## Fixed Point Theory and Applications

Open Access

Fixed Point Theory and Applications20112011:66

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

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.

## 1. Introduction and Preliminaries

Assume that X is a real inner product space and f : X 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.

Pinsker [1] characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan [2] generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonally Cauchy functional equation
$f\left(x+y\right)=f\left(x\right)+f\left(y\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}x\perp y,$

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:

(O1) totality of for zero: x 0, 0 x for all x X;

(O2) independence: if x, y X - {0}, x y, then x, y are linearly independent;

(O3) homogeneity: if x, y X, x y, then αx βy for all α, β ;

(O4) the Thalesian property: if P is a 2-dimensional subspace of X, x P and λ +, which is the set of nonnegative real numbers, then there exists y0 P such that x y0 and x + y0 λx - y0.

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

Some interesting examples are
1. (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.

2. (ii)

The ordinary orthogonality on an inner product space (X, 〈., .〉) given by x y if and only if 〈x, y〉 = 0.

3. (iii)

The Birkhoff-James orthogonality on a normed space (X, ||·||) defined by x y if and only if ||x + λy|| ≥ ||x|| for all λ .

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 [612]).

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 [1620] 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 : XY such that $||f\left(x\right)-g\left(x\right)\parallel \le \frac{16}{3}\epsilon$ 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) - 2f(x) - 2f(y)|| ≤ ε for some ε > 0, then there is a unique quadratic mapping g : XY such that $||f\left(x\right)-g\left(x\right)\parallel \le \frac{\epsilon }{2}$. 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 [2528]).

$f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right),x\perp y$

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.

Let X be a set. A function m : X × X → [0, ∞] is called a generalized metric on X if m satisfies
1. (1)

m(x, y) = 0 if and only if x = y;

2. (2)

m(x, y) = m(y, x) for all x, y X;

3. (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 : XX be a strictly contractive mapping with Lipschitz constant α < 1. Then, for each given element x X, either
$m\left({J}^{n}x,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{J}^{n+1}x\right)=\infty$
for all nonnegative integers n or there exists a positive integer n 0 such that
1. (1)

m(J n x, j n+1 x) < ∞, nn 0 ;

2. (2)

the sequence {J n x} converges to a fixed point y* of J;

3. (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. (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 [4046]).

This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the following orthogonally additive-quadratic functional equation
$2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right)=\frac{3f\left(x\right)}{2}-\frac{f\left(-x\right)}{2}+\frac{f\left(y\right)}{2}+\frac{f\left(-y\right)}{2}$
(1.1)

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

For a given mapping f : XY, we define
$\begin{array}{ll}\hfill Df\left(x,y\right):\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right)\phantom{\rule{2em}{0ex}}\\ -\phantom{\rule{1em}{0ex}}\frac{3f\left(x\right)}{2}+\frac{f\left(-x\right)}{2}-\frac{f\left(y\right)}{2}-\frac{f\left(-y\right)}{2}\phantom{\rule{2em}{0ex}}\end{array}$

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

Let f : XY 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 : X2 → [0, ∞) be a function such that there exists an α < 1 with
$\phi \left(x,y\right)\le 4\alpha \phi \left(\frac{x}{2},\frac{y}{2}\right)$
(2.1)
for all x, y X with x Y. Let f : XY be an even mapping satisfying f(0) = 0 and
$\parallel Df\left(x,y\right){\parallel }_{Y}\le \phi \left(x,y\right)$
(2.2)
for all x, y X with x y. Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$||f\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \frac{\alpha }{1-\alpha }\phi \left(x,0\right)$
(2.3)

for all x X.

Proof. Letting y = 0 in (2.2), we get
${∥4f\left(\frac{x}{2}\right)-f\left(x\right)∥}_{Y}\le \phi \left(x,0\right)$
(2.4)
for all x X, since x 0. Thus
${∥f\left(x\right)-\frac{1}{4}f\left(2x\right)∥}_{Y}\le \frac{1}{4}\phi \left(2x,0\right)\le \frac{4\alpha }{4}\phi \left(x,0\right)$
(2.5)

for all x X.

