# Functional equations and inequalities in matrix paranormed spaces

## Abstract

In this paper, we prove the Hyers-Ulam stability of the Cauchy additive functional inequality, the Cauchy additive functional equation and the quadratic functional equation in matrix paranormed spaces.

MSC:47L25, 39B82, 39B72, 46L07, 39B52, 39B62.

## 1 Introduction and preliminaries

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently, and since then several generalizations and applications of this notion have been investigated by various authors (see [3â€“7]). This notion was defined in normed spaces by Kolk [8].

We recall some basic facts concerning FrÃ©chet spaces.

Definition 1.1 [9]

Let X be a vector space. A paranorm $P\left(â‹\dots \right):Xâ†’\left[0,\mathrm{âˆž}\right)$ is a function on X such that

1. (1)

$P\left(0\right)=0$;

2. (2)

$P\left(âˆ’x\right)=P\left(x\right)$;

3. (3)

$P\left(x+y\right)â‰¤P\left(x\right)+P\left(y\right)$ (triangle inequality);

4. (4)

If $\left\{{t}_{n}\right\}$ is a sequence of scalars with ${t}_{n}â†’t$ and $\left\{{x}_{n}\right\}âŠ‚X$ with $P\left({x}_{n}âˆ’x\right)â†’0$, then $P\left({t}_{n}{x}_{n}âˆ’tx\right)â†’0$ (continuity of multiplication).

The pair $\left(X,P\left(â‹\dots \right)\right)$ is called a paranormed space if $P\left(â‹\dots \right)$ is a paranorm on X.

The paranorm is called total if, in addition, we have

1. (5)

$P\left(x\right)=0$ implies $x=0$.

A FrÃ©chet space is a total and complete paranormed space.

The stability problem of functional equations originated from the question of Ulam [10] concerning the stability of group homomorphisms.

The functional equation

$f\left(x+y\right)=f\left(x\right)+f\left(y\right)$

is called the Cauchy additive functional equation. In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping. Hyers [11] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyersâ€™ theorem was generalized by Aoki [12] for additive mappings and by Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by GÄƒvruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassiasâ€™ approach.

In 1990, during the 27th International Symposium on Functional Equations, Rassias [15] asked the question whether such a theorem can also be proved for $pâ‰¥1$. In 1991, Gajda [16], following the same approach as in Rassias [13], gave an affirmative solution to this question for $p>1$. It was shown by Gajda [16], as well as by Rassias and Å emrl [17], that one cannot prove a Rassias-type theorem when $p=1$ (cf. the books of Czerwik [18] and Hyers et al. [19]).

The functional equation

$f\left(x+y\right)+f\left(xâˆ’y\right)=2f\left(x\right)+2f\left(y\right)$

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [20] for mappings $f:Xâ†’Y$, where X is a normed space and Y is a Banach space. Cholewa [21] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [22] proved the Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [23â€“27]).

In [28], GilÃ¡nyi showed that if f satisfies the functional inequality

$âˆ¥2f\left(x\right)+2f\left(y\right)âˆ’f\left(x{y}^{âˆ’1}\right)âˆ¥â‰¤âˆ¥f\left(xy\right)âˆ¥,$
(1.1)

then f satisfies the Jordan-von Neumann functional equation

$2f\left(x\right)+2f\left(y\right)=f\left(xy\right)+f\left(x{y}^{âˆ’1}\right).$

See also [29]. GilÃ¡nyi [30] and Fechner [31] proved the Hyers-Ulam stability of functional inequality (1.1).

Park et al. [32] proved the Hyers-Ulam stability of the following functional inequality:

$âˆ¥f\left(x\right)+f\left(y\right)+f\left(z\right)âˆ¥â‰¤âˆ¥f\left(x+y+z\right)âˆ¥.$
(1.2)

The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces [33] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory (see [34]).

The proof given in [33] appealed to the theory of ordered operator spaces [35]. Effros and Ruan [36] showed that one can give a purely metric proof of this important theorem by using the technique of Pisier [37] and Haagerup [38] (as modified in [39]).

We will use the following notations:

${M}_{n}\left(X\right)$ is the set of all $nÃ—n$-matrices in X;

${e}_{j}âˆˆ{M}_{1,n}\left(\mathbb{C}\right)$ is that j th component is 1 and the other components are zero;

${E}_{ij}âˆˆ{M}_{n}\left(\mathbb{C}\right)$ is that $\left(i,j\right)$-component is 1 and the other components are zero;

${E}_{ij}âŠ—xâˆˆ{M}_{n}\left(X\right)$ is that $\left(i,j\right)$-component is x and the other components are zero.

For $xâˆˆ{M}_{n}\left(X\right)$, $yâˆˆ{M}_{k}\left(X\right)$,

$xâŠ•y=\left(\begin{array}{cc}x& 0\\ 0& y\end{array}\right).$

Note that $\left(X,\left\{{âˆ¥â‹\dots âˆ¥}_{n}\right\}\right)$ is a matrix normed space if and only if $\left({M}_{n}\left(X\right),{âˆ¥â‹\dots âˆ¥}_{n}\right)$ is a normed space for each positive integer n and ${âˆ¥AxBâˆ¥}_{k}â‰¤âˆ¥Aâˆ¥âˆ¥Bâˆ¥{âˆ¥xâˆ¥}_{n}$ holds for $Aâˆˆ{M}_{k,n}$, $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$ and $Bâˆˆ{M}_{n,k}$, and that $\left(X,\left\{{âˆ¥â‹\dots âˆ¥}_{n}\right\}\right)$ is a matrix Banach space if and only if X is a Banach space and $\left(X,\left\{{âˆ¥â‹\dots âˆ¥}_{n}\right\}\right)$ is a matrix normed space.

Definition 1.2 Let $\left(X,P\left(â‹\dots \right)\right)$ be a paranormed space.

