# Nearly Radical Quadratic Functional Equations inp-2-Normed Spaces

Authors: M. Eshaghi Gordji; H. Khodaei; A. Ebadian; G. H. Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We establish some stability results in 2-normed spaces for the radical quadratic functional equationf(∑i=1n(xi+yi)2)+f(∑i=1n(xi-yi)2)=2∑i=1n(f(xi)+f(yi)) and then use subadditive functions to prove its stability in p-2-normed spaces.


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## Body

## 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 equationf(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
(1.1)f(x+y)+f(x-y)=2f(x)+2f(y).
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]).
Let𝒳  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{xn} 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, limn,m→∞∥xn-xm,y∥=0 and limn,m→∞∥xn-xm,z∥=0.A sequence{xn} in a linear 2-normed space 𝒳 is called a convergent sequence if there is an x∈𝒳 such that limn→∞∥xn-x,y∥=0, for all y∈𝒳. If {xn} converges to x, write xn→x as n→∞ and call x the limit of {xn}. In this case, we also write limn→∞xn=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{xn} in a 2-normed space 𝒳, limn→∞∥xn,y∥=∥limn→∞xn,y∥, for all y∈𝒳 (see [48]).We fix a real numberp 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 functionφa:A→B, having a domain A and a codomain (B,≤) that are both closed under addition, with the following property:
(1.2)φa(x+y)≤φa(x)+φa(y),
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 φa:A→B satisfies
(1.3)lφa(x+y)≤lLl(φa(x)+φa(y)),
for all x,y∈A, then we say that φa is contractively subadditive if ℓ=1, and φa is expansively superadditive if ℓ=-1. It follows by the last inequality that φa satisfies the following properties:
(1.4)φa(2lx)≤2lLφa(x),φa(2lkx)≤(2lL)kφa(x),
for all x∈A and integers k≥1.Now, we consider the radical quadratic functional equation(1.5)f(∑i=1n(xi+yi)2)+f(∑i=1n(xi-yi)2)=2∑i=1n(f(xi)+f(yi)),
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, letX 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 xi=yi=0(1≤i≤n) in (1.5), we get f(0)=0. Setting xi=yi=x(1≤i≤n) in (1.5) and using f(0)=0, we get
(2.1)f(4nx2)=4nf(x),
for all x∈ℝ. Putting xi=2x,yi=0(1≤i≤n) in (1.5) and using f(0)=0, we get
(2.2)2f(4nx2)=2nf(2x),
for all x∈ℝ. It follows from (2.1) and (2.2) that
(2.3)f(2mx)=4mf(x),
for all x∈ℝ and integers m≥1. Setting yn=-yn in (1.5) and then comparing it with (1.5), we obtain f(-yn)=f(yn), for all yn∈ℝ. Letting xi=yi=0(2≤i≤n) in (1.5), we get
(2.4)f((x1+y1)2)+f((x1-y1)2)=2f(x1)+2f(y1),
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 mappingf:ℝ→X satisfies the functional equation  (1.5), then  f is quadratic.Corollary 2.2.
If a mappingf: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 that𝒳 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.Letϕ be a function from ℝ2n+1 to ℝ+∪{0}. A mapping f:ℝ→𝒳 is called a ϕ-approximatively radical quadratic function if
(2.5)‖f(∑i=1n(xi+yi)2)+f(∑i=1n(xi-yi)2)-2∑i=1n(f(xi)+f(yi)),z‖X≤ϕ(x1,…,xn,y1,…,yn,z),
for all x1,…,xn,y1,…,yn,z∈ℝ, where n∈ℕ is a fixed integer.Theorem 2.3.