Consider the set
$S:=\left\{h:X\to Y\right\}$
and introduce the generalized metric on S:
$m\left(g,h\right)=inf\left\{\mu \in {ℝ}_{+}:\parallel g\left(x\right)-h\left(x\right){\parallel }_{Y}\le \mu \phi \left(x,0\right),\phantom{\rule{2.77695pt}{0ex}}\forall x\in X\right\},$

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

Now we consider the linear mapping J : SS such that
$Jg\left(x\right):=\frac{1}{4}g\left(2x\right)$

for all x X.

Let g, h S be given such that m(g, h) = ε. Then,
$\parallel g\left(x\right)-h\left(x\right){\parallel }_{Y}\le \phi \left(x,0\right)$
for all x X. Hence
$\parallel Jg\left(x\right)-Jh\left(x\right){\parallel }_{Y}={∥\frac{1}{4}g\left(2x\right)-\frac{1}{4}h\left(2x\right)∥}_{Y}\le \alpha \phi \left(x,0\right)$
for all x X. So m(g, h) = ε implies that m(Jg, Jh) ≤ αε. This means that
$m\left(Jg,Jh\right)\le \alpha m\left(g,h\right)$

for all g, h S.

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

By Theorem 1.1, there exists a mapping Q : XY satisfying the following:
1. (1)
Q is a fixed point of J, i.e.,
$Q\left(2x\right)=4Q\left(x\right)$
(2.6)

for all x X. The mapping Q is a unique fixed point of J in the set
$M=\left\{g\in S:m\left(h,g\right)<\infty \right\}.$
This implies that Q is a unique mapping satisfying (2.6) such that there exists a μ (0, ∞) satisfying
$||f\left(x\right)-Q\left(x\right){\parallel }_{Y}\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}\mu \phi \left(x,0\right)$
for all x X;
1. (2)
m(J n f, Q) → 0 as n. This implies the equality
$\underset{n\to \infty }{lim}\frac{1}{{4}^{n}}f\left({2}^{n}x\right)=Q\left(x\right)$

for all x X;
1. (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.

It follows from (2.1) and (2.2) that
$\begin{array}{ll}\hfill {∥DQ\left(x,y\right)∥}_{Y}\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}\underset{n\to \infty }{lim}\frac{1}{{4}^{n}}\parallel Df\left({2}^{n}x,{2}^{n}y\right){\parallel }_{Y}\phantom{\rule{2em}{0ex}}\\ \le \phantom{\rule{1em}{0ex}}\underset{n\to \infty }{lim}\frac{1}{{4}^{n}}\phi \left({2}^{n}x,{2}^{n}y\right)\le \underset{n\to \infty }{lim}\frac{{4}^{n}{\alpha }^{n}}{{4}^{n}}\phi \left(x,y\right)=0\phantom{\rule{2em}{0ex}}\end{array}$

for all x, y X with x y. So DQ(x, y) = 0 for all x, y X with x y. Hence Q : XY 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 : XY be an even mapping satisfying f(0) = 0 and
$\parallel Df\left(x,y\right){\parallel }_{Y}\le \theta \left(\parallel x{\parallel }^{p}+\parallel y{\parallel }^{p}\right)$
(2.7)
for all x, y X with x y. Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$\parallel f\left(x\right)-Q\left(x\right){\parallel }_{Y}\phantom{\rule{2.77695pt}{0ex}}\le \frac{{2}^{p}\theta }{4-{2}^{p}}||x|{|}^{p}$

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 = 2p-2in Theorem 2.1, we get the desired result.   □

Theorem 2.3. Let φ : X2 → [0, ∞) be a function such that there exists an α < 1 with
$\phi \left(x,y\right)\le \frac{\alpha }{4}\phi \left(2x,2y\right)$
for all x, y X with x y. Let f : XY be an even mapping satisfying f(0) = 0 and (2.2). Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$\parallel f\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \frac{1}{1-\alpha }\phi \left(x,0\right)$

for all x X.

