# Orthogonal Stability of an Additive-Quadratic Functional Equation

- Choonkil Park
^{1}Email author

**2011**:66

https://doi.org/10.1186/1687-1812-2011-66

© Park; licensee Springer. 2011

**Received: **17 March 2011

**Accepted: **25 October 2011

**Published: **25 October 2011

## Abstract

Using the fixed point method and using the direct method, we prove the Hyers-Ulam stability of an orthogonally additive-quadratic functional equation in orthogonality spaces.

**(2010) Mathematics Subject Classification:** Primary 39B55; 47H10; 39B52; 46H25.

## Keywords

## 1. Introduction and Preliminaries

Assume that X is a real inner product space and *f* : *X* → ℝ is a solution of the orthogonally Cauchy functional equation *f*(*x* + *y*) = *f*(*x*) + *f*(*y*), 〈*x*, *y*〉 = 0. By the Pythagorean theorem, *f*(*x*) = ||*x*||^{2} is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus, orthogonally Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space.

in which ⊥ is an abstract orthogonality relation, was first investigated by Gudder and Strawther [3]. They defined ⊥ by a system consisting of five axioms and described the general semi-continuous real-valued solution of conditional Cauchy functional equation. In 1985, Rätz [4] introduced a new definition of orthogonality by using more restrictive axioms than of Gudder and Strawther. Moreover, he investigated the structure of orthogonally additive mappings. Rätz and Szabó [5] investigated the problem in a rather more general framework.

Let us recall the orthogonality in the sense of Rätz; cf. [4].

Suppose *X* is a real vector space (algebraic module) with dim *X* ≥ 2 and ⊥ is a binary relation on *X* with the following properties:

(*O*_{1}) totality of ⊥ for zero: *x* ⊥ 0, 0 ⊥ *x* for all *x* ∈ *X*;

(*O*_{2}) independence: if *x*, *y* ∈ *X* - {0}, *x* ⊥ *y*, then *x*, *y* are linearly independent;

(*O*_{3}) homogeneity: if *x*, *y* ∈ *X*, *x* ⊥ *y*, then *αx* ⊥ *βy* for all *α*, *β* ∈ ℝ;

(*O*_{4}) the Thalesian property: if *P* is a 2-dimensional subspace of *X*, *x* ∈ *P* and *λ* ∈ ℝ_{+}, which is the set of nonnegative real numbers, then there exists *y*_{0} ∈ *P* such that *x* ⊥ *y*_{0} and *x* + *y*_{0} ⊥ *λx* - *y*_{0}.

The pair (*X*, ⊥) is called an orthogonality space. By an orthogonality normed space, we mean an orthogonality space having a normed structure.

- (i)
The trivial orthogonality on a vector space

*X*defined by (*O*_{1}), and for non-zero elements*x*,*y*∈*X*,*x*⊥*y*if and only if*x*,*y*are linearly independent. - (ii)
The ordinary orthogonality on an inner product space (

*X*, 〈., .〉) given by*x*⊥*y*if and only if 〈*x*,*y*〉 = 0. - (iii)
The Birkhoff-James orthogonality on a normed space (

*X*, ||·||) defined by*x*⊥*y*if and only if ||*x*+*λy*|| ≥ ||*x*|| for all*λ*∈ ℝ.

The relation ⊥ is called symmetric if *x* ⊥ *y* implies that *y* ⊥ *x* for all *x*, *y* ∈ *X*. Clearly examples (i) and (ii) are symmetric but example (iii) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [6–12]).

The stability problem of functional equations originated from the following question of Ulam [13]: *Under what condition does there exist an additive mapping near an approximately additive mapping?* In 1941, Hyers [14] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1978, Th.M. Rassias [15] extended the theorem of Hyers by considering the unbounded Cauchy difference ||*f*(*x* + *y*) - *f*(*x*) - *f*(*y*)|| ≤ *ε*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ), (*ε* > 0, *p* ∈ [0, 1)). The result of Th.M. Rassias has provided a lot of influence in the development of what we now call *generalized Hyers-Ulam stability* or *Hyers-Ulam stability* of functional equations. During the last decades, several stability problems of functional equations have been investigated in the spirit of Hyers-Ulam-Rassias. The reader is referred to [16–20] and references therein for detailed information on stability of functional equations.

