# Approximately cubic functional equations and cubic multipliers

## Abstract

In this paper, we prove the Hyers-Ulam stability and the superstability for cubic functional equation by using the fixed point alternative theorem. As a consequence, we show that the cubic multipliers are superstable under some conditions.

2000 Mathematics Subject Classification: 39B82; 39B52.

## 1. Introduction

The stability problem for functional equations is related to the following question originated by Ulam  in 1940, concerning the stability of group homomorphisms: Let (G1, .) be a group and let (G2, *) be a metric group with the metric d(., .). Given ε > 0, does there exist δ > 0 such that, if a mapping h : G1→ G2 satisfies the inequality d(h(x.y), h(x) * h(y)) < δ for all x, y G1, then there exists a homomorphism H : G1→ G2 with d(h(x), H(x)) < ε for all x G1?

In 1941, Hyers  gave a first affirmative answer to the question of Ulam for Banach spaces. Later, Rassias in  provided a remarkable generalization of the Hyers' result by allowing the Cauchy difference to be bounded for the first time, in the subject of functional equations and inequalities. Gǎvruta then generalized the Rassias' result in  for the unbounded Cauchy difference.

The functional equation

$f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$
(1.1)

is called quadratic functional equation. Also, every solution (for example f(x) = ax2) of functional Equation (1.1) is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof  for mappings $f:\mathcal{X}\to \mathcal{Y}$, where $\mathcal{X}$ is a normed space and $\mathcal{Y}$ is a Banach space. Cholewa  noticed that the theorem of Skof is still true if the relevant domain $\mathcal{X}$ is replaced by an abelian group. In  Czerwik proved the Hyers-Ulam-Rassias stability of the quadratic functional equation. Several functional equations have been extensively investigated by a number of authors (for instances, ).

Jun and Kim  introduced the functional equation

$f\left(2x+y\right)+f\left(2x-y\right)=2f\left(x+y\right)+2f\left(x-y\right)+12f\left(x\right)$
(1.2)

which is somewhat different from (1.1). It is easy to see that function f(x) = ax3 is a solution of (1.2). Thus, it is natural that Equation (1.2) is called a cubic functional equation and every solution of this cubic functional equation is said to be a cubic function. One year after that, they solved the generalized Hyers-Ulam-Rassias stability of a cubic functional equation f(x + 2y) + f(x - 2y) + 6f(x) = f(x + y) + 4f(x - y) in Jun and Kim . Since then, a number of authors (for details see [13, 14]) proved the stability problems for cubic functional equation.

Recently, Bodaghi et al in  proved the superstability of quadratic double centralizers and of quadratic multipliers on Banach algebras by fixed point methods. Also, the stability and the superstability of cubic double centralizers of Banach algebras which are strongly without order had been established in Eshaghi Gordji et al. .

In this paper, we remove the condition strongly without order and investigate the generalized Hyers-Ulam-Rassias stability and the superstability by using the alternative fixed point for cubic functional Equation (1.2) and their correspondent cubic multipliers.

## 2. Stability of cubic function equations

Throughout this section, X is a normed vector space and Y is a Banach space. For the given mapping f : XY , we consider

$D\phantom{\rule{0.3em}{0ex}}f\left(x,y\right):=f\left(\mathsf{\text{2}}x+y\right)+f\left(\mathsf{\text{2}}x-y\right)-\mathsf{\text{2}}f\left(x+y\right)-\mathsf{\text{2}}f\left(x-y\right)-\mathsf{\text{12}}f\left(x\right)$

for all x, y X.

We need the following known fixed point theorem, which is useful for our goals (an extension of the result was given in Turinici ).

