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Orthogonal stability of functional equations with the fixed point alternative
Advances in Difference Equations volume 2012, Article number: 173 (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 HyersUlam 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.
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.
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 BirkhoffJames orthogonality. The orthogonal Cauchy functional equation
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 semicontinuous realvalued 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 2dimensional 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.
Some interesting examples are as follows:

(1)
The trivial orthogonality on a vector space X defined by ({O}_{1}) and, for any nonzero 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 BirkhoffJames 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 BirkhoffJames orthogonality is symmetric. There are several orthogonality notions on a real normed space such as BirkhoffJames, Boussouis, Singer, Carlsson, unitaryBoussouis, 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 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 HyersUlamRassias. The readers refer to [16–20] and references therein for detailed information on the 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
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.
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
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 HyersUlamRassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [25–31]).
The orthogonally quadratic equation
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]).
In [41], Jun and Kim considered the following cubic functional equation:
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.
Let 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.
In [42], Lee 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).
Let 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.
Let 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 HyersUlam stability of the orthogonally Jensen additive functional equation in orthogonality modules over a unital Banach algebra. In Section 3, we prove the HyersUlam stability of the orthogonally Jensen quadratic functional equation in orthogonality modules over a unital Banach algebra. In Section 4, we prove the HyersUlam stability of the orthogonally cubic functional equation (1.2) in orthogonality modules over a unital Banach algebra. In Section 5, we prove the HyersUlam 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
In this section, applying some ideas from [18, 21], we deal with the stability problem for 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 Aadditive 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 Aadditive mapping L:X\to Y such that
for all x\in X.
Proof Putting y=0 and a=e in (2.2), we get
for all x\in X since x\perp 0. So, we have
for all x\in X. Consider the set
and introduce the generalized metric on S:
where, as usual, inf\varphi =+\mathrm{\infty}. It is easy to show that (S,d) is complete (see [50]).
Now, we consider the linear mapping J:S\to S such that
for all x\in X. Let g,h\in S be given such that d(g,h)=\epsilon. Then we have
for all x\in X. Hence,
for all x\in X. So, d(g,h)=\epsilon implies that d(Jg,Jh)\le \alpha \epsilon. This means that
for all g,h\in S. It follows from (2.5) that d(f,Jf)\le \alpha. By Theorem 1.1, there exists a mapping L:X\to Y satisfying the following:

(1)
L is a fixed point of J, i.e.,
L(2x)=2L(x)(2.6)
for all x\in X. The mapping L is a unique fixed point of J in the set
This implies that L is a unique mapping satisfying (2.6) such that there exists a \mu \in (0,\mathrm{\infty}) satisfying
for all x\in X;

(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)
for all x\in 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.
Let a=e in (2.2). It follows from (2.1) and (2.2) that
for all x,y\in X with x\perp y. So,
for all x,y\in X with x\perp y. Hence, L:X\to Y is an orthogonally Jensen additive mapping.
Let y=0 in (2.2). It follows from (2.1) and (2.2) that
for all x\in X. So, we have
for all x\in X, and hence
for all a\in {A}_{1} and x\in X. By the same reasoning as in the proof of Theorem in [15], we can show that L:X\to Y is \mathbb{R}linear since the mapping f(tx) is continuous in t\in \mathbb{R} for each x\in X. For each a\in A with a\ne 0, we have
for all x\in X. Thus L:X\to Y is a unique orthogonally Jensen Aadditive 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 Aadditive 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}^{p1} 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 Aadditive 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.
Now, we consider the linear mapping J:S\to S such that
for all x\in X. It follows from (2.4) that d(f,Jf)\le 1. So,
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 Aadditive 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}^{1p} and we get the desired result. □
3 Stability of the orthogonally Jensen quadratic functional equation
In this section, applying some ideas from [18, 21], we deal with the stability problem for 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 Aquadratic 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 Aquadratic 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, since x\perp 0. So, we have
for all x\in X. By the same reasoning as in the proof of Theorem 2.2, one can obtain an orthogonally Jensen quadratic mapping Q:X\to Y defined by
for all x\in X.
Let (S,d) be the generalized metric space defined in the proof of Theorem 2.2.
Now, we consider the linear mapping J:S\to S such that
for all x\in X. It follows from (3.5) that d(f,Jf)\le \alpha. So,
Thus we obtain inequality (3.3). Let y=0 in (3.2). It follows from (3.1) and (3.2) that
for all x\in X. So, we have
for all x\in X, and hence
for all a\in {A}_{1} and x\in X. By the same reasoning as in the proof of [[15], Theorem], we can show that, for each t\in \mathbb{R}, Q:X\to Y satisfies Q(tx)={t}^{2}Q(x) all x\in X since the mapping f(tx) is continuous in t\in \mathbb{R} for each x\in X. For each a\in A with a\ne 0, we have
for all x\in X. Thus Q:X\to Y is a unique orthogonally Jensen Aquadratic 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 Aquadratic 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}^{p2} 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 Aquadratic 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.
Now, we consider the linear mapping J:S\to S such that
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 Aquadratic 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}^{2p} and we get the desired result. □
4 Stability of the orthogonally cubic functional equation
In this section, applying some ideas from [18, 21], we deal with the stability problem for 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 Acubic 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 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 Acubic 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 since x\perp 0. So, we have
for all x\in X.
Let (S,d) be the generalized metric space defined in the proof of Theorem 2.2.
Now, we consider the linear mapping J:S\to S such that
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 Acubic 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}^{p3} 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 Acubic 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.
Now, we consider the linear mapping J:S\to S such that
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 Acubic 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}^{3p} and we get the desired result. □
5 Stability of the orthogonally quartic functional equation
Applying some ideas from [18, 21], we deal with the stability problem for 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 Aquartic 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 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 Aquartic 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, since x\perp 0. So, we have
for all x\in X.
Let (S,d) be the generalized metric space defined in the proof of Theorem 2.2.
Now, we consider the linear mapping J:S\to S such that
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 Aquartic 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}^{p4} 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 Aquartic 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.
Now, we consider the linear mapping J:S\to S such that
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 Aquartic 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}^{4p} and we get the desired result. □
6 Conclusions
Using a fixed point method, we have proved the HyersUlam 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.
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Park, C., Cho, Y.J. & Lee, J.R. Orthogonal stability of functional equations with the fixed point alternative. Adv Differ Equ 2012, 173 (2012). https://doi.org/10.1186/168718472012173
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DOI: https://doi.org/10.1186/168718472012173
Keywords
 HyersUlam stability
 orthogonally (Jensen additive, Jensen quadratic, cubic, quartic) functional equation
 fixed point
 orthogonality module over Banach algebra
 orthogonality space