---
abstract: |
    Consider the functional equation ${\mathcal E}_1(f) = {\mathcal
    E}_2(f)~~({\mathcal E})$  in a certain framework. We say a function $f_0$ is an approximate solution of $({\mathcal E})$ if ${\mathcal E}_1(f_0)$ and ${\mathcal E}_2(f_0)$ are close in some sense. The stability problem is whether or not there is an exact solution of $({\mathcal E})$ near $f_0$.

    In this paper, the stability of derivations on Hilbert $C^*$-modules is investigated in the spirit of Hyers–Ulam–Rassias.
address:
- 'Department of Mathematics, Is. Azad University, Rahnamaei Ave., Mashhad 91735, Iran, and Banach Mathematical Research Group (BMRG)'
- 'Department Mathematics, Ferdowsi University, P. O. Box 1159, Mashhad 91775, Iran, and Centre of Excellency in Analysis on Algebraic Structures (CEAAS), Ferdowsi Univ., Iran.'
author:
- Maryam Amyari
- Mohammad Sal Moslehian
title: 'Hyers–Ulam–Rassias Stability of Derivations on Hilbert $C^*$-Modules'
---

[^1]

Introduction
============

One of the interesting questions in the theory of functional equations is the following (see [@GRU]):

[When is it true that a function which approximately satisfies a functional equation ${\mathcal E}$ must be close to an exact solution of ${\mathcal E}$?]{}

If there exists an affirmative answer we say that the equation ${\mathcal E}$ is stable.

The first stability problem was raised by S. M. Ulam during his talk before a Mathematical Colloquium at the University of Wisconsin in 1940 [@ULA]:

[Given a group ${\mathcal G}_1$, a metric group $({\mathcal G}_2, d)$ and a positive number $\epsilon$, does there exist a number $\delta>0$ such that if a function $f :
{\mathcal G}_1 \to {\mathcal G}_2$ satisfies the inequality $d(f(xy), f(x)f(y))<\delta$ for all $x, y \in {\mathcal G}_1$ then there exists a homomorphism $T : {\mathcal G}_1 \to
{\mathcal G}_2$ such that $d(f(x), T(x))<\epsilon$ for all $x\in
{\mathcal G}_1$?]{}

Ulam’s problem was partially solved by D. H. Hyers in 1941 in the context of Banach spaces with $\delta = \epsilon$ in the following form [@HYE]:

[Suppose that ${\mathcal X}_1$ and ${\mathcal X}_2$ are Banach spaces and $f:{\mathcal X}_1 \to {\mathcal X}_2$ satisfies the following condition: If there is $\epsilon>0$ such that $\|f(x+y)-f(x)-f(y)\|<\epsilon$ for all $x, y \in {\mathcal
X}_1$, then there is a unique additive mapping $T:{\mathcal
X}_1\to {\mathcal X}_2$ defined by $T(x) =
\displaystyle{\lim_{n\to\infty}}\frac{f(2^nx)}{2^n}$ such that $\|f(x)-T(x)\|<\epsilon$ for all $x\in {\mathcal X}_1$.]{}

Th. M. Rassias [@RAS1] extended Hyers’ theorem in the following form where Cauchy difference is allowed to be unbounded:

[Assume that ${\mathcal X}_1$ and ${\mathcal X}_2$ are real normed spaces with ${\mathcal X}_2$ complete, $f: {\mathcal
X}_1\to {\mathcal X}_2$ is a mapping such that for each fixed $x\in {\mathcal X}_1$ the mapping $t\mapsto f(tx)$ is continuous on ${\mathbb R}$, and let there exist $\varepsilon\ge 0$ and $p\in[0,1)$ such that $$\|f(x+y)-f(x)-f(y)\|\le \varepsilon(\|x\|^p+\|y\|^p)$$ for all $x, y \in {\mathcal X}_1$. Then there exists a unique linear mapping $T: {\mathcal X}_1 \to {\mathcal X}_2$ such that $$\|f(x)-T(x)\|\le\frac{\epsilon \|x\|^p}{1-2^{p - 1}}$$ for all $x \in {\mathcal X}_1$.]{}