Theorem 2.1. (The fixed point alternative ) Suppose that (Ω, d) a complete generalized metric space and let$\mathcal{J}:\Omega \to \Omega$be a strictly contractive mapping with Lipschitz constant L < 1. Then for each element x Ω, either$d\left({\mathcal{J}}^{n}x,{\mathcal{J}}^{n+1}x\right)=\infty$for all n ≥ 0, or there exists a natural number n0such that:

(i) $d\left({\mathcal{J}}^{n}x,{\mathcal{J}}^{n+1}x\right)<\infty$for all n ≥ n0;

(ii) the sequence$\left\{{\mathcal{J}}^{n}x\right\}$is convergent to a fixed point y* of$\mathcal{J}$;

(iii) y* is the unique fixed point of$\mathcal{J}$in the set

$\Lambda =\left\{y\in \Omega :d\left({\mathcal{J}}^{{n}_{0}}x,\phantom{\rule{2.77695pt}{0ex}}y\phantom{\rule{2.77695pt}{0ex}}\right)\phantom{\rule{2.77695pt}{0ex}}<\infty \right\};$

(iv) $d\left(y,{y}^{*}\right)\le \frac{1}{1-L}d\left(y,\phantom{\rule{2.77695pt}{0ex}}\mathcal{J}\phantom{\rule{0.3em}{0ex}}y\right)$for all y Λ.

Theorem 2.2. Let f : X → Y be a mapping with f(0) = 0, and let ψ : X × X → [0, ∞) be a function satisfying

$\underset{n\to \infty }{lim}\frac{\psi \left({2}^{n}x,{2}^{n}y\right)}{{8}^{n}}=0$
(2.1)

and

$||D\phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)||\phantom{\rule{2.77695pt}{0ex}}\le \psi \left(x,y\right)$
(2.2)

for all x, y X. If there exists L (0, 1) such that

$\psi \left(2x,\phantom{\rule{2.77695pt}{0ex}}0\right)\le 8L\psi \left(x,\phantom{\rule{2.77695pt}{0ex}}0\right)$
(2.3)

for all x X, then there exists a unique cubic mapping C : X → Y such that

$||f\left(x\right)-C\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le \frac{L}{2\left(1-L\right)}\psi \left(x,0\right)$
(2.4)

for all x X.

Proof: We consider the set Ω := {g : XY | g(0) = 0} and introduce the generalized metric on Ω as follows:

$d\left({g}_{1},\phantom{\rule{2.77695pt}{0ex}}{g}_{2}\right):=inf\left\{C\in \left(0,\infty \right):\phantom{\rule{2.77695pt}{0ex}}||{g}_{1}\left(x\right)-{g}_{2}\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le C\psi \left(x,0\right)\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{all}}\phantom{\rule{2.77695pt}{0ex}}x\in X\right\}$

if there exists such constant C, and d(g1, g2) = ∞, otherwise. One can prove that the metric space (Ω, d) is complete. Now, we define the mapping $\mathcal{J}:\Omega \to \Omega$ by

$\mathcal{J}\phantom{\rule{0.3em}{0ex}}g\left(x\right)=\frac{1}{8}g\left(2x\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(x\in X\right).$

If g1, g2 Ω such that d(g1, g2) < C, by definition of d and $\mathcal{J}$, we have

for all x X. By using (2.3), we get

$∥\frac{1}{8}{g}_{1}\left(2x\right)-\frac{1}{8}{g}_{2}\left(2x\right)∥\le C\phantom{\rule{0.3em}{0ex}}L\psi \left(x,0\right)$

for all x X. The above inequality shows that $d\left(\mathcal{J}{g}_{1},\mathcal{J}{g}_{2}\right)\le Ld\left({g}_{1},{g}_{2}\right)$ for all g1, g2 Ω. Hence, $\mathcal{J}$ is a strictly contractive mapping on Ω with Lipschitz constant L. Putting y = 0 in (2.2), using (2.3), and dividing both sides of the resulting inequality by 16, we have

$∥\frac{1}{8}f\left(2x\right)-f\left(x\right)∥\le \frac{1}{16}\psi \left(x,\phantom{\rule{2.77695pt}{0ex}}0\right)\le \frac{1}{2}L\psi \left(\frac{x}{2},0\right)$

for all x X. Thus,$d\left(f,\phantom{\rule{2.77695pt}{0ex}}\mathcal{J}\phantom{\rule{0.3em}{0ex}}f\right)\le \frac{L}{2}<\infty$. By Theorem 2.1, the sequence $\left\{{\mathcal{J}}^{n}f\right\}$ converges to a fixed point C : XY in the set Ω1 = {g Ω; d(f, g) < ∞}, that is