1. (1)

$\left(X,\left\{{P}_{n}\left(â‹\dots \right)\right\}\right)$ is a matrix paranormed space if $\left({M}_{n}\left(X\right),{P}_{n}\left(â‹\dots \right)\right)$ is a paranormed space for each positive integer n, ${P}_{n}\left({E}_{kl}âŠ—x\right)=P\left(x\right)$ for $xâˆˆX$, and $P\left({x}_{kl}\right)â‰¤{P}_{n}\left(\left[{x}_{ij}\right]\right)$ for $\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$.

2. (2)

$\left(X,\left\{{P}_{n}\left(â‹\dots \right)\right\}\right)$ is a matrix FrÃ©chet space if X is a FrÃ©chet space and $\left(X,\left\{{P}_{n}\left(â‹\dots \right)\right\}\right)$ is a matrix paranormed space.

Let E, F be vector spaces. For a given mapping $h:Eâ†’F$ and a given positive integer n, define ${h}_{n}:{M}_{n}\left(E\right)â†’{M}_{n}\left(F\right)$ by

${h}_{n}\left(\left[{x}_{ij}\right]\right)=\left[h\left({x}_{ij}\right)\right]$

for all $\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(E\right)$.

In Section 2, we prove the Hyers-Ulam stability of Cauchy additive functional inequality (1.2) in matrix paranormed spaces. In Section 3, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in matrix paranormed spaces. In Section 4, we prove the Hyers-Ulam stability of the quadratic functional equation in matrix paranormed spaces.

Throughout this paper, let $\left(X,\left\{{âˆ¥â‹\dots âˆ¥}_{n}\right\}\right)$ be a matrix Banach space and $\left(Y,\left\{{P}_{n}\left(â‹\dots \right)\right\}\right)$ be a matrix FrÃ©chet space.

## 2 Hyers-Ulam stability of additive functional inequality (1.2) in matrix paranormed spaces

In this section, we prove the Hyers-Ulam stability of additive functional inequality (1.2) in matrix paranormed spaces.

Lemma 2.1 Let $\left(X,\left\{{P}_{n}\left(â‹\dots \right)\right\}\right)$ be a matrix paranormed space. Then

1. (1)

$P\left({x}_{kl}\right)â‰¤{P}_{n}\left(\left[{x}_{ij}\right]\right)â‰¤{âˆ‘}_{i,j=1}^{n}P\left({x}_{ij}\right)$ for $\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$;

2. (2)

${lim}_{sâ†’\mathrm{âˆž}}{x}_{s}=x$ if and only if ${lim}_{sâ†’\mathrm{âˆž}}{x}_{sij}={x}_{ij}$ for ${x}_{s}=\left[{x}_{sij}\right],x=\left[{x}_{ij}\right]âˆˆ{M}_{k}\left(X\right)$.

Proof (1) By Definition 1.2, $P\left({x}_{kl}\right)â‰¤{P}_{n}\left(\left[{x}_{ij}\right]\right)$.

Since $\left[{x}_{ij}\right]={âˆ‘}_{i,j=1}^{n}{E}_{ij}âŠ—{x}_{ij}$,

${P}_{n}\left(\left[{x}_{ij}\right]\right)={P}_{n}\left(\underset{i,j=1}{\overset{n}{âˆ‘}}{E}_{ij}âŠ—{x}_{ij}\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}{P}_{n}\left({E}_{ij}âŠ—{x}_{ij}\right)=\underset{i,j=1}{\overset{n}{âˆ‘}}P\left({x}_{ij}\right).$
1. (2)

By (1), we have

$P\left({x}_{skl}âˆ’{x}_{kl}\right)â‰¤{P}_{n}\left(\left[{x}_{sij}âˆ’{x}_{ij}\right]\right)={P}_{n}\left(\left[{x}_{sij}\right]âˆ’\left[{x}_{ij}\right]\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}P\left({x}_{sij}âˆ’{x}_{ij}\right).$

So, we get the result.â€ƒâ–¡

Lemma 2.2 Let $\left(X,\left\{{âˆ¥â‹\dots âˆ¥}_{n}\right\}\right)$ be a matrix normed space. Then

(1) ${âˆ¥{E}_{kl}âŠ—xâˆ¥}_{n}=âˆ¥xâˆ¥$ for $xâˆˆX$;

(2) $âˆ¥{x}_{kl}âˆ¥â‰¤{âˆ¥\left[{x}_{ij}\right]âˆ¥}_{n}â‰¤{âˆ‘}_{i,j=1}^{n}âˆ¥{x}_{ij}âˆ¥$ for $\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$;

(3) ${lim}_{nâ†’\mathrm{âˆž}}{x}_{n}=x$ if and only if ${lim}_{nâ†’\mathrm{âˆž}}{x}_{ijn}={x}_{ij}$ for ${x}_{n}=\left[{x}_{ijn}\right],x=\left[{x}_{ij}\right]âˆˆ{M}_{k}\left(X\right)$.

Proof (1) Since ${E}_{kl}âŠ—x={e}_{k}^{âˆ—}x{e}_{l}$ and $âˆ¥{e}_{k}^{âˆ—}âˆ¥=âˆ¥{e}_{l}âˆ¥=1$, ${âˆ¥{E}_{kl}âŠ—xâˆ¥}_{n}â‰¤âˆ¥xâˆ¥$. Since ${e}_{k}\left({E}_{kl}âŠ—x\right){e}_{l}^{âˆ—}=x$, $âˆ¥xâˆ¥â‰¤{âˆ¥{E}_{kl}âŠ—xâˆ¥}_{n}$. So, ${âˆ¥{E}_{kl}âŠ—xâˆ¥}_{n}=âˆ¥xâˆ¥$.

