Theory and Modern Applications

# Fixed points and quadratic equations connected with homomorphisms and derivations on non-Archimedean algebras

## Abstract

We apply the fixed point method to prove the stability of the systems of functional equations

on non-Archimedean Banach algebras. Moreover, we give some applications of our results in non-Archimedean Banach algebras over p-adic numbers.

MSC:34K36, 46S40, 47S40, 39B82, 39B52, 26E50.

## 1 Introduction and preliminaries

The story of the stability of functional equations dates back to 1925 when a stability result appeared in the celebrated book by Gy. PÃ³lya and G. SzegÅ‘ [1]. In 1940, Ulam [2] posed the famous Ulam stability problem which was partially solved by Hyers [3] in the framework of Banach spaces. Later, Aoki [4] considered the stability problem with unbounded Cauchy differences. In 1978, Th. M. Rassias [5] provided a generalization of the Hyersâ€™ theorem by proving the existence of unique linear mappings near approximate additive mappings. GÇŽvruta [6] obtained a generalized result of Th. M. Rassiasâ€™ theorem which allows the Cauchy difference to be controlled by a general unbounded function. On the other hand, J. M. Rassias [7â€“10] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity, a counterexample was given by GÇŽvruta [11].

Bourgin [12] proved the stability of ring homomorphisms in two unital Banach algebras. Badora [13] gave a generalization of the Bourginâ€™s result. The stability result concerning derivations on operator algebras was first obtained by Å emrl [14]. In [15], Badora proved the stability of functional equation $f\left(xy\right)=xf\left(y\right)+f\left(x\right)y$, where f is a mapping on normed algebra A with unit.

Let $\mathcal{A}$, $\mathcal{B}$ be two algebras. A mapping $f:\mathcal{A}â†’\mathcal{B}$ is called a quadratic homomorphism if f is a quadratic mapping satisfying $f\left(xy\right)=f\left(x\right)f\left(y\right)$ for all $x,yâˆˆ\mathcal{A}$. For instance, let $\mathcal{A}$ be commutative. Then the mapping $f:\mathcal{A}â†’\mathcal{A}$, defined by $f\left(x\right)={x}^{2}$ ($xâˆˆ\mathcal{A}$), is a quadratic homomorphism.

A mapping $f:\mathcal{A}â†’\mathcal{A}$ is called a quadratic derivation if f is a quadratic mapping satisfying $f\left(xy\right)={x}^{2}f\left(y\right)+f\left(x\right){y}^{2}$ for all $x,yâˆˆ\mathcal{A}$. For instance, consider the algebra of $2Ã—2$ matrices

$\mathcal{A}=\left\{\left[\begin{array}{cc}{c}_{1}& {c}_{2}\\ 0& 0\end{array}\right]:{c}_{1},{c}_{2}âˆˆ\mathbb{C}\phantom{\rule{0.25em}{0ex}}\left(\text{complex field}\right)\right\}.$

Then it is easy to see that the mapping $f:\mathcal{A}â†’\mathcal{A}$, defined by $f\left(\left[\begin{array}{cc}{c}_{1}& {c}_{2}\\ 0& 0\end{array}\right]\right)=\left[\begin{array}{cc}0& {c}_{2}^{2}\\ 0& 0\end{array}\right]$, is a quadratic derivation. We note that quadratic derivations and ring derivations are different.

Arriola and Beyer in [16] initiated the stability of functional equations in non-Archimedean spaces. In fact they established the stability of Cauchy functional equations over p-adic fields. After their results, some papers (see for instance [17â€“24]) on the stability of other equations in such spaces have been published. Although different methods are known for establishing the stability of functional equations, almost all proofs depend on the Hyersâ€™s method in [3]. In 2003, Radu [25] employed the alternative fixed point theorem, due to Diaz and Margolis [26], to prove the stability of the Cauchy additive functional equation. Subsequently, this method was applied to investigate the Hyers-Ulam stability for the Jensen functional equation [27], as well as for the Cauchy functional equation [28], by considering a general control function $\mathrm{Ï†}\left(x,y\right)$ with suitable properties. Using such an elegant idea, several authors applied the method to investigate the stability of some functional equations, see [29â€“33].