$C\left(x\right)=\underset{n\to \infty }{lim}\frac{f\left({2}^{n}x\right)}{{8}^{n}}$
(2.5)

for all x X. By Theorem 2.1, we have

$d\left(f,\phantom{\rule{2.77695pt}{0ex}}C\right)\le \frac{d\left(f,\mathcal{J}\phantom{\rule{2.77695pt}{0ex}}f\right)}{1-L}\le \frac{L}{2\left(1-L\right)}.$
(2.6)

It follows from (2.6) that (2.4) holds for all x X. Substituting x, y by 2 nx, 2 ny in (2.2), respectively, and applying (2.1) and (2.5), we have

$\begin{array}{lll}\hfill ||DC\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)||\phantom{\rule{2.77695pt}{0ex}}& =\underset{n\to \infty }{lim}\frac{1}{{8}^{n}}||D\phantom{\rule{0.3em}{0ex}}f\left({2}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{n}y\right)||\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le \underset{n\to \infty }{lim}\frac{1}{{8}^{n}}\psi \left({2}^{n}x,\phantom{\rule{2.77695pt}{0ex}}{2}^{n}y\right)=0\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$

for all x X. Therefore C is a cubic mapping, which is unique. □

Corollary 2.3. Let p and λ be non-negative real numbers such that p < 3. Suppose that f : XY is a mapping satisfying

$||Df\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)||\phantom{\rule{2.77695pt}{0ex}}\le \lambda \left(||x|{|}^{p}+||y|{|}^{p}\right)$
(2.7)

for all x, y X. Then, there exists a unique cubic mapping C : XY such that

$||f\left(x\right)-C\left(x\right)||\phantom{\rule{0.3em}{0ex}}\le \frac{{2}^{p}\lambda }{2\left(8-{2}^{p}\right)}||x|{|}^{p}$
(2.8)

for all x X.

Proof: The result follows from Theorem 2.2 by using ψ(x, y) = λ(||x|| p + ||y|| p ). □

Now, we establish the superstability of cubic mapping on Banach spaces.

Corollary 2.4. Let p, q, λ be non-negative real numbers such that p, q (3, ∞). Suppose a mapping f : XY satisfies

$||D\phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)||\phantom{\rule{2.77695pt}{0ex}}\le \lambda ||x|{|}^{q}||y|{|}^{p}$
(2.9)

for all x, y X. Then, f is a cubic mapping on X.

Proof: Letting x = y = 0 in (2.9), we get f(0) = 0. Once more, if we put x = 0 in (2.9), we have f(2x) = 8f(x) for all x X. It is easy to see that by induction, we have f(2 nx) = 8 nf(x), and so $f\left(x\right)=\frac{f\left({2}^{n}x\right)}{{8}^{n}}$ for all x X and n . Now, it follows from Theorem 2.2 that f is a cubic mapping. □

Note that in Corollary 2.4, if p + q (0, 3) and p > 0 such that the inequality (2.9) holds, then by applying ψ(x, y) = λ||x|| p ||y|| q in Theorem 2.2, f is again a cubic mapping.

Theorem 2.5. Let f : XY be a mapping with f(0) = 0, and let ψ : X × X → [0, ∞) be a function satisfying

$\underset{n\to \infty }{lim}{8}^{n}\psi \left(\frac{x}{{2}^{n}},\phantom{\rule{2.77695pt}{0ex}}\frac{y}{{2}^{n}}\right)=0$
(2.10)

and

$||Df\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)||\phantom{\rule{2.77695pt}{0ex}}\le \psi \left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)$
(2.11)

for all x, y X. If there exists L (0, 1) such that

$\psi \left(x,\phantom{\rule{2.77695pt}{0ex}}0\right)\le \frac{1}{8}L\psi \left(2x,\phantom{\rule{2.77695pt}{0ex}}0\right)$
(2.12)

for all x X, then there exists a unique cubic mapping C : XY such that

$||f\left(x\right)-C\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le \frac{L}{16\left(1-L\right)}\psi \left(x,0\right)$
(2.13)

for all x X.