1. (2)

Since ${e}_{k}x{e}_{l}^{âˆ—}={x}_{kl}$ and $âˆ¥{e}_{k}âˆ¥=âˆ¥{e}_{l}^{âˆ—}âˆ¥=1$, $âˆ¥{x}_{kl}âˆ¥â‰¤{âˆ¥\left[{x}_{ij}\right]âˆ¥}_{n}$. Since $\left[{x}_{ij}\right]={âˆ‘}_{i,j=1}^{n}{E}_{ij}âŠ—{x}_{ij}$,

${âˆ¥\left[{x}_{ij}\right]âˆ¥}_{n}={âˆ¥\underset{i,j=1}{\overset{n}{âˆ‘}}{E}_{ij}âŠ—{x}_{ij}âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}{âˆ¥{E}_{ij}âŠ—{x}_{ij}âˆ¥}_{n}=\underset{i,j=1}{\overset{n}{âˆ‘}}âˆ¥{x}_{ij}âˆ¥.$
2. (3)

By

$âˆ¥{x}_{kln}âˆ’{x}_{kl}âˆ¥â‰¤{âˆ¥\left[{x}_{ijn}âˆ’{x}_{ij}\right]âˆ¥}_{n}={âˆ¥\left[{x}_{ijn}\right]âˆ’\left[{x}_{ij}\right]âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}âˆ¥{x}_{ijn}âˆ’{x}_{ij}âˆ¥,$

we get the result.â€ƒâ–¡

We need the following result.

Lemma 2.3 Let $f:Xâ†’Y$ be an odd mapping such that

$P\left(f\left(a\right)+f\left(b\right)+f\left(c\right)\right)â‰¤P\left(f\left(a+b+c\right)\right)$
(2.1)

for all $a,b,câˆˆX$. Then $f:Xâ†’Y$ is additive.

Proof Letting $c=âˆ’aâˆ’b$ in (2.1), we get $P\left(f\left(a\right)+f\left(b\right)+f\left(âˆ’aâˆ’b\right)\right)â‰¤P\left(f\left(0\right)\right)=0$ for all $a,bâˆˆX$. So,

$f\left(a\right)+f\left(b\right)âˆ’f\left(a+b\right)=f\left(a\right)+f\left(b\right)+f\left(âˆ’aâˆ’b\right)=0$

for all $a,bâˆˆX$. Thus $f:Xâ†’Y$ is additive.â€ƒâ–¡

Note that $P\left(2x\right)â‰¤2P\left(x\right)$ for all $xâˆˆY$.

Theorem 2.4 Let r, Î¸ be positive real numbers with $r>1$. Let $f:Xâ†’Y$ be an odd mapping such that

$\begin{array}{rcl}{P}_{n}\left({f}_{n}\left(\left[{x}_{ij}\right]\right)+{f}_{n}\left(\left[{y}_{ij}\right]\right)+{f}_{n}\left(\left[{z}_{ij}\right]\right)\right)& â‰¤& {P}_{n}\left({f}_{n}\left(\left[{x}_{ij}\right]+\left[{y}_{ij}\right]+\left[{z}_{ij}\right]\right)\right)\\ +\underset{i,j=1}{\overset{n}{âˆ‘}}\mathrm{Î¸}\left({âˆ¥{x}_{ij}âˆ¥}^{r}+{âˆ¥{y}_{ij}âˆ¥}^{r}+{âˆ¥{z}_{ij}âˆ¥}^{r}\right)\end{array}$
(2.2)

for all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right],z=\left[{z}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$. Then there exists a unique additive mapping $A:Xâ†’Y$ such that

${P}_{n}\left({f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{{2}^{r}+2}{{2}^{r}âˆ’2}\mathrm{Î¸}{âˆ¥{x}_{ij}âˆ¥}^{r}$
(2.3)

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$.

Proof When $n=1$, (2.2) is equivalent to

$P\left(f\left(a\right)+f\left(b\right)+f\left(c\right)\right)â‰¤P\left(f\left(a+b+c\right)\right)+\mathrm{Î¸}\left({âˆ¥aâˆ¥}^{r}+{âˆ¥bâˆ¥}^{r}+{âˆ¥câˆ¥}^{r}\right)$
(2.4)

for all $a,b,câˆˆX$.

Letting $b=a$ and $c=âˆ’2a$ in (2.4), we get

$P\left(f\left(2a\right)âˆ’2f\left(a\right)\right)â‰¤\left(2+{2}^{r}\right)\mathrm{Î¸}{âˆ¥aâˆ¥}^{r},$

and so

$P\left(f\left(a\right)âˆ’2f\left(\frac{a}{2}\right)\right)â‰¤\frac{2+{2}^{r}}{{2}^{r}}\mathrm{Î¸}{âˆ¥aâˆ¥}^{r}$

for all $a,bâˆˆX$.

One can easily show that

$P\left({2}^{p}f\left(\frac{a}{{2}^{p}}\right)âˆ’{2}^{q}f\left(\frac{a}{{2}^{q}}\right)\right)â‰¤\underset{l=p}{\overset{qâˆ’1}{âˆ‘}}\frac{\left(2+{2}^{r}\right){2}^{l}}{{2}^{\left(l+1\right)r}}\mathrm{Î¸}{âˆ¥aâˆ¥}^{r}$
(2.5)

for all $a,bâˆˆX$ and nonnegative integers p, q with $p. It follows from (2.5) that the sequence $\left\{{2}^{l}f\left(\frac{a}{{2}^{l}}\right)\right\}$ is Cauchy for all $aâˆˆX$. Since Y is complete, the sequence $\left\{{2}^{l}f\left(\frac{a}{{2}^{l}}\right)\right\}$ converges. So, one can define the mapping $A:Xâ†’Y$ by

$A\left(a\right)=\underset{lâ†’\mathrm{âˆž}}{lim}{2}^{l}f\left(\frac{a}{{2}^{l}}\right)$

for all $aâˆˆX$.

Moreover, letting $p=0$ and passing the limit $qâ†’\mathrm{âˆž}$ in (2.5), we get

$P\left(f\left(a\right)âˆ’A\left(a\right)\right)â‰¤\frac{{2}^{r}+2}{{2}^{r}âˆ’2}\mathrm{Î¸}{âˆ¥aâˆ¥}^{r}$
(2.6)

for all $aâˆˆX$.

