Generalized Stability of Euler-Lagrange Quadratic Functional Equation

Abstract

The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equation 𝑓(𝑎𝑥+𝑏𝑦)+𝑎𝑓(𝑥−𝑏𝑦)=(𝑎+1)𝑏²𝑓(𝑦)+𝑎(𝑎+1)𝑓(𝑥), in (𝛽,𝑝)-Banach space, where 𝑎,𝑏 are fixed rational numbers such that 𝑎≠−1,0 and 𝑏≠0.

1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

Let 𝐺 be a group and let 𝐺′ be a metric group with metric 𝜌(⋅,⋅). Given 𝜖>0, does there exist a 𝛿>0 such that if 𝑓∶𝐺→𝐺′ satisfies 𝜌(𝑓(𝑥𝑦),𝑓(𝑥)𝑓(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺, then a homomorphism ℎ∶𝐺→𝐺′ exists with 𝜌(𝑓(𝑥),ℎ(𝑥))<𝜖 for all 𝑥∈𝐺?

In 1941, the first result concerning the stability of functional equations was presented by Hyers [2]. He has answered the question of Ulam for the case where 𝐺₁ and 𝐺₂ are Banach spaces.

Let 𝐸₁ and 𝐸₂ be real vector spaces. A function 𝑓∶𝐸₁→𝐸₂ is called a quadratic function if and only if 𝑓 is a solution function of the quadratic functional equation 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦. (1.1)

It is well known that a function 𝑓 between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function 𝐵 such that 𝑓(𝑥)=𝐵(𝑥,𝑥) for all 𝑥, where the mapping 𝐵 is given by 𝐵(𝑥,𝑦)=(1/4)(𝑓(𝑥+𝑦)−𝑓(𝑥−𝑦)). See [3, 4] for the details. The Hyers-Ulam stability of the quadratic functional equation (1.1) was first proved by Skof [5] for functions 𝑓∶𝐸₁→𝐸₂, where 𝐸₁ is a normed space and 𝐸₂ is a Banach space. Cholewa [6] demonstrated that Skof’s theorem is also valid if 𝐸₁ is replaced by an Abelian group 𝐺. Assume that a function 𝑓∶𝐺→𝐸 satisfies the inequality 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦≤𝛿, (1.2) for some 𝛿≥0 and for all 𝑥,𝑦∈𝐺. Then there exists a unique quadratic function 𝑄∶𝐺→𝐸 such that 𝑓(𝑥)(𝑥≤𝛿−𝑄)2, (1.3) for all 𝑥∈𝐺. Czerwik [7] proved the Hyers-Ulam-Rassias stability of quadratic functional equation (1.1). Let 𝐸₁ and 𝐸₂ be a real normed space and a real Banach space, respectively, and let 𝑝≠2 be a positive constant. If a function 𝑓∶𝐸₁→𝐸₂ satisfies the inequality 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦≤𝜖‖𝑥‖^𝑝+𝑦^𝑝, (1.4) for some 𝜖>0 and for all 𝑥,𝑦∈𝐸₁, then there exists a unique quadratic function 𝑞∶𝐸₁→𝐸₂ such that 𝑓(𝑥)−𝑞(𝑥)≤2𝜖|4−2^𝑝|‖𝑥‖^𝑝, (1.5) for all 𝑥∈𝐸₁. Furthermore, according to the theorem of Borelli and Forti [8], we know the following generalization of stability theorem for quadratic functional equation. Let 𝐺 be an Abelian group and 𝐸 a Banach space, and let 𝑓∶𝐺→𝐸 be a mapping with 𝑓(0)=0 satisfying the inequality 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦≤𝜑𝑥,𝑦, (1.6) for all 𝑥,𝑦∈𝐺. Assume that one of the series Φ𝑥,𝑦∶=^∞_𝑘∑₌₀12^2(𝑘+1)𝜑2^𝑘𝑥,2^𝑘𝑦<∞,_∞𝑘∑₌₀2^2𝑘𝜑𝑥2^(𝑘+1),𝑦2^(𝑘+1)<∞, (1.7) then there exists a unique quadratic function 𝑄∶𝐺→𝐸 such that 𝑓(𝑥)−𝑄(𝑥)≤Φ(𝑥,𝑥), (1.8) for all 𝑥∈𝐺. During the last three decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability of several functional equations, and there are many interesting results concerning this problem [9–13].

The notion of quasi-𝛽-normed space was introduced by Rassias and Kim in [14]. This notion is a generalization of that of quasi-normed space. We consider some basic concepts concerning quasi-𝛽-normed space. We fix a real number 𝛽 with 0<𝛽≤1 and let 𝕂 denote either ℝ or ℂ. Let 𝑋 be a linear space over 𝕂. A quasi-𝛽-norm ‖⋅‖ is a real-valued function on 𝑋 satisfying the following:(1)‖𝑥‖≥0 for all 𝑥∈𝑋 and ‖𝑥‖=0 if and only if 𝑥=0,(2)‖𝜆𝑥‖=|𝜆|^𝛽‖𝑥‖ for all 𝜆∈𝕂 and all 𝑥∈𝑋,(3)there is a constant 𝐾≥1 such that ‖𝑥+𝑦‖≤𝐾(‖𝑥‖+‖𝑦‖) for all 𝑥,𝑦∈𝑋.

