# Functional Equations Related to  Inner Product Spaces

Authors: Choonkil Park; Won-Gil Park; Abbas Najati
Journal: Abstract and Applied Analysis
 (2009)

---

## Abstract

LetV,W be real vector spaces. It is shown that an odd mapping
f:V→W satisfies ∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi) for all x1,…,x2n∈V if and only if the odd mapping f:V→W is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation
in real Banach spaces.


---

## Body

## 1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of Th. M. Rassias' theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function.The functional equation,(1.1)f(x+y)+f(x-y)=2f(x)+2f(y),
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings f:X→Y, where X is a normed space and Y is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. The generalized Hyers-Ulam stability of the quadratic functional equation has been proved by Czerwik [8], J. M. Rassias [9], Găvruta [10], and others [11]. In [12], Th. M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer n≥2(1.2)∑i=1n‍∥xi-1n∑j=1n‍xj∥2=∑i=1n‍∥xi∥2-n∥1n∑i=1n‍xi∥2
holds for all x1,…,xn∈V. An operator extension of this norm equality is presented in [13]. For more information on the recent results on the stability of quadratic functional equation, see [14]. Inner product spaces, Cauchy equation, Euler-Lagrange-Rassias equations, and Ulam-Găvruta-Rassias stability have been studied by several authors (see [15–27]).In [28], C. Park, Lee, and Shin proved that if an even mapping f:V→W satisfies(1.3)∑i=12n‍f(xi-12n∑j=12n‍xj)=∑i=12n‍f(xi)-2nf(12n∑i=12n‍xi),
then the even mapping f:V→W is quadratic. Moreover, they proved the generalized Hyers-Ulam stability of the quadratic functional equation  (1.3) in real Banach spaces.Throughout this paper, assume thatn is a fixed positive integer, X and Y are real normed vector spaces.In this paper, we investigate the functional equation(1.4)∑i=12n‍f(xi-12n∑j=12n‍xj)=∑i=12n‍f(xi)-2nf(12n∑i=12n‍xi),
and prove the generalized Hyers-Ulam stability of the functional equation  (1.4) in real Banach spaces.


