A Fixed Point Approach to the Stability of the Cauchy Additive and Quadratic Type Functional Equation

Abstract

We investigate the stability of the functional equation 2𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)+𝑓(𝑦−𝑥)−3𝑓(𝑥)−𝑓(−𝑥)−3𝑓(𝑦)−𝑓(−𝑦)=0 by using the fixed point theory in the sense of Cădariu and Radu.

1. Introduction

In 1940, Ulam [1] raised a question concerning the stability of homomorphisms as follow. Given a group 𝐺₁, a metric group 𝐺₂ with the metric 𝑑(⋅,⋅), and a positive number 𝜀, does there exist a 𝛿>0 such that if a mapping 𝑓∶𝐺₁→𝐺₂ satisfies the inequality 𝑑𝑓𝑥𝑦,𝑓(𝑥)𝑓𝑦<𝛿, (1.1) for all 𝑥,𝑦∈𝐺₁ then there exists a homomorphism 𝐹∶𝐺₁→𝐺₂ with 𝑑𝑓(𝑥),𝐹(𝑥)<𝜀, (1.2) for all 𝑥∈𝐺₁? When this problem has a solution, we say that the homomorphisms from 𝐺₁ to 𝐺₂ are stable. In the next year, Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings under the assumption that 𝐺₁ and 𝐺₂ are Banach spaces. Hyers' result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering the stability problem with unbounded Cauchy’s differences. The paper of Rassias had much influence in the development of stability problems. The terminology Hyers-Ulam-Rassias stability originated from this historical background. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5–12].

Almost all subsequent proofs, in this very active area, have used Hyers' method of [2]. Namely, the mapping 𝐹, which is the solution of a functional equation, is explicitly constructed, starting from the given mapping 𝑓, by the formulae 𝐹(𝑥)=lim_𝑛→∞(1/2^𝑛)𝑓(2^𝑛𝑥) or 𝐹(𝑥)=lim_𝑛→∞2^𝑛𝑓(𝑥/2^𝑛). We call it a direct method. In 2003, Cădariu and Radu [13] observed that the existence of the solution 𝐹 for a functional equation and the estimation of the difference with the given mapping 𝑓 can be obtained from the fixed point theory alternative. This method is called a fixed point method. In 2004, they applied this method [14] to prove stability theorems of the Cauchy functional equation: 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦=0. (1.3) In 2003, they [15] obtained the stability of the quadratic functional equation: 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦=0, (1.4) by using the fixed point method. Notice that if we consider the functions 𝑓₁,𝑓₂∶ℝ→ℝ defined by 𝑓₁(𝑥)=𝑎𝑥 and 𝑓₂(𝑥)=𝑎𝑥², where 𝑎 is a real constant, then 𝑓₁ satisfies (1.3), and 𝑓₂ holds (1.4), respectively. We call a solution of (1.3) an additive map, and a mapping satisfying (1.4) is called a quadratic map. Now we consider the functional equation: 2𝑓𝑥+𝑦+𝑓𝑥−𝑦+𝑓𝑦−𝑥−3𝑓(𝑥)−𝑓(−𝑥)−3𝑓𝑦−𝑓−𝑦=0, (1.5) which is called the Cauchy additive and quadratic-type functional equation. The function 𝑓∶ℝ→ℝ defined by 𝑓(𝑥)=𝑎𝑥²+𝑏𝑥 satisfies this functional equation, where 𝑎,𝑏 are real constants. We call a solution of (1.5) a quadratic-additive mapping.

In this paper, we will prove the stability of the functional equation (1.5) by using the fixed point theory. Precisely, we introduce a strictly contractive mapping with the Lipschitz constant 0<𝐿<1. Using the fixed point theory in the sense of Cădariu and Radu, together with suitable conditions, we can show that the contractive mapping has the fixed point. Actually the fixed point 𝐹 becomes the precise solution of (1.5). In Section 2, we prove several stability results of the functional equation (1.5) using the fixed point theory, see Theorems 2.3, 2.4, and 2.5. In Section 3, we use the results in the previous sections to get a stability of the Cauchy functional equation (1.3) and that of the quadratic functional equation (1.4), respectively.

2. Main Results

We recall the following result of the fixed point theory by Margolis and Diaz.

Theorem 2.1 (see [16] or [17]). Suppose that a complete generalized metric space (𝑋,𝑑), which means that the metric 𝑑 may assume infinite values, and a strictly contractive mapping 𝐽∶𝑋→𝑋 with the Lipschitz constant 0<𝐿<1 are given. Then, for each given element 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=+∞,∀𝑛∈ℕ∪{0} (2.1) or there exists a nonnegative integer 𝑘 such that (1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<+∞ for all 𝑛≥𝑘, (2)the sequence {𝐽^𝑛𝑥} is convergent to a fixed point 𝑦^∗ of 𝐽, (3)𝑦^∗ is the unique fixed point of 𝐽 in 𝑌∶={𝑦∈𝑋,𝑑(𝐽^𝑘𝑥,𝑦)<+∞}, (4)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝐽𝑦) for all 𝑦∈𝑌.

Throughout this paper, let 𝑉 be a (real or complex) linear space, and let 𝑌 be a Banach space. For a given mapping 𝑓∶𝑉→𝑌, we use the following abbreviation: 𝐷𝑓𝑥,𝑦∶=2𝑓𝑥+𝑦+𝑓𝑥−𝑦+𝑓𝑦−𝑥−3𝑓(𝑥)−𝑓(−𝑥)−3𝑓𝑦−𝑓−𝑦, (2.2) for all 𝑥,𝑦∈𝑉. If 𝑓 is a solution of the functional equation 𝐷𝑓≡0, see (1.5), we call it a quadratic-additive mapping. We first prove the following lemma.