Recently, Eshaghi and Khodaei [34] considered the following quadratic functional equation:

$f\left(ax+by\right)+f\left(axâˆ’by\right)=2{a}^{2}f\left(x\right)+2{b}^{2}f\left(y\right)$
(1.1)

and proved the Hyers-Ulam stability of the above functional equation in classical Banach spaces. Recently, Eshaghi [35] proved the Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean Banach algebras.

In the present paper, we adopt the idea of CÇŽdariu and Radu [28] to establish the stability of quadratic homomorphisms and quadratic derivations related to the quadratic functional equation (1.1) over non-Archimedean Banach algebras. Some applications of our results in non-Archimedean Banach algebras over p-adic numbers will be exhibited.

Now, we recall some notations and basic definitions used later on in the paper.

In 1897, Hensel [36] discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. During the last three decades p-adic numbers have gained the interest of physicists for their research, in particular in problems coming from quantum physics, p-adic strings and superstrings [37, 38]. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: For $x,y>0$, there exists such that $x (see [39, 40]).

Example 1.1 Let p be a prime number. For any nonzero rational number $a={p}^{r}\frac{m}{n}$ such that m and n are coprime to the prime number p, define the p-adic absolute value ${|a|}_{p}={p}^{âˆ’r}$. Then âˆ¥ is a non-Archimedean norm on $\mathbb{Q}$. The completion of $\mathbb{Q}$ with respect to âˆ¥ is denoted by ${\mathbb{Q}}_{p}$ and is called the p-adic number field. Note that if $pâ‰¥3$, then $|{2}^{n}|=1$ in for each integer n.

Let $\mathbb{K}$ denote a field and function (valuation absolute) $|â‹\dots |$ from $\mathbb{K}$ into $\left[0,\mathrm{âˆž}\right)$. A non-Archimedean valuation is a function $|â‹\dots |$ that satisfies the strong triangle inequality; namely, $|x+y|â‰¤max\left\{|x|,|y|\right\}â‰¤|x|+|y|$ for all $x,yâˆˆ\mathbb{K}$. The associated field $\mathbb{K}$ is referred to as a non-Archimedean field. Clearly, $|1|=|âˆ’1|=1$ and $|n|â‰¤1$ for all $nâ‰¥1$. A trivial example of a non-Archimedean valuation is the function $|â‹\dots |$ taking everything except 0 into 1 and $|0|=0$. We always assume in addition that $|â‹\dots |$ is nontrivial, i.e., there is a $zâˆˆ\mathbb{K}$ such that .

Let X be a linear space over a field $\mathbb{K}$ with a non-Archimedean nontrivial valuation $|â‹\dots |$. A function $âˆ¥â‹\dots âˆ¥:Xâ†’\left[0,\mathrm{âˆž}\right)$ is said to be a non-Archimedean norm if it is a norm over $\mathbb{K}$ with the strong triangle inequality (ultrametric); namely, $âˆ¥x+yâˆ¥â‰¤max\left\{âˆ¥xâˆ¥,âˆ¥yâˆ¥\right\}$ for all $x,yâˆˆX$. Then $\left(X,âˆ¥â‹\dots âˆ¥\right)$ is called a non-Archimedean space. In any such a space a sequence ${\left\{{x}_{n}\right\}}_{nâˆˆ\mathbb{N}}$ is Cauchy if and only if ${\left\{{x}_{n+1}âˆ’{x}_{n}\right\}}_{nâˆˆ\mathbb{N}}$ converges to zero. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. A non-Archimedean Banach algebra is a complete non-Archimedean algebra $\mathcal{A}$ which satisfies $âˆ¥xyâˆ¥â‰¤âˆ¥xâˆ¥âˆ¥yâˆ¥$ for all $x,yâˆˆ\mathcal{A}$. For more details the reader is referred to [41, 42].