## 2. Functional Equations Related to Inner Product Spaces

We investigate the functional equation  (1.4).Lemma 2.1.
LetV and W be real vector spaces. An odd mapping f:V→W satisfies
(2.1)∑i=12n‍f(xi-12n∑j=12n‍xj)=∑i=12n‍f(xi)-2nf(12n∑i=12n‍xi),
for all x1,…,x2n∈V if and only if the odd mapping f:V→W is Cauchy additive, that is,
(2.2)f(x+y)=f(x)+f(y),
for all x,y∈V.Proof.
Assume thatf:V→W satisfies (2.1).
Lettingx1=⋯=xn=x, xn+1=⋯=x2n=y in  (2.1), we get
(2.3)nf(x-x+y2)+nf(y-x+y2)=nf(x)+nf(y)-2nf(x+y2),
for allx,y∈V. Since f:V→W is odd,
(2.4)0=nf(x)+nf(y)-2nf(x+y2),
for all x,y∈V and f(0)=0. So
(2.5)2f(x+y2)=f(x)+f(y),
for all x,y∈V. Letting y=0 in (2.5), we get 2f(x/2)=f(x) for all x∈V. Thus
(2.6)f(x+y)=2f(x+y2)=f(x)+f(y),
for all x,y∈V.
It is easy to prove the converse.For a given mappingf:X→Y, we define
(2.7)Df(x1,…,x2n):=∑i=12n‍f(xi-12n∑j=12n‍xj)-∑i=12n‍f(xi)+2nf(12n∑i=12n‍xi),
for all x1,…,x2n∈X.We are going to prove the generalized Hyers-Ulam stability of the functional equationDf(x1,…,x2n)=0 in real Banach spaces.Theorem 2.2.
Letf:X→Y be a mapping satisfying f(0)=0 for which there exists a function φ:X2n→[0,∞) such that
(2.8)φ̃(x1,…,x2n):=∑j=0∞‍2jφ(x12j,…,x2n2j)<∞,(2.9)∥Df(x1,…,x2n)∥≤φ(x1,…,x2n),
for all x1,…,x2n∈X. Then there exists a unique Cauchy additive mapping A:X→Y satisfying  (2.1) such that(2.10)∥f(x)-f(-x)-A(x)∥≤1nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+1nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X.Proof.
Lettingx1=⋯=xn=x and xn+1=⋯=x2n=0 in  (2.9), we get
(2.11)∥3nf(x2)+nf(-x2)-nf(x)∥≤φ(x,…,x︸ntimes,0,…,0︸ntimes),
for all x∈X. Replacing x by -x in (2.11), we get
(2.12)∥3nf(-x2)+nf(x2)-nf(-x)∥≤φ(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X. Let g(x):=f(x)-f(-x) for all x∈X. It follows from  (2.11) and  (2.12) that
(2.13)∥2ng(x2)-ng(x)∥≤φ(x,…,x︸ntimes,0,…,0︸ntimes)+φ(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X. So
(2.14)∥g(x)-2g(x2)∥≤1nφ(x,…,x︸ntimes,0,…,0︸ntimes)+1nφ(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X. Hence
(2.15)∥2lg(x2l)-2mg(x2m)∥≤∑j=lm-1‍2jnφ(x2j,…,x2j︸ntimes,0,…,0︸ntimes)+∑j=lm-1‍2jnφ(-x2j,…,-x2j︸ntimes,0,…,0︸ntimes),
for all nonnegative integers m and l with m>l and all x∈X. It follows from  (2.8) and  (2.15) that the sequence {2kg(x/2k)} is Cauchy for all x∈X. Since Y is complete, the sequence {2kg(x/2k)} converges. So one can define the mapping A:X→Y by
(2.16)A(x):=limk→∞2kg(x2k),
for all x∈X.
By (2.8) and (2.9),
(2.17)∥DA(x1,…,x2n)∥=limk→∞2k∥Dg(x12k,…,x2n2k)∥≤limk→∞2k[φ(x12k,…,x2n2k)+φ(-x12k,…,-x2n2k)]=0,
for all x1,…,x2n∈X. So DA(x1,…,x2n)=0. By Lemma 2.1, the mapping A:X→Y is Cauchy additive. Moreover, letting l=0 and passing the limit m→∞ in (2.15), we get (2.10). So there exists a Cauchy additive mapping A:X→Y satisfying (2.1) and (2.10).
Now, letL:X→Y be another Cauchy additive mapping satisfying  (2.1) and  (2.10). Then we have
(2.18)∥A(x)-L(x)∥=2q∥A(x2q)-L(x2q)∥≤2q(∥A(x2q)-f(x2q)+f(-x2q)∥+∥L(x2q)-f(x2q)+f(-x2q)∥)≤2·2qnφ̃(x2q,…,x2q︸ntimes,0,…,0︸ntimes)+2·2qnφ̃(-x2q,…,-x2q︸ntimes,0,…,0︸ntimes),