Lemma 2.2. If 𝑓∶𝑉→𝑌 is a mapping such that 𝐷𝑓(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑉⧵{0}, then 𝑓 is a quadratic-additive mapping.

Proof. By choosing 𝑥∈𝑉⧵{0}, we get (0)()()()()()()()()()8𝑓=𝐷𝑓3𝑥,2𝑥−𝐷𝑓4𝑥,𝑥−2𝐷𝑓3𝑥,𝑥−𝐷𝑓2𝑥,𝑥+𝐷𝑓−3𝑥,−2𝑥−𝐷𝑓−4𝑥,−𝑥−2𝐷𝑓−3𝑥,−𝑥−𝐷𝑓−2𝑥,−𝑥+4𝐷𝑓𝑥,−𝑥=0. (2.3)
Since 𝑓(0)=0, we easily obtain 𝐷𝑓(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑉.

Now we can prove some stability results of the functional equation (1.5).

Theorem 2.3. Let 𝜑∶(𝑉⧵{0})²→[0,∞) be a given function. Suppose that the mapping 𝑓∶𝑉→𝑌 satisfies 𝐷𝑓𝑥,𝑦≤𝜑𝑥,𝑦, (2.4) for all 𝑥,𝑦∈𝑉⧵{0}. If there exist constants 0<𝐿,𝐿^′<1 such that 𝜑 has the property 1𝐿^′𝜑𝑥,𝑦≤𝜑2𝑥,2𝑦≤2𝐿𝜑𝑥,𝑦, (2.5) for all 𝑥,𝑦∈𝑉⧵{0}, then there exists a unique quadratic-additive mapping 𝐹∶𝑉→𝑌 such that 𝑓(𝑥)(𝑥≤3𝜑()()−𝐹)𝑥,𝑥+𝜑−𝑥,−𝑥16(1−𝐿), (2.6) for all 𝑥∈𝑉⧵{0}. In particular, 𝐹 is represented by 𝐹(𝑥)=lim_𝑛→∞𝑓2^𝑛𝑥+𝑓−2^𝑛𝑥2⋅4^𝑛+𝑓2^𝑛𝑥−𝑓−2^𝑛𝑥2^𝑛+1, (2.7) for all 𝑥∈𝑉.