Let X be a nonempty set and $d:XÃ—Xâ†’\left[0,\mathrm{âˆž}\right]$ satisfying: $d\left(x,y\right)=0$ if and only if $x=y$, $d\left(x,y\right)=d\left(y,x\right)$ and $d\left(x,z\right)â‰¤max\left\{d\left(x,y\right),d\left(y,z\right)\right\}$ (strong triangle inequality), for all $x,y,zâˆˆX$. Then $\left(X,d\right)$ is called a non-Archimedean generalized metric space. $\left(X,d\right)$ is called complete if every d-Cauchy sequence in X is d-convergent.

Using the strong triangle inequality in the proof of the main result of [26], we get to the following result:

Theorem 1.2 (Non-Archimedean Alternative Contraction Principle)

If $\left(\mathrm{Î©},d\right)$ is a non-Archimedean generalized complete metric space and $T:\mathrm{Î©}â†’\mathrm{Î©}$ a strictly contractive mapping (that is $d\left(T\left(x\right),T\left(y\right)\right)â‰¤Ld\left(x,y\right)$, for all $x,yâˆˆT$ and a Lipschitz constant $L<1$). Let $xâˆˆ\mathrm{Î©}$, then either

1. (i)

$d\left({T}^{n}\left(x\right),{T}^{n+1}\left(x\right)\right)=\mathrm{âˆž}$ for all $nâ‰¥0$, or

2. (ii)

there exists some ${n}_{0}â‰¥0$ such that $d\left({T}^{n}\left(x\right),{T}^{n+1}\left(x\right)\right)<\mathrm{âˆž}$ for all $nâ‰¥{n}_{0}$; the sequence $\left\{{T}^{n}\left(x\right)\right\}$ is convergent to a fixed point ${x}^{âˆ—}$ of T; ${x}^{âˆ—}$ is the unique fixed point of T in the set

$\mathrm{Î›}=\left\{yâˆˆ\mathrm{Î©}:d\left({T}^{{n}_{0}}\left(x\right),y\right)<\mathrm{âˆž}\right\}$

and $d\left(y,{x}^{âˆ—}\right)â‰¤d\left(y,T\left(y\right)\right)$ for all y in this set.

Hereafter, unless otherwise stated, we will assume that $\mathcal{A}$ and $\mathcal{B}$ are two non-Archimedean Banach algebras. Also, let $|4|<1$; and we assume that in $\mathbb{K}$ (i.e., the characteristic of $\mathbb{K}$ is not 4).

Theorem 2.1 Let $\mathrm{â„“}âˆˆ\left\{âˆ’1,1\right\}$ be fixed and let $f:\mathcal{A}â†’\mathcal{B}$ be a mapping with $f\left(0\right)=0$ for which there exists a function $\mathrm{Ï†}:{\mathcal{A}}^{4}â†’\left[0,\mathrm{âˆž}\right)$ such that

$âˆ¥f\left(ax+by\right)+f\left(axâˆ’by\right)+f\left(zw\right)âˆ’2{a}^{2}f\left(x\right)âˆ’2{b}^{2}f\left(y\right)âˆ’f\left(z\right)f\left(w\right)âˆ¥â‰¤\mathrm{Ï†}\left(x,y,z,w\right)$
(2.1)

for all $x,y,z,wâˆˆ\mathcal{A}$ and nonzero fixed integers a, b. If there exists an $L<1$ such that

$\mathrm{Ï†}\left(x,y,z,w\right)â‰¤{|4|}^{\mathrm{â„“}\left(\mathrm{â„“}+2\right)}L\mathrm{Ï†}\left(\frac{x}{{2}^{\mathrm{â„“}}},\frac{y}{{2}^{\mathrm{â„“}}},\frac{z}{{2}^{\mathrm{â„“}}},\frac{w}{{2}^{\mathrm{â„“}}}\right)$
(2.2)

for all $x,y,z,wâˆˆ\mathcal{A}$. Then there exists a unique quadratic homomorphism $\mathcal{H}:\mathcal{A}â†’\mathcal{B}$ such that