The pair (𝑋,‖⋅‖) is called a quasi-𝛽-normed space if ‖⋅‖ is a quasi-𝛽-norm on 𝑋. The smallest possible 𝐾 is called the modulus of concavity of ‖⋅‖. A quasi-𝛽-Banach space is a complete quasi-𝛽-normed space. A quasi-𝛽-norm ‖⋅‖ is called a (𝛽,𝑝)-norm (0<𝑝≤1) if 𝑥+𝑦^𝑝≤‖𝑥‖^𝑝+𝑦^𝑝, (1.9) for all 𝑥,𝑦∈𝑋. In this case, the quasi-𝛽-Banach space is called a (𝛽,𝑝)-Banach space. We observe that if 𝑥₁,𝑥₂,…,𝑥_𝑛 are nonnegative real numbers, then ^𝑛_𝑖=1𝑥_𝑖^𝑝≤^𝑛_𝑖=1𝑥^𝑝_𝑖, (1.10) where 0<𝑝≤1 [15].

J. M. Rassias investigated the stability of Ulam for the Euler-Lagrange functional equation 𝑓𝑎𝑥+𝑏𝑦+𝑓𝑏𝑥−𝑎𝑦=𝑎²+𝑏²𝑓(𝑥)+𝑓𝑦 (1.11) in the paper of [16]. Gordji and Khodaei investigated the generalized Hyers-Ulam stability of other Euler-Lagrange quadratic functional equations [17]. Jun et al. [18] introduced a new quadratic Euler-Lagrange functional equation 𝑓𝑎𝑥+𝑦+𝑎𝑓𝑥−𝑦=(𝑎+1)𝑓𝑦+𝑎(𝑎+1)𝑓(𝑥), (1.12) for any fixed 𝑎∈ℤ with 𝑎≠0,−1, which was a modified and instrumental equation for [19], and solved the generalized stability of (1.12). Now, we improve the functional equation (1.12) to the following functional equations: 𝑓=()𝑓()𝑓(𝑥),𝑓=()𝑏𝑎𝑥+𝑏𝑦+𝑎𝑓𝑥−𝑏𝑦𝑎+1𝑏𝑦+𝑎𝑎+1𝑎𝑥+𝑏𝑦+𝑎𝑓𝑥−𝑏𝑦𝑎+1²𝑓𝑦+𝑎(𝑎+1)𝑓(𝑥), (1.13)(1.14) for any fixed rational numbers 𝑎,𝑏∈ℚ with 𝑎≠0,−1 and 𝑏≠0, which are generalized versions of (1.12). In this paper, we establish the general solution of (1.13) and (1.14) and then prove the generalized Hyers-Ulam stability of (1.13) and (1.14). We remark that there are some interesting papers concerning the stability of functional equations in quasi-Banach spaces [15, 20–23] and quasi-𝛽-normed spaces [14, 24, 25].

2. General Solution of (1.13) and (1.14)

First, we present the general solution of (1.14) in the class of all functions between vector spaces.

Lemma 2.1. Let 𝑋 and 𝑌 be vector spaces over 𝕂. Then a mapping 𝑓∶𝑋→𝑌 is a solution of the functional equation (1.12) for any fixed rational number 𝑎∈ℚ with 𝑎≠0,−1 if and only if 𝑓 is quadratic.

Proof. See the same proof in [18].

Lemma 2.2. Let 𝑋 and 𝑌 be vector spaces over 𝕂. Then a mapping 𝑓∶𝑋→𝑌 is a solution of the functional equation (1.13) if and only if 𝑓 is quadratic.

Proof. We assume that a mapping 𝑓∶𝑋→𝑌 satisfies the functional equation (1.13). Letting 𝑏𝑦=𝑢 in (1.13), then (1.13) is equivalent to (1.12). Then by Lemma 2.1, 𝑓 is quadratic. Conversely, if 𝑓 is quadratic, then it is obvious that 𝑓 satisfies (1.13).

Theorem 2.3. Let 𝑋 and 𝑌 be vector spaces over 𝕂. Then a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional equation (1.14) if and only if 𝑓 is quadratic. In this case, 𝑓(𝑎𝑥)=𝑎²𝑓(𝑥) and 𝑓(𝑏𝑥)=𝑏²𝑓(𝑥) hold for all 𝑥∈𝑋.