Proof. It follows from (2.5) that lim_𝑛→∞𝜑2^𝑛𝑥,2^𝑛𝑦2^𝑛=0, (2.8) for all 𝑥,𝑦∈𝑉⧵{0}, and 8𝑓(0)=lim_𝑛→∞𝐷𝑓3𝑥2^𝑛,2𝑥2^𝑛−𝐷𝑓4𝑥2^𝑛,𝑥2^𝑛−2𝐷𝑓3𝑥2^𝑛,𝑥2^𝑛−𝐷𝑓2𝑥2^𝑛,𝑥2^𝑛+𝐷𝑓−3𝑥2^𝑛,−2𝑥2^𝑛−𝐷𝑓−4𝑥2^𝑛𝑥,−2^𝑛−2𝐷𝑓−3𝑥2^𝑛𝑥,−2^𝑛−𝐷𝑓−2𝑥2^𝑛𝑥,−2^𝑛𝑥+4𝐷𝑓2^𝑛𝑥,−2^𝑛≤lim_𝑛→∞𝜑3𝑥2^𝑛,2𝑥2^𝑛+𝜑4𝑥2^𝑛,𝑥2^𝑛+2𝜑3𝑥2^𝑛,𝑥2^𝑛+𝜑2𝑥2^𝑛,𝑥2^𝑛+𝜑−3𝑥2^𝑛,−2𝑥2^𝑛+𝜑−4𝑥2^𝑛𝑥,−2^𝑛+2𝜑−3𝑥2^𝑛𝑥,−2^𝑛+𝜑−2𝑥2^𝑛𝑥,−2^𝑛𝑥+4𝜑2^𝑛𝑥,−2^𝑛≤lim_𝑛→∞𝐿^′𝑛𝜑()()()()()()()()()3𝑥,2𝑥+𝜑4𝑥,𝑥+2𝜑3𝑥,𝑥+𝜑2𝑥,𝑥+𝜑−3𝑥,−2𝑥+𝜑−4𝑥,−𝑥+2𝜑−3𝑥,−𝑥+𝜑−2𝑥,−𝑥+4𝜑𝑥,−𝑥=0, (2.9) for all 𝑥∈𝑉⧵{0}. From this, we know that 𝑓(0)=0. Let 𝑆 be the set of all mappings   𝑔∶𝑉→𝑌 with 𝑔(0)=0. We introduce a generalized metric on 𝑆 by 𝑑𝑔,ℎ=inf𝐾∈ℝ⁺∣𝑔(𝑥)−ℎ(𝑥)≤𝐾𝜑(𝑥,𝑥)+𝜑(−𝑥,−𝑥)∀𝑥∈𝑉⧵{0}. (2.10) It is easy to show that (𝑆,𝑑) is a generalized complete metric space. Now we consider the mapping 𝐽∶𝑆→𝑆, which is defined by (𝑥)𝑔()()𝐽𝑔∶=2𝑥−𝑔−2𝑥4+𝑔(2𝑥)+𝑔(−2𝑥)8, (2.11) for all 𝑥∈𝑉. Notice 𝐽^𝑛𝑔(𝑥)=𝑔2^𝑛𝑥−𝑔−2^𝑛𝑥2^𝑛+1+𝑔2^𝑛𝑥+𝑔−2^𝑛𝑥2⋅4^𝑛, (2.12) for all 𝑛∈ℕ and 𝑥∈𝑉. Let 𝑔,ℎ∈𝑆, and let 𝐾∈[0,∞] be an arbitrary constant with 𝑑(𝑔,ℎ)≤𝐾. From the definition of 𝑑, we have (𝑥)(𝑥=3𝐽𝑔−𝐽ℎ)8𝑔()(+12𝑥−ℎ2𝑥)8𝑔()(≤1−2𝑥−ℎ−2𝑥)2𝐾𝜑()()𝜑()(),2𝑥,2𝑥+𝜑−2𝑥,−2𝑥≤𝐾𝐿𝑥,𝑥+𝜑−𝑥,−𝑥 (2.13) for all 𝑥∈𝑉⧵{0}, which implies that 𝑑𝐽𝑔,𝐽ℎ≤𝐿𝑑𝑔,ℎ, (2.14) for any 𝑔,ℎ∈𝑆. That is, 𝐽 is a strictly contractive self-mapping of 𝑆 with the Lipschitz constant 𝐿. Moreover, by (2.4), we see that 𝑓(𝑥)(𝑥=1−𝐽𝑓)()(≤316−3𝐷𝑓𝑥,𝑥+𝐷𝑓−𝑥,−𝑥)𝜑()(16𝑥,𝑥+𝜑−𝑥,−𝑥), (2.15) for all 𝑥∈𝑉⧵{0}. It means that 𝑑(𝑓,𝐽𝑓)≤3/16<∞ by the definition of 𝑑. Therefore, according to Theorem 2.1, the sequence {𝐽^𝑛𝑓} converges to the unique fixed point 𝐹∶𝑉→𝑌 of 𝐽 in the set 𝑇={𝑔∈𝑆∣𝑑(𝑓,𝑔)<∞}, which is represented by (2.7) for all 𝑥∈𝑉. By the definition of 𝐹, together with (2.4) and (2.7), it follows that 𝐷𝐹𝑥,𝑦=lim_𝑛→∞𝐷𝑓2^𝑛𝑥,2^𝑛𝑦−𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦2^𝑛+1+𝐷𝑓2^𝑛𝑥,2^𝑛𝑦+𝐷𝑓−2^𝑛𝑥,−2^𝑛𝑦2⋅4^𝑛≤lim_𝑛→∞2^𝑛+12⋅4^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦+𝜑−2^𝑛𝑥,−2^𝑛𝑦=0, (2.16) for all 𝑥,𝑦∈𝑉⧵{0}. By Lemma 2.2, we have proved that 𝐷𝐹𝑥,𝑦=0, (2.17) for all 𝑥,𝑦∈𝑉.

Theorem 2.4. Let 𝜑∶(𝑉⧵{0})²→[0,∞) be a given function. Suppose that the mapping 𝑓∶𝑉→𝑌 satisfies (2.5) for all 𝑥,𝑦∈𝑉⧵{0}. If there exists a constant 0<𝐿<1/2 such that 𝜑 has the property 𝜑2𝑥,2𝑦≤2𝐿𝜑𝑥,𝑦 (2.18) for all 𝑥,𝑦∈𝑉⧵{0}, then there exists a unique quadratic-additive mapping 𝐹∶𝑉→𝑌 satisfying (2.6) for all 𝑥∈𝑉⧵{0}. Moreover, if 𝜑(𝑥,𝑦) is continuous, then 𝑓 itself is a quadratic-additive mapping.