Letℓ∈{-1,1} be fixed, and let f:ℝ→𝒳 be a ϕ- approximatively radical quadratic function with f(0)=0.  If the function ϕ:ℝ2n+1→ℝ+∪{0} satisfies
(2.6)Φ(x,z)∶=∑j=(1-l)/2∞14lj(ϕ(2ljx,…,2ljx︷2n,z)+12ϕ(21+ljx,…,21+ljx︷n,0,…,0︷n,z))<∞,
and limm→∞(1/4ℓm)ϕ(2ℓmx1,…,2ℓmxn,2ℓmy1,…,2ℓmyn,z)=0, for all x,x1,…,xn,y1,…,yn,z∈ℝ, then there exists a unique quadratic mapping ℱ:ℝ→𝒳, satisfies (1.5) and the inequality
(2.7)‖f(x)-F(x),y‖X≤14nΦ(x,y),
for all x,y∈ℝ.Proof.
Lettingxi=x+y,yi=x-y(1≤i≤n) in (2.5), we get
(2.8)‖f(4nx2)+f(4ny2)-2nf(x+y)-2nf(x-y),z‖X≤ϕ(x+y,…,x+y︷n,x-y,…,x-y︷n,z),
for all x,y,z∈ℝ. Setting xi=yi=x(1≤i≤n) in (2.5), we get
(2.9)‖f(4nx2)-4nf(x),z‖X≤ϕ(x,…,x︷2n,z),
for all x,z∈ℝ. Replacing y by x in (2.8), we obtain
(2.10)‖f(4nx2)-nf(2x),z‖X≤12ϕ(2x,…,2x︷n,0,…,0︷n,z),
for all x,z∈ℝ. It follows from (2.9) and (2.10) that
(2.11)‖4f(x)-f(2x),y‖X≤1nϕ(x,…,x︷2n,y)+12nϕ(2x,…,2x︷n,0,…,0︷n,y),
for all x,y∈ℝ. Thus,
(2.12)‖f(x)-14f(2x),y‖X≤14nϕ(x,…,x︷2n,y)+18nϕ(2x,…,2x︷n,0,…,0︷n,y),‖f(x)-4f(x2),y‖X≤1nϕ(x2,…,x2︷2n,y)+12nϕ(x,…,x︷n,0,…,0︷n,y),
for all x,y∈ℝ. Hence,
(2.13)‖14lkf(2lkx)-14lrf(2lrx),y‖X≤14n∑j=k+(1-l)/2r-(1+l)/214lj(ϕ(2ljx,…,2ljx︷2n,y)+12ϕ(21+ljx,…,21+ljx︷n,0,…,0︷n,y))
for all x,y∈ℝ and integers r>k≥0. Thus, {(1/4ℓm)f(2ℓmx)} is a Cauchy sequence in the 2-Banach space 𝒳. Hence, we can define a mapping ℱ:ℝ→𝒳 by ℱ(x):=limm→∞(1/4ℓm)f(2ℓmx), for all x∈ℝ. That is,
(2.14)lim⁡m→∞‖14lmf(2lmx)-F(x),y‖X=0,
for all x,y∈ℝ. In addition, it is clear from (2.5) that the following inequality:
(2.15)‖F(∑i=1n(xi+yi)2)+F(∑i=1n(xi-yi)2)-2∑i=1n(F(xi)+F(yi)),z‖X=lim⁡m→∞14lm‖f(4lm∑i=1n(xi+yi)2)+f(4lm∑i=1n(xi-yi)2)-2∑i=1n(f(2lmxi)+f(2lmyi)),z‖X≤lim⁡m→∞14lmϕ(2lmx1,…,2lmxn,2lmy1,…,2lmyn,z)=0
holds for all x1,…,xn,y1,…,yn,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
(2.16)G(2lx)=4lG(x),G(2lmx)=4lmG(x)
for all x∈ℝ and integers m≥1. Thus, one proves by the last equality and (2.7) that
(2.17)‖14lmf(2lmx)-G(x),y‖X=14lm‖f(2lmx)-G(2lmx),y‖X≤14ml+1nΦ(2lmx,y),
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.