It follows from (2.4) that

$\begin{array}{c}P\left({2}^{l}\left(f\left(\frac{a}{{2}^{l}}\right)+f\left(\frac{b}{{2}^{l}}\right)+f\left(\frac{c}{{2}^{l}}\right)\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}â‰¤{2}^{l}P\left(f\left(\frac{a+b+c}{{2}^{l}}\right)\right)+\frac{{2}^{l}}{{2}^{lr}}\mathrm{Î¸}\left({âˆ¥aâˆ¥}^{r}+{âˆ¥bâˆ¥}^{r}+{âˆ¥câˆ¥}^{r}\right)\hfill \end{array}$

for all $a,b,câˆˆX$. Passing the limit $lâ†’\mathrm{âˆž}$ in the above inequality, we get

$P\left(A\left(a\right)+A\left(b\right)+A\left(c\right)\right)â‰¤P\left(A\left(a+b+c\right)\right)$

for all $a,b,câˆˆX$. Since $f:Xâ†’Y$ is an odd mapping, the mapping $A:Xâ†’Y$ is odd. By Lemma 2.3, $A:Xâ†’Y$ is additive.

Now, let $T:Xâ†’Y$ be another additive mapping satisfying (2.6). Then we have

$\begin{array}{rcl}P\left(A\left(a\right)âˆ’T\left(a\right)\right)& =& P\left({2}^{q}A\left(\frac{a}{{2}^{q}}\right)âˆ’{2}^{q}T\left(\frac{a}{{2}^{q}}\right)\right)\\ â‰¤& P\left({2}^{q}\left(A\left(\frac{a}{{2}^{q}}\right)âˆ’g\left(\frac{a}{{2}^{q}}\right)\right)\right)+P\left({2}^{q}\left(T\left(\frac{a}{{2}^{q}}\right)âˆ’g\left(\frac{a}{{2}^{q}}\right)\right)\right)\\ â‰¤& 2\frac{{2}^{r}+2}{{2}^{r}âˆ’2}\frac{{2}^{q}}{{2}^{qr}}\mathrm{Î¸}{âˆ¥aâˆ¥}^{r},\end{array}$

which tends to zero as $qâ†’\mathrm{âˆž}$ for all $aâˆˆX$. So, we can conclude that $A\left(a\right)=T\left(a\right)$ for all $aâˆˆX$. This proves the uniqueness of A.

By Lemma 2.1 and (2.6),

${P}_{n}\left({f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}P\left(f\left({x}_{ij}\right)âˆ’A\left({x}_{ij}\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{\left(2+{2}^{r}\right)}{{2}^{r}âˆ’2}\mathrm{Î¸}{âˆ¥{x}_{ij}âˆ¥}^{r}$

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$. Thus $A:Xâ†’Y$ is a unique additive mapping satisfying (2.3), as desired.â€ƒâ–¡

Theorem 2.5 Let r, Î¸ be positive real numbers with $r<1$. Let $f:Yâ†’X$ be an odd mapping such that

$\begin{array}{rcl}{âˆ¥{f}_{n}\left(\left[{x}_{ij}\right]\right)+{f}_{n}\left(\left[{y}_{ij}\right]\right)+{f}_{n}\left(\left[{z}_{ij}\right]\right)âˆ¥}_{n}& â‰¤& {âˆ¥{f}_{n}\left(\left[{x}_{ij}\right]+\left[{y}_{ij}\right]+\left[{z}_{ij}\right]\right)âˆ¥}_{n}\\ +\underset{i,j=1}{\overset{n}{âˆ‘}}\mathrm{Î¸}\left(P{\left({x}_{ij}\right)}^{r}+P{\left({y}_{ij}\right)}^{r}+P{\left({z}_{ij}\right)}^{r}\right)\end{array}$
(2.7)

for all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right],z=\left[{z}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$. Then there exists a unique additive mapping $A:Yâ†’X$ such that

${âˆ¥{f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2+{2}^{r}}{2âˆ’{2}^{r}}\mathrm{Î¸}P{\left({x}_{ij}\right)}^{r}$
(2.8)

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$.

Proof Let $n=1$ in (2.7). Then (2.7) is equivalent to

$âˆ¥f\left(a\right)+f\left(b\right)+f\left(c\right)âˆ¥â‰¤âˆ¥f\left(a+b+c\right)âˆ¥+\mathrm{Î¸}\left(P{\left(a\right)}^{r}+P{\left(b\right)}^{r}+P{\left(c\right)}^{r}\right)$
(2.9)

for all $a,b,câˆˆY$.

Letting $b=a$ and $c=âˆ’2a$ in (2.9), we get

$âˆ¥f\left(2a\right)âˆ’2f\left(a\right)âˆ¥â‰¤\left(2+{2}^{r}\right)\mathrm{Î¸}P{\left(a\right)}^{r},$

and so

$âˆ¥f\left(a\right)âˆ’\frac{1}{2}f\left(2a\right)âˆ¥â‰¤\frac{2+{2}^{r}}{2}\mathrm{Î¸}P{\left(a\right)}^{r}$

for all $aâˆˆY$.

One can easily show that

$âˆ¥\frac{1}{{2}^{p}}f\left({2}^{p}a\right)âˆ’\frac{1}{{2}^{q}}f\left({2}^{q}a\right)âˆ¥â‰¤\underset{l=p}{\overset{qâˆ’1}{âˆ‘}}\frac{{2}^{lr}}{{2}^{l}}\frac{2+{2}^{r}}{2}\mathrm{Î¸}P{\left(a\right)}^{r}$
(2.10)

for all $aâˆˆY$ and nonnegative integers p, q with $p. It follows from (2.10) that the sequence $\left\{\frac{1}{{2}^{l}}f\left({2}^{l}a\right)\right\}$ is Cauchy for all $aâˆˆY$. Since X is complete, the sequence $\left\{\frac{1}{{2}^{l}}f\left({2}^{l}a\right)\right\}$ converges. So, one can define the mapping $A:Yâ†’X$ by

$A\left(a\right)=\underset{lâ†’\mathrm{âˆž}}{lim}\frac{1}{{2}^{l}}f\left({2}^{l}a\right)$

for all $aâˆˆY$.

