# Orthogonal stability of functional equations with the fixed point alternative

- Choonkil Park
^{1}, - Yeol Je Cho
^{2}and - Jung Rye Lee
^{3}Email author

**2012**:173

https://doi.org/10.1186/1687-1847-2012-173

© Park et al.; licensee Springer 2012

**Received: **15 March 2012

**Accepted: **10 September 2012

**Published: **4 October 2012

## Abstract

In this paper, we investigate the orthogonal stability of functional equations in orthogonality modules over a unital Banach algebra. Using a fixed point method, we prove the Hyers-Ulam stability of the orthogonally Jensen additive functional equation

the orthogonally Jensen quadratic functional equation

the orthogonally cubic functional equation

and the orthogonally quartic functional equation

for all *x*, *y* with $x\perp y$, where ⊥ is the orthogonality in the sense of Rätz.

**MSC:**39B55, 47H10, 39B52, 46H25.

### Keywords

Hyers-Ulam stability orthogonally (Jensen additive, Jensen quadratic, cubic, quartic) functional equation fixed point orthogonality module over Banach algebra orthogonality space## 1 Introduction and preliminaries

Assume that *X* is a real inner product space and $f:X\to \mathbb{R}$ is a solution of the orthogonal Cauchy functional equation $f(x+y)=f(x)+f(y)$, where $\u3008x,y\u3009=0$. By the Pythagorean theorem, $f(x)={\parallel x\parallel}^{2}$ is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus the orthogonal 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 the conditional Cauchy functional equation. In 1985, Rätz [4] introduced a new definition of orthogonality by using more restrictive axioms than 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 [4].

Suppose that *X* is a real vector space (algebraic module) with $dimX\ge 2$, and ⊥ is a binary relation on *X* with the following properties:

(${O}_{1}$) *Totality* of ⊥ for zero: $x\perp 0$ and $0\perp x$ for all $x\in X$;

(${O}_{2}$) *Independence*: if $x,y\in X-\{0\}$ and $x\perp y$, then *x* and *y* are linearly independent;

(${O}_{3}$) *Homogeneity*: if $x,y\in X$ and $x\perp y$, then $\alpha x\perp \beta y$ for all $\alpha ,\beta \in \mathbb{R}$;

(${O}_{4}$) *Thalesian property*: if *P* is a 2-dimensional subspace of *X*, $x\in P$ and $\lambda \in {\mathbb{R}}_{+}$, which is the set of nonnegative real numbers, then there exists ${y}_{0}\in P$ such that $x\perp {y}_{0}$ and $x+{y}_{0}\perp \lambda x-{y}_{0}$.

The pair $(X,\perp )$ is called an *orthogonality space* (resp., *module*). By an *orthogonality normed space* (*normed module*) we mean an orthogonality space (resp., module) having a normed (resp., normed module) structure.

- (1)
The trivial orthogonality on a vector space

*X*defined by (${O}_{1}$) and, for any non-zero elements $x,y\in X$, $x\perp y$ if and only if*x*,*y*are linearly independent. - (2)
The ordinary orthogonality on an inner product space $(X,\u3008\cdot ,\cdot \u3009)$ given by $x\perp y$ if and only if $\u3008x,y\u3009=0$.

- (3)
The Birkhoff-James orthogonality on a normed space $(X,\parallel \cdot \parallel )$ defined by $x\perp y$ if and only if $\parallel x+\lambda y\parallel \ge \parallel x\parallel $ for all $\lambda \in \mathbb{R}$.

The relation ⊥ is called *symmetric* if $x\perp y$ implies that $y\perp x$ for all $x,y\in X$. Clearly, Examples (1) and (2) are symmetric, but Example (3) is not. It is remarkable to note, however, that a real normed space of a 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]).

*Under what condition is there 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, Rassias [15] extended the theorem of Hyers by considering the unbounded Cauchy difference

During the last decades, several stability problems of functional equations have been investigated in the spirit of Hyers-Ulam-Rassias. The readers refer to [16–20] and references therein for detailed information on the stability of functional equations.

*f*is a mapping from an orthogonality space

*X*into a real Banach space

*Y*and

for all $x,y\in X$ with $x\perp y$ and for some $\epsilon >0$, then there exists exactly one orthogonally additive mapping $g:X\to Y$ such that $\parallel f(x)-g(x)\parallel \le \frac{16}{3}\epsilon $ for all $x\in X$.