Letℓ∈{-1,1} be fixed. If there exist nonnegative real numbers  pi,qi,q with  ℓ∑i=1n(pi+qi)<2ℓ such that a mapping  f:ℝ→𝒳 satisfies the inequality
(2.18)‖f(∑i=1n(xi+yi)2)+f(∑i=1n(xi-yi)2)-2∑i=1n(f(xi)+f(yi)),z‖X≤θ∏i=1n|xi|pi|yi|qi|z|q,
for all x1,…,xn,y1,…,yn,z∈ℝ and some θ≥0, then there exists a unique quadratic mapping ℱ:ℝ→𝒳, satisfies (1.5) and the inequality
(2.19)‖f(x)-F(x),y‖X≤1ln(4-2λ)θ|x|λ|y|q,
for all x,y∈ℝ, where λ:=∑i=1n(pi+qi).Corollary 2.5.
Letℓ∈{-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
(2.20)‖f(∑i=1n(xi+yi)2)+f(∑i=1n(xi-yi)2)-2∑i=1n(f(xi)+f(yi)),z‖X≤θ∑i=1n(|xi|s+|yi|s)|z|t,
for all x1,…,xn,y1,…,yn,z∈ℝ and some θ≥0, then there exists a unique quadratic mapping ℱ:ℝ→𝒳 satisfies (1.5) and the inequality
(2.21)‖f(x)-F(x),y‖X≤1+2s-2l(2-2s-1)θ|x|s|y|t,
for all x,y∈ℝ.Now, we are going to establish the modified Hyers-Ulam stability of (1.5).Theorem 2.6.
Letℓ∈{-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  2ℓ(1-3p)L<1, where  3ℓp≤ℓ, then there exists a unique quadratic mapping  ℱ:ℝ→𝒴 which satisfies  (1.5) and the inequality
(2.22)‖f(x)-F(x),y‖Y≤1l(4p-21-pLl)Ψ(x,y),
for all x,y∈ℝ, where
(2.23)Ψ(x,y)∶=1npϕ(x,…,x︷2n,y)+1(2n)pϕ(2x,…,2x︷n,0,…,0︷n,y).Proof.
Using the same method as in the proof of Theorem2.3, we have
(2.24)‖f(x)-14f(2x),y‖Y≤14pΨ(x,y),‖f(x)-4f(x2),y‖Y≤2pΨ(x2,y2),
for all x,y∈ℝ. Hence
(2.25)‖14lkf(2lkx)-14lrf(2lrx),y‖Y≤14p∑j=k+(1-l)/2r-(1+l)/2123lpjΨ(2ljx,2ljy)≤14p∑j=k+(1-l)/2r-(1+l)/2(2lL)j23lpjΨ(x,y)=Ψ(x,y)4p∑j=k+(1-l)/2r-(1+l)/2(2l(1-3p)L)j,
for all x,y∈ℝ and integers r>k≥0. Thus, {(1/4ℓm)f(2ℓmx)} is a Cauchy sequence in the p-2-Banach space 𝒴. Hence, we can define a mapping ℱ:ℝ→𝒴 by ℱ(x):=limn→∞(1/4ℓn)f(2ℓnx), for all x∈ℝ. Also
(2.26)‖F(∑i=1n(xi+yi)2)+F(∑i=1n(xi-yi)2)-2∑i=1n(F(xi)+F(yi)),z‖Y=lim⁡m→∞‖14lmf(4lm∑i=1n(xi+yi)2)+14lmf(4lm∑i=1n(xi-yi)2)-24lm∑i=1n(f(2lmxi)+f(2lmyi)),z‖Y≤lim⁡m→∞123lpmϕ(2lmx1,…,2lmxn,2lmy1,…,2lmyn,2lmz)≤lim⁡m→∞(2l(1-3p)L)mϕ(x1,…,xn,y1,…,yn,z)=0
holds for all x1,…,xn,y1,…,yn,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.



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