Moreover, letting $p=0$ and passing the limit $qâ†’\mathrm{âˆž}$ in (2.10), we get

$âˆ¥f\left(a\right)âˆ’A\left(a\right)âˆ¥â‰¤\frac{2+{2}^{r}}{2âˆ’{2}^{r}}\mathrm{Î¸}P{\left(a\right)}^{r}$
(2.11)

for all $aâˆˆY$.

It follows from (2.9) that

$âˆ¥\frac{1}{{2}^{l}}\left(f\left({2}^{l}a\right)+f\left({2}^{l}b\right)+f\left({2}^{l}c\right)\right)âˆ¥â‰¤âˆ¥\frac{1}{{2}^{l}}f\left({2}^{l}\left(a+b+c\right)\right)âˆ¥+\frac{{2}^{lr}}{{2}^{l}}\mathrm{Î¸}\left({âˆ¥aâˆ¥}^{r}+{âˆ¥bâˆ¥}^{r}+{âˆ¥câˆ¥}^{r}\right)$

for all $a,b,câˆˆY$. Passing the limit $lâ†’\mathrm{âˆž}$ in the above inequality, we get

$âˆ¥A\left(a\right)+A\left(b\right)+A\left(c\right)âˆ¥â‰¤âˆ¥A\left(a+b+c\right)âˆ¥$

for all $a,b,câˆˆY$. By [[32], Lemma 3.1], the mapping $A:Yâ†’X$ is additive.

Now, let $T:Yâ†’X$ be another additive mapping satisfying (2.11). Let $n=1$. Then we have

$\begin{array}{rcl}âˆ¥A\left(a\right)âˆ’T\left(a\right)âˆ¥& =& âˆ¥\frac{1}{{2}^{q}}A\left({2}^{q}a\right)âˆ’\frac{1}{{2}^{q}}T\left({2}^{q}a\right)âˆ¥\\ â‰¤& âˆ¥\frac{1}{{2}^{q}}\left(A\left({2}^{q}a\right)âˆ’g\left({2}^{q}a\right)\right)âˆ¥+âˆ¥\frac{1}{{2}^{q}}\left(T\left({2}^{q}a\right)âˆ’g\left({2}^{q}a\right)\right)âˆ¥\\ â‰¤& 2\frac{2+{2}^{r}}{2âˆ’{2}^{r}}\frac{{2}^{qr}}{{2}^{q}}\mathrm{Î¸}P{\left(a\right)}^{r},\end{array}$

which tends to zero as $qâ†’\mathrm{âˆž}$ for all $aâˆˆY$. So, we can conclude that $A\left(a\right)=T\left(a\right)$ for all $aâˆˆY$. This proves the uniqueness of A.

By Lemma 2.2 and (2.11),

${âˆ¥{f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}âˆ¥f\left({x}_{ij}\right)âˆ’A\left({x}_{ij}\right)âˆ¥â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2+{2}^{r}}{{2}^{r}âˆ’2}\mathrm{Î¸}P{\left({x}_{ij}\right)}^{r}$

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$. Thus $A:Yâ†’X$ is a unique additive mapping satisfying (2.8), as desired.â€ƒâ–¡

## 3 Hyers-Ulam stability of the Cauchy additive functional equation in matrix paranormed spaces

In this section, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in matrix paranormed spaces.

For a mapping $f:Xâ†’Y$, define $Df:{X}^{2}â†’Y$ and $D{f}_{n}:{M}_{n}\left({X}^{2}\right)â†’{M}_{n}\left(Y\right)$ by

$\begin{array}{c}Df\left(a,b\right)=f\left(a+b\right)âˆ’f\left(a\right)âˆ’f\left(b\right),\hfill \\ D{f}_{n}\left(\left[{x}_{ij}\right],\left[{y}_{ij}\right]\right):={f}_{n}\left(\left[{x}_{ij}+{y}_{ij}\right]\right)âˆ’{f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{f}_{n}\left(\left[{y}_{ij}\right]\right)\hfill \end{array}$

for all $a,bâˆˆX$ and all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$.

Theorem 3.1 Let r, Î¸ be positive real numbers with $r>1$. Let $f:Xâ†’Y$ be a mapping such that

${P}_{n}\left(D{f}_{n}\left(\left[{x}_{ij}\right],\left[{y}_{ij}\right]\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\mathrm{Î¸}\left({âˆ¥{x}_{ij}âˆ¥}^{r}+{âˆ¥{y}_{ij}âˆ¥}^{r}\right)$
(3.1)

for all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$. Then there exists a unique additive mapping $A:Xâ†’Y$ such that

${P}_{n}\left({f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2\mathrm{Î¸}}{{2}^{r}âˆ’2}{âˆ¥{x}_{ij}âˆ¥}^{r}$
(3.2)

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$.

Proof Let $n=1$ in (3.1). Then (3.1) is equivalent to

$P\left(f\left(a+b\right)âˆ’f\left(a\right)âˆ’f\left(b\right)\right)â‰¤\mathrm{Î¸}\left({âˆ¥aâˆ¥}^{r}+{âˆ¥bâˆ¥}^{r}\right)$
(3.3)

for all $a,bâˆˆX$.

Letting $b=a$ in (3.3), we get

$P\left(f\left(2a\right)âˆ’2f\left(a\right)\right)â‰¤2\mathrm{Î¸}{âˆ¥aâˆ¥}^{r},$

and so

$P\left(f\left(a\right)âˆ’2f\left(\frac{a}{2}\right)\right)â‰¤\frac{2}{{2}^{r}}\mathrm{Î¸}{âˆ¥aâˆ¥}^{r}$

for all $a,bâˆˆX$.