which tends to zero as q→∞ for all x∈X. So we can conclude that A(x)=L(x) for all x∈X. This proves the uniqueness of A.Corollary 2.3.
Letp>1 and θ be positive real numbers, and let f:X→Y be a mapping such that
(2.19)∥Df(x1,…,x2n)∥≤θ∑j=12n‍∥xj∥p,
for all x1,…,x2n∈X. Then there exists a unique Cauchy additive mapping A:X→Y satisfying  (2.1) such that
(2.20)∥f(x)-f(-x)-A(x)∥≤2p+1θ2p-2∥x∥p,
for all x∈X.Proof.
Defineφ(x1,…,x2n)=θ∑j=12n‍∥xj∥p, and apply Theorem 2.2 to get the desired result.Corollary 2.4.
Letf:X→Y be an odd mapping for which there exists a function φ:X2n→[0,∞) satisfying (2.8) and (2.9). Then there exists a unique Cauchy additive mapping A:X→Y satisfying (2.1) such that
(2.21)∥2f(x)-A(x)∥≤1nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+1nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes),
or (alternative approximation)
(2.22)∥f(x)-A(x)∥≤12nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+12nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X, where φ̃ is defined in  (2.8).Theorem 2.5.
Letf:X→Y be a mapping satisfying f(0)=0 for which there exists a function φ:X2n→[0,∞) satisfying (2.9) such that
(2.23)φ̃(x1,…,x2n):=∑j=1∞‍2-jφ(2jx1,…,2jx2n)<∞,
for all x1,…,x2n∈X. Then there exists a unique Cauchy additive mapping A:X→Y satisfying (2.1) such that
(2.24)∥f(x)-f(-x)-A(x)∥≤1nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+1nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X.Proof.
It follows from (2.13) that
(2.25)∥g(x)-12g(2x)∥≤12nφ(2x,…,2x︸ntimes,0,…,0︸ntimes)+12nφ(-2x,…,-2x︸ntimes,0,…,0︸ntimes),
for all x∈X. So
(2.26)∥12lg(2lx)-12mg(2mx)∥≤∑j=l+1m‍12jnφ(2jx,…,2jx︸ntimes,0,…,0︸ntimes)+∑j=l+1m‍12jnφ(-2jx,…,-2jx︸ntimes,0,…,0︸ntimes),
for all nonnegative integers m and l with m>l and all x∈X. It follows from  (2.23) and  (2.26) that the sequence {(1/2k)g(2kx)} is Cauchy for all x∈X. Since Y is complete, the sequence {(1/2k)g(2kx)} converges. So one can define the mapping A:X→Y by
(2.27)A(x):=limk→∞12kg(2kx),
for all x∈X.
By (2.9) and (2.23),
(2.28)∥DA(x1,…,x2n)∥=limk→∞12k∥Dg(2kx1,…,2kx2n)∥≤limk→∞12k(φ(2kx1,…,2kx2n)+φ(-2kx1,…,-2kx2n))=0,
for all x1,…,x2n∈X. So DA(x1,…,x2n)=0. By Lemma 2.1, the mapping A:X→Y is Cauchy additive. Moreover, letting l=0 and passing the limit m→∞ in (2.26), we get (2.24). So there exists a Cauchy additive mapping A:X→Y satisfying (2.1) and (2.24).
The rest of the proof is similar to the proof of Theorem2.2.Corollary 2.6.
Letp<1 and θ be positive real numbers, and let f:X→Y be a mapping satisfying  (2.19). Then there exists a unique Cauchy additive mapping A:X→Y satisfying (2.1) such that
(2.29)∥f(x)-f(-x)-A(x)∥≤2p+1θ2-2p∥x∥p,
for all x∈X.Proof.
Defineφ(x1,…,x2n)=θ∑j=12n‍∥xj∥p, and apply Theorem 2.5 to get the desired result.Corollary 2.7.
Letf:X→Y be an odd mapping for which there exists a function φ:X2n→[0,∞) satisfying (2.9) and (2.23). Then there exists a unique Cauchy additive mapping A:X→Y satisfying (2.1) such that
(2.30)∥2f(x)-A(x)∥≤1nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+1nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes),
or (alternative approximation),