Proof: We consider the set Ω := {g : XY | g(0) = 0} and introduce the generalized metric on Ω:

$d\left({g}_{1},\phantom{\rule{2.77695pt}{0ex}}{g}_{2}\right):=inf\left\{C\in \left(0,\infty \right):\phantom{\rule{2.77695pt}{0ex}}||{g}_{1}\left(x\right)-{g}_{2}\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le C\psi \left(x,0\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x\in X\right\}$

if there exists such constant C, and d(g1, g2) = ∞, otherwise. It is easy to show that (Ω, d) is complete. We will show that the mapping $\mathcal{J}:\Omega \to \Omega$ defined by $\mathcal{J}\phantom{\rule{0.3em}{0ex}}g\left(x\right)=8g\left(\frac{x}{2}\right)$; (x X) is strictly contractive. For given g1, g2 Ω such that d(g1, g2) < C, we have

$∥8{g}_{1}\left(\frac{x}{2}\right)-8{g}_{2}\left(\frac{x}{2}\right)∥\le \frac{1}{8}C\psi \left(2x,\phantom{\rule{2.77695pt}{0ex}}0\right)$

for all x X. By using (2.12), we obtain

$∥8{g}_{1}\left(\frac{x}{2}\right)-8{g}_{2}\left(\frac{x}{2}\right)∥\le CL\psi \left(x,\phantom{\rule{2.77695pt}{0ex}}0\right)$

for all x X. It follows from the last inequality that $d\left(\mathcal{J}{g}_{1},\mathcal{J}{g}_{2}\right)\le Ld\left({g}_{1},{g}_{2}\right)$ for all g1, g2 Ω. Hence, $\mathcal{J}$ is a strictly contractive mapping on Ω with Lipschitz constant L. By putting y = 0 and replacing x by $\frac{x}{2}$ in (2.11) and using (2.12), then by dividing both sides of the resulting inequality by 2, we have

$∥8f\left(\frac{x}{2}\right)-f\left(x\right)∥\le \frac{1}{2}\psi \left(\frac{x}{2},0\right)\le \frac{1}{16}L\psi \left(x,\phantom{\rule{2.77695pt}{0ex}}0\right)$

for all x X. Hence, $d\left(f,\mathcal{J}f\right)\le \frac{L}{16}<\infty$. By applying the fixed point alternative, there exists a unique mapping C : XY in the set Ω1 = {g Ω; d(f, g) < ∞} such that

$C\left(x\right)=\underset{n\to \infty }{lim}{8}^{n}f\left(\frac{x}{{2}^{n}}\right)$
(2.14)

for all x X. Again, Theorem 2.1 shows that

$d\left(f,\phantom{\rule{2.77695pt}{0ex}}C\right)\le \frac{d\left(f,\mathcal{J}\phantom{\rule{0.3em}{0ex}}f\right)}{1-L}\le \frac{L}{16\left(1-L\right)}$
(2.15)

where the inequality (2.15) implies the relation (2.13). Replacing x, y by 2 nx, 2 ny in (2.11), respectively, and using (2.10) and (2.14), we conclude

$\begin{array}{lll}\hfill ||DC\left(x,y\right)||\phantom{\rule{2.77695pt}{0ex}}& =\underset{n\to \infty }{lim}{8}^{n}∥D\phantom{\rule{0.3em}{0ex}}f\left(\frac{x}{{2}^{n}},\frac{y}{{2}^{n}}\right)∥\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le \underset{n\to \infty }{lim}{8}^{n}\psi \phantom{\rule{2.77695pt}{0ex}}\left(\frac{x}{{2}^{n}},\phantom{\rule{2.77695pt}{0ex}}\frac{y}{{2}^{n}}\right)=0\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$

for all x X. Therefore C is a cubic mapping. □

Corollary 2.6. Let p and λ be non-negative real numbers such that p > 3. Suppose that f : XY is a mapping satisfying