One can easily show that

$P\left({2}^{p}f\left(\frac{a}{{2}^{p}}\right)âˆ’{2}^{q}f\left(\frac{a}{{2}^{q}}\right)\right)â‰¤\underset{l=p}{\overset{qâˆ’1}{âˆ‘}}\frac{2â‹\dots {2}^{l}}{{2}^{\left(l+1\right)r}}\mathrm{Î¸}{âˆ¥aâˆ¥}^{r}$
(3.4)

for all $a,bâˆˆX$ and nonnegative integers p, q with $p. It follows from (3.4) that the sequence $\left\{{2}^{l}f\left(\frac{a}{{2}^{l}}\right)\right\}$ is Cauchy for all $aâˆˆX$. Since Y is complete, the sequence $\left\{{2}^{l}f\left(\frac{a}{{2}^{l}}\right)\right\}$ converges. So, one can define the mapping $A:Xâ†’Y$ by

$A\left(a\right)=\underset{lâ†’\mathrm{âˆž}}{lim}{2}^{l}f\left(\frac{a}{{2}^{l}}\right)$

for all $aâˆˆX$.

Moreover, letting $p=0$ and passing the limit $qâ†’\mathrm{âˆž}$ in (3.4), we get

$P\left(f\left(a\right)âˆ’A\left(a\right)\right)â‰¤\frac{2\mathrm{Î¸}}{{2}^{r}âˆ’2}{âˆ¥aâˆ¥}^{r}$
(3.5)

for all $aâˆˆX$.

It follows from (3.3) that

$\begin{array}{rcl}P\left({2}^{l}\left(f\left(\frac{a+b}{{2}^{l}}\right)âˆ’f\left(\frac{a}{{2}^{l}}\right)âˆ’f\left(\frac{b}{{2}^{l}}\right)\right)\right)& â‰¤& {2}^{l}P\left(f\left(\frac{a+b}{{2}^{l}}\right)âˆ’f\left(\frac{a}{{2}^{l}}\right)âˆ’f\left(\frac{b}{{2}^{l}}\right)\right)\\ â‰¤& \frac{{2}^{l}}{{2}^{lr}}\mathrm{Î¸}\left({âˆ¥aâˆ¥}^{r}+{âˆ¥bâˆ¥}^{r}\right),\end{array}$

which tends to zero as $lâ†’\mathrm{âˆž}$. So, $P\left(A\left(a+b\right)âˆ’A\left(a\right)âˆ’A\left(b\right)\right)=0$, i.e., $A\left(a+b\right)=A\left(a\right)+A\left(b\right)$ for all $a,bâˆˆX$. Hence $A:Xâ†’Y$ is additive.

The proof of the uniqueness of A is similar to the proof of Theorem 2.4.

By Lemma 2.1 and (3.5),

${P}_{n}\left({f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}P\left(f\left({x}_{ij}\right)âˆ’A\left({x}_{ij}\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2\mathrm{Î¸}}{{2}^{r}âˆ’2}{âˆ¥{x}_{ij}âˆ¥}^{r}$

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$. Thus $A:Xâ†’Y$ is a unique additive mapping satisfying (3.2), as desired.â€ƒâ–¡

Theorem 3.2 Let r, Î¸ be positive real numbers with $r<1$. Let $f:Yâ†’X$ be a mapping such that

${âˆ¥D{f}_{n}\left(\left[{x}_{ij}\right],\left[{y}_{ij}\right]\right)âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\mathrm{Î¸}\left(P{\left({x}_{ij}\right)}^{r}+P{\left({y}_{ij}\right)}^{r}\right)$
(3.6)

for all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$. Then there exists a unique additive mapping $A:Yâ†’X$ such that

${âˆ¥{f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2\mathrm{Î¸}}{2âˆ’{2}^{r}}P{\left({x}_{ij}\right)}^{r}$
(3.7)

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$.

Proof Let $n=1$ in (3.6). Then (3.6) is equivalent to

$âˆ¥f\left(a+b\right)âˆ’f\left(a\right)âˆ’f\left(b\right)âˆ¥â‰¤\mathrm{Î¸}\left(P{\left(a\right)}^{r}+P{\left(b\right)}^{r}\right)$
(3.8)

for all $a,bâˆˆY$.

Letting $b=a$ in (3.8), we get

$âˆ¥f\left(2a\right)âˆ’2f\left(a\right)âˆ¥â‰¤2\mathrm{Î¸}P{\left(a\right)}^{r},$

and so

$âˆ¥f\left(a\right)âˆ’\frac{1}{2}f\left(2a\right)âˆ¥â‰¤\mathrm{Î¸}P{\left(a\right)}^{r}$

for all $aâˆˆY$.

One can easily show that

$âˆ¥\frac{1}{{2}^{p}}f\left({2}^{p}a\right)âˆ’\frac{1}{{2}^{q}}f\left({2}^{q}a\right)âˆ¥â‰¤\underset{l=p}{\overset{qâˆ’1}{âˆ‘}}\frac{{2}^{lr}}{{2}^{l}}\mathrm{Î¸}P{\left(a\right)}^{r}$
(3.9)

for all $aâˆˆY$ and nonnegative integers p, q with $p. It follows from (3.9) that the sequence $\left\{\frac{1}{{2}^{l}}f\left({2}^{l}a\right)\right\}$ is Cauchy for all $aâˆˆY$. Since X is complete, the sequence $\left\{\frac{1}{{2}^{l}}f\left({2}^{l}a\right)\right\}$ converges. So, one can define the mapping $A:Yâ†’X$ by

$A\left(a\right)=\underset{lâ†’\mathrm{âˆž}}{lim}\frac{1}{{2}^{l}}f\left({2}^{l}a\right)$

for all $aâˆˆY$.