$âˆ¥f\left(x\right)âˆ’\mathcal{H}\left(x\right)âˆ¥â‰¤\frac{{L}^{\frac{1âˆ’\mathrm{â„“}}{2}}}{|4|}\mathrm{Ïˆ}\left(x\right)$
(2.3)

for all $xâˆˆ\mathcal{A}$, where

$\begin{array}{rcl}\mathrm{Ïˆ}\left(x\right)& :=& max\left\{\mathrm{Ï†}\left(\frac{x}{a},\frac{x}{b},0,0\right),\mathrm{Ï†}\left(\frac{x}{a},0,0,0\right),\frac{1}{|2{b}^{2}|}\mathrm{Ï†}\left(x,x,0,0\right),\\ \frac{1}{|2{b}^{2}|}\mathrm{Ï†}\left(x,âˆ’x,0,0\right),\mathrm{Ï†}\left(0,\frac{x}{b},0,0\right)\right\}.\end{array}$

Proof Letting $z=w=0$ in (2.1), we get

$âˆ¥f\left(ax+by\right)+f\left(axâˆ’by\right)âˆ’2{a}^{2}f\left(x\right)âˆ’2{b}^{2}f\left(y\right)âˆ¥â‰¤\mathrm{Ï†}\left(x,y,0,0\right)$
(2.4)

for all $x,yâˆˆ\mathcal{A}$. Setting $y=âˆ’y$ in (2.4), we get

$âˆ¥f\left(axâˆ’by\right)+f\left(ax+by\right)âˆ’2{a}^{2}f\left(x\right)âˆ’2{b}^{2}f\left(âˆ’y\right)âˆ¥â‰¤\mathrm{Ï†}\left(x,âˆ’y,0,0\right)$
(2.5)

for all $x,yâˆˆ\mathcal{A}$. It follows from (2.4) and (2.5) that

$âˆ¥2{b}^{2}f\left(y\right)âˆ’2{b}^{2}f\left(âˆ’y\right)âˆ¥â‰¤max\left\{\mathrm{Ï†}\left(x,y,0,0\right),\mathrm{Ï†}\left(x,âˆ’y,0,0\right)\right\}$
(2.6)

for all $x,yâˆˆ\mathcal{A}$. Putting $y=0$ in (2.4), we get

$âˆ¥2f\left(ax\right)âˆ’2{a}^{2}f\left(x\right)âˆ¥â‰¤\mathrm{Ï†}\left(x,0,0,0\right)$
(2.7)

for all $xâˆˆ\mathcal{A}$. Setting $x=0$ in (2.4), we get

$âˆ¥f\left(by\right)+f\left(âˆ’by\right)âˆ’2{b}^{2}f\left(y\right)âˆ¥â‰¤\mathrm{Ï†}\left(0,y,0,0\right)$
(2.8)

for all $yâˆˆ\mathcal{A}$. Putting $y=by$ in (2.6), we get

$âˆ¥f\left(by\right)âˆ’f\left(âˆ’by\right)âˆ¥â‰¤max\left\{\frac{1}{|2{b}^{2}|}\mathrm{Ï†}\left(x,by,0,0\right),\frac{1}{|2{b}^{2}|}\mathrm{Ï†}\left(x,âˆ’by,0,0\right)\right\}$
(2.9)

for all $x,yâˆˆ\mathcal{A}$. It follows from (2.8) and (2.9) that

$âˆ¥2f\left(by\right)âˆ’2{b}^{2}f\left(y\right)âˆ¥â‰¤max\left\{\frac{1}{|2{b}^{2}|}\mathrm{Ï†}\left(x,by,0,0\right),\frac{1}{|2{b}^{2}|}\mathrm{Ï†}\left(x,âˆ’by,0,0\right),\mathrm{Ï†}\left(0,y,0,0\right)\right\}$
(2.10)

for all $x,yâˆˆ\mathcal{A}$. Replacing x and y by $\frac{x}{a}$ and $\frac{x}{b}$ in (2.4), respectively, we get