(2.31)∥f(x)-A(x)∥≤12nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+12nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X, where φ̃ is defined in (2.23).The following was proved in [28].Remark 2.8 ([28]).
Letf:X→Y be a mapping satisfying f(0)=0 for which there exists a function φ:X2n→[0,∞) satisfying  (2.9) such that
(2.32)Φ(x1,…,x2n):=∑j=0∞‍4jφ(x12j,…,x2n2j)<∞,
for all x1,…,x2n∈X. Then there exists a unique quadratic mapping Q:X→Y satisfying  (2.1) such that
(2.33)∥f(x)+f(-x)-Q(x)∥≤1nΦ(x,…,x︸ntimes,0,…,0︸ntimes)+1nΦ(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X.Note that(2.34)∑j=0∞‍2jφ(x12j,…,x2n2j)≤∑j=0∞‍4jφ(x12j,…,x2n2j).
Combining Theorem 2.2 and Remark 2.8, we obtain the following result.Theorem 2.9.
Letf:X→Y be a mapping satisfying f(0)=0 for which there exists a function φ:X2n→[0,∞) satisfying (2.9) and (2.32). Then there exist a unique Cauchy additive mapping A:X→Y satisfying (2.1) and a unique quadratic mapping Q:X→Y satisfying  (2.1) such that
(2.35)∥2f(x)-A(x)-Q(x)∥≤1nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+1nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes)+1nΦ(x,…,x︸ntimes,0,…,0︸ntimes)+1nΦ(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X, where φ̃ and Φ are defined in  (2.8) and  (2.32), respectively.Corollary 2.10.
Letp>2 and θ be positive real numbers, and let f:X→Y be a mapping satisfying (2.19). Then there exist a unique Cauchy additive mapping A:X→Y satisfying (2.1) and a unique quadratic mapping Q:X→Y satisfying (2.1) such that
(2.36)∥2f(x)-A(x)-Q(x)∥≤(2p+12p-2+2p+12p-4)θ∥x∥p,
for all x∈X.Proof.
Defineφ(x1,…,x2n)=θ∑j=12n‍∥xj∥p, and apply Theorem 2.9 to get the desired result.The following was proved in [28].Remark 2.11 (see [28]).
Letf:X→Y be a mapping satisfying f(0)=0 for which there exists a function φ:X2n→[0,∞) satisfying (2.9) such that
(2.37)Φ(x1,…,x2n):=∑j=1∞‍4-jφ(2jx1,…,2jx2n)<∞,
for all x1,…,x2n∈X. Then there exists a unique quadratic mapping Q:X→Y satisfying  (2.1) such that
(2.38)∥f(x)+f(-x)-Q(x)∥≤1nΦ(x,…,x︸ntimes,0,…,0︸ntimes)+1nΦ(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X.Note that(2.39)∑j=1∞‍4-jφ(2jx1,…,2jx2n)≤∑j=1∞‍2-jφ(2jx1,…,2jx2n).Combining Theorem2.5 and Remark 2.11, we obtain the following result.Theorem 2.12.
Letf:X→Y be a mapping satisfying f(0)=0 for which there exists a function φ:X2n→[0,∞) satisfying  (2.9) and  (2.23). Then there exist a unique Cauchy additive mapping A:X→Y satisfying  (2.1) and a unique quadratic mapping Q:X→Y satisfying  (2.1) such that
(2.40)∥2f(x)-A(x)-Q(x)∥≤1nφ̃(x,…,x︸ntimes,0,…,0︸ntimes)+1nφ̃(-x,…,-x︸ntimes,0,…,0︸ntimes)+1nΦ(x,…,x︸ntimes,0,…,0︸ntimes)+1nΦ(-x,…,-x︸ntimes,0,…,0︸ntimes),
for all x∈X, where φ̃ and Φ are defined in  (2.23) and  (2.37), respectively.Corollary 2.13.
Letp<1 and θ be positive real numbers, and let f:X→Y be a mapping satisfying (2.19). Then there exist a unique Cauchy additive mapping A:X→Y satisfying (2.1) and a unique quadratic mapping Q:X→Y satisfying (2.1) such that
(2.41)∥2f(x)-A(x)-Q(x)∥≤(2p+12-2p+2p+14-2p)θ∥x∥p,
for all x∈X.Proof.
Defineφ(x1,…,x2n)=θ∑j=12n‍∥xj∥p, and apply Theorem 2.12 to get the desired result.



---

*Source: 907121-2009-07-14.xml*