Moreover, letting $p=0$ and passing the limit $qâ†’\mathrm{âˆž}$ in (3.9), we get

$âˆ¥f\left(a\right)âˆ’A\left(a\right)âˆ¥â‰¤\frac{2\mathrm{Î¸}}{2âˆ’{2}^{r}}P{\left(a\right)}^{r}$
(3.10)

for all $aâˆˆY$.

It follows from (3.8) that

$âˆ¥\frac{1}{{2}^{l}}\left(f\left({2}^{l}\left(a+b\right)\right)âˆ’f\left({2}^{l}a\right)âˆ’f\left({2}^{l}b\right)\right)âˆ¥â‰¤\frac{{2}^{lr}}{{2}^{l}}\mathrm{Î¸}\left({âˆ¥aâˆ¥}^{r}+{âˆ¥bâˆ¥}^{r}\right),$

which tends to zero as $lâ†’\mathrm{âˆž}$. So, $âˆ¥A\left(a+b\right)âˆ’A\left(a\right)âˆ’A\left(b\right)âˆ¥=0$, i.e., $A\left(a+b\right)=A\left(a\right)+A\left(b\right)$ for all $a,bâˆˆY$. Hence $A:Yâ†’X$ is additive.

The proof of the uniqueness of A is similar to the proof of Theorem 2.5.

By Lemma 2.2 and (3.10),

${âˆ¥{f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{A}_{n}\left(\left[{x}_{ij}\right]\right)âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}âˆ¥f\left({x}_{ij}\right)âˆ’A\left({x}_{ij}\right)âˆ¥â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2\mathrm{Î¸}}{{2}^{r}âˆ’2}P{\left({x}_{ij}\right)}^{r}$

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$. Thus $A:Yâ†’X$ is a unique additive mapping satisfying (3.7), as desired.â€ƒâ–¡

## 4 Hyers-Ulam stability of the quadratic functional equation in matrix paranormed spaces

In this section, we prove the Hyers-Ulam stability of the quadratic functional equation in matrix paranormed spaces.

For a mapping $f:Xâ†’Y$, define $Df:{X}^{2}â†’Y$ and $D{f}_{n}:{M}_{n}\left({X}^{2}\right)â†’{M}_{n}\left(Y\right)$ by

$\begin{array}{c}Df\left(a,b\right)=f\left(a+b\right)+f\left(aâˆ’b\right)âˆ’2f\left(a\right)âˆ’2f\left(b\right),\hfill \\ D{f}_{n}\left(\left[{x}_{ij}\right],\left[{y}_{ij}\right]\right):={f}_{n}\left(\left[{x}_{ij}+{y}_{ij}\right]\right)+{f}_{n}\left(\left[{x}_{ij}âˆ’{y}_{ij}\right]\right)âˆ’2{f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’2{f}_{n}\left(\left[{y}_{ij}\right]\right)\hfill \end{array}$

for all $a,bâˆˆX$ and all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$.

Theorem 4.1 Let r, Î¸ be positive real numbers with $r>2$. Let $f:Xâ†’Y$ be a mapping such that

${P}_{n}\left(D{f}_{n}\left(\left[{x}_{ij}\right],\left[{y}_{ij}\right]\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\mathrm{Î¸}\left({âˆ¥{x}_{ij}âˆ¥}^{r}+{âˆ¥{y}_{ij}âˆ¥}^{r}\right)$
(4.1)

for all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$. Then there exists a unique quadratic mapping $Q:Xâ†’Y$ such that

${P}_{n}\left({f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{Q}_{n}\left(\left[{x}_{ij}\right]\right)\right)â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2\mathrm{Î¸}}{{2}^{r}âˆ’4}{âˆ¥{x}_{ij}âˆ¥}^{r}$

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(X\right)$.

Proof Let $n=1$ in (4.1). Then (4.1) is equivalent to

$P\left(f\left(a+b\right)+f\left(aâˆ’b\right)âˆ’2f\left(a\right)âˆ’2f\left(b\right)\right)â‰¤\mathrm{Î¸}\left({âˆ¥aâˆ¥}^{r}+{âˆ¥bâˆ¥}^{r}\right)$
(4.2)

for all $a,bâˆˆX$.

Letting $a=b=0$ in (4.2), we get $P\left(2f\left(0\right)\right)â‰¤0$ and so $f\left(0\right)=0$.

Letting $b=a$ in (4.2), we get

$P\left(f\left(2a\right)âˆ’4f\left(a\right)\right)â‰¤2\mathrm{Î¸}{âˆ¥aâˆ¥}^{r},$

and so

$P\left(f\left(a\right)âˆ’4f\left(\frac{a}{2}\right)\right)â‰¤\frac{2}{{2}^{r}}\mathrm{Î¸}{âˆ¥aâˆ¥}^{r}$

for all $a,bâˆˆX$.

The rest of the proof is similar to the proof of Theorem 3.1.â€ƒâ–¡

Theorem 4.2 Let r, Î¸ be positive real numbers with $r<2$. Let $f:Yâ†’X$ be a mapping such that

${âˆ¥D{f}_{n}\left(\left[{x}_{ij}\right],\left[{y}_{ij}\right]\right)âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\mathrm{Î¸}\left(P{\left({x}_{ij}\right)}^{r}+P{\left({y}_{ij}\right)}^{r}\right)$
(4.3)

for all $x=\left[{x}_{ij}\right],y=\left[{y}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$. Then there exists a unique quadratic mapping $Q:Yâ†’X$ such that

${âˆ¥{f}_{n}\left(\left[{x}_{ij}\right]\right)âˆ’{Q}_{n}\left(\left[{x}_{ij}\right]\right)âˆ¥}_{n}â‰¤\underset{i,j=1}{\overset{n}{âˆ‘}}\frac{2\mathrm{Î¸}}{4âˆ’{2}^{r}}P{\left({x}_{ij}\right)}^{r}$

for all $x=\left[{x}_{ij}\right]âˆˆ{M}_{n}\left(Y\right)$.