$âˆ¥f\left(2x\right)âˆ’2{a}^{2}f\left(\frac{x}{a}\right)âˆ’2{b}^{2}f\left(\frac{x}{b}\right)âˆ¥â‰¤\mathrm{Ï†}\left(\frac{x}{a},\frac{x}{b},0,0\right)$
(2.11)

for all $xâˆˆ\mathcal{A}$. Setting $x=\frac{x}{a}$ in (2.7), we get

$âˆ¥2{a}^{2}f\left(\frac{x}{a}\right)âˆ’2f\left(x\right)âˆ¥â‰¤\mathrm{Ï†}\left(\frac{x}{a},0,0,0\right)$
(2.12)

for all $xâˆˆ\mathcal{A}$. Putting $y=\frac{x}{b}$ in (2.10), we get

(2.13)

for all $xâˆˆ\mathcal{A}$. It follows from (2.11), (2.12) and (2.13) that

$âˆ¥f\left(2x\right)âˆ’4f\left(x\right)âˆ¥â‰¤\mathrm{Ïˆ}\left(x\right)$
(2.14)

for all $xâˆˆ\mathcal{A}$. For every $g,h:\mathcal{A}â†’\mathcal{B}$, define

$d\left(g,h\right):=inf\left\{Câˆˆ\left(0,\mathrm{âˆž}\right):âˆ¥g\left(x\right)âˆ’h\left(x\right)âˆ¥â‰¤C\mathrm{Ïˆ}\left(x\right),\mathrm{âˆ€}xâˆˆ\mathcal{A}\right\}.$

Hence, d defines a complete generalized non-Archimedean metric on $\mathrm{Î©}:=\left\{g|g:\mathcal{A}â†’\mathcal{B},g\left(0\right)=0\right\}$ (see [27, 28, 33]). Let $T:\mathrm{Î©}â†’\mathrm{Î©}$ be defined by $Tg\left(x\right)=\frac{1}{{4}^{\mathrm{â„“}}}g\left({2}^{\mathrm{â„“}}x\right)$ for all $xâˆˆ\mathcal{A}$ and all $gâˆˆ\mathrm{Î©}$. If for some $g,hâˆˆ\mathrm{Î©}$ and $C>0$,

$âˆ¥g\left(x\right)âˆ’h\left(x\right)âˆ¥â‰¤C\mathrm{Ïˆ}\left(x\right)$

for all $xâˆˆ\mathcal{A}$, then

$âˆ¥Tg\left(x\right)âˆ’Th\left(x\right)âˆ¥â‰¤\frac{1}{{|4|}^{\mathrm{â„“}}}âˆ¥g\left({2}^{\mathrm{â„“}}x\right)âˆ’h\left({2}^{\mathrm{â„“}}x\right)âˆ¥â‰¤\frac{C}{{|4|}^{\mathrm{â„“}}}\mathrm{Ïˆ}\left({2}^{\mathrm{â„“}}x\right)â‰¤LC\mathrm{Ïˆ}\left(x\right)$

for all $xâˆˆ\mathcal{A}$, so

$d\left(Tg,Th\right)â‰¤Ld\left(g,h\right).$

Hence T is a strictly contractive mapping on Î© with the Lipschitz constant L. It follows from (2.14) by using (2.2) that

$âˆ¥f\left(x\right)âˆ’4f\left(\frac{x}{2}\right)âˆ¥â‰¤\mathrm{Ïˆ}\left(\frac{x}{2}\right)â‰¤\frac{L}{|4|}\mathrm{Ïˆ}\left(x\right)$

and

$âˆ¥f\left(x\right)âˆ’\frac{1}{4}f\left(2x\right)âˆ¥â‰¤\frac{1}{|4|}\mathrm{Ïˆ}\left(x\right)$

for all $xâˆˆ\mathcal{A}$, that is, $d\left(f,Tf\right)â‰¤\frac{{L}^{\frac{1âˆ’\mathrm{â„“}}{2}}}{|4|}<\mathrm{âˆž}$.