Ger and Sikorska [21] investigated the orthogonal stability of the Cauchy functional equation *f*(*x* + *y*) = *f*(*x*) + *f*(*y*), namely, they showed that if *f* is a mapping from an orthogonality space *X* into a real Banach space *Y* and ||*f*(*x* + *y*) - *f*(*x*) - *f*(*y*)|| ≤ *ε* for all *x*, *y* ∈ *X* with *x* ⊥ *y* and some *ε* > 0, then there exists exactly one orthogonally additive mapping *g* : *X* → *Y* such that $\left|\right|f\left(x\right)-g\left(x\right)\parallel \le \frac{16}{3}\epsilon $ for all *x* ∈ *X*.

The first author treating the stability of the quadratic equation was Skof [22] by proving that if *f* is a mapping from a normed space *X* into a Banach space *Y* satisfying ||*f*(*x* + *y*) + *f*(*x* - *y*) - 2*f*(*x*) - 2*f*(*y*)|| ≤ *ε* for some *ε* > 0, then there is a unique quadratic mapping *g* : *X* → *Y* such that $\left|\right|f\left(x\right)-g\left(x\right)\parallel \le \frac{\epsilon}{2}$. Cholewa [23] extended the Skof's theorem by replacing *X* by an abelian group *G*. The Skof's result was later generalized by Czerwik [24] in the spirit of Hyers-Ulam-Rassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [25–28]).

was first investigated by Vajzović [29] when *X* is a Hilbert space, *Y* is the scalar field, *f* is continuous and ⊥ means the Hilbert space orthogonality. Later, Drljević [30], Fochi [31], Moslehian [32, 33], Szabó [34], Moslehian and Th.M. Rassias [35] and Paganoni and Rätz [36] have investigated the orthogonal stability of functional equations.

*X*be a set. A function

*m*:

*X*×

*X*→ [0, ∞] is called a

*generalized metric*on

*X*if

*m*satisfies

- (1)
*m*(*x*,*y*) = 0 if and only if*x*=*y*; - (2)
*m*(*x*,*y*) =*m*(*y*,*x*) for all*x*,*y*∈*X*; - (3)
*m*(*x*,*z*) ≤*m*(*x*,*y*) +*m*(*y*,*z*) for all*x*,*y*,*z*∈*X*.

We recall a fundamental result in fixed point theory.

**Theorem 1.1**. [37, 38]

*Let*(

*X*,

*m*)

*be a complete generalized metric space and let J*:

*X*→

*X be a strictly contractive mapping with Lipschitz constant α*< 1.

*Then, for each given element x*∈

*X, either*

*for all nonnegative integers n or there exists a positive integer n*

_{0}

*such that*

- (1)
*m*(*J*^{ n }*x*,*j*^{n+1}*x*) < ∞, ∀*n*≥*n*_{0}*;* - (2)
*the sequence*{*J*^{ n }*x*}*converges to a fixed point**y***of**J;* - (3)
*y***is the unique fixed point of**J in the set*$Y=\left\{y\in X|m\left({J}^{{n}_{0}}x,y\right)<\infty \right\}$*;* - (4)
$m\left(y,{y}^{*}\right)\le \frac{1}{1-\alpha}m\left(y,Jy\right)$

*for all**y*∈*Y*.

In 1996, Isac and Th.M. Rassias [39] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [40–46]).

in orthogonality spaces by using the fixed point method. In Section 3, we prove the Hyers-Ulam stability of the orthogonally additive-quadratic functional equation (1.1) in orthogonality spaces by using the direct method.

Throughout this paper, assume that (*X*, ⊥) is an orthogonality space and that (*Y*, ||.||_{
Y
}) is a real Banach space.

