𝑥,…,^{𝑛0𝑥0𝑓𝑒37}2^1+ℓ𝑗𝑥,0,…,^{𝑛0𝑥0𝑓𝑒37}0,𝑦 (2.13) for all 𝑥,𝑦∈ℝ and integers 𝑟>𝑘≥0. Thus, {(1/4^ℓ𝑚)𝑓(2^ℓ𝑚𝑥)} is a Cauchy sequence in the 2-Banach space 𝒳. Hence, we can define a mapping ℱ∶ℝ→𝒳 by ℱ(𝑥)∶=lim_𝑚→∞(1/4^ℓ𝑚)𝑓(2^ℓ𝑚𝑥), for all 𝑥∈ℝ. That is, lim_𝑚→∞14^ℓ𝑚𝑓2^ℓ𝑚𝑥−ℱ(𝑥),𝑦_𝒳=0, (2.14) for all 𝑥,𝑦∈ℝ. In addition, it is clear from (2.5) that the following inequality: ℱ^𝑛_𝑖=1𝑥_𝑖+𝑦_𝑖²+ℱ^𝑛_𝑖=1𝑥_𝑖−𝑦_𝑖²−2^𝑛_𝑖=1ℱ(𝑥_𝑖)+ℱ(𝑦_𝑖),𝑧_𝒳=lim_𝑚→∞14^ℓ𝑚𝑓4^𝑛ℓ𝑚_𝑖=1𝑥_𝑖+𝑦_𝑖²+𝑓4^𝑛ℓ𝑚_𝑖=1𝑥_𝑖−𝑦_𝑖²−2^𝑛_𝑖=1𝑓2^ℓ𝑚𝑥_𝑖+𝑓2^ℓ𝑚𝑦_𝑖,𝑧_𝒳≤lim_𝑚→∞14^ℓ𝑚𝜙2^ℓ𝑚𝑥₁,…,2^ℓ𝑚𝑥_𝑛,2^ℓ𝑚𝑦₁,…,2^ℓ𝑚𝑦_𝑛,𝑧=0 (2.15) holds for all 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧∈ℝ, and so by Lemma 2.1, the mapping ℱ∶ℝ→𝒳 is quadratic. Taking the limit 𝑟→∞ in (2.13) with 𝑘=0, we find that the mapping ℱ is quadratic mapping satisfying the inequality (2.7) near the approximate mapping 𝑓∶ℝ→𝒳 of (1.5). To prove the aforementioned uniqueness, we assume now that there is another quadratic mapping 𝒢∶ℝ→𝒳 which satisfies (1.5) and the inequality (2.7). Since the mapping 𝒢∶ℝ→𝒳 satisfies (1.5), then 𝒢2^ℓ𝑥=4^ℓ𝒢(𝑥),𝒢2^ℓ𝑚𝑥=4^ℓ𝑚𝒢(𝑥) (2.16) for all 𝑥∈ℝ and integers 𝑚≥1. Thus, one proves by the last equality and (2.7) that 14^ℓ𝑚𝑓(2^ℓ𝑚𝑥)−𝒢(𝑥),𝑦_𝒳=14^ℓ𝑚𝑓(2^ℓ𝑚𝑥)−𝒢(2^ℓ𝑚𝑥),𝑦_𝒳≤14^𝑚ℓ+1𝑛Φ2^ℓ𝑚𝑥,𝑦, (2.17) for all 𝑥,𝑦∈ℝ and integers 𝑚≥1. Therefore, from 𝑚→∞, one establishes ℱ(𝑥)−𝒢(𝑥)=0 for all 𝑥∈ℝ.

Corollary 2.4. Let ℓ∈{−1,1} be fixed. If there exist nonnegative real numbers 𝑝_𝑖,𝑞_𝑖,𝑞 with ℓ∑^𝑛_𝑖=1(𝑝_𝑖+𝑞_𝑖)<2ℓ such that a mapping 𝑓∶ℝ→𝒳 satisfies the inequality 𝑓^𝑛_𝑖=1𝑥_𝑖+𝑦_𝑖²+𝑓^𝑛_𝑖=1𝑥_𝑖−𝑦_𝑖²−2^𝑛_𝑖=1𝑓(𝑥_𝑖)+𝑓(𝑦_𝑖),𝑧_𝒳≤𝜃^𝑛_𝑖=1𝑥_𝑖^𝑝𝑖𝑦_𝑖^𝑞𝑖|𝑧|^𝑞, (2.18) for all 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧∈ℝ and some 𝜃≥0, then there exists a unique quadratic mapping ℱ∶ℝ→𝒳, satisfies (1.5) and the inequality 𝑓(𝑥)−ℱ(𝑥),𝑦_𝒳≤1ℓ𝑛4−2^𝜆𝜃|𝑥|^𝜆𝑦^𝑞, (2.19) for all 𝑥,𝑦∈ℝ, where 𝜆∶=∑^𝑛_𝑖=1(𝑝_𝑖+𝑞_𝑖).