*f*is a mapping from a normed space

*X*into a Banach space

*Y*satisfying

for some $\epsilon >0$, then there is a unique quadratic mapping $g:X\to Y$ such that $\parallel f(x)-g(x)\parallel \le \frac{\epsilon}{2}$. Cholewa [23] extended Skof’s theorem by replacing *X* by an abelian group *G*. 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 [25–31]).

was first investigated by Vajzović [32] when *X* is a Hilbert space, *Y* is the scalar field, *f* is continuous and ⊥ means the Hilbert space orthogonality. Later, Drljević [33], Fochi [34], Moslehian [35, 36] and Szabó [37] generalized this result (see also [38–40]).

It is easy to show that the function $f(x)={x}^{3}$ satisfies the functional equation (1.1), which is called a *cubic functional equation*, and every solution of the cubic functional equation is said to be a *cubic mapping*.

*X*be an orthogonality space and

*Y*be a real Banach space. A mapping $f:X\to Y$ is called

*orthogonally cubic*if it satisfies the orthogonally cubic functional equation

for all *x*, *y* with $x\perp y$.

*et al.*considered the following quartic functional equation:

It is easy to show that the function $f(x)={x}^{4}$ satisfies the functional equation (1.3), which is called a *quartic functional equation*, and every solution of the quartic functional equation is said to be a *quartic mapping* (for the stability of the ACQ and quartic functional equations, see [26, 31] and others).

*X*be an orthogonality space and

*Y*be a Banach space. A mapping $f:X\to Y$ is called

*orthogonally quartic*if it satisfies the orthogonally quartic functional equation

for all *x*, *y* with $x\perp y$.

*X*be a set. A function $d:X\times X\to [0,\mathrm{\infty}]$ is called a

*generalized metric*on

*X*if

*d*satisfies the following conditions:

- (1)
$d(x,y)=0$ if and only if $x=y$;

- (2)
$d(x,y)=d(y,x)$ for all $x,y\in X$;

- (3)
$d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$.

We recall a fundamental result in a fixed point theory.

*Let*$(X,d)$

*be a complete generalized metric space and*$J:X\to X$

*be a strictly contractive mapping with the Lipschitz constant*$\alpha <1$.

*Then*,

*for each given element*$x\in X$,

*either*

*for all nonnegative integers*

*n*

*or there exists a positive integer*${n}_{0}$

*such that*

- (1)
$d({J}^{n}x,{J}^{n+1}x)<\mathrm{\infty}$

*for all*$n\ge {n}_{0}$; - (2)
*the sequence*$\{{J}^{n}x\}$*converges to a fixed point*${y}^{\ast}$*of**J*; - (3)
${y}^{\ast}$

*is the unique fixed point of**J**in the set*$Y=\{y\in X\mid d({J}^{{n}_{0}}x,y)<\mathrm{\infty}\}$; - (4)
$d(y,{y}^{\ast})\le \frac{1}{1-\alpha}d(y,Jy)$

*for all*$y\in Y$.

In 1996, Isac and Rassias [45] were the first to provide applications of the 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 [46–55]).

This paper is organized as follows. In Section 2, we prove the Hyers-Ulam stability of the orthogonally Jensen additive functional equation in orthogonality modules over a unital Banach algebra. In Section 3, we prove the Hyers-Ulam stability of the orthogonally Jensen quadratic functional equation in orthogonality modules over a unital Banach algebra. In Section 4, we prove the Hyers-Ulam stability of the orthogonally cubic functional equation (1.2) in orthogonality modules over a unital Banach algebra. In Section 5, we prove the Hyers-Ulam stability of the orthogonally quartic functional equation (1.4) in orthogonality modules over a unital Banach algebra.

Throughout this paper, assume that $(X,\perp )$ is an orthogonality module over a unital Banach algebra *A* and that $(Y,{\parallel \cdot \parallel}_{Y})$ is a real Banach module over *A*. Let ${A}_{1}:=\{u\in A\mid \parallel u\parallel =1\}$, and *e* be the unity of *A*.

## 2 Stability of the orthogonally Jensen additive functional equation

for all $x,y\in X$ with $x\perp y$.

**Definition 2.1** An additive mapping $f:X\to Y$ is called an *A-additive mapping* if $f(ax)=af(x)$ for all $a\in A$ and $x\in X$.