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

Now we consider the linear mapping J : SS such that
$Jg\left(x\right):=4g\left(\frac{x}{2}\right)$

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 : XY be an even mapping satisfying f(0) = 0 and (2.7). Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$\parallel f\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \frac{{2}^{p}\theta }{{2}^{p}-4}||x|{|}^{p}$

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 = 22-pin Theorem 2.3, we get the desired result.   □

Let f : XY 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 : X2 → [0, ∞) be a function such that there exists an α < 1 with
$\phi \left(x,y\right)\le 2\alpha \phi \left(\frac{x}{2},\frac{y}{2}\right)$
for all x, y X with x y. Let f : XY be an odd mapping satisfying (2.2). Then, there exists a unique orthogonally additive mapping A : XY such that
$\parallel f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{\alpha }{2-2\alpha }\phi \left(x,0\right)$

for all x X.

Proof. Letting y = 0 in (2.2), we get
${∥4f\left(\frac{x}{2}\right)-2f\left(x\right)∥}_{Y}\le \phi \left(x,0\right)$
(2.8)
for all x X, since x 0. Thus,
${∥f\left(x\right)-\frac{1}{2}f\left(2x\right)∥}_{Y}\le \frac{1}{4}\phi \left(2x,0\right)\le \frac{2\alpha }{4}\phi \left(x,0\right)$
(2.9)

for all x X.

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

Now we consider the linear mapping J : SS such that
$Jg\left(x\right):=\frac{1}{2}g\left(2x\right)$

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 : XY be an odd mapping satisfying (2.7). Then, there exists a unique orthogonally additive mapping A : XY such that
$\parallel f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{{2}^{p}\theta }{2\left(2-{2}^{p}\right)}||x|{|}^{p}$

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 = 2p-1in Theorem 2.5, we get the desired result.   □

Theorem 2.7. Let φ : X2 → [0, ∞) be a function such that there exists an α < 1 with
$\phi \left(x,y\right)\le \frac{\alpha }{2}\phi \left(2x,2y\right)$
for all x, y X with x y. Let f : XY be an odd mapping satisfying (2.2). Then, there exists a unique orthogonally additive mapping A : XY such that
$\parallel f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{1}{2-2\alpha }\phi \left(x,0\right)$

for all x X.

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

Now we consider the linear mapping J : SS such that
$Jg\left(x\right):=2g\left(\frac{x}{2}\right)$

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 : XY be an odd mapping satisfying (2.7). Then, there exists a unique orthogonally additive mapping A : XY such that
$\parallel f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{{2}^{p}\theta }{{2\left(2}^{p}-2\right)}||x{||}^{p}$

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 = 21-pin Theorem 2.7, we get the desired result.   □

Let f : XY 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 : XY be a mapping satisfying f(0) = 0 and (2.7). Then, there exist an orthogonally additive mapping A : XY and an orthogonally quadratic mapping Q : XY such that
$\parallel f\left(x\right)-A\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \left(\frac{{2}^{p}}{2|2-{2}^{p}|}+\frac{{2}^{p}}{|4-{2}^{p}|}\right)\theta ||x|{|}^{p}$

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 : XY be an even mapping satisfying f(0) = 0 for which there exists a function φ : X2 → [0, ∞) satisfying (2.2) and
$\stackrel{̃}{\phi }\left(x,y\right):=\sum _{j=0}^{\infty }{4}^{j}\phi \left(\frac{x}{{2}^{j}},\frac{y}{{2}^{j}}\right)<\infty$
(3.1)
for all x, y X with x y. Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$\parallel f\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \stackrel{̃}{\phi }\left(x,0\right)$
(3.2)

for all x X.

Proof. It follows from (2.4) that
${∥{4}^{l}f\left(\frac{x}{{2}^{l}}\right)-{4}^{m}f\left(\frac{x}{{2}^{m}}\right)∥}_{Y}\le \sum _{j=1}^{m-1}{4}^{j}\phi \left(\frac{x}{{2}^{j}},0\right)$
(3.3)
for all nonnegative integers 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 : XY by
$Q\left(x\right):=\underset{n\to \infty }{lim}{4}^{n}f\left(\frac{x}{{2}^{n}}\right)$

for all x X.