$||D\phantom{\rule{0.3em}{0ex}}f\phantom{\rule{2.77695pt}{0ex}}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\phantom{\rule{2.77695pt}{0ex}}\right)\phantom{\rule{2.77695pt}{0ex}}||\phantom{\rule{2.77695pt}{0ex}}\le \lambda \left(||x|{|}^{p}+||y|{|}^{p}\right)$

for all x, y X. Then there exists a unique cubic mapping C : XY such that

$||f\left(x\right)-C\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le \frac{\lambda }{2\left({2}^{p}-8\right)}||x|{|}^{p}$

for all x X.

Proof: It is enough to let ψ(x, y) = λ(||x|| p + ||y|| p ) in Theorem 2.5. □

Corollary 2.7. Let p, q, λ be non-negative real numbers such that p + q (0, 3) and p > 0. Suppose a mapping f : X → Y satisfies

$||D\phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)||\phantom{\rule{2.77695pt}{0ex}}\le \lambda ||x|{|}^{q}||y|{|}^{p}$
(2.16)

for all x, y X. Then, f is a cubic mapping on X.

Proof: If we put x = y = 0 in (2.16), we get f(0) = 0. Again, putting x = 0 in (2.16), we conclude that $f\left(x\right)=8f\left(\frac{x}{2}\right)$, and thus $f\left(x\right)={8}^{n}f\left(\frac{x}{{2}^{n}}\right)$ for all x X and n . Now, we can obtain the desired result by Theorem 2.5. □

One should remember that if a mapping f : XY satisfies the inequality (2.16), where p, q, λ be non-negative real numbers such that p + q > 3 and p > 0, then it is obvious that f is a cubic mapping on X by putting ψ(x, y) = λ||x|| p ||y|| q in Theorem 2.5.

## 3. Stability of cubic multipliers

In this section, we investigate the Hyers-Ulam stability and the superstability of cubic multipliers.

Definition 3.1. A cubic multiplier on an algebra A is a cubic mapping T : AA such that aT(b) = T(a)b for all a, b A.

The following example introduces a cubic multiplier on Banach algebras.

Example. Let (A, ||·||) be a Banach algebra. Then, we take B = A × A × A × A × A × A = A6. Let a = (a1, a2, a3, a4, a5, a6) be an arbitrary member of B where we define $|\phantom{\rule{0.3em}{0ex}}||a||\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{2.77695pt}{0ex}}={\sum }_{i=1}^{6}||{a}_{i}||$. It is easy to see (B, |||·|||)) is a Banach space. For two elements a = (a1, a2, a3, a4, a5, a6) and b = (b1, b2, b3, b4, b5, b6) of B, we define ab = (0, a1b4, a1b5 + a2b6, 0, a4b6, 0). It is easy to show that B is a Banach algebra. We define T : BB by T(a) = a3 for all a B. It is shown in Eshaghi Gordji et al.  that T is a cubic multiplier on A.

Theorem 3.2. Let f : AA be a mapping with f(0) = 0 and let ψ: A4 → [0, ∞) be a function such that

$||D\phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)+f\left(z\right)w-z\phantom{\rule{0.3em}{0ex}}f\left(w\right)||\phantom{\rule{2.77695pt}{0ex}}\le \psi \left(x,y,\phantom{\rule{2.77695pt}{0ex}}z,\phantom{\rule{2.77695pt}{0ex}}w\right)$
(3.1)

for all x, y, z, w A. If there exists a constant L (0, 1) such that

$\psi \left(2x,\phantom{\rule{2.77695pt}{0ex}}2y,\phantom{\rule{2.77695pt}{0ex}}2z,\phantom{\rule{2.77695pt}{0ex}}2w\right)\le 8L\psi \left(x,y,\phantom{\rule{2.77695pt}{0ex}}z,\phantom{\rule{2.77695pt}{0ex}}w\right)$
(3.2)

for all x, y, z, w A, then there exists a unique cubic multiplier T on A satisfying

$||f\left(x\right)-T\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le \frac{L}{2\left(1-L\right)}\psi \left(x,x,\phantom{\rule{2.77695pt}{0ex}}0,0\right)$
(3.3)

for all x A.