Proof. It follows from (2.18) that lim_𝑛→∞𝜑2^𝑛𝑥,2^𝑛𝑦=lim_𝑛→∞(2𝐿)^𝑛𝜑𝑥,𝑦=0, (2.19) for all 𝑥,𝑦∈𝑉⧵{0}, and 8𝑓(0)=lim_𝑛→∞𝐷𝑓3⋅2^𝑛𝑥,2^𝑛+1𝑥−𝐷𝑓2^𝑛+2𝑥,2^𝑛𝑥−2𝐷𝑓3⋅2^𝑛𝑥,2^𝑛𝑥−𝐷𝑓2^𝑛+1𝑥,2^𝑛𝑥+𝐷𝑓−3⋅2^𝑛𝑥,−2^𝑛+1𝑥−𝐷𝑓−2^𝑛+2𝑥,−2^𝑛𝑥−2𝐷𝑓−3⋅2^𝑛𝑥,−2^𝑛𝑥−𝐷𝑓−2^𝑛+1𝑥,−2^𝑛𝑥+4𝐷𝑓2^𝑛𝑥,−2^𝑛𝑥≤lim_𝑛→∞𝜑3⋅2^𝑛𝑥,2^𝑛+1𝑥+𝜑2^𝑛+2𝑥,2^𝑛𝑥+2𝜑3⋅2^𝑛𝑥,2^𝑛𝑥+𝜑2^𝑛+1𝑥,2^𝑛𝑥+𝜑−3⋅2^𝑛𝑥,−2^𝑛+1𝑥+𝜑−2^𝑛+2𝑥,−2^𝑛𝑥+2𝜑−3⋅2^𝑛𝑥,−2^𝑛𝑥+𝜑−2^𝑛+1𝑥,−2^𝑛𝑥+4𝜑2^𝑛𝑥,−2^𝑛𝑥≤lim_𝑛→∞(2𝐿)^𝑛𝜑()()()()()()()()(3𝑥,2𝑥+𝜑4𝑥,𝑥+2𝜑3𝑥,𝑥+𝜑2𝑥,𝑥+𝜑−3𝑥,−2𝑥+𝜑−4𝑥,−𝑥+2𝜑−3𝑥,−𝑥+𝜑−2𝑥,−𝑥+4𝜑𝑥,−𝑥)=0, (2.20) for all 𝑥∈𝑉⧵{0}. By the same method used in Theorem 2.3, we know that there exists a unique quadratic-additive mapping 𝐹∶𝑉→𝑌 satisfying (2.6) for all 𝑥∈𝑉⧵{0}. Since 𝜑 is continuous, we get lim_𝑛→∞𝜑𝑎₁⋅2^𝑛+𝑎₂𝑥,𝑏₁⋅2^𝑛+𝑏₂𝑦≤lim_𝑛→∞(2𝐿)^𝑛𝜑𝑎₁+𝑎₂2^𝑛𝑥,𝑏₁+𝑏₂2^𝑛𝑦=0⋅𝜑𝑎₁𝑥,𝑏₁𝑦=0, (2.21) for all 𝑥,𝑦∈𝑉⧵{0} and for any fixed integers 𝑎₁,𝑎₂,𝑏₁,𝑏₂ with 𝑎₁,𝑏₁≠0. Therefore, we obtain 2𝑓(𝑥)−𝐹(𝑥)≤lim_𝑛→∞𝐷𝑓2^𝑛+1𝑥,−2^𝑛𝑥−𝐷𝐹2^𝑛+1𝑥,−2^𝑛𝑥+𝐹−𝑓2^𝑛+1+1𝑥+𝐹−𝑓−2^𝑛+1𝑥2+1+3𝑓−𝐹^𝑛+1𝑥+𝑓−𝐹−2^𝑛+1𝑥+3𝑓−𝐹−2^𝑛𝑥+𝑓−𝐹2^𝑛𝑥≤lim_𝑛→∞𝜑2^𝑛+1𝑥,−2^𝑛𝑥+38(1−𝐿)lim_𝑛→∞𝜓2^𝑛+1𝑥2+1+2𝜓^𝑛+1𝑥+2𝜓2^𝑛𝑥=0, (2.22) for all 𝑥∈𝑉⧵{0}, where 𝜓(𝑥) is defined by 𝜓(𝑥)=𝜑(𝑥,𝑥)+𝜑(−𝑥,−𝑥). Since 𝑓(0)=0=𝐹(0), we have shown that 𝑓≡𝐹. This completes the proof of this theorem.

We continue our investigation with the next result.

Theorem 2.5. Let 𝜑∶(𝑉⧵{0})²→[0,∞). Suppose that 𝑓∶𝑉→𝑌 satisfies the inequality ‖𝐷𝑓(𝑥,𝑦)‖≤𝜑(𝑥,𝑦) for all 𝑥,𝑦∈𝑉⧵{0}. If there exists 0<𝐿<1 such that the mapping 𝜑 has the property 𝐿𝜑2𝑥,2𝑦≥4𝜑𝑥,𝑦 (2.23) for all 𝑥,𝑦∈𝑉⧵{0}, then there exists a unique quadratic-additive mapping 𝐹∶𝑉→𝑌 such that 𝑓(𝑥)(𝑥≤𝐿−𝐹)8()𝜑()()1−𝐿𝑥,𝑥+𝜑−𝑥,−𝑥, (2.24) for all 𝑥∈𝑉⧵{0}. In particular, 𝐹 is represented by 𝐹(𝑥)=lim_𝑛→∞2^𝑛−1𝑓𝑥2^𝑛−𝑥−𝑓2^𝑛+4^𝑛2𝑓𝑥2^𝑛−𝑥+𝑓2^𝑛, (2.25) for all 𝑥∈𝑉.