Proof. We assume that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional equation (1.14). Then replacing 𝑦 in (1.14) by 0, we also get the equality 𝑓(𝑎𝑥)=𝑎²𝑓(𝑥) for all 𝑥∈𝑋. Now, we decompose 𝑓 into the even part and the odd part by setting 𝑓_𝑒(𝑥)=12𝑓(𝑥)+𝑓(−𝑥),𝑓_𝑜(𝑥)=12𝑓(𝑥)−𝑓(−𝑥), (2.1) for all 𝑥∈𝑋. Then 𝑓_𝑒 and 𝑓_𝑜 satisfy the functional equation (1.14). Therefore, we may assume without loss of generality that 𝑓 is even and satisfies (1.14) for all 𝑥,𝑦∈𝑋. If we replace 𝑥 in (1.14) by 0, then we get 𝑓𝑏𝑦+𝑎𝑓−𝑏𝑦=(𝑎+1)𝑏²𝑓𝑦, (2.2) for all 𝑦∈𝑋. From this equality, we have 𝑓(𝑏𝑦)=𝑏²𝑓(𝑦) for all 𝑦∈𝑋. Therefore, (1.14) implies (1.13) for all 𝑥,𝑦∈𝑋. By Lemma 2.2,  𝑓 is quadratic.
Now, we assume that 𝑓 is odd and satisfies (1.14) for all 𝑥,𝑦∈𝑋. For the case 𝑎=1, we have 𝑓𝑥+𝑏𝑦+𝑓𝑥−𝑏𝑦=2𝑏²𝑓𝑦+2𝑓(𝑥), (2.3) for all 𝑥,𝑦∈𝑋. Setting 𝑥 by 0 in (2.3), one obtains 𝑓≡0. Let 𝑎≠1. Replacing 𝑥 by 0 in (1.14), we have (1−𝑎)𝑓𝑏𝑦=(𝑎+1)𝑏²𝑓𝑦, (2.4) for all 𝑦∈𝑋. From (1.14) and (2.4), we get 𝑓𝑎𝑥+𝑏𝑦+𝑎𝑓𝑥−𝑏𝑦=(1−𝑎)𝑓𝑏𝑦+𝑎(𝑎+1)𝑓(𝑥), (2.5) for all 𝑥,𝑦∈𝑋. Putting 𝑏𝑦=𝑢 in (2.5), then we obtain 𝑓(𝑎𝑥+𝑢)+𝑎𝑓(𝑥−𝑢)=(1−𝑎)𝑓(𝑢)+𝑎(𝑎+1)𝑓(𝑥), (2.6) for all 𝑥,𝑢∈𝑋. Replacing 𝑢 by 𝑎𝑢 in (2.6), we get 𝑓(𝑎𝑥+𝑎𝑢)+𝑎𝑓(𝑥−𝑎𝑢)=(1−𝑎)𝑓(𝑎𝑢)+𝑎(𝑎+1)𝑓(𝑥), (2.7) for all 𝑥,𝑢∈𝑋. Since 𝑓(𝑎𝑥)=𝑎²𝑓(𝑥), (2.7) yields 𝑎𝑓(𝑥+𝑢)+𝑓(𝑥−𝑎𝑢)=(1−𝑎)𝑎𝑓(𝑢)+(𝑎+1)𝑓(𝑥), (2.8) for all 𝑥,𝑢∈𝑋. Interchanging 𝑥 and 𝑢 in (2.8), we have by oddness of 𝑓−𝑓(𝑎𝑥−𝑢)+𝑎𝑓(𝑥+𝑢)=(1−𝑎)𝑎𝑓(𝑥)+(𝑎+1)𝑓(𝑢), (2.9) for all 𝑥,𝑢∈𝑋. Replacing 𝑢 by −𝑢 in (2.6), we get 𝑓(𝑎𝑥−𝑢)+𝑎𝑓(𝑥+𝑢)=−(1−𝑎)𝑓(𝑢)+𝑎(𝑎+1)𝑓(𝑥), (2.10) for all 𝑥,𝑢∈𝑋. Adding (2.9) and (2.10) side by side, this leads to 𝑓(𝑥+𝑢)=𝑓(𝑥)+𝑓(𝑢), (2.11) for all 𝑥,𝑢∈𝑋. Therefore, 𝑓 is additive and so 𝑓(𝑎𝑥)=𝑎𝑓(𝑥) for all 𝑥∈𝑋 and for any odd function satisfying (1.14). Using the equality 𝑓(𝑎𝑥)=𝑎²𝑓(𝑥), we obtain 𝑓(𝑥)=0 for all 𝑥∈𝑋. Therefore, 𝑓(𝑥)=𝑓_𝑒(𝑥)+𝑓_𝑜(𝑥) is a quadratic mapping, as desired.
Conversely, if 𝑓 is quadratic, then it is obvious that 𝑓 satisfies (1.14).

We note that 𝑓(0)=0 if 𝑎+𝑏²≠1 and 𝑓 satisfies (1.14).

3. Generalized Stability of (1.14) for 𝑎≠1

For convenience, we use the following abbreviation: for any fixed rational numbers 𝑎 and 𝑏 with 𝑎≠−1,0,1 and 𝑏≠0, 𝐷_𝑓𝑥,𝑦∶=𝑓𝑎𝑥+𝑏𝑦+𝑎𝑓𝑥−𝑏𝑦−(𝑎+1)𝑏²𝑓𝑦−𝑎(𝑎+1)𝑓(𝑥), (3.1) for all 𝑥,𝑦∈𝑋, which is called the approximate remainder of the functional equation (1.14) and acts as a perturbation of the equation.

From now on, let 𝑋 be a vector space, and let 𝑌 be a (𝛽,𝑝)-Banach space unless we give any specific reference. We will investigate the generalized Hyers-Ulam stability problem for the functional equation (1.14). Thus, we find some conditions such that there exists a true quadratic function near an approximate solution of (1.14).

Theorem 3.1. Let 𝜑∶𝑋×𝑋→[0,∞) be a function such that Φ(𝑥)∶=^∞_𝑛=01|𝑎|^{2𝛽𝑛𝑝}𝜑𝑎^𝑛𝑥,0^𝑝<∞,lim_𝑛→∞1|𝑎|^2𝛽𝑛𝜑𝑎^𝑛𝑥,𝑎^𝑛𝑦=0, (3.2)(3.3) for all 𝑥,𝑦∈𝑋. Suppose that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies 𝐷_𝑓𝑥,𝑦_𝑌≤𝜑𝑥,𝑦, (3.4) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 satisfying 𝑓(𝑥)−𝑄(𝑥)_𝑌≤1|𝑎|^2𝛽[Φ(𝑥)]^1/𝑝, (3.5) for all 𝑥∈𝑋. The function 𝑄 is given by 𝑄(𝑥)=lim_𝑘→∞1𝑎^2𝑘𝑓𝑎^𝑘𝑥, (3.6) for all 𝑥∈𝑋.