## 2. Hyers-Ulam Stability of the Orthogonally Additive-Quadratic Functional Equation (1.1): Fixed Point Method

*f*:

*X*→

*Y*, we define

for all *x*, *y* ∈ *X* with *x* ⊥ *y*, where ⊥ is the orthogonality in the sense of Rätz.

Let *f* : *X* → *Y* be an even mapping satisfying *f*(0) = 0 and (1.1). Then, *f* is a quadratic mapping, i.e., $2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right)=f\left(x\right)+f\left(y\right)$ holds.

Using the fixed point method and applying some ideas from [18, 21], we prove the Hyers-Ulam stability of the additive-quadratic functional equation *Df*(*x*, *y* = 0) in orthogonality spaces.

**Theorem 2.1**.

*Let φ*:

*X*

^{2}→ [0, ∞)

*be a function such that there exists an α*< 1

*with*

*for all x*,

*y*∈

*X*

*with*

*x*⊥

*Y. Let*

*f*:

*X*→

*Y be an even mapping satisfying*

*f*(0) = 0

*and*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Letting

*y*= 0 in (2.2), we get

*x*∈

*X*, since

*x*⊥ 0. Thus

for all *x* ∈ *X*.

*S*:

where, as usual, inf *ϕ* = +∞. It is easy to show that (*S*, *m*) is complete (see [[47], Lemma 2.1]).

*J*:

*S*→

*S*such that

for all *x* ∈ *X*.

*g*,

*h*∈

*S*be given such that

*m*(

*g*,

*h*) =

*ε*. Then,

*x*∈

*X*. Hence

*x*∈

*X*. So

*m*(

*g*,

*h*) =

*ε*implies that

*m*(

*Jg*,

*Jh*) ≤

*αε*. This means that

for all *g*, *h* ∈ *S*.

It follows from (2.5) that *m*(*f*, *Jf* ) ≤ *α*.

*Q*:

*X*→

*Y*satisfying the following:

- (1)
*Q*is a fixed point of*J*, i.e.,$Q\left(2x\right)=4Q\left(x\right)$(2.6)

*x*∈

*X*. The mapping

*Q*is a unique fixed point of

*J*in the set

*Q*is a unique mapping satisfying (2.6) such that there exists a

*μ*∈ (0, ∞) satisfying

*x*∈

*X*;

- (2)
*m*(*J*^{ n }*f*,*Q*) → 0 as*n*→*∞*. This implies the equality$\underset{n\to \infty}{lim}\frac{1}{{4}^{n}}f\left({2}^{n}x\right)=Q\left(x\right)$

*x*∈

*X*;

- (3)$m\left(f,Q\right)\le \frac{1}{1-\alpha}m\left(f,Jf\right)$, which implies the inequality$m\left(f,Q\right)\le \frac{\alpha}{1-\alpha}.$

This implies that the inequality (2.3) holds.

for all *x*, *y* ∈ *X* with *x* ⊥ *y*. So *DQ*(*x*, *y*) = 0 for all *x*, *y* ∈ *X* with *x* ⊥ *y*. Hence *Q* : *X* → *Y* is an orthogonally quadratic mapping, as desired. □

**Corollary 2.2**.

*Assume that*(

*X*, ⊥)

*is an orthogonality normed space. Let θ be a positive real number and p a real number with*0 <

*p*< 2.

*Let f*:

*X*→

*Y be an even mapping satisfying f*(0) = 0

*and*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y* and choosing *α* = 2^{p-2}in Theorem 2.1, we get the desired result. □

**Theorem 2.3**.

*Let φ*:

*X*

^{2}→ [0, ∞)

*be a function such that there exists an α*< 1

*with*

*for all*

*x*,

*y*∈

*X*

*with*

*x*⊥

*y. Let*

*f*:

*X*→

*Y be an even mapping satisfying*

*f*(0) = 0

*and*(2.2).

*Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Let (*S*, *m*) be the generalized metric space defined in the proof of Theorem 2.1.

*J*:

*S*→

*S*such that

for all *x* ∈ *X*.

It follows from (2.4) that *m*(*f*, *Jf*) ≤ 1.