Proof. It follows from the relation (3.2) that

$\underset{n\to \infty }{lim}\frac{\psi \left({2}^{n}x,{2}^{n}y,{2}^{n}z,{2}^{n}w\right)}{{8}^{n}}=0$
(3.4)

for all x, y, z, w A.

Putting y = z = w = 0 in (3.1), we obtain

$||2f\left(2x\right)-16f\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le \psi \left(x,\phantom{\rule{2.77695pt}{0ex}}0,0,0\right)$

for all x A. Thus,

$∥\frac{1}{8}f\left(2x\right)-f\left(x\right)∥\le \frac{1}{16}\psi \left(x,\phantom{\rule{2.77695pt}{0ex}}0,0,0\right)$
(3.5)

for all x A.

Now, similar to the proof of theorems in previous section, we consider the set X := {h : AA | h(0) = 0} and introduce the generalized metric on X as:

if there exists such constant C, and d(h1, h2) = ∞, otherwise. The metric space (X, d) is complete, and by the same reasoning as in the proof of Theorem 2.2, the mapping Φ: XX defined by $\Phi \left(h\right)\left(x\right)=\frac{1}{8}h\left(2x\right)$; (x A) is strictly contractive on X and has a unique fixed point T such that $\underset{n\to \infty }{\mathrm{lim}}d\left({\Phi }^{n}f,T\right)=0$, i.e.,

$T\left(x\right)=\underset{n\to \infty }{lim}\frac{f\left({2}^{n}x\right)}{{8}^{n}}$
(3.6)

for all x A. By Theorem 2.1, we have

$d\left(f,\phantom{\rule{2.77695pt}{0ex}}T\right)\le \frac{d\left(f,\Phi \phantom{\rule{0.3em}{0ex}}f\right)}{1-L}\le \frac{L}{2\left(1-L\right)}.$
(3.7)

The proof of Theorem 2.2 shows that T is a cubic mapping. If we substitute z and w by 2 nz and 2 nw in (3.1), respectively, and put x = y = 0 and we divide the both sides of the obtained inequality by 24n, we get

$∥z\frac{f\left({2}^{n}w\right)}{{8}^{n}}-\frac{f\left({2}^{n}z\right)}{{8}^{n}}w∥\le \frac{\psi \left(0,0,{2}^{n}z,{2}^{n}w\right)}{{2}^{4n}}.$

Passing to the limit as n and from (3.4), we conclude that zT(w) = T(z)w for all z, w A. □

Corollary 3.3. Let r, θ be non-negative real numbers with r < 3 and let f : AA be a mapping with f(0) = 0 such that

$||D\phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)+f\left(z\right)w-z\phantom{\rule{0.3em}{0ex}}f\left(w\right)||\phantom{\rule{2.77695pt}{0ex}}\le \theta \left(||x|{|}^{r}+||y|{|}^{r}+||z|{|}^{r}+||w|{|}^{r}\right)$

for all x, y, z, w A. Then, there exists a unique cubic multiplier T on A satisfying

$||f\left(x\right)-T\left(x\right)||\phantom{\rule{2.77695pt}{0ex}}\le \frac{{2}^{r-1}\theta }{8-{2}^{r}}||x|{|}^{r}$

for all x A.

Proof. The proof follows from Theorem 3.2 by taking

$\psi \left(x,y,z,w\right)=\theta \left(||x|{|}^{r}+||y|{|}^{r}+||z|{|}^{r}+||w|{|}^{r}\right)$

for all x, y, z, w A. □

Now, we have the following result for the superstability of cubic multipliers.