Proof. Letting 𝑦 by 0 in (3.4), we get 𝑓(𝑎𝑥)−𝑎²𝑓(𝑥)_𝑌≤𝜑(𝑥,0), (3.7) for all 𝑥∈𝑋. Multiplying both sides by 1/|𝑎|^2𝛽 in (3.7), we have 1𝑎²𝑓(𝑎𝑥)−𝑓(𝑥)_𝑌≤1|𝑎|^2𝛽𝜑(𝑥,0), (3.8) for all 𝑥∈𝑋. Replacing 𝑥 by 𝑎^𝑛𝑥 and multiplying both sides by 1/|𝑎|^2𝑛𝛽 in (3.8), we have 1𝑎^2(𝑛+1)𝑓𝑎^𝑛+1𝑥−1𝑎^2𝑛𝑓𝑎^𝑛𝑥_𝑌≤1|𝑎|^{2𝛽(𝑛+1)}𝜑𝑎^𝑛𝑥,0, (3.9) for all 𝑥∈𝑋. Next we show that the sequence {(1/𝑎^2𝑛)𝑓(𝑎^𝑛𝑥)} is a Cauchy sequence. For any 𝑚,𝑛∈ℕ, 𝑚>𝑛≥0, and 𝑥∈𝑋, it follows from (3.9) that 1𝑎^2(𝑚+1)𝑓𝑎^𝑚+1𝑥−1𝑎^2𝑛𝑓𝑎^𝑛𝑥^𝑝_𝑌=^𝑚_𝑖=𝑛1𝑎^2(𝑖+1)𝑓𝑎^𝑖+1𝑥−1𝑎^2𝑖𝑓𝑎^𝑖𝑥^𝑝_𝑌≤^𝑚_𝑖=𝑛1𝑎^2(𝑖+1)𝑓𝑎^𝑖+1𝑥−1𝑎^2𝑖𝑓𝑎^𝑖𝑥^𝑝_𝑌≤^𝑚_𝑖=𝑛1|𝑎|^{2𝛽𝑝(𝑖+1)}𝜑𝑎^𝑖𝑥,0^𝑝=1|𝑎|^{𝑚2𝛽𝑝}_𝑖=𝑛1|𝑎|^{2𝛽𝑝𝑖}𝜑𝑎^𝑖𝑥,0^𝑝, (3.10) for all 𝑥∈X. It follows from (3.2) and (3.10) that the sequence {(1/𝑎^2𝑛)𝑓(𝑎^𝑛𝑥)} is a Cauchy sequence in 𝑌 for all 𝑥∈𝑋. Since 𝑌 is a (𝛽,𝑝)-Banach space, the sequence {(1/𝑎^2𝑛)𝑓(𝑎^𝑛𝑥)} converges for all 𝑥∈𝑋. Therefore, we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)=lim_𝑛→∞1𝑎^2𝑛𝑓𝑎^𝑛𝑥, (3.11) for all 𝑥∈𝑋. Taking 𝑚→∞ and 𝑛=0 in (3.10), we have 𝑄(𝑥)−𝑓(𝑥)^𝑝_𝑌≤1|𝑎|^∞2𝛽𝑝_𝑖=01|𝑎|^{2𝛽𝑝𝑖}𝜑𝑎^𝑖𝑥,0^𝑝=1|𝑎|^2𝛽𝑝Φ(𝑥), (3.12) for all 𝑥∈𝑋. Therefore, 𝑄(𝑥)−𝑓(𝑥)_𝑌≤1|𝑎|^2𝛽[Φ(𝑥)]^1/𝑝, (3.13) for all 𝑥∈𝑋, that is, the mapping 𝑄 satisfies (3.5). It follows from (3.3) and (3.4) that 𝐷_𝑄𝑥,𝑦_𝑌=lim_𝑛→∞1𝑎^2𝑛𝐷_𝑓𝑎^𝑛𝑥,𝑎^𝑛𝑦_𝑌=lim_𝑛→∞1|𝑎|^2𝛽𝑛𝐷_𝑓𝑎^𝑛𝑥,𝑎^𝑛𝑦_𝑌≤lim_𝑛→∞1|𝑎|^2𝛽𝑛𝜑𝑎^𝑛𝑥,𝑎^𝑛𝑦=0, (3.14) for all 𝑥,𝑦∈𝑋. Therefore, 𝑄 satisfies (1.14), and so the function 𝑄 is quadratic.
To prove the uniqueness of the quadratic function 𝑄, let us assume that there exists a quadratic function 𝑄′∶𝑋→𝑌 satisfying the inequality (3.5). Then we have 𝑄(𝑥)−𝑄^′(𝑥)^𝑝_𝑌=1𝑎^2𝑛𝑄𝑎^𝑛𝑥−1𝑎^2𝑛𝑄^′𝑎^𝑛𝑥^𝑝_𝑌=1𝑎^{2𝑛𝛽𝑝}𝑄𝑎^𝑛𝑥−𝑄^′𝑎^𝑛𝑥^𝑝_𝑌≤1𝑎^{2𝑛𝛽𝑝}𝑄𝑎^𝑛𝑥−𝑓𝑎^𝑛𝑥^𝑝_𝑌+𝑄^′𝑎^𝑛𝑥−𝑓𝑎^𝑛𝑥^𝑝_𝑌≤1|𝑎|^{2𝑛𝛽𝑝}2|𝑎|^2𝛽𝑝Φ𝑎^𝑛𝑥=2|𝑎|^{∞2𝛽𝑝(𝑛+1)}_𝑖=01|𝑎|^{2𝛽𝑝𝑖}𝜑𝑎^𝑖+𝑛𝑥,0^𝑝=2|𝑎|^∞2𝛽𝑝_𝑖=𝑛1|𝑎|^{2𝛽𝑝𝑖}𝜑𝑎^𝑖𝑥,0^𝑝, (3.15) for all 𝑥∈𝑋 and 𝑛∈ℕ. Therefore, letting 𝑛→∞, one has 𝑄(𝑥)−𝑄′(𝑥)=0 for all 𝑥∈𝑋, completing the proof of uniqueness.