Now, by the non-Archimedean alternative contraction principle, T has a unique fixed point $\mathcal{H}:\mathcal{A}â†’\mathcal{B}$ in the set $\mathrm{Î›}=\left\{gâˆˆ\mathrm{Î©}:d\left(f,g\right)<\mathrm{âˆž}\right\}$, which $\mathcal{H}$ is defined by

$\mathcal{H}\left(x\right)=\underset{nâ†’\mathrm{âˆž}}{lim}{T}^{n}f\left(x\right)=\underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{{4}^{\mathrm{â„“}n}}f\left({2}^{\mathrm{â„“}n}x\right)$
(2.15)

for all $xâˆˆ\mathcal{A}$. By (2.2),

$\underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{{|4|}^{\mathrm{â„“}\left(\mathrm{â„“}+2\right)n}}\mathrm{Ï†}\left({2}^{\mathrm{â„“}n}x,{2}^{\mathrm{â„“}n}y,{2}^{\mathrm{â„“}n}z,{2}^{\mathrm{â„“}n}w\right)=0$
(2.16)

for all $x,y,z,wâˆˆ\mathcal{A}$. It follows from (2.1), (2.15) and (2.16) that

for all $x,yâˆˆ\mathcal{A}$. This shows that $\mathcal{H}$ is quadratic. Also,

$\begin{array}{rcl}âˆ¥\mathcal{H}\left(zw\right)âˆ’\mathcal{H}\left(z\right)\mathcal{H}\left(w\right)âˆ¥& =& \underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{{|4|}^{2\mathrm{â„“}n}}âˆ¥f\left({4}^{\mathrm{â„“}n}zw\right)âˆ’f\left({2}^{\mathrm{â„“}n}z\right)f\left({2}^{\mathrm{â„“}n}w\right)âˆ¥\\ â‰¤& \underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{{|4|}^{2\mathrm{â„“}n}}\mathrm{Ï†}\left(0,0,{2}^{\mathrm{â„“}n}z,{2}^{\mathrm{â„“}n}w\right)\\ â‰¤& \underset{nâ†’\mathrm{âˆž}}{lim}\frac{1}{{|4|}^{\mathrm{â„“}\left(\mathrm{â„“}+2\right)n}}\mathrm{Ï†}\left(0,0,{2}^{\mathrm{â„“}n}z,{2}^{\mathrm{â„“}n}w\right)=0\end{array}$

for all $z,wâˆˆ\mathcal{A}$. Therefore, $\mathcal{H}$ is a quadratic homomorphism. Moreover, by the non-Archimedean alternative contraction principle,

$d\left(f,\mathcal{H}\right)â‰¤d\left(f,Tf\right)â‰¤\frac{{L}^{\frac{1âˆ’\mathrm{â„“}}{2}}}{|4|}.$

This implies the inequality (2.3) holds.â€ƒâ–¡

Corollary 2.2 Let $\mathcal{A}$, $\mathcal{B}$ be non-Archimedean Banach algebras over ${\mathbb{Q}}_{2}$, $\mathrm{â„“}âˆˆ\left\{âˆ’1,1\right\}$ be fixed and Î¸, r be non-negative real numbers such that $\mathrm{â„“}r>2\mathrm{â„“}\left(\mathrm{â„“}+2\right)$. Suppose that a mapping $f:\mathcal{A}â†’\mathcal{B}$ satisfies $f\left(0\right)=0$ and

for all $x,y,z,wâˆˆ\mathcal{A}$, where a, b are positive fixed integers. Then there exists a unique quadratic homomorphism $H:\mathcal{A}â†’\mathcal{B}$ such that

$âˆ¥f\left(x\right)âˆ’\mathcal{H}\left(x\right)âˆ¥â‰¤{2}^{\frac{\mathrm{â„“}\left(4âˆ’r\right)+r}{2}}\mathrm{Î¸}{âˆ¥xâˆ¥}^{r}\left\{\begin{array}{cc}4,\hfill & gcd\left(a,2\right)=gcd\left(b,2\right)=1;\hfill \\ max\left\{1+{2}^{ir},4\right\},\hfill & a=k{2}^{i},gcd\left(b,2\right)=1;\hfill \\ max\left\{1+{2}^{jr},{2}^{2j+2}\right\},\hfill & gcd\left(a,2\right)=1,b=m{2}^{j};\hfill \\ max\left\{{2}^{