This result is still valid in the case where $p < 0$ by the same approach given in [@RAS1] if we assume that $\|0\|^p =
\infty$. In 1990, Th. M. Rassias during the 27th International Symposium on Functional Equations asked the question whether his theorem can be proved for $p\geq 1$. In 1991, Z. Gajda [@GAJ] following the same approach as in [@RAS1] provided an affirmative solution to this question for $p>1$. Using Hyers’ method, indeed, $T(x)$ is defined by $\lim_{n\to\infty}2\sp{-n}f(2^nx)$ if $p<1$, and $\lim_{n\to\infty}2^nf(2\sp{-n}x)$ if $p>1$. It is shown that there is no analogue of Th. M. Rassias’ result for $p = 1$ (see [@GAJ] and [@R-S]). This phenomenon of stability that was introduced by Th. M. Rassias [@RAS1] is called [*H*yers–Ulam–Rassias stability]{}. Thus the Hyers–Ulam stability will be regarded as a special case of the Hyers–Ulam–Rassias stability. A number of Rassias type results related to the stability of various functional equations are presented in [@RAS2; @RAS3].

In 1992, a generalization of Rassias’ theorem was obtained by Găvruta as follows [@GAV]:

[Suppose $({\mathcal G},+)$ is an abelian group, ${\mathcal X}$ is a Banach space and the so-called admissible control function $\varphi:{\mathcal G}\times {\mathcal G}\to [0,
\infty)$ satisfies $$\widetilde{\varphi}(x, y):
 = \frac{1}{2}\displaystyle{\sum_{n = 0}^\infty}2^{-n}\varphi(2^n
x,2^n y)<\infty$$ for all $x,y\in {\mathcal G}$. If $f :
{\mathcal G} \to {\mathcal X}$ is a mapping with $$\|f(x+y)-f(x)-f(y)\|\leq \varphi(x,y)$$ for all $x, y \in {\mathcal G}$, then there exists a unique mapping $T : {\mathcal G} \to
{\mathcal X}$ such that $T(x+y) = T(x)+T(y)$ and $\|f(x)-T(x)\|\leq\widetilde{\varphi}(x, x)$ for all $x,y\in
{\mathcal G}$.]{}

There are four methods in the study of stability of functional equations. The first method is the [*d*irect method]{} in which one uses an iteration process producing the so-called [*H*yers type sequences]{} [@HYE]. Another method is based on [*s*andwich theorems]{} which are generalizations of the Hahn-Banach separation theorems; cf. [@PAL]. The third technique focuses on using [*i*nvariant means]{}; cf. [@SZE], and the foundation of the forth method is [*f*ixed point techniques]{}; cf. [@C-R; @I-R]. The reader is referred to [@CZE; @H-I-R; @JUN] and references therein for further information on stability.

The notion of Hilbert $C^*$-module is a generalization of the notion of Hilbert space.

Let ${\mathcal A}$ be a $C^*$-algebra and ${\mathcal M}$ be a linear space which is a left ${\mathcal A}$-module with a scalar multiplication satisfying $\lambda(xa) = x(\lambda a) = (\lambda
x)a$ for $x \in {\mathcal M},a \in {\mathcal A}, \lambda \in {\mathbb C}$.\
The space ${\mathcal M}$ is called a [*p*re-Hilbert ${\mathcal A}$-module]{} or [*i*nner product ${\mathcal
A}$-module]{} if there exists an inner product $\langle .,.\rangle  :{\mathcal M} \times {\mathcal M}\to {\mathcal A}$ with the following properties:\
(i) $\langle x,x\rangle\geq 0$; and $\langle x,x\rangle = 0~~~  {\rm iff} ~~~ x = 0;\\
(ii) \langle \lambda x + y, z\rangle = \lambda \langle x, y\rangle + \langle y,z\rangle;\\
(iii) \langle ax,y\rangle = a\langle x,y\rangle;\\
(iv) \langle x,y\rangle^* = \langle y,x\rangle$.\
${\mathcal M}$ is called a (left) [*H*ilbert ${\mathcal
A}$-module]{} if it is complete with respect to the norm $\| x \| =
\| \langle x,x\rangle\| ^{1/2}$.

[**(i)**]{} Every inner product space is a left Hilbert ${\mathbb
C}$-module.

[**(ii)**]{} Let ${\mathcal A}$ be a $C^*$-algebra Then every closed left ideal $I$ of ${\mathcal A}$ is a Hilbert ${\mathcal
A}$-module if one defines $\langle a,b\rangle = ab^*\;\;(a, b\in
I)$.

Assume that ${\mathcal N}$ is another Hilbert ${\mathcal
A}$-module. Recall that a mapping $T: {\mathcal M}\to {\mathcal
N}$ is said to be adjointable if there exists a mapping $S:
{\mathcal N} \to {\mathcal M}$ such that $\langle Tx, y\rangle =
\langle x, Sy\rangle$ for all $x \in {\mathcal M}, y \in
{\mathcal N}$. The mapping $S$ is denoted by $T^*$ and called the adjoint of $T$. If $T$ is adjointable, then it is ${\mathcal
A}$-linear and automatically continuous; cf. [@LAN].

Following [@L-X], a linear mapping $d: {\mathcal M} \to
{\mathcal M}$ is called a [*d*erivation]{} on the Hilbert $C^*$-module ${\mathcal M}$ if it satisfies the condition $d(\langle x, y\rangle z) = \langle d(x), y\rangle z + \langle x,
d(y)\rangle z + \langle x, y\rangle d(z)$ for every $x, y, z \in
{\mathcal M}$. It is clear that every adjointable mapping $T$ satisfying $T^* = -T$ is a derivation. The converse is not true in general. For example, let $u_0$ be a bounded linear operator acting on a Hilbert space $H$ of dimension greater than one such that $u_0^* = -u_0$ and $u_0$ is not an element of the center of $B(H)$ (e.g. consider a fixed vector $\xi \in H$ and put $u_0(\zeta) = 2{\bf i}\langle \zeta, \xi \rangle \xi, \;(\zeta
\in H)$). Obviously, the mapping $d : B(H) \to B(H)$ defined by $d(v) = u_0v - vu_0$ is an ordinary $*$-derivation on $B(H)$ and so it is a derivation on $B(H)$ regarded as a Hilbert $C^*$-module over itself. If $d$ were a $B(H)$-module map, in the sense that $d(vw) = vd(w)$ for all $v, w \in B(H)$, then $u_0(vw)
- (vw)u_0 = v(u_0w - wu_0)$ and so $u_0vw = vu_0w$ for each $v, w
\in B(H)$. In particular, $u_0v = vu_0$ for all $v \in B(H)$ which is a contradiction. Since adjointable mappings are module map, we infer that $d$ is not adjointable.

Recently, several extended notions of derivations such as $(\sigma-\tau)$-derivations and generalized derivations have been treated in the Banach algebra theory (see [@B-V; @M-M]). In addition, the stability of these derivations in the spirit of Hyers–Ulam–Rassias has extensively studied by many mathematicians; see [@MOS1; @MOS2; @PAR1; @PAR2]. In this paper we establish the stability of derivations on Hilbert $C^*$-modules. Throughout the paper, ${\mathcal M}$ denotes a Hilbert module over a $C^*$-algebra ${\mathcal A}$.

Main results
============

Recently, the stability of several mappings on Hilbert $C^*$-modules was investigated (see [@AMY; @MOS3]). Using some ideas from [@C-R] we investigate the stability of derivations on Hilbert $C^*$-modules. Our results may be regarded as an extension of those of [@PAR2] when we consider a unital $C^*$-algebra ${\mathcal A}$ as an ${\mathcal A}$-bimodule via its multiplication. We start our work with a known fixed point theorem.

\[The alternative of fixed point\] Suppose $({\mathcal S},d)$ is a complete generalized metric space and $J:{\mathcal S}\to {\mathcal S}$ is a strictly contractive mapping with the Lipschitz constant L. Then, for each given element $x\in {\mathcal S}$,either\
(A1) $d(J^nx, J^{n+1}x) = \infty$\
for all $n\geq 0$, or\
(A2) There exists a natural number $n_0$ such that:\
(A20) $d(J^nx, J^{n+1}x)<\infty$, for all $n\geq n_0$;\
(A21) The sequence $\{J^nx\}$ is convergent to a fixed point $y^*$ of $J$;\
(A22) $y^*$ is the unique fixed point of $J$ in the set $U = \{y\in {\mathcal S}: d(J^{n_0}x,y) < \infty\}$;\
(A23) $d(y,y^*)\leq\frac{1}{1-L}d(y,Jy)$ for all $y\in U$.

The following lemma gives us a useful strictly contractive mapping.

\[lemma\] Suppose that ${\mathcal X}$ is a Banach space, $0 \leq L < 1$ and $\lambda \geq 0$ are given numbers and $\psi(x): {\mathcal X} \to
[0, \infty)$ has the property $$\begin{aligned}
\psi(x)\leq \lambda L\psi(\frac{x}{\lambda}),\end{aligned}$$ for all $x \in {\mathcal X}$. Assume that ${\mathcal S}: =
\{g:{\mathcal X}\to {\mathcal X}: g(0) = 0\}$ and the generalized metric $d$ on ${\mathcal S}$ is defined by $$d(g,h) = \inf\{c\in (0,\infty): \|g(x)-h(x)\|\leq c\psi(x),
\forall x\in {\mathcal X}\}.$$ Then the mapping $J:{\mathcal S}\to
{\mathcal S}$ given by $(Jg)(x): = \frac{1}{\lambda}g(\lambda x)$ is an strictly contractive mapping.

It is easy to see that $({\mathcal S},d)$ is complete. For arbitrary elements $g, h\in {\mathcal S}$ we have $$\begin{split}
d(g,h)<c &\Rightarrow \|g(x)-h(x)\|\leq c\psi(x),~~~x\in {\mathcal X}\\
&\Rightarrow \|\frac{1}{\lambda}g(\lambda
x)-\frac{1}{\lambda}h(\lambda
x)\|\leq\frac{1}{\lambda}c\psi(\lambda x),~~~x\in
{\mathcal X}\\
&\Rightarrow \|\frac{1}{\lambda}g(\lambda x)-\frac{1}{\lambda}h(\lambda x)\|
\leq Lc\psi(x),~~~x\in {\mathcal X}\\
&\Rightarrow d(Jg,Jh)\leq Lc.
\end{split}$$

Therefore $$d(Jg,Jh)\leq Ld(g,h),~~~g,h\in {\mathcal S}.$$ Hence $J$ is a strictly contractive mapping on ${\mathcal S}$ with the Lipschitz constant $L$.

\[main1\] Let $\varphi:{\mathcal
M}^5 \to [0,\infty)$ be a control function with the property $$\lim_{n\to\infty}\frac{\varphi(2^nx, 2^ny, 2^nu, 2^nv, 2^nw
)}{2^n} = 0,$$ for all $x, y, u, v, w \in {\mathcal M}$. Suppose that $f:{\mathcal M}\to {\mathcal M}$ is a mapping satisfying $f(0) =
0$ and $$\begin{gathered}
\label{f}
\|f(\mu x+y)-\mu f(x)-f(y) + f(\langle u, v\rangle w) -\\
\langle f(u), v\rangle w - \langle u, f(v)\rangle w - \langle u,
v\rangle f(w)\|\leq \varphi(x, y, u, v, w),\end{gathered}$$ for all $\mu \in {\mathbb T} = \{z\in {\mathbb C}: |z| = 1\}$ and $x, y, u, v, w \in {\mathcal M}$. Assume that there exists $0
\leq L_0 <1$ such that the mapping $\psi(x) =
\varphi(\frac{x}{2}, \frac{x}{2}, 0, 0, 0)$ has the property $$\begin{aligned}
\label{property1}
\psi(x)\leq 2L_0\psi(\frac{x}{2}),\end{aligned}$$ for all $x \in {\mathcal M}$.\
Then there exists a unique derivation $T_0:{\mathcal M}\to
{\mathcal M}$ such that $$\begin{aligned}
\|f(x)-T_0(x)\|\leq \frac{L_0}{1-L_0}\psi(x),\end{aligned}$$ for all $x\in {\mathcal M}$.

Setting $\mu = 1, y = x$ and $u = v = w = 0$ in (\[f\]) we obtain $$\label{2f}
\|f(2x)- 2f(x)\|\leq \varphi(x, x, 0, 0, 0).$$ It follows from (\[property1\]) and (\[2f\]) that $$\|\frac{1}{2}f(2x)-f(x)\|\leq \frac{1}{2}\psi(2x) \leq L_0\psi(x).$$ Hence $d(f,J_0f)\leq L_0 < \infty$, where the mapping $J_0$ is defined on ${\mathcal S}: = \{g:{\mathcal M}\to {\mathcal M}:
g(0) = 0\}$ by $(J_0g)(x): = \frac{1}{2}g(2x)$ as in Lemma \[lemma\]. Applying the fixed point alternative we deduce the existence of a mapping $T_0: {\mathcal M}\to {\mathcal M}$ such that $T_0$ is a fixed point of $J_0$ that is $T_0(2x) = 2T_0(x)$ for all $x \in {\mathcal M}$. Since $\displaystyle{\lim_{n \to
\infty}}d(J_0^nf,T_0) = 0$ we easily conclude that $$\lim_{n\to\infty}\frac{f(2^nx)}{2^n} = T_0(x),$$ for all $x \in {\mathcal M}$. Note that the sequence $\{\frac{f(2^nx)}{2^n}\}$ is the Hyers sequence when one use the direct method in establishing stability.

The mapping $T_0$ is the unique fixed point of $J_0$ in the set $U = \{g\in {\mathcal S}: d(f,g) < \infty\}$. Hence $T_0$ is the unique fixed point of $J_0$ such that $\|f(x) - T_0(x)\|\leq K
\psi(x)$ for some $K > 0$ and for all $x \in {\mathcal M}$. Again, by applying the fixed point alternative theorem we obtain $$d(f,T_0)\leq\frac{1}{1-L_0}d(f,J_0f)\leq\frac{L_0}{1-L_0},$$ and so $$\|f(x) - T_0(x)\| \leq \frac{L_0}{1-L_0}\varphi(\frac{x}{2}, \frac{x}{2}, 0, 0, 0),$$ for all $x\in {\mathcal M}$.

It follows from (\[f\]) that $$\label{fphi}
\|f(\mu x + y) - \mu f(x) - f(y)\|\leq \varphi(x, y, 0, 0, 0).$$ Let us replace $x$ and $y$ in (\[fphi\]) by $2^nx$ and $2^ny$, respectively, and divide the both sides by $2^n$. Passing the limit as $n \to \infty$ we get $$T_0(\mu x+y) = \mu T_0(x)+T_0(y),$$ for all $\mu \in {\mathbb T}$ and all $x, y \in {\mathcal M}$.

Next, let $\lambda\in {\mathbb C}$ ($\lambda \neq 0$) and let $K$ be a natural number greater than $4|\lambda|$. Then $|\frac{\lambda}{K}| < \frac{1}{4} < 1 - \frac{2}{3} = 1/3$. By Theorem 1 of [@K-P], there exist three numbers $\mu_1, \mu_2,
\mu_3\in {\mathbb T}$ such that $3\frac{\lambda}{K} = \mu_1 +
\mu_2 + \mu_3$. By the additivity of $T_0$ we get $T_0(\frac{1}{3}x) = \frac{1}{3}T_0(x)$ for all $ x\in {\mathcal
M}$. Therefore, $$\begin{aligned}
T_0(\lambda x) &=& T_0(\frac{K}{3}\cdot 3 \cdot
\frac{\lambda}{K}x) = \frac{K}{3}T_0(3\cdot \frac{\lambda}{K}x)\\
&=& \frac{K}{3}T_0(\mu_1x+\mu_2x+\mu_3x) = \frac{K}{3}(T_0(\mu_1x)+T_0(\mu_2x)+T_0(\mu_3x))\\
&=& \frac{K}{3}(\mu_1+\mu_2+\mu_3)T_0(x)\\
&=& \lambda T_0(x),\end{aligned}$$ for all $x \in {\mathcal M}$. So that $T_0$ is ${\mathbb
C}$-linear.

Putting $x = y = 0$ and replacing $u, v, w$ by $2^nu, 2^nv, 2^nw$ in (\[f\]) we get $$\begin{gathered}
\|2^{-3n}f(2^{3n}\langle u, v\rangle w) -
\langle 2^{-n}f(2^nu), v\rangle w \\
- \langle u, 2^{-n}f(2^nv)\rangle w - \langle u, v\rangle
2^{-n}f(2^nw)\| \leq 2^{-3n}\varphi(0, 0, 2^nu, 2^nv, 2^nw).\end{gathered}$$ Letting $n$ tend to infinity we obtain $$T_0(\langle u, v\rangle w) = \langle T_0(u),
v\rangle w + \langle u, T_0(v)\rangle w + \langle u, v\rangle
T_0(w),$$ for all $u, v, w \in {\mathcal M}$. Hence $T_0$ is a derivation on ${\mathcal M}$.

As a consequence of Theorem \[main1\], we show the Rassias stability of derivations on Hilbert $C^*$-modules.

\[c1\] Suppose that $p \in [0, 1)$, $\alpha, \beta, \gamma >0$ and $f:{\mathcal M} \to {\mathcal M}$ is a mapping satisfying $f(0) =
0$ and $$\begin{split}
\|f(\mu x + y) - \mu f(x) - f(y) - f(\langle u, v\rangle w) +
\langle f(u), v\rangle w - \langle u, f(v)\rangle w - \langle u,
v\rangle f(w)\|\\
\leq \alpha + \beta(\|x\|^p + \|y\|^p + \|u\|^p + \|v\|^p +
\|w\|^p)\\
+ \gamma
\|x\|^{p/2}\;\|y\|^{p/2}\;\|u\|^{p/2}\;\|v\|^{p/2}\;\|w\|^{p/2},
\end{split}$$ for all $\mu \in {\mathbb T}$ and all $x, y, u, v, w \in
{\mathcal M}$. Then there exists a unique derivation $T_0:{\mathcal M}\to {\mathcal M}$ such that $$\|f(x)-T_0(x)\|\leq \frac{\alpha + \beta 2^{1-p}\|x\|^p + \gamma
2^{-p}\|x\|^p}{2^{1-p} - 1},$$ for all $x \in {\mathcal M}$.

Put $\varphi(x, y, u, v, w)  = \alpha + \beta(\|x\|^p + \|y\|^p +
\|u\|^p + \|v\|^p + \|w\|^p) + \gamma
\|x\|^{p/2}\;\|y\|^{p/2}\;\|u\|^{p/2}\;\|v\|^{p/2}\;\|w\|^{p/2}$, and let $L_0 = \frac{1}{2^{1-p}}$ in Theorem \[main1\]. Then $\psi(x) = \alpha + 2^{1-p}\beta\|x\|^p + 2^{-p}\gamma\|x\|^p$ and there exists a derivation $T_0$ with the required property.

\[main2\] Let $\varphi:{\mathcal
M}^5 \to [0,\infty)$ be a control function with the property $$\lim_{n\to\infty} 2^n \varphi(2^{-n}x, 2^{-n}y, 2^{-n}u, 2^{-n}v,
2^{-n}w ) = 0,$$ for all $x, y, u, v, w \in {\mathcal M}$. Suppose that the mapping $f:{\mathcal M}\to {\mathcal M}$ is a mapping satisfying $f(0) =
0$ and $$\begin{gathered}
\label{f2}
\|f(\mu x+y)-\mu f(x)-f(y) + f(\langle u, v\rangle w) -\\
\langle f(u), v\rangle w - \langle u, f(v)\rangle w - \langle u,
v\rangle f(w)\|\leq \varphi(x, y, u, v, w),\end{gathered}$$ for all $\mu \in {\mathbb T}$ and $x, y, u, v, w \in {\mathcal
M}$. Assume that there exists $0 \leq L_1 < 1$ such that the mapping $\psi(x) = \varphi(\frac{x}{2}, \frac{x}{2}, 0, 0, 0)$ has the property $$\begin{aligned}
\psi(x)\leq \frac{1}{2}L_1\psi(2x),\end{aligned}$$ for all $x \in {\mathcal M}$.\
Then there exists a unique derivation $T_1:{\mathcal M}\to
{\mathcal M}$ such that $$\begin{aligned}
\|f(x)-T_1(x)\|\leq \frac{1}{1-L_1}\psi(x),\end{aligned}$$ for all $x\in {\mathcal M}$.

Putting $\mu = 1, y = x$ and $u = v = w = 0$ in (\[f2\]) we get $$\label{2f2}
\|f(2x)- 2f(x)\|\leq \varphi(x, x, 0, 0, 0).$$ Replacing $x$ by $\frac{x}{2}$ in (\[2f2\]), we obtain $$\|f(x)-2f(\frac{x}{2})\|\leq \psi(x).$$ Hence $d(f,J_1f)\leq L_1 < \infty$, where the mapping $J_1$ is defined on ${\mathcal S}: = \{g:{\mathcal M}\to {\mathcal M}:
g(0) = 0\}$ by $(J_1g)(x): = 2g(\frac{1}{2}x)$ as in Lemma \[lemma\]. Applying the fixed point alternative we deduce the existence of a mapping $T_1: {\mathcal M}\to {\mathcal M}$ such that $T_1$ is a fixed point of $J_1$ that is $T_1(\frac{1}{2}x) =
\frac{1}{2}T_1(x)$ for all $x \in {\mathcal M}$. Since $\displaystyle{\lim_{n \to \infty}}d(J_1^nf,T_1) = 0$ we easily conclude that $$\lim_{n\to\infty} 2^n f(2^{-n}x) = T_1(x),$$ for all $x \in {\mathcal M}$.

The mapping $T_1$ is the unique fixed point of $J_1$ in the set $U = \{g\in {\mathcal S}: d(f,g) < \infty\}$. Hence $T_1$ is the unique fixed point of $J_1$ such that $\|f(x) - T_1(x)\|\leq K
\psi(x)$ for some $K > 0$ and for all $x \in {\mathcal M}$. Again, by applying the fixed point alternative theorem we obtain $$d(f,T_1)\leq \frac{1}{1-L_1}d(f,J_1f)\leq \frac{1}{1-L_1},$$ and so $$\|f(x) - T_1(x)\| \leq \frac{1}{1-L_1}\varphi(\frac{x}{2}, \frac{x}{2}, 0, 0, 0),$$ for all $x\in {\mathcal M}$. The rest is similar to the proof of Theorem \[main1\].

The following corollary is similar to Corollary \[c1\] for the case where $p > 1$.

Suppose that $p > 1$, $\beta, \gamma >0$ and $f:{\mathcal M} \to {\mathcal M}$ is a mapping satisfying $f(0) =
0$ and $$\begin{split}
\|f(\mu x+y)-\mu f(x)-f(y) + f(\langle u, v\rangle w) - \langle
f(u), v\rangle w - \langle u, f(v)\rangle w - \langle u,
v\rangle f(w)\|\\
\leq \beta(\|x\|^p + \|y\|^p + \|u\|^p + \|v\|^p + \|w\|^p) +
\gamma
\|x\|^{p/2}\;\|y\|^{p/2}\;\|u\|^{p/2}\;\|v\|^{p/2}\;\|w\|^{p/2},
\end{split}$$ for all $\mu \in {\mathbb T}$ and all $x, y, u, v, w \in
{\mathcal M}$. Then there exists a unique derivation $T_1 :
{\mathcal M} \to {\mathcal M}$ such that $$\|f(x)-T_1(x)\|\leq \frac{(\beta + \gamma/2)\|x\|^p}{2^{p-1}-1},$$ for all $x\in {\mathcal M}$.

Put $\varphi(x, y, u, v, w)  = \alpha + \beta(\|x\|^p + \|y\|^p +
\|u\|^p + \|v\|^p + \|w\|^p) + \gamma
\|x\|^{p/2}\;\|y\|^{p/2}\;\|u\|^{p/2}\;\|v\|^{p/2}\;\|w\|^{p/2}$, and let $L_1 = \frac{1}{2^{p-1}}$ in Theorem \[main2\]. Then $\psi(x) = 2^{1-p}\beta\|x\|^p + 2^{-p}\gamma\|x\|^p$ and there exists a derivation $T_1$ with the required property.

The case $p = 1$ remains unsolved.

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[^1]: The authors would like to sincerely thank Professor Marina Haralampidou for the great hospitality during their stay in Greece.