In the following corollary, we get a stability result of (1.14).

Corollary 3.2. Let 𝑋 be a quasi-𝛼-normed space for fixed real number 𝛼 with 0<𝛼≤1. Let 𝜃₁,𝜃₂,𝜃₃,𝛼₁,𝛼₂,𝛾₁,𝛾₂ be positive reals such that either (1)  |𝑎|>1, (𝛼₁+𝛼₂)𝛼<2𝛽, and 𝛾_𝑖𝛼<2𝛽 or (2)  |𝑎|<1, (𝛼₁+𝛼₂)𝛼>2𝛽, and 𝛾_𝑖𝛼>2𝛽, for 𝑖=1,2. Assume that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓𝑥,𝑦_𝑌≤𝜃₁‖𝑥‖^𝛼1𝑦^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃𝑦^𝛾2, (3.16) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 which satisfies the inequality 𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝜃₂‖𝑥‖^𝛾1|𝑎|^2𝛽𝑝−|𝑎|^{𝛾1𝛼𝑝1/𝑝}, (3.17) for all 𝑥∈𝑋. The function 𝑄 is given by 𝑄(𝑥)=lim_𝑘→∞𝑓𝑎^𝑘𝑥𝑎^2𝑘, (3.18) for all 𝑥∈𝑋.

Proof. Let 𝜑(𝑥,𝑦)=𝜃₁‖𝑥‖^𝛼1‖𝑦‖^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃‖𝑦‖^𝛾2. Then Φ(𝑥)=^∞_𝑛=01|𝑎|^{2𝛽𝑛𝑝}𝜑𝑎^𝑛𝑥,0^𝑝=^∞_𝑛=01|𝑎|^{2𝛽𝑛𝑝}𝜃^𝑝₂𝑎^𝑛𝑥^𝛾1𝑝=𝜃^𝑝₂‖𝑥‖^𝛾1𝑝∞_𝑛=0|𝑎|^{(𝛾1𝛼−2𝛽)𝑛𝑝}<∞,lim_𝑛→∞1|𝑎|^2𝛽𝑛𝜑𝑎^𝑛𝑥,𝑎^𝑛𝑦=lim_𝑛→∞1|𝑎|^2𝛽𝑛𝜃₁𝑎^𝑛𝑥^𝛼1𝑎^𝑛𝑦^𝛼2+𝜃₂𝑎^𝑛𝑥^𝛾1+𝜃₃𝑎^𝑛𝑦^𝛾2=𝜃₁‖𝑥‖^𝛼1𝑦^𝛼2lim_𝑛→∞|𝑎|^{((𝛼1+𝛼2)𝛼−2𝛽)𝑛}+𝜃₂‖𝑥‖^𝛾1lim_𝑛→∞|𝑎|^{(𝛾1𝛼−2𝛽)𝑛}+𝜃₃𝑦^𝛾2lim_𝑛→∞|𝑎|^{(𝛾2𝛼−2𝛽)𝑛}=0. (3.19)(3.20) By Theorem 3.1, there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)_𝑌≤1|𝑎|^2𝛽[Φ(𝑥)]^1/𝑝=𝜃₂‖𝑥‖^𝛾1|𝑎|^2𝛽∞_𝑛=0|𝑎|^{(𝛾1𝛼−2𝛽)𝑛𝑝1/𝑝}=𝜃₂‖𝑥‖^𝛾1|𝑎|^2𝛽𝑝−|𝑎|^{𝛾1𝛼𝑝1/𝑝}, (3.21) for all 𝑥∈𝑋.

Theorem 3.3. Let 𝜑∶𝑋×𝑋→[0,∞) be a function such that Ψ(𝑥)∶=^∞_𝑛=0|𝑎|^{2𝛽𝑛𝑝}𝜑𝑥𝑎^𝑛+1,0^𝑝<∞,lim_𝑛→∞|𝑎|^2𝛽𝑛𝜑𝑥𝑎^𝑛,𝑦𝑎^𝑛=0, (3.22)(3.23) for all 𝑥,𝑦∈𝑋. Suppose that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies 𝐷_𝑓𝑥,𝑦_𝑌≤𝜑𝑥,𝑦, (3.24) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 satisfying 𝑓(𝑥)−𝑄(𝑥)_𝑌≤[Ψ(𝑥)]^1/𝑝, (3.25) for all 𝑥∈𝑋. The function Q is given by 𝑄(𝑥)=lim_𝑘→∞𝑎^2𝑘𝑓𝑥𝑎^𝑘, (3.26) for all 𝑥∈𝑋.