Corollary 2.5. Let ℓ∈{−1,1} be fixed. If there exist nonnegative real numbers 𝑠,𝑡 with ℓ𝑠<2ℓ such that a mapping 𝑓∶ℝ→𝒳 satisfies the inequality 𝑓^𝑛_𝑖=1𝑥_𝑖+𝑦_𝑖²+𝑓^𝑛_𝑖=1𝑥_𝑖−𝑦_𝑖²−2^𝑛_𝑖=1𝑓(𝑥_𝑖)+𝑓(𝑦_𝑖),𝑧_𝒳≤𝜃^𝑛_𝑖=1𝑥_𝑖^𝑠+𝑦_𝑖^𝑠|𝑧|^𝑡, (2.20) for all 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧∈ℝ and some 𝜃≥0, then there exists a unique quadratic mapping ℱ∶ℝ→𝒳 satisfies (1.5) and the inequality 𝑓(𝑥)−ℱ(𝑥),𝑦_𝒳≤1+2^𝑠−2ℓ2−2^𝑠−1𝜃|𝑥|^𝑠𝑦^𝑡, (2.21) for all 𝑥,𝑦∈ℝ.

Now, we are going to establish the modified Hyers-Ulam stability of (1.5).

Theorem 2.6. Let ℓ∈{−1,1} be fixed, let 𝒴 be a 𝑝-2-Banach space, and, 𝑓∶ℝ→𝒴 be a 𝜙-approximatively radical quadratic function with 𝑓(0)=0. Assume that the map 𝜙 is contractively subadditive if ℓ=1 and is expansively superadditive if ℓ=−1 with a constant 𝐿 satisfying 2^{ℓ(1−3𝑝)}𝐿<1, where 3ℓ𝑝≤ℓ, then there exists a unique quadratic mapping ℱ∶ℝ→𝒴 which satisfies (1.5) and the inequality 𝑓(𝑥)−ℱ(𝑥),𝑦_𝒴≤1ℓ4^𝑝−2^1−𝑝𝐿^ℓΨ𝑥,𝑦, (2.22) for all 𝑥,𝑦∈ℝ, where Ψ1𝑥,𝑦∶=𝑛^𝑝𝜙^{2𝑛0𝑥0𝑓𝑒37}+1𝑥,…,𝑥,𝑦(2𝑛)^𝑝𝜙2𝑥,…,^{𝑛0𝑥0𝑓𝑒37}2𝑥,0,…,^{𝑛0𝑥0𝑓𝑒37}0,𝑦. (2.23)

Proof. Using the same method as in the proof of Theorem 2.3, we have 𝑓(𝑥)−14𝑓(2𝑥),𝑦_𝒴≤14^𝑝Ψ,𝑓(𝑥)𝑥𝑥,𝑦−4𝑓2,𝑦_𝒴≤2^𝑝Ψ𝑥2,𝑦2, (2.24) for all 𝑥,𝑦∈ℝ. Hence 14^ℓ𝑘𝑓2^ℓ𝑘𝑥−14^ℓ𝑟𝑓2^ℓ𝑟𝑥,𝑦_𝒴≤14^{𝑝𝑟−(1+ℓ)/2}_{()𝑗=𝑘+1−ℓ/2}12^3ℓ𝑝𝑗Ψ2^ℓ𝑗𝑥,2^ℓ𝑗𝑦≤14^{𝑝𝑟−(1+ℓ)/2}_{()𝑗=𝑘+1−ℓ/2}2^ℓ𝐿^𝑗2^3ℓ𝑝𝑗Ψ=Ψ𝑥,𝑦𝑥,𝑦4^{𝑝𝑟−(1+ℓ)/2}_{()𝑗=𝑘+1−ℓ/2}2^{ℓ(1−3𝑝)}𝐿^𝑗, (2.25) for all 𝑥,𝑦∈ℝ and integers 𝑟>𝑘≥0. Thus, {(1/4^ℓ𝑚)𝑓(2^ℓ𝑚𝑥)} is a Cauchy sequence in the 𝑝-2-Banach space 𝒴. Hence, we can define a mapping ℱ∶ℝ→𝒴 by ℱ(𝑥)∶=lim_𝑛→∞(1/4^ℓ𝑛)𝑓(2^ℓ𝑛𝑥), for all 𝑥∈ℝ. Also ℱ^𝑛_𝑖=1𝑥_𝑖+𝑦_𝑖²+ℱ^𝑛_𝑖=1𝑥_𝑖−𝑦_𝑖²−2^𝑛_𝑖=1ℱ(𝑥_𝑖)+ℱ(𝑦_𝑖),𝑧_𝒴=lim_𝑚→∞14^ℓ𝑚𝑓4^𝑛ℓ𝑚_𝑖=1𝑥_𝑖+𝑦_𝑖²+14^ℓ𝑚𝑓4^𝑛ℓ𝑚_𝑖=1𝑥_𝑖−𝑦_𝑖²−24^𝑛ℓ𝑚_𝑖=1𝑓2^ℓ𝑚𝑥_𝑖+𝑓2^ℓ𝑚𝑦_𝑖,𝑧_𝒴≤lim_𝑚→∞12^3ℓ𝑝𝑚𝜙2^ℓ𝑚𝑥₁,…,2^ℓ𝑚𝑥_𝑛,2^ℓ𝑚𝑦₁,…,2^ℓ𝑚𝑦_𝑛,2^ℓ𝑚𝑧≤lim_𝑚→∞2^{ℓ(1−3𝑝)}𝐿^𝑚𝜙𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧=0 (2.26) holds for all 𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,𝑧∈ℝ, and so by Lemma 2.1, the mapping ℱ∶ℝ→𝒴 is quadratic. Taking the limit 𝑟→∞ in (2.25) with 𝑘=0, we find that the mapping ℱ is quadratic mapping satisfying the inequality (2.22) near the approximate mapping 𝑓∶ℝ→𝒴 of (1.5). The remaining assertion goes through in a similar way to the corresponding part of Theorem 2.3.