**Theorem 2.2**

*Let*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*be a function such that there exists an*$\alpha <1$

*with*

*for all*$x,y\in X$

*with*$x\perp y$.

*Let*$f:X\to Y$

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

*and*

*for all*$a\in {A}_{1}$

*and all*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*additive mapping*$L:X\to Y$

*such that*

*for all* $x\in X$.

*Proof*Putting $y=0$ and $a=e$ in (2.2), we get

*S*:

where, as usual, $inf\varphi =+\mathrm{\infty}$. It is easy to show that $(S,d)$ is complete (see [50]).

- (1)
*L*is a fixed point of*J*,*i.e.*,$L(2x)=2L(x)$(2.6)

*L*is a unique fixed point of

*J*in the set

*L*is a unique mapping satisfying (2.6) such that there exists a $\mu \in (0,\mathrm{\infty})$ satisfying

- (2)$d({J}^{n}f,L)\to 0$ as $n\to \mathrm{\infty}$. This implies the equality$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{{2}^{n}}f\left({2}^{n}x\right)=L(x)$

- (3)$d(f,L)\le \frac{1}{1-\alpha}d(f,Jf)$, which implies the inequality$d(f,L)\le \frac{\alpha}{1-\alpha}.$

This implies that inequality (2.3) holds.

for all $x,y\in X$ with $x\perp y$. Hence, $L:X\to Y$ is an orthogonally Jensen additive mapping.

for all $x\in X$. Thus $L:X\to Y$ is a unique orthogonally Jensen *A*-additive mapping satisfying (2.3). This completes the proof. □

From now on, in corollaries, assume that $(X,\perp )$ is an orthogonality normed module over a unital Banach algebra *A*.

**Corollary 2.3**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$0<p<1$.

*Let*$f:X\to Y$

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

*and*

*for all*$a\in {A}_{1}$

*and*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*additive mapping*$L:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 2.2 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{p-1}$ and we get the desired result. □

**Theorem 2.4**

*Let*$f:X\to Y$

*be a mapping satisfying*(2.2)

*and*$f(0)=0$

*for which there exists a function*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*such that*

*for all*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*additive mapping*$L:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.2.

Thus we obtain inequality (2.8). The rest of the proof is similar to the proof of Theorem 2.2. This completes the proof. □

**Corollary 2.5**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$p>1$.

*Let*$f:X\to Y$

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

*and*(2.7).

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*additive mapping*$L:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 2.4 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{1-p}$ and we get the desired result. □

## 3 Stability of the orthogonally Jensen quadratic functional equation

for all $x,y\in X$ with $x\perp y$.

**Definition 3.1** A quadratic mapping $f:X\to Y$ is called an *A*-*quadratic mapping* if $f(ax)={a}^{2}f(x)$ for all $a\in A$ and $x\in X$.

**Theorem 3.2**

*Let*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*be a function such that there exists an*$\alpha <1$

*with*

*for all*$x,y\in X$

*with*$x\perp y$.

*Let*$f:X\to Y$

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

*and*

*for all*$a\in {A}_{1}$

*and*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*quadratic mapping*$Q:X\to Y$

*such that*

*for all* $x\in X$.

*Proof*Putting $y=0$ and $a=e$ in (3.2), we get

for all $x\in X$.

Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.2.

for all $x\in X$. Thus $Q:X\to Y$ is a unique orthogonally Jensen *A*-quadratic mapping satisfying (3.3). This completes the proof. □

**Corollary 3.3**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$0<p<2$.

*Let*$f:X\to Y$

*be a mapping satisfying*

*for all*$a\in {A}_{1}$

*and*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*quadratic mapping*$Q:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 3.2 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{p-2}$ and we get the desired result. □

**Theorem 3.4**

*Let*$f:X\to Y$

*be a mapping satisfying*(3.2)

*and*$f(0)=0$

*for which there exists a function*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*such that*

*for all*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*quadratic mapping*$Q:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.2.

for all $x\in X$. It follows from (3.4) that $d(f,Jf)\le 1$. So, we obtain inequality (3.7). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2. □

**Corollary 3.5**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$p>2$.

*Let*$f:X\to Y$

*be a mapping satisfying*(3.6).