Proof. Letting 𝑦 by 0 in (3.24), we get 𝑓(𝑎𝑥)−𝑎²𝑓(𝑥)_𝑌≤𝜑(𝑥,0), (3.27) for all 𝑥∈𝑋. Replacing 𝑥 by 𝑥/𝑎 in (3.27), we have 𝑓(𝑥)−𝑎²𝑓𝑥𝑎_𝑌𝑥≤𝜑𝑎,0, (3.28) for all 𝑥∈𝑋. Replacing 𝑥 by 𝑥/𝑎^𝑛 and multiplying both sides by |𝑎|^2𝛽𝑛 in (3.28), we have 𝑎^2𝑛𝑓𝑥𝑎^𝑛−𝑎^2(𝑛+1)𝑓𝑥𝑎^𝑛+1_𝑌≤|𝑎|^2𝛽𝑛𝜑𝑥𝑎^𝑛+1,0, (3.29) for all 𝑥∈𝑋. Next we show that the sequence {𝑎^2𝑛𝑓(𝑥/𝑎^𝑛)} is a Cauchy sequence. For any 𝑚,𝑛∈ℕ, 𝑚>𝑛≥0, and 𝑥∈𝑋, it follows from (3.29) that 𝑎^2𝑛𝑓𝑥𝑎^𝑛−𝑎^2(𝑚+1)𝑓𝑥𝑎^𝑚+1𝑝_𝑌=^𝑚_𝑖=𝑛𝑎^2𝑖𝑓𝑥𝑎^𝑖−𝑎^2(𝑖+1)𝑓𝑥𝑎^𝑖+1𝑝_𝑌≤^𝑚_𝑖=𝑛𝑎^2𝑖𝑓𝑥𝑎^𝑖−𝑎^2(𝑖+1)𝑓𝑥𝑎^𝑖+1𝑝_𝑌≤^𝑚_𝑖=𝑛|𝑎|^{2𝛽𝑝𝑖}𝜑𝑥𝑎^𝑖+1,0^𝑝. (3.30) It follows from (3.22) and (3.30) that the sequence {𝑎^2𝑛𝑓(𝑥/𝑎^𝑛)} is a Cauchy sequence in 𝑌 for all 𝑥∈𝑋. Since 𝑌 is a (𝛽,𝑝)-Banach space, the sequence {𝑎^2𝑛𝑓(𝑥/𝑎^𝑛)} converges for all 𝑥∈𝑋. Therefore, we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)=lim_𝑛→∞𝑎^2𝑛𝑓𝑥𝑎^𝑛, (3.31) for all 𝑥∈𝑋. The rest of the proof is similar to the corresponding proof of Theorem 3.1.

Corollary 3.4. Let 𝑋 be a quasi-𝛼-normed space for fixed real number 𝛼 with 0<𝛼≤1. Let 𝜃₁,𝜃₂,𝜃₃,𝛼₁,𝛼₂,𝛾₁,𝛾₂ be positive reals such that either (1)  |𝑎|>1, (𝛼₁+𝛼₂)𝛼>2𝛽, and 𝛾_𝑖𝛼>2𝛽 or (2)  |𝑎|<1, (𝛼₁+𝛼₂)𝛼<2𝛽, and 𝛾_𝑖𝛼<2𝛽, for 𝑖=1,2. Assume that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓𝑥,𝑦_𝑌≤𝜃₁‖𝑥‖^𝛼1𝑦^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃𝑦^𝛾2, (3.32) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 which satisfies the inequality 𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝜃₂‖𝑥‖^𝛾1|𝑎|^{𝛾1𝛼𝑝}−|𝑎|^{2𝛽𝑝1/𝑝}, (3.33) for all 𝑥∈𝑋. The function 𝑄 is given by 𝑄(𝑥)=lim_𝑘→∞𝑎^2𝑘𝑓𝑥𝑎^𝑘, (3.34) for all 𝑥∈𝑋.

Proof. Let 𝜑(𝑥,𝑦)=𝜃₁‖𝑥‖^𝛼1‖𝑦‖^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃‖𝑦‖^𝛾2. Then 𝜑 satisfies the conditions (3.22) and (3.23). Applying Theorem 3.3, we obtain the results, as desired.

4. Generalized Stability of (1.13)

For convenience, we use the following abbreviation: for any fixed rational numbers 𝑎 and 𝑏 with 𝑎≠−1,0 and 𝑏≠0, 𝐸_𝑓𝑥,𝑦∶=𝑓𝑎𝑥+𝑏𝑦+𝑎𝑓𝑥−𝑏𝑦−(𝑎+1)𝑓𝑏𝑦−𝑎(𝑎+1)𝑓(𝑥), (4.1) for all 𝑥,𝑦∈𝑋, which is called the approximate remainder of the functional equation (1.13) and acts as a perturbation of the equation.

We will investigate the generalized Hyers-Ulam stability problem for the functional equation (1.13).

Theorem 4.1. Let 𝜑∶𝑋×𝑋→[0,∞) be a function such that Φ(𝑥)∶=^∞_𝑛=01|𝑎+1|^{2𝛽𝑛𝑝}𝜑(𝑎+1)^𝑛()𝑥,𝑎+1^𝑛𝑥𝑏^𝑝<∞,lim_𝑛→∞1|𝑎+1|^2𝛽𝑛𝜑(𝑎+1)^𝑛𝑥,(𝑎+1)^𝑛𝑦=0, (4.2)(4.3) for all 𝑥,𝑦∈𝑋. Suppose that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies 𝐸_𝑓𝑥,𝑦_𝑌≤𝜑𝑥,𝑦, (4.4) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 satisfying 𝑓(𝑥)−𝑄(𝑥)_𝑌≤1|𝑎+1|^2𝛽[Φ(𝑥)]^1/𝑝, (4.5) for all 𝑥∈𝑋. The function Q is given by 𝑄(𝑥)=lim_𝑘→∞1(𝑎+1)^2𝑘𝑓(𝑎+1)^𝑘𝑥, (4.6) for all 𝑥∈𝑋.