The rest of the proof is similar to the proof of Theorem 2.1. □

**Corollary 2.4**.

*Assume that*(

*X*, ⊥)

*is an orthogonality normed space. Let θ be a positive real number and p a real number with p*> 2.

*Let f*:

*X*→

*Y be an even mapping satisfying f*(0) = 0

*and*(2.7).

*Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y* and choosing *α* = 2^{2-p}in Theorem 2.3, we get the desired result. □

Let *f* : *X* → *Y* be an odd mapping satisfying (1.1). Then, *f* is an additive mapping, i.e., $2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right)=2f\left(x\right)$ holds.

**Theorem 2.5**.

*Let φ*:

*X*

^{2}→ [0, ∞)

*be a function such that there exists an α*< 1

*with*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Let*

*f*:

*X*→

*Y be an odd mapping satisfying*(2.2).

*Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Letting

*y*= 0 in (2.2), we get

*x*∈

*X*, since

*x*⊥ 0. Thus,

for all *x* ∈ *X*.

Let (*S*, *m*) be the generalized metric space defined in the proof of Theorem 2.1.

*J*:

*S*→

*S*such that

for all *x* ∈ *X*.

It follows from (2.9) that $m\left(f,Jf\right)\le \frac{\alpha}{2}$.

The rest of the proof is similar to the proof of Theorem 2.1. □

**Corollary 2.6**.

*Assume that*(

*X*, ⊥)

*is an orthogonality normed space. Let θ be a positive real number and p a real number with*0 <

*p*< 1.

*Let f*:

*X*→

*Y be an odd mapping satisfying*(2.7).

*Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y* and choosing *α* = 2^{p-1}in Theorem 2.5, we get the desired result. □

**Theorem 2.7**.

*Let φ*:

*X*

^{2}→ [0, ∞)

*be a function such that there exists an α*< 1

*with*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Let*

*f*:

*X*→

*Y be an odd mapping satisfying*(2.2).

*Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Let (*S*, *m*) be the generalized metric space defined in the proof of Theorem 2.1.

*J*:

*S*→

*S*such that

for all *x* ∈ *X*.

It follows from (2.8) that $m\left(f,Jf\right)\le \frac{1}{2}$.

The rest of the proof is similar to the proof of Theorem 2.1. □

**Corollary 2.8**.

*Assume that*(

*X*, ⊥)

*is an orthogonality normed space. Let θ be a positive real number and p a real number with p*> 1.

*Let f*:

*X*→

*Y be an odd mapping satisfying*(2.7).

*Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y* and choosing *α* = 2^{1-p}in Theorem 2.7, we get the desired result. □

Let *f* : *X* → *Y* be a mapping satisfying *f*(0) = 0 and (1.1). Let ${f}_{e}\left(x\right):=\frac{f\left(x\right)+f\left(-x\right)}{2}$ and ${f}_{o}\left(x\right)=\frac{f\left(x\right)-f\left(-x\right)}{2}$. Then, *f*_{
e
} is an even mapping satisfying (1.1) and *f*_{
o
} is an odd mapping satisfying (1.1) such that *f*(*x*) = *f*_{
e
} (*x*) + *f*_{
o
} (*x*). So we obtain the following.

**Theorem 2.9**.

*Assume that*(

*X*, ⊥)

*is an orthogonality normed space. Let θ be a positive real number and p a positive real number with p*≠ 1.

*Let f*:

*X*→

*Y be a mapping satisfying f*(0) = 0

*and*(2.7).

*Then, there exist an orthogonally additive mapping A*:

*X*→

*Y and an orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

## 3. Hyers-Ulam Stability of the Orthogonally Additive-Quadratic Functional Equation (1.1): Direct Method

In this section, using the direct method and applying some ideas from [18, 21], we prove the Hyers-Ulam stability of the additive-quadratic functional equation *Df*(*x*, *y*) = 0 in orthogonality spaces.