Proof. By the similar method used to prove 𝑓(0)=0 in the proof of Theorem 2.3, we can easily show that 𝑓(0)=0. Let the set (𝑆,𝑑) be as in the proof of Theorem 2.3. Now we consider the mapping 𝐽∶𝑆→𝑆 defined by (𝑥)𝑥𝐽𝑔∶=𝑔2−𝑥−𝑔2𝑔𝑥+22−𝑥+𝑔2, (2.26) for all 𝑔∈𝑆 and 𝑥∈𝑉. Notice that 𝐽^𝑛𝑔(𝑥)=2^𝑛−1𝑔𝑥2^𝑛−𝑥−𝑔2^𝑛+4^𝑛2𝑔𝑥2^𝑛−𝑥+𝑔2^𝑛 (2.27) and 𝐽⁰𝑔(𝑥)=𝑔(𝑥), for all 𝑥∈𝑉. Let 𝑔,ℎ∈𝑆, and let 𝐾∈[0,∞] be an arbitrary constant with 𝑑(𝑔,ℎ)≤𝐾. From the definition of 𝑑, we have (𝑥)(𝑥𝑔𝑥𝐽𝑔−𝐽ℎ)=32𝑥−ℎ2+𝑔−𝑥2−𝑥−ℎ2𝜑𝑥≤4𝐾2,𝑥2−𝑥+𝜑2𝑥,−2≤𝐿𝐾𝜑(𝑥,𝑥)+𝜑(−𝑥,−𝑥), (2.28) for all 𝑥∈𝑉⧵{0}. So 𝑑𝐽𝑔,𝐽ℎ≤𝐿𝑑𝑔,ℎ, (2.29) for any 𝑔,ℎ∈𝑆. That is, 𝐽 is a strictly contractive self-mapping of 𝑆 with the Lipschitz constant 𝐿. Also we see that 𝑓(𝑥)(𝑥=1−𝐽𝑓)2𝑥𝐷𝑓2,𝑥2≤12𝜑𝑥2,𝑥2−𝑥+𝜑2𝑥,−2≤𝐿8𝜑(𝑥,𝑥)+𝜑(−𝑥,−𝑥), (2.30) for all 𝑥∈𝑉⧵{0}, which implies that 𝑑(𝑓,𝐽𝑓)≤𝐿/8<∞. Therefore, according to Theorem 2.1, the sequence {𝐽^𝑛𝑓} converges to the unique fixed point 𝐹 of 𝐽 in the set 𝑇∶={𝑔∈𝑆∣𝑑(𝑓,𝑔)<∞}, which is represented by (2.25). Since 𝑑≤1𝑓,𝐹𝑑≤𝐿1−𝐿𝑓,𝐽𝑓8(1−𝐿), (2.31) the inequality (2.24) holds. From the definition of 𝐹(𝑥), (2.4), and (2.23), we have 𝐷𝐹𝑥,𝑦=lim_𝑛→∞2^𝑛−1𝑥𝐷𝑓2^𝑛,𝑦2^𝑛−𝑥−𝐷𝑓2^𝑛𝑦,−2^𝑛+4^𝑛2𝑥𝐷𝑓2^𝑛,𝑦2^𝑛−𝑥+𝐷𝑓2^𝑛𝑦,−2^𝑛≤lim_𝑛→∞2^𝑛+4^𝑛2𝜑𝑥2^𝑛,𝑦2^𝑛−𝑥+𝜑2^𝑛𝑦,−2^𝑛=0, (2.32) for all 𝑥,𝑦∈𝑉⧵{0}. By Lemma 2.2, 𝐹 is quadratic additive.

Remark 2.6. If 𝜑 satisfies the equality 𝜑(𝑥,𝑦)=𝜑(−𝑥,−𝑦) for all 𝑥,𝑦∈𝑉⧵{0} in Theorems 2.3, 2.4, and 2.5, then the inequalities (2.6) and (2.24) can be replaced by 𝑓(𝑥)(𝑥≤𝜑()−𝐹)𝑥,𝑥4(),𝑓(𝑥)(𝑥≤()1−𝐿−𝐹)𝐿𝜑𝑥,𝑥8(),1−𝐿 (2.33) for all 𝑥∈𝑉⧵{0}, respectively.

3. Applications

For a given mapping 𝑓∶𝑉→𝑌, we use the following abbreviations: (𝑥)𝑦,(𝑥)𝑦,𝐴𝑓𝑥,𝑦∶=𝑓𝑥+𝑦−𝑓−𝑓𝑄𝑓𝑥,𝑦∶=𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓−2𝑓 (3.1) for all 𝑥,𝑦∈𝑉. Using Theorems 2.3, 2.4, and 2.5 we will show the stability results of the additive functional equation 𝐴𝑓≡0 and the quadratic functional equation 𝑄𝑓≡0 in the following corollaries.

Corollary 3.1. Let 𝑓_𝑖∶𝑉→𝑌,𝑖=1,2,3, be mappings for which there exist functions 𝜙_𝑖∶(𝑉⧵{0})²→[0,∞),𝑖=1,2,3, such that 𝐴𝑓_𝑖𝑥,𝑦≤𝜙_𝑖𝑥,𝑦, (3.2) for all 𝑥,𝑦∈𝑉⧵{0}. If there exists 0<𝐿<1 such that 1𝐿𝜙₁𝑥,𝑦≤𝜙₁2𝑥,2𝑦≤2𝐿𝜙₁𝑥,𝑦, (3.3)𝜙₂2𝑥,2𝑦≤𝐿𝜙₂𝑥,𝑦, (3.4)4𝜙₃𝑥,𝑦≤𝐿𝜙₃2𝑥,2𝑦 (3.5) for all 𝑥,𝑦∈𝑉⧵{0}, then there exist unique additive mappings 𝐹_𝑖∶𝑉→𝑌,𝑖=1,2,3, such that 𝑓₁(𝑥)−𝐹₁(𝑥≤3𝜙)¹(𝑥,𝑥)+𝜙₁(𝑥,−𝑥)+𝜙₁(−𝑥,𝑥)+𝜙₁(−𝑥,−𝑥)8(1−𝐿), (3.6)𝑓₂(𝑥)−𝐹₂(𝑥≤3𝜙)²(𝑥,𝑥)+𝜙₂(𝑥,−𝑥)+𝜙₂(−𝑥,𝑥)+𝜙₂(−𝑥,−𝑥)8(1−𝐿/2), (3.7)𝑓₃(𝑥)−𝐹₃(𝑥≤𝐿𝜙)³(𝑥,𝑥)+𝜙₃(𝑥,−𝑥)+𝜙₃(−𝑥,𝑥)+𝜙₃(−𝑥,−𝑥)4(1−𝐿), (3.8) for all 𝑥∈𝑉⧵{0}. In particular, the mappings 𝐹_𝑖,𝑖=1,2,3, are represented by 𝐹₁(𝑥)=lim_𝑛→∞𝑓₁2^𝑛𝑥2^𝑛, (3.9)𝐹₂(𝑥)=lim_𝑛→∞𝑓₂2^𝑛𝑥2^𝑛, (3.10)𝐹₃(𝑥)=lim_𝑛→∞2^𝑛𝑓₃𝑥2^𝑛, (3.11) for all 𝑥∈𝑉. Moreover, if 𝜙₂(𝑥,𝑦) is continuous, then 𝑓₂ is itself an additive mapping.