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

Now, let Q′: XY be another orthogonally quadratic mapping satisfying (3.2). Then, we have
$\begin{array}{ll}\hfill \parallel Q\left(x\right)-& {Q}^{\prime }\left(x\right){\parallel }_{Y}={4}^{n}{∥Q\left(\frac{x}{{2}^{n}}\right)-{Q}^{\prime }\left(\frac{x}{{2}^{n}}\right)∥}_{Y}\phantom{\rule{2em}{0ex}}\\ \le {4}^{n}\left({∥Q\left(\frac{x}{{2}^{n}}\right)-f\left(\frac{x}{{2}^{n}}\right)∥}_{Y}+{∥{Q}^{\prime }\left(\frac{x}{{2}^{n}}\right)-f\left(\frac{x}{{2}^{n}}\right)∥}_{Y}\right)\phantom{\rule{2em}{0ex}}\\ \le 2\cdot {4}^{n}\stackrel{̃}{\phi }\left(\frac{x}{{2}^{n}},0\right),\phantom{\rule{2em}{0ex}}\end{array}$

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 : XY be an even mapping satisfying f(0) = 0 and (2.7). Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$\parallel f\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \frac{{2}^{p}\theta }{{2}^{p}-4}||x|{|}^{p}$

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 : XY be an even mapping satisfying f(0) = 0 for which there exists a function φ : X2 → [0, ∞) satisfying (2.2) and
$\stackrel{˜}{\phi }\left(x,y\right):=\sum _{j=1}^{\infty }\frac{1}{{4}^{j}}\phi {\left(2}^{j}x{,2}^{j}y\right)<\infty$
for all x, y X with x y. Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$\parallel f\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \stackrel{̃}{\phi }\left(x,0\right)$

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 : XY be an even mapping satisfying f(0) = 0 and (2.7). Then, there exists a unique orthogonally quadratic mapping Q : XY such that
$||f\left(x\right)-Q\left(x\right){\parallel }_{Y}\le \frac{{2}^{p}\theta }{4-{2}^{p}}||x|{|}^{p}$

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 : XY be an odd mapping for which there exists a function φ : X2 → [0, ∞) satisfying (2.2) and
$\stackrel{̃}{\phi }\left(x,y\right):=\sum _{j=0}^{\infty }{2}^{j}\phi \left(\frac{x}{{2}^{j}},\frac{y}{{2}^{j}}\right)<\infty$
(3.4)
for all x, y X with x y. Then, there exists a unique orthogonally additive mapping A : XY such that
$||f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{1}{2}\stackrel{̃}{\phi }\left(x,0\right)$
(3.5)

for all x X.

Proof. It follows from (2.8) that
${∥f\left(x\right)-2f\left(\frac{x}{2}\right)∥}_{Y}\le \frac{1}{2}\phi \left(x,0\right)$

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 : XY be an odd mapping satisfying (2.7). Then, there exists a unique orthogonally additive mapping A : XY such that
$\parallel f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{{2}^{p}\theta }{{2\left(2}^{p}-2\right)}||x{||}^{p}$

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 : XY be an odd mapping for which there exists a function φ : X2 → [0, ∞) satisfying (2.2) and
$\stackrel{˜}{\phi }\left(x,y\right):\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\sum _{j=1}^{\infty }\frac{1}{{2}^{j}}\phi {\left(2}^{j}x{,2}^{j}y\right)<\infty$
for all x, y X with x y. Then, there exists a unique orthogonally additive mapping A : XY such that
$\parallel f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{1}{2}\stackrel{̃}{\phi }\left(x,0\right)$

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 : XY be an odd mapping satisfying (2.7). Then, there exists a unique orthogonally additive mapping A : XY such that
$\parallel f\left(x\right)-A\left(x\right){\parallel }_{Y}\le \frac{{2}^{p}\theta }{2\left(2-{2}^{p}\right)}||x|{|}^{p}$