Proof. Replacing 𝑥 by 𝑏𝑦 in (4.4), we get 𝑓(𝑎+1)𝑏𝑦−(𝑎+1)²𝑓𝑏𝑦_𝑌≤𝜑𝑏𝑦,𝑦, (4.7) for all 𝑦∈𝑋. Letting 𝑏𝑦 be 𝑥 in (4.7), we have 𝑓((𝑎+1)𝑥)−(𝑎+1)²𝑓(𝑥)_𝑌𝑥≤𝜑𝑥,𝑏, (4.8) for all 𝑥∈𝑋. Multiplying both sides by 1/|𝑎+1|^2𝛽 in (4.8), we have 1(𝑎+1)²𝑓((𝑎+1)𝑥)−𝑓(𝑥)_𝑌≤1|𝑎+1|^2𝛽𝜑𝑥𝑥,𝑏, (4.9) for all 𝑥∈𝑋. Replacing 𝑥 by (𝑎+1)^𝑖𝑥 and multiplying both sides by 1/|𝑎+1|^2𝑖𝛽 in (4.9), we have 1(𝑎+1)^2(𝑖+1)𝑓(𝑎+1)^𝑖+1𝑥−1(𝑎+1)^2𝑖𝑓(𝑎+1)^𝑖𝑥_𝑌≤1|𝑎+1|^{2𝛽(𝑖+1)}𝜑(𝑎+1)^𝑖()𝑥,𝑎+1^𝑖𝑥𝑏, (4.10) for all 𝑥∈𝑋. Next we show that the sequence {(1/(𝑎+1)^2𝑛)𝑓((𝑎+1)^𝑛𝑥)} is a Cauchy sequence. For any 𝑚,𝑛∈ℕ, 𝑚>𝑛≥0, and 𝑥∈𝑋, it follows from (4.10) that 1(𝑎+1)^2(𝑚+1)𝑓(𝑎+1)^𝑚+1𝑥−1(𝑎+1)^2𝑛𝑓(𝑎+1)^𝑛𝑥^𝑝_𝑌=^𝑚_𝑖=𝑛1(𝑎+1)^2(𝑖+1)𝑓(𝑎+1)^𝑖+1𝑥−1(𝑎+1)^2𝑖𝑓(𝑎+1)^𝑖𝑥^𝑝_𝑌≤^𝑚_𝑖=𝑛1(𝑎+1)^2(𝑖+1)𝑓(𝑎+1)^𝑖+1𝑥−1(𝑎+1)^2𝑖𝑓(𝑎+1)^𝑖𝑥^𝑝_𝑌≤^𝑚_𝑖=𝑛1|𝑎+1|^{2𝛽𝑝(𝑖+1)}𝜑(𝑎+1)^𝑖()𝑥,𝑎+1^𝑖𝑥𝑏^𝑝=1|𝑎+1|^{𝑚2𝛽𝑝}_𝑖=𝑛1|𝑎+1|^{2𝛽𝑝𝑖}𝜑(𝑎+1)^𝑖()𝑥,𝑎+1^𝑖𝑥𝑏^𝑝, (4.11) for all 𝑥∈𝑋. It follows from (4.2) and (4.11) that the sequence {𝑓((𝑎+1)^𝑛𝑥)/(𝑎+1)^2𝑛} is a Cauchy sequence in 𝑌 for all 𝑥∈𝑋. Since 𝑌 is a (𝛽,𝑝)-Banach space, the sequence {𝑓((𝑎+1)^𝑛𝑥)/(𝑎+1)^2𝑛} converges for all 𝑥∈𝑋. Therefore, we can define a mapping 𝑄∶𝑋→𝑌 by 𝑄(𝑥)=lim_𝑛→∞1(𝑎+1)^2𝑛𝑓(𝑎+1)^𝑛𝑥, (4.12) for all 𝑥∈𝑋. The rest of the proof is similar to the corresponding proof of Theorem 3.1.

In the following corollary, we get a stability result of (1.13).

Corollary 4.2. Let 𝑋 be a quasi-𝛼-normed space for fixed real number 𝛼 with 0<𝛼≤1. Let 𝜃₁,𝜃₂,𝜃₃,𝛼₁,𝛼₂,𝛾₁,𝛾₂ be positive reals such that either (1)  |𝑎+1|>1, (𝛼₁+𝛼₂)𝛼<2𝛽, and 𝛾_𝑖𝛼<2𝛽 or (2)  |𝑎+1|<1, (𝛼₁+𝛼₂)𝛼>2𝛽, and 𝛾_𝑖𝛼>2𝛽, for 𝑖=1,2. Assume that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐸_𝑓𝑥,𝑦_𝑌≤𝜃₁‖𝑥‖^𝛼1𝑦^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃𝑦^𝛾2, (4.13) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 which satisfies the inequality 𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝜃^𝑝₁‖𝑥‖^{(𝛼1+𝛼2)𝑝}|𝑏|^{𝛼𝛼2𝑝}|𝑎+1|^2𝛽𝑝−|𝑎+1|^{(𝛼1+𝛼2)𝛼𝑝}+𝜃^𝑝₂‖𝑥‖^𝛾1𝑝|𝑎+1|^2𝛽𝑝−|𝑎+1|^{𝛾1𝛼𝑝}+𝜃^𝑝₃‖𝑥‖^𝛾2𝑝|𝑏|^{𝛾2𝛼𝑝}|𝑎+1|^2𝛽𝑝−|𝑎+1|^{𝛾2𝛼𝑝1/𝑝}, (4.14) for all 𝑥∈𝑋. The function 𝑄 is given by 𝑄(𝑥)=lim_𝑘→∞1(𝑎+1)^2𝑘𝑓(𝑎+1)^k𝑥, (4.15) for all 𝑥∈𝑋.