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally Jensen*

*A*-

*quadratic mapping*$Q:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 3.4 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{2-p}$ and we get the desired result. □

## 4 Stability of the orthogonally cubic functional equation

for all $x,y\in X$ with $x\perp y$.

**Definition 4.1** A cubic mapping $f:X\to Y$ is called an *A*-*cubic mapping* if $f(ax)={a}^{3}f(x)$ for all $a\in A$ and $x\in X$.

**Theorem 4.2**

*Let*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*be a function such that there exists an*$\alpha <1$

*with*

*for all*$a\in {A}_{1}$

*and*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*cubic mapping*$C:X\to Y$

*such that*

*for all* $x\in X$.

*Proof*Putting $y=0$ and $a=e$ in (4.1), we get

for all $x\in X$.

Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.2.

for all $x\in X$. The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2. This completes the proof. □

**Corollary 4.3**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$0<p<3$.

*Let*$f:X\to Y$

*be a mapping satisfying*

*for all*$a\in {A}_{1}$

*and*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*cubic mapping*$C:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 4.2 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{p-3}$ and we get the desired result. □

**Theorem 4.4**

*Let*$f:X\to Y$

*be a mapping satisfying*(4.1)

*and*$f(0)=0$

*for which there exists a function*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*such that*

*for all*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*cubic mapping*$C:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.2.

for all $x\in X$. It follows from (4.2) that $d(f,Jf)\le \frac{\alpha}{16}$. So, we obtain inequality (4.4). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2. □

**Corollary 4.5**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$p>3$.

*Let*$f:X\to Y$

*be a mapping satisfying*(4.3).

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*cubic mapping*$C:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 4.4 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{3-p}$ and we get the desired result. □

## 5 Stability of the orthogonally quartic functional equation

for all $x,y\in X$ with $x\perp y$.

**Definition 5.1** A quartic mapping $f:X\to Y$ is called an *A*-*quartic mapping* if $f(ax)={a}^{4}f(x)$ for all $a\in A$ and $x\in X$.

**Theorem 5.2**

*Let*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*be a function such that there exists an*$\alpha <1$

*with*

*for all*$a\in {A}_{1}$

*and*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*quartic mapping*$P:X\to Y$

*such that*

*for all* $x\in X$.

*Proof*Putting $y=0$ and $a=e$ in (5.1), we get

for all $x\in X$.

Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.2.

for all $x\in X$. The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2. □

**Corollary 5.3**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$0<p<4$.

*Let*$f:X\to Y$

*be a mapping satisfying*

*for all*$a\in {A}_{1}$

*and*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*quartic mapping*$P:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 5.2 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{p-4}$ and we get the desired result. □

**Theorem 5.4**

*Let*$f:X\to Y$

*be a mapping satisfying*(5.1)

*and*$f(0)=0$

*for which there exists a function*$\phi :{X}^{2}\to [0,\mathrm{\infty})$

*such that*

*for all*$x,y\in X$

*with*$x\perp y$.

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*quartic mapping*$P:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* Let $(S,d)$ be the generalized metric space defined in the proof of Theorem 2.2.

for all $x\in X$. It follows from (5.2) that $d(f,Jf)\le \frac{\alpha}{32}$. So, we obtain inequality (5.4). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2. □

**Corollary 5.5**

*Let*

*θ*

*be a positive real number and*

*p*

*be a real number with*$p>4$.

*Let*$f:X\to Y$

*be a mapping satisfying*(5.3).

*If*,

*for each*$x\in X$,

*the mapping*$f(tx)$

*is continuous in*$t\in \mathbb{R}$,

*then there exists a unique orthogonally*

*A*-

*quartic mapping*$P:X\to Y$

*such that*

*for all* $x\in X$.

*Proof* The proof follows from Theorem 5.4 by taking $\phi (x,y)=\theta ({\parallel x\parallel}^{p}+{\parallel y\parallel}^{p})$ for all $x,y\in X$ with $x\perp y$. Then we can choose $\alpha ={2}^{4-p}$ and we get the desired result. □

## 6 Conclusions

Using a fixed point method, we have proved the Hyers-Ulam stability of the orthogonally Jensen additive functional equation, of the orthogonally Jensen quadratic functional equation, of the orthogonally cubic functional equation and of the orthogonally quartic functional equation in orthogonality modules over a unital Banach algebra.

## Declarations

## Authors’ Affiliations

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