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.   □

## Authors’ Affiliations

(1)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, Republic of Korea

## References

1. 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
2. Sundaresan K: Orthogonality and nonlinear functionals on Banach spaces. Proc Amer Math Soc 1972, 34: 187–190. 10.1090/S0002-9939-1972-0291835-X
3. Gudder S, Strawther D: Orthogonally additive and orthogonally increasing functions on vector spaces. Pac J Math 1975, 58: 427–436.
4. Rätz J: On orthogonally additive mappings. Aequationes Math 1985, 28: 35–49. 10.1007/BF02189390
5. Rätz J, Szabó Gy: On orthogonally additive mappings IV. Aequationes Math 1989, 38: 73–85. 10.1007/BF01839496
6. Alonso J, Benítez C: Orthogonality in normed linear spaces: a survey I. Main properties. Extr Math 1988, 3: 1–15.Google Scholar
7. Alonso J, Benítez C: Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities. Extr Math 1989, 4: 121–131.Google Scholar
8. Birkhoff G: Orthogonality in linear metric spaces. Duke Math J 1935, 1: 169–172. 10.1215/S0012-7094-35-00115-6
9. Carlsson SO: Orthogonality in normed linear spaces. Ark Mat 1962, 4: 297–318. 10.1007/BF02591506
10. Diminnie CR: A new orthogonality relation for normed linear spaces. Math Nachr 1983, 114: 197–203. 10.1002/mana.19831140115
11. James RC: Orthogonality in normed linear spaces. Duke Math J 1945, 12: 291–302. 10.1215/S0012-7094-45-01223-3
12. James RC: Orthogonality and linear functionals in normed linear spaces. Trans Amer Math Soc 1947, 61: 265–292. 10.1090/S0002-9947-1947-0021241-4
13. Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1960.Google Scholar
14. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222
15. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1
16. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey; 2002.Google Scholar
17. Czerwik S: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor; 2003.Google Scholar
18. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.
19. Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.Google Scholar
20. Rassias ThM, ed: Functional Equations, Inequalities and Applications. Kluwer, Dordrecht; 2003.Google Scholar
21. Ger R, Sikorska J: Stability of the orthogonal additivity. Bull Polish Acad Sci Math 1995, 43: 143–151.
22. Skof F: Proprietà locali e approssimazione di operatori. Rend Sem Mat Fis Milano 1983, 53: 113–129. 10.1007/BF02924890
23. Cholewa PW: Remarks on the stability of functional equations. Aequationes Math 1984, 27: 76–86. 10.1007/BF02192660
24. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh Math Sem Univ Hamburg 1992, 62: 59–64. 10.1007/BF02941618
25. Park C, Park J: Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping. J Differ Equat Appl 2006, 12: 1277–1288. 10.1080/10236190600986925
26. 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
27. Rassias ThM: The problem of S.M. Ulam for approximately multiplicative mappings. J Math Anal Appl 2000, 246: 352–378. 10.1006/jmaa.2000.6788
28. Rassias ThM: On the stability of functional equations in Banach spaces. J Math Anal Appl 2000, 251: 264–284. 10.1006/jmaa.2000.7046
29. 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.
30. Drljević F: On a functional which is quadratic on A -orthogonal vectors. Publ Inst Math (Beograd) 1986, 54: 63–71.Google Scholar
31. Fochi M: Functional equations in A -orthogonal vectors. Aequationes Math 1989, 38: 28–40. 10.1007/BF01839491
32. Moslehian MS: On the orthogonal stability of the Pexiderized quadratic equation. J Differ Equat Appl 2005, 11: 999–1004. 10.1080/10236190500273226
33. Moslehian MS: On the stability of the orthogonal Pexiderized Cauchy equation. J Math Anal Appl 2006, 318: 211–223. 10.1016/j.jmaa.2005.05.052
34. Szabó Gy: Sesquilinear-orthogonally quadratic mappings. Aequationes Math 1990, 40: 190–200. 10.1007/BF02112295
35. Moslehian MS, Rassias ThM: Orthogonal stability of additive type equations. Aequationes Math 2007, 73: 249–259. 10.1007/s00010-006-2868-0
36. Paganoni L, Rätz J: Conditional function equations and orthogonal additivity. Aequationes Math 1995, 50: 135–142. 10.1007/BF01831116
37. 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
38. 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/S0002-9904-1968-11933-0
39. Isac G, Rassias ThM: Stability of ψ -additive mappings: applications to nonlinear analysis. Intern J Math Math Sci 1996, 19: 219–228. 10.1155/S0161171296000324
40. 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
41. 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
42. 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.2004.10.017
43. 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-z
44. 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.
45. Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl 2008, 2008: Art ID 493751.
46. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.
47. 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.