Proof. Let 𝜑(𝑥,𝑦)=𝜃₁‖𝑥‖^𝛼1‖𝑦‖^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃‖𝑦‖^𝛾2. Then 𝜑 satisfies the conditions (4.2) and (4.3). By Theorem 4.1, there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)_𝑌≤1|𝑎+1|^2𝛽∞_𝑛=01|𝑎+1|^{2𝛽𝑛𝑝}𝜑(𝑎+1)^𝑛()𝑥,𝑎+1^𝑛𝑥𝑏^𝑝1/𝑝≤𝜃^𝑝₁‖𝑥‖^{(𝛼1+𝛼2)𝑝}|𝑏|^{𝛼𝛼2𝑝}|𝑎+1|^2𝛽𝑝−|𝑎+1|^{(𝛼1+𝛼2)𝛼𝑝}+𝜃^𝑝₂‖𝑥‖^𝛾1𝑝|𝑎+1|^2𝛽𝑝−|𝑎+1|^{𝛾1𝛼𝑝}+𝜃^𝑝₃‖𝑥‖^𝛾2𝑝|𝑏|^{𝛾2𝛼𝑝}|𝑎+1|^2𝛽𝑝−|𝑎+1|^{𝛾2𝛼𝑝1/𝑝}, (4.16) for all 𝑥∈𝑋.

Theorem 4.3. Let 𝜑∶𝑋×𝑋→[0,∞) be a function such that Ψ(𝑥)∶=^∞_𝑛=0|𝑎+1|^{2𝛽𝑛𝑝}𝜑𝑥(𝑎+1)^𝑛+1,𝑥(𝑎+1)^𝑛+1𝑏^𝑝<∞,lim_𝑛→∞|𝑎+1|^2𝛽𝑛𝜑𝑥(𝑎+1)^𝑛,𝑦(𝑎+1)^𝑛=0, (4.17) for all 𝑥,𝑦∈𝑋. Suppose that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies 𝐸_𝑓(𝑥,𝑦)_𝑌≤𝜑𝑥,𝑦, (4.18) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 satisfying 𝑓(𝑥)−𝑄(𝑥)_𝑌≤[Ψ(𝑥)]^1/𝑝, (4.19) for all 𝑥∈𝑋. The function Q is given by 𝑄(𝑥)=lim_𝑘→∞(𝑎+1)^2𝑘𝑓𝑥(𝑎+1)^𝑘, (4.20) for all 𝑥∈𝑋.

Proof. Replacing 𝑥 by 𝑥/(𝑎+1) in (4.8), we have 𝑓(𝑥)−(𝑎+1)²𝑓𝑥𝑎+1_𝑌𝑥≤𝜑,𝑥𝑎+1()𝑏,𝑎+1 (4.21) for all 𝑥∈𝑋. The rest of the proof is similar to the corresponding proof of Theorem 3.3.

Corollary 4.4. Let 𝑋 be a quasi-𝛼-normed space for fixed real number 𝛼 with 0<𝛼≤1. Let 𝜃₁,𝜃₂,𝜃₃,𝛼₁,𝛼₂,𝛾₁,𝛾₂ be positive reals such that either (1)  |𝑎+1|>1 and (𝛼₁+𝛼₂)𝛼>2𝛽, 𝛾_𝑖𝛼>2𝛽 or (2)  |𝑎+1|<1 and (𝛼₁+𝛼₂)𝛼<2𝛽, 𝛾_𝑖𝛼<2𝛽, for 𝑖=1,2. Assume that a function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝐸_𝑓𝑥,𝑦_𝑌≤𝜃₁‖𝑥‖^𝛼1𝑦^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃𝑦^𝛾2, (4.22) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 which satisfies the inequality 𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝜃^𝑝₁‖𝑥‖^{(𝛼1+𝛼2)𝑝}|𝑏|^{𝛼𝛼2𝑝}|𝑎+1|^{(𝛼1+𝛼2)𝛼𝑝}−|𝑎+1|^2𝛽𝑝+𝜃^𝑝₂‖𝑥‖^𝛾1𝑝|𝑎+1|^{𝛾1𝛼𝑝}−|𝑎+1|^2𝛽𝑝+𝜃^𝑝₃‖𝑥‖^𝛾2𝑝|𝑏|^{𝛼𝛾2𝑝}|𝑎+1|^{𝛾2𝛼𝑝}−|𝑎+1|^{2𝛽𝑝1/𝑝}, (4.23) for all 𝑥∈𝑋. The function 𝑄 is given by 𝑄(𝑥)=lim_𝑘→∞(𝑎+1)^2𝑘𝑓𝑥(𝑎+1)^𝑘, (4.24) for all 𝑥∈𝑋.

Proof. Let 𝜑(𝑥,𝑦)=𝜃₁‖𝑥‖^𝛼1‖𝑦‖^𝛼2+𝜃₂‖𝑥‖^𝛾1+𝜃₃‖𝑦‖^𝛾2. Then 𝜑 satisfies the conditions (4.17). Applying Theorem 4.3, we obtain the results, as desired.

Acknowledgment

This study was supported by the Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (no. 2012R1A1A2008139).