**Theorem 3.1**.

*Let f*:

*X*→

*Y be an even mapping satisfying f*(0) = 0

*for which there exists a function φ*:

*X*

^{2}→ [0, ∞)

*satisfying*(2.2)

*and*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. It follows from (2.4) that

*m*and

*l*with

*m*>

*l*and all

*x*∈

*X*. It follows from (3.1) and (3.3) that the sequence $\left\{{4}^{n}f\left(\frac{x}{{2}^{n}}\right)\right\}$ is a Cauchy sequence for all

*x*∈

*X*. Since

*Y*is complete, the sequence $\left\{{4}^{n}f\left(\frac{x}{{2}^{n}}\right)\right\}$ converges. So one can define the mapping

*Q*:

*X*→

*Y*by

for all *x* ∈ *X*.

By the same reasoning as in the proof of Theorem 2.1, one can show that the mapping *Q* : *X* → *Y* is an orthogonally quadratic mapping satisfying (3.2).

*Q*′:

*X*→

*Y*be another orthogonally quadratic mapping satisfying (3.2). Then, we have

which tends to zero as *n* → ∞ for all *x* ∈ *X*. So we can conclude that *Q*(*x*) = *Q*′(*x*) for all *x* ∈ *X*. This proves the uniqueness of *Q*. □

**Corollary 3.2**.

*Assume that*(

*X*, ⊥)

*is an orthogonality space. Let θ be a positive real number and p a real number with p*> 2.

*Let f*:

*X*→

*Y be an even mapping satisfying f*(0) = 0

*and*(2.7).

*Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all* *x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y*, and applying Theorem 3.1, we get the desired result. □

Similarly, we can obtain the following. We will omit the proof.

**Theorem 3.3**.

*Let f*:

*X*→

*Y be an even mapping satisfying f*(0) = 0

*for which there exists a function φ*:

*X*

^{2}→ [0, ∞)

*satisfying*(2.2)

*and*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

**Corollary 3.4**.

*Assume that*(

*X*, ⊥)

*is an orthogonality space. Let θ be a positive real number and p a real number with*0 <

*p*< 2.

*Let f*:

*X*→

*Y be an even mapping satisfying f*(0) = 0

*and*(2.7).

*Then, there exists a unique orthogonally quadratic mapping Q*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y*, and applying Theorem 3.3, we get the desired result. □

**Theorem 3.5**.

*Let f*:

*X*→

*Y be an odd mapping for which there exists a function φ*:

*X*

^{2}→ [0, ∞)

*satisfying*(2.2)

*and*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. It follows from (2.8) that

for all *x* ∈ *X*.

The rest of the proof is similar to the proofs of Theorems 2.5 and 3.1. □

**Corollary 3.6**.

*Assume that*(

*X*, ⊥)

*is an orthogonality space. Let θ be a positive real number and p a real number with p*> 1.

*Let f*:

*X*→

*Y be an odd mapping satisfying*(2.7).

*Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y*, and applying Theorem 3.5, we get the desired result. □

Similarly, we can obtain the following. We will omit the proof.

**Theorem 3.7**.

*Let f*:

*X*→

*Y be an odd mapping for which there exists a function φ*:

*X*

^{2}→ [0, ∞)

*satisfying*(2.2)

*and*

*for all x*,

*y*∈

*X with*

*x*⊥

*y. Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all* *x* ∈ *X*.

**Corollary 3.8**.

*Assume that*(

*X*, ⊥)

*is an orthogonality space. Let θ be a positive real number and p a real number with*0 <

*p*< 1.

*Let f*:

*X*→

*Y be an odd mapping satisfying*(2.7).

*Then, there exists a unique orthogonally additive mapping A*:

*X*→

*Y such that*

*for all x* ∈ *X*.