Proof. Notice that 𝐷𝑓_𝑖𝑥,𝑦=2𝐴𝑓_𝑖𝑥,𝑦+𝐴𝑓_𝑖𝑥,−𝑦+𝐴𝑓_𝑖𝑦,−𝑥, (3.12) for all 𝑥,𝑦∈𝑉 and 𝑖=1,2,3. Put 𝜑_𝑖𝑥,𝑦∶=2𝜙_𝑖𝑥,𝑦+𝜙_𝑖𝑥,−𝑦+𝜙_𝑖𝑦,−𝑥, (3.13) for all 𝑥,𝑦∈𝑉 and 𝑖=1,2,3, then 𝜑₁ satisfies (2.5), 𝜑₂ satisfies (2.18), and 𝜑₃ satisfies (2.23). Therefore, ‖𝐷𝑓_𝑖(𝑥,𝑦)‖≤𝜑_𝑖(𝑥,𝑦), for all 𝑥,𝑦∈𝑉⧵{0} and 𝑖=1,2,3. According to Theorem 2.3, there exists a unique mapping 𝐹₁∶𝑉→𝑌 satisfying (3.6), which is represented by (2.7). Observe that, by (3.2) and (3.3), lim_𝑛→∞𝑓₁2^𝑛𝑥+𝑓₁−2^𝑛𝑥2^𝑛+1=lim_𝑛→∞𝑓₁2^𝑛𝑥+𝑓₁−2^𝑛𝑥−𝑓₁(0)2^𝑛+1=lim_𝑛→∞12^𝑛+1𝐴𝑓₁2^𝑛𝑥,−2^𝑛𝑥≤lim_𝑛→∞12^𝑛+1𝜙₁2^𝑛𝑥,−2^𝑛𝑥≤lim_𝑛→∞𝐿^𝑛2𝜙₁(𝑥,−𝑥)=0, (3.14) as well as lim_𝑛→∞𝑓₁2^𝑛𝑥+𝑓₁−2^𝑛𝑥2⋅4^𝑛≤lim_𝑛→∞2^𝑛𝐿^𝑛2⋅4^𝑛𝜙₁(𝑥,−𝑥)=0, (3.15) for all 𝑥∈𝑉⧵{0}. From this and (2.7), we get (3.9). Moreover, we have 𝐴𝑓₁2^𝑛𝑥,2^𝑛𝑦2^𝑛≤𝜙₁2^𝑛𝑥,2^𝑛𝑦2^𝑛≤𝐿^𝑛𝜙₁𝑥,𝑦, (3.16) for all 𝑥,𝑦∈𝑉⧵{0}. Taking the limit as 𝑛→∞ in the above inequality and using 𝐹₁(0)=0, we get 𝐴𝐹₁𝑥,𝑦=0, (3.17) for all 𝑥,𝑦∈𝑉. According to Theorem 2.4, there exists a unique mapping 𝐹₂∶𝑉→𝑌 satisfying (3.7), which is represented by (2.7). By using the similar method to prove (3.9), we can show that 𝐹₂ is represented by (3.10). In particular, if 𝜙₂(𝑥,𝑦) is continuous, then 𝜑₂ is continuous on (𝑉⧵{0})², and we can say that 𝑓₂ is an additive map by Theorem 2.4. On the other hand, according to Theorem 2.5, there exists a unique mapping 𝐹₃∶𝑉→𝑌 satisfying (3.8) which is represented by (2.25). Observe that, by (3.2) and (3.5), lim_𝑛→∞2^2𝑛−1𝑓₃𝑥2^𝑛+𝑓₃−𝑥2^𝑛=lim_𝑛→∞2^2𝑛−1𝐴𝑓₃𝑥2^𝑛𝑥,−2^𝑛≤lim_𝑛→∞2^2𝑛−1𝜙₃𝑥2^𝑛𝑥,−2^𝑛≤lim_𝑛→∞𝐿^𝑛2𝜙₃(𝑥,−𝑥)=0, (3.18) as well as lim_𝑛→∞2^𝑛−1𝑓₃𝑥2^𝑛+𝑓₃−𝑥2^𝑛≤lim_𝑛→∞𝐿^𝑛2^𝑛+1𝜙₃(𝑥,−𝑥)=0, (3.19) for all 𝑥∈𝑉⧵{0}. From these and (2.25), we get (3.11). Moreover, we have 2^𝑛𝐴𝑓₃𝑥2^𝑛,𝑦2^𝑛≤2^𝑛𝜙₃𝑥2^𝑛,𝑦2^𝑛≤𝐿^𝑛2^𝑛𝜙₃𝑥,𝑦, (3.20) for all 𝑥,𝑦∈𝑉⧵{0}. Taking the limit as 𝑛→∞ in the above inequality and using 𝐹₃(0)=0, we get 𝐴𝐹₃𝑥,𝑦=0, (3.21) for all 𝑥,𝑦∈𝑉.

Corollary 3.2. Let 𝑓_𝑖∶𝑉→𝑌,𝑖=1,2,3, be mappings for which there exist functions 𝜙_𝑖∶(𝑉⧵{0})²→[0,∞),𝑖=1,2,3, such that 𝑄𝑓_𝑖𝑥,𝑦≤𝜙_𝑖𝑥,𝑦, (3.22) for all 𝑥,𝑦∈𝑉⧵{0}. If there exists 0<𝐿<1 such that the mapping 𝜙₁ satisfies (3.3), 𝜙₂ satisfies (3.4), and 𝜙₃ satisfies (3.5) for all 𝑥,𝑦∈𝑉⧵{0}, then there exist unique quadratic mappings 𝐹_𝑖∶𝑉→𝑌,𝑖=1,2,3, such that 𝑓₁(𝑥)−𝐹₁(𝑥≤3𝜙)¹(𝑥,𝑥)+𝜙₁(𝑥,−𝑥)+𝜙₁(−𝑥,𝑥)+𝜙₁(−𝑥,−𝑥)16(1−𝐿), (3.23)𝑓₂(𝑥)−𝐹₂(𝑥≤3𝜙)²(𝑥,𝑥)+𝜙₂(𝑥,−𝑥)+𝜙₂(−𝑥,𝑥)+𝜙₂(−𝑥,−𝑥)16(1−𝐿/2), (3.24)𝑓₃(𝑥)−𝐹₃(𝑥≤𝐿𝜙)³(𝑥,𝑥)+𝜙₃(𝑥,−𝑥)+𝜙₃(−𝑥,𝑥)+𝜙₃(−𝑥,−𝑥)8(1−𝐿), (3.25) for all 𝑥∈𝑉⧵{0}. In particular, the mappings 𝐹_𝑖,𝑖=1,2,3, are represented by 𝐹₁(𝑥)=lim_𝑛→∞𝑓₁2^𝑛𝑥4^𝑛, (3.26)𝐹₂(𝑥)=lim_𝑛→∞𝑓₂2^𝑛𝑥4^𝑛, (3.27)𝐹₃(𝑥)=lim_𝑛→∞4^𝑛𝑓₃𝑥2^𝑛, (3.28) for all 𝑥∈𝑉. Moreover, if 𝜙₂(𝑥,𝑦) is continuous, then 𝑓₂ itself is a quadratic mapping.

Proof. Notice that 𝐷𝑓_𝑖𝑥,𝑦=𝑄𝑓_𝑖+1𝑥,𝑦2𝑄𝑓_𝑖𝑥,−𝑦+𝑄𝑓_𝑖𝑦,−𝑥, (3.29) for all 𝑥,𝑦∈𝑉 and 𝑖=1,2,3. Put 𝜑_𝑖(𝑥,𝑦)∶=𝜙_𝑖(𝑥,𝑦)+(1/2)(𝜙_𝑖(𝑥,−𝑦)+𝜙_𝑖(𝑦,−𝑥)), for all 𝑥,𝑦∈𝑉 and 𝑖=1,2,3, then 𝜑₁ satisfies (2.5), 𝜑₂ satisfies (2.18), and 𝜑₃ satisfies (2.23). Moreover, 𝐷𝑓_𝑖𝑥,𝑦≤𝜑_𝑖𝑥,𝑦, (3.30) for all 𝑥,𝑦∈𝑉⧵{0} and 𝑖=1,2,3. According to Theorem 2.3, there exists a unique mapping 𝐹₁∶𝑉→𝑌 satisfying (3.23) which is represented by (2.7). Observe that lim_𝑛→∞𝑓₁2^𝑛𝑥−𝑓₁−2^𝑛𝑥2^𝑛+1=lim_𝑛→∞12^𝑛+1𝑄𝑓₁2^𝑛−1𝑥,−2^𝑛−1𝑥−𝑄𝑓₁−2^𝑛−1𝑥,2^𝑛−1𝑥≤lim_𝑛→∞12^𝑛+1𝜙₁2^𝑛−1𝑥,−2^𝑛−1𝑥+𝜙₁−2^𝑛−1𝑥,2^𝑛−1𝑥≤lim_𝑛→∞𝐿^𝑛2𝜙₁𝑥2𝑥,−2+𝜙₁−𝑥2,𝑥2=0, (3.31) as well as lim_𝑛→∞𝑓₁2^𝑛𝑥−𝑓₁−2^𝑛𝑥2⋅4^𝑛≤lim_𝑛→∞𝐿^𝑛2^𝑛+1𝜙₁𝑥2𝑥,−2+𝜙₁−𝑥2,𝑥2=0, (3.32) for all 𝑥∈𝑉⧵{0}. From this and (2.7), we get (3.26) for all 𝑥∈𝑉. Moreover, we have 𝑄𝑓₁2^𝑛𝑥,2^𝑛𝑦4^𝑛≤𝜙₁2^𝑛𝑥,2^𝑛𝑦4^𝑛≤𝐿^𝑛2^𝑛𝜙₁𝑥,𝑦, (3.33) for all 𝑥,𝑦∈𝑉⧵{0}. Taking the limit as 𝑛→∞ in the above inequality, we get 𝑄𝐹₁𝑥,𝑦=0 (3.34) for all 𝑥,𝑦∈𝑉⧵{0}. Using 𝐹₁(0)=0, we have 𝑄𝐹₁(𝑥,0)=0,𝑄𝐹₁0,𝑦=−𝐹₁𝑦2𝑦,−2+𝐹₁−𝑦2,𝑦2=0, (3.35) for all 𝑥,𝑦∈𝑉⧵{0}. Therefore, 𝑄𝐹₁(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑉.
Next, by Theorem 2.4, there exists a unique mapping 𝐹₂∶𝑉→𝑌 satisfying (3.24), which is represented by (2.7). By using the similar method to prove (3.26), we can show that 𝐹₂ is represented by (3.27). In particular, 𝜙₂(𝑥,𝑦) is continuous, then 𝜑₂ is continuous on (𝑉⧵{0})², and we can say that 𝑓₂ is a quadratic map by Theorem 2.4. On the other hand, according to Theorem 2.5, there exists a unique mapping 𝐹₃∶𝑉→𝑌 satisfying (3.25) which is represented by (2.25). Observe that 4^𝑛𝑓₃𝑥2^𝑛−𝑓₃−𝑥2^𝑛=4^𝑛𝑄𝑓₃𝑥2^𝑛+1𝑥,−2^𝑛+1−𝑄𝑓₃−𝑥2^𝑛+1,𝑥2^𝑛+1≤4^𝑛𝜙₃𝑥2^𝑛+1𝑥,−2^𝑛+1+𝜙₃−𝑥2^𝑛+1,𝑥2^𝑛+1≤𝐿^𝑛𝜙₃𝑥2𝑥,−2+𝜙₃−𝑥2,𝑥2, (3.36) for all 𝑥∈𝑉⧵{0}. It leads us to get lim_𝑛→∞4^𝑛𝑓₃𝑥2^𝑛−𝑓₃−𝑥2^𝑛=0,lim_𝑛→∞2^𝑛𝑓₃𝑥2^𝑛−𝑓₃−𝑥2^𝑛=0, (3.37) for all 𝑥∈𝑉⧵{0}. From these and (2.25), we obtain (3.28). Moreover, we have 4^𝑛𝑄𝑓₃𝑥2^𝑛,𝑦2^𝑛≤4^𝑛𝜙₃𝑥2^𝑛,𝑦2^𝑛≤𝐿^𝑛𝜙₃𝑥,𝑦, (3.38) for all 𝑥,𝑦∈𝑉⧵{0}. Taking the limit as 𝑛→∞ in the above inequality and using 𝐹₃(0)=0, we get 𝑄𝐹₃𝑥,𝑦=0, (3.39) for all 𝑥,𝑦∈𝑉.