Proof Let $n=1$ in (4.3). Then (4.3) is equivalent to

$âˆ¥f\left(a+b\right)+f\left(aâˆ’b\right)âˆ’2f\left(a\right)âˆ’2f\left(b\right)âˆ¥â‰¤\mathrm{Î¸}\left(P{\left(a\right)}^{r}+P{\left(b\right)}^{r}\right)$
(4.4)

for all $a,bâˆˆY$.

Letting $a=b=0$ in (4.4), we get $âˆ¥2f\left(0\right)âˆ¥â‰¤0$ and so $f\left(0\right)=0$.

Letting $b=a$ in (4.4), we get

$âˆ¥f\left(2a\right)âˆ’4f\left(a\right)âˆ¥â‰¤2\mathrm{Î¸}P{\left(a\right)}^{r},$

and so

$âˆ¥f\left(a\right)âˆ’\frac{1}{4}f\left(2a\right)âˆ¥â‰¤\frac{\mathrm{Î¸}}{2}P{\left(a\right)}^{r}$

for all $a,bâˆˆY$.

The rest of the proof is similar to the proof of Theorem 3.2.â€ƒâ–¡

## References

1. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241â€“244.

2. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 73â€“74.

3. Fridy JA: On statistical convergence. Analysis 1985, 5: 301â€“313.

4. Karakus S: Statistical convergence on probabilistic normed spaces. Math. Commun. 2007, 12: 11â€“23.

5. Mursaleen M: Î» -Statistical convergence. Math. Slovaca 2000, 50: 111â€“115.

6. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233: 142â€“149. 10.1016/j.cam.2009.07.005

7. Å alÃ¡t T: On the statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139â€“150.

8. Kolk E: The statistical convergence in Banach spaces. Tartu Ãœlik. Toim. 1991, 928: 41â€“52.

9. Wilansky A: Modern Methods in Topological Vector Space. McGraw-Hill, New York; 1978.

10. Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

11. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222â€“224. 10.1073/pnas.27.4.222

12. Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64â€“66. 10.2969/jmsj/00210064

13. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297â€“300. 10.1090/S0002-9939-1978-0507327-1

14. GÇŽvruta G: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431â€“436. 10.1006/jmaa.1994.1211

15. Rassias TM: Problem 16; 2. Report of the 27th international symp. on functional equations. Aequ. Math. 1990, 39: 292â€“293. 309

16. Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431â€“434. 10.1155/S016117129100056X

17. Rassias TM, Å emrl P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 1992, 114: 989â€“993. 10.1090/S0002-9939-1992-1059634-1

18. Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore; 2002.

19. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. BirkhÃ¤user, Basel; 1998.

20. Skof F: Proprietaâ€™ locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 1983, 53: 113â€“129. 10.1007/BF02924890

21. Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76â€“86. 10.1007/BF02192660

22. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 59â€“64. 10.1007/BF02941618

23. Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.

24. Eshaghi Gordji M, Savadkouhi MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 2010, 23: 1198â€“1202. 10.1016/j.aml.2010.05.011

25. Isac G, Rassias TM: On the Hyers-Ulam stability of Ïˆ -additive mappings. J. Approx. Theory 1993, 72: 131â€“137. 10.1006/jath.1993.1010

26. Jun K, Lee Y: A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations. J. Math. Anal. Appl. 2004, 297: 70â€“86. 10.1016/j.jmaa.2004.04.009

27. Park C: Homomorphisms between Poisson $J{C}^{âˆ—}$-algebras. Bull. Braz. Math. Soc. 2005, 36: 79â€“97. 10.1007/s00574-005-0029-z

28. GilÃ¡nyi A: Eine zur Parallelogrammgleichung Ã¤quivalente Ungleichung. Aequ. Math. 2001, 62: 303â€“309. 10.1007/PL00000156

29. RÃ¤tz J: On inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math. 2003, 66: 191â€“200. 10.1007/s00010-003-2684-8

30. GilÃ¡nyi A: On a problem by K. Nikodem. Math. Inequal. Appl. 2002, 5: 707â€“710.

31. Fechner W: Stability of a functional inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math. 2006, 71: 149â€“161. 10.1007/s00010-005-2775-9

32. Park C, Cho Y, Han M: Functional inequalities associated with Jordan-von Neumann-type additive functional equations. J. Inequal. Appl. 2007., 2007: Article ID 41820

33. Ruan ZJ: Subspaces of ${C}^{âˆ—}$-algebras. J. Funct. Anal. 1988, 76: 217â€“230. 10.1016/0022-1236(88)90057-2

34. Effros E, Ruan ZJ: On approximation properties for operator spaces. Int. J. Math. 1990, 1: 163â€“187. 10.1142/S0129167X90000113

35. Choi MD, Effros E: Injectivity and operator spaces. J. Funct. Anal. 1977, 24: 156â€“209. 10.1016/0022-1236(77)90052-0

36. Effros E, Ruan ZJ: On the abstract characterization of operator spaces. Proc. Am. Math. Soc. 1993, 119: 579â€“584. 10.1090/S0002-9939-1993-1163332-4

37. Pisier G: Grothendieckâ€™s theorem for non-commutative ${C}^{âˆ—}$-algebras with an appendix on Grothendieckâ€™s constants. J. Funct. Anal. 1978, 29: 397â€“415. 10.1016/0022-1236(78)90038-1

38. Haagerup, U: Decomp. of completely bounded maps. Unpublished manuscript

39. Effros E: On multilinear completely bounded module maps. Contemp. Math. 62. In Operator Algebras and Mathematical Physics. Am. Math. Soc., Providence; 1987:479â€“501.

## Acknowledgements

CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

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Correspondence to Dong Yun Shin.

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The authors declare that they have no competing interests.

### Authorsâ€™ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Park, C., Lee, J.R. & Shin, D.Y. Functional equations and inequalities in matrix paranormed spaces. J Inequal Appl 2013, 547 (2013). https://doi.org/10.1186/1029-242X-2013-547