*Proof*. Taking *φ*(*x*, *y*) = *θ*(||*x*|| ^{
p
} + ||*y*|| ^{
p
} ) for all *x*, *y* ∈ *X* with *x* ⊥ *y*, and applying Theorem 3.7, we get the desired result. □

## Declarations

## Authors’ Affiliations

## References

- Pinsker AG:
**Sur une fonctionnelle dans l'espace de Hilbert.**In*Acad Sci URSS n Ser*Edited by: Dokl CR. 1938,**20:**411–414.Google Scholar - Sundaresan K:
**Orthogonality and nonlinear functionals on Banach spaces.***Proc Amer Math Soc*1972,**34:**187–190. 10.1090/S0002-9939-1972-0291835-XMathSciNetView ArticleGoogle Scholar - Gudder S, Strawther D:
**Orthogonally additive and orthogonally increasing functions on vector spaces.***Pac J Math*1975,**58:**427–436.MathSciNetView ArticleGoogle Scholar - Rätz J:
**On orthogonally additive mappings.***Aequationes Math*1985,**28:**35–49. 10.1007/BF02189390MathSciNetView ArticleGoogle Scholar - Rätz J, Szabó Gy:
**On orthogonally additive mappings IV.***Aequationes Math*1989,**38:**73–85. 10.1007/BF01839496MathSciNetView ArticleGoogle Scholar - Alonso J, Benítez C:
**Orthogonality in normed linear spaces: a survey I. Main properties.***Extr Math*1988,**3:**1–15.Google Scholar - Alonso J, Benítez C:
**Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities.***Extr Math*1989,**4:**121–131.Google Scholar - Birkhoff G:
**Orthogonality in linear metric spaces.***Duke Math J*1935,**1:**169–172. 10.1215/S0012-7094-35-00115-6MathSciNetView ArticleGoogle Scholar - Carlsson SO:
**Orthogonality in normed linear spaces.***Ark Mat*1962,**4:**297–318. 10.1007/BF02591506MathSciNetView ArticleGoogle Scholar - Diminnie CR:
**A new orthogonality relation for normed linear spaces.***Math Nachr*1983,**114:**197–203. 10.1002/mana.19831140115MathSciNetView ArticleGoogle Scholar - James RC:
**Orthogonality in normed linear spaces.***Duke Math J*1945,**12:**291–302. 10.1215/S0012-7094-45-01223-3MathSciNetView ArticleGoogle Scholar - James RC:
**Orthogonality and linear functionals in normed linear spaces.***Trans Amer Math Soc*1947,**61:**265–292. 10.1090/S0002-9947-1947-0021241-4MathSciNetView ArticleGoogle Scholar - Ulam SM:
*Problems in Modern Mathematics.*Wiley, New York; 1960.Google Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proc Natl Acad Sci USA*1941,**27:**222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proc Amer Math Soc*1978,**72:**297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleGoogle Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables.*World Scientific Publishing Company, New Jersey; 2002.Google Scholar - Czerwik S:
*Stability of Functional Equations of Ulam-Hyers-Rassias Type.*Hadronic Press, Palm Harbor; 2003.Google Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables.*Birkhäuser, Basel; 1998.View ArticleGoogle Scholar - Jung S:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis.*Hadronic Press, Palm Harbor; 2001.Google Scholar - Rassias ThM, ed: Functional Equations, Inequalities and Applications. Kluwer, Dordrecht; 2003.Google Scholar
- Ger R, Sikorska J:
**Stability of the orthogonal additivity.***Bull Polish Acad Sci Math*1995,**43:**143–151.MathSciNetGoogle Scholar - Skof F:
**Proprietà locali e approssimazione di operatori.***Rend Sem Mat Fis Milano*1983,**53:**113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar - Cholewa PW:
**Remarks on the stability of functional equations.***Aequationes Math*1984,**27:**76–86. 10.1007/BF02192660MathSciNetView ArticleGoogle Scholar - Czerwik S:
**On the stability of the quadratic mapping in normed spaces.***Abh Math Sem Univ Hamburg*1992,**62:**59–64. 10.1007/BF02941618MathSciNetView ArticleGoogle Scholar - Park C, Park J:
**Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping.***J Differ Equat Appl*2006,**12:**1277–1288. 10.1080/10236190600986925View ArticleGoogle Scholar - Rassias ThM:
*On the stability of the quadratic functional equation and its applications.**Volume 43*. Studia Univ Babeş-Bolyai Math; 1998:89–124.Google Scholar - Rassias ThM:
**The problem of S.M. Ulam for approximately multiplicative mappings.***J Math Anal Appl*2000,**246:**352–378. 10.1006/jmaa.2000.6788MathSciNetView ArticleGoogle Scholar - Rassias ThM:
**On the stability of functional equations in Banach spaces.***J Math Anal Appl*2000,**251:**264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleGoogle Scholar - Vajzović F:
**Über das Funktional**H**mit der Eigenschaft: (**x**,**y**) = 0 ⇒**H**(**x**+**y**) +**H**(**x**-**y**) = 2**H**(**x**) + 2**H**(**y**).***Glasnik Mat Ser III*1967,**2**(22):73–81.MathSciNetGoogle Scholar - Drljević F:
**On a functional which is quadratic on**A**-orthogonal vectors.***Publ Inst Math (Beograd)*1986,**54:**63–71.Google Scholar - Fochi M:
**Functional equations in**A**-orthogonal vectors.***Aequationes Math*1989,**38:**28–40. 10.1007/BF01839491MathSciNetView ArticleGoogle Scholar - Moslehian MS:
**On the orthogonal stability of the Pexiderized quadratic equation.***J Differ Equat Appl*2005,**11:**999–1004. 10.1080/10236190500273226MathSciNetView ArticleGoogle Scholar - Moslehian MS:
**On the stability of the orthogonal Pexiderized Cauchy equation.***J Math Anal Appl*2006,**318:**211–223. 10.1016/j.jmaa.2005.05.052MathSciNetView ArticleGoogle Scholar - Szabó Gy:
**Sesquilinear-orthogonally quadratic mappings.***Aequationes Math*1990,**40:**190–200. 10.1007/BF02112295MathSciNetView ArticleGoogle Scholar - Moslehian MS, Rassias ThM:
**Orthogonal stability of additive type equations.***Aequationes Math*2007,**73:**249–259. 10.1007/s00010-006-2868-0MathSciNetView ArticleGoogle Scholar - Paganoni L, Rätz J:
**Conditional function equations and orthogonal additivity.***Aequationes Math*1995,**50:**135–142. 10.1007/BF01831116MathSciNetView ArticleGoogle Scholar - Cădariu L, Radu V:
**Fixed points and the stability of Jensen's functional equation.***J Inequal Pure Appl Math*2003,**4**(1):Art. ID 4.Google Scholar - Diaz J, Margolis B:
**A fixed point theorem of the alternative for contractions on a generalized complete metric space.***Bull Amer Math Soc*1968,**74:**305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleGoogle Scholar - Isac G, Rassias ThM:
**Stability of**ψ**-additive mappings: applications to nonlinear analysis.***Intern J Math Math Sci*1996,**19:**219–228. 10.1155/S0161171296000324MathSciNetView ArticleGoogle Scholar - Cădariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.***Grazer Math Ber*2004,**346:**43–52.Google Scholar - Cădariu L, Radu V:
**Fixed point methods for the generalized stability of functional equations in a single variable.***Fixed Point Theory Appl*2008,**2008:**Art ID 749392.Google Scholar - Jung Y, Chang I:
**The stability of a cubic type functional equation with the fixed point alternative.***J Math Anal Appl*2005,**306:**752–760. 10.1016/j.jmaa.2004.10.017MathSciNetView ArticleGoogle Scholar - Mirzavaziri M, Moslehian MS:
**A fixed point approach to stability of a quadratic equation.***Bull Braz Math Soc*2006,**37:**361–376. 10.1007/s00574-006-0016-zMathSciNetView ArticleGoogle Scholar - Park C:
**Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras.***Fixed Point Theory Appl*2007,**2007:**Art ID 50175.View ArticleGoogle Scholar - Park C:
**Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach.***Fixed Point Theory Appl*2008,**2008:**Art ID 493751.View ArticleGoogle Scholar - Radu V:
**The fixed point alternative and the stability of functional equations.***Fixed Point Theory*2003,**4:**91–96.MathSciNetGoogle Scholar - Miheţ D, Radu V:
**On the stability of the additive Cauchy functional equation in random normed spaces.***J Math Anal Appl*2008,**343:**567–572.MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.