Now, we obtain Hyers-Ulam-Rassias stability results in the framework of normed spaces using Theorems 2.3 and 2.4.

Corollary 3.3. Let 𝑋 be a normed space, and let 𝑌 be a Banach space. Suppose that the mapping 𝑓∶𝑋→𝑌 satisfies the inequality 𝐷𝑓𝑥,𝑦≤𝜃‖𝑥‖^𝑝+𝑦^𝑝 (3.40) for all 𝑥,𝑦∈𝑋⧵{0}, where 𝜃≥0 and 𝑝∈(−∞,0)∪(0,1)∪(2,∞). Then there exists a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤𝜃−𝐹)2^𝑝‖𝑥‖−4^𝑝𝜃if𝑝>2,2−2^𝑝‖𝑥‖^𝑝if0<𝑝<1 (3.41) for all 𝑥∈𝑋⧵{0}. Moreover if 𝑝<0, then 𝑓 is itself a quadratic-additive mapping.

Proof. This corollary follows from Theorems 2.3, 2.4, and 2.5, and Remark 2.6, by putting 𝜑𝑥,𝑦∶=𝜃‖𝑥‖^𝑝+𝑦^𝑝, (3.42) for all 𝑥,𝑦∈𝑋⧵{0} with 𝐿=2^𝑝−1<1 if 𝑝<1, 𝐿=2^2−𝑝<1 if 𝑝>2, and 𝐿^′=2^−𝑝<1 if 𝑝>0.

Corollary 3.4. Let 𝑋 be a normal space let and 𝑌 be a Banach space. Suppose that the mapping 𝑓∶𝑋→𝑌 satisfies the inequality 𝐷𝑓𝑥,𝑦≤𝜃‖𝑥‖^𝑝𝑦^𝑞, (3.43) for all 𝑥,𝑦∈𝑋⧵{0}, where 𝜃≥0 and 𝑝+𝑞∈(−∞,0)∪(0,1)∪(2,∞). Then there exists a unique quadratic-additive mapping 𝐹∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤𝜃‖𝑥‖−𝐹)^𝑝+𝑞22^𝑝+𝑞𝜃‖𝑥‖−4if𝑝+𝑞>2,_𝑝+𝑞22−2^𝑝+𝑞if0<𝑝+𝑞<1, (3.44) for all 𝑥∈𝑋⧵{0}. Moreover, if 𝑝+𝑞<0, then 𝑓 is itself a quadratic-additive mapping.

Proof. This corollary follows from Theorems 2.3, 2.4, 2.5, and Remark 2.6, by putting 𝜑𝑥,𝑦∶=𝜃‖𝑥‖^𝑝𝑦^𝑞, (3.45) for all 𝑥,𝑦∈𝑋⧵{0} with 𝐿=2^{𝑝+𝑞−1}<1 if 𝑝+𝑞<1, 𝐿=2^{2−𝑝−𝑞}<1 if 𝑝+𝑞>2, and 𝐿^′=2^{−𝑝−𝑞}<1 if 𝑝+𝑞>0.