Corollary 3.4. Let r j (1 ≤ j ≤ 4). θ be non-negative real numbers with ${\sum }_{j=1}^{4}{r}_{j}<3$ and let f: AA be a mapping with f (0) = 0 such that

$||D\phantom{\rule{0.3em}{0ex}}f\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)+f\left(z\right)w-z\phantom{\rule{0.3em}{0ex}}f\left(w\right)||\phantom{\rule{2.77695pt}{0ex}}\le \theta \left(||x|{|}^{{r}_{1}}||y|{|}^{{r}_{2}}||z|{|}^{{r}_{3}}||w|{|}^{{r}_{4}}\right)$

for all x, y, z, w A. Then, f is a cubic multiplier on A.

Proof. It is enough to let $\psi \left(x,y,z,w\right)=\theta \left(||x|{|}^{{r}_{1}}||y|{|}^{{r}_{2}}||z|{|}^{{r}_{3}}||w|{|}^{{r}_{4}}\right)$ in Theorem 3.2. □

## References

1. Ulam SM: Problems in Modern Mathematics. Volume chapt. VI. science edition. Wiley, New York; 1940.

2. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci 1941, 27: 222–224. 10.1073/pnas.27.4.222

3. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

4. Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431–436. MR1281518 (95e:47089) 10.1006/jmaa.1994.1211

5. Skof F: Propriet locali e approssimazione di operatori. Rend Sem Math Fis Milano 1983, 53: 113–129. 10.1007/BF02924890

6. Cholewa PW: Remarks on the stability of functional equations. Aequ Math 1984, 27: 76–86. 10.1007/BF02192660

7. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh Math Sem Univ Hamburg 1992, 62: 59–64. MR1182841 (94e:39026) 10.1007/BF02941618

8. Eshaghi Gordji M, Bodaghi A: On the Hyers-Ulam-Rasias stability problem for quadratic functional equations. East Journal of Approximations 2010,16(2):123–130.

9. Eshaghi Gordji M, Bodaghi A: On the stability of quadratic double centralizers on Banach algebras. J Comput Anal Appl 2011,13(4):724–729.

10. Lee J, An J, Park C: On the stability of qudratic functional equations. Abstr Appl Anal 2008, 2008: 1–9.

11. Jun KW, Kim HM: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J Math Anal Appl 2002, 274: 867–878. 10.1016/S0022-247X(02)00415-8

12. Jun KW, Kim HM: On the Hyers-Ulam-Rassias stability of a general cubic functional equation. Math Inequal Appl 2003,6(2):289–302.

13. Chang IS, Jun KW, Jung YS: The modified Hyers-Ulam-Rassias stability of a cubic type functional equation. Math Inequal Appl 2005,8(4):675–683.

14. Najati A: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. Turk J Math 2007, 31: 395–408.

15. Bodaghi A, Alias IA, Eshaghi Gordji M: On the stability of quadratic double centralizers and quadratic multipliers: a fixed point approach. J Inequal Appl 2011,2011(Article ID 957541):9.

16. Eshaghi Gordji M, Shams S, Ramezani M, Ebadian A: Approximately cubic double centralizers. Nonlinear Func Anal Appl 2010,15(3):503–512.

17. Turinici M: Sequentially iterative processes and applications to Volterra functional equations. Ann Univ Mariae-Curie Sklodowska (Sect A) 1978, 32: 127–134.

18. Diaz JB, Margolis B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull Am Math Soc 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0

## Acknowledgements

The authors sincerely thank the anonymous reviewer for his careful reading, constructive comments and fruitful suggestions to improve the quality of the manuscript. The first and third author would like to thank Islamic Azad University of Garmsar for financial support.

## Author information

Authors

### Corresponding author

Correspondence to Abasalt Bodaghi.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The work presented here was carried out in collaboration between all authors. AB suggested to write the current paper. All authors read and approved the final manuscript.

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Bodaghi, A., Alias, I.A. & Ghahramani, M.H. Approximately cubic functional equations and cubic multipliers. J Inequal Appl 2011, 53 (2011). https://doi.org/10.1186/1029-242X-2011-53

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• DOI: https://doi.org/10.1186/1029-242X-2011-53

### Keywords

• cubic functional equation
• multiplier
• Hyers-Ulam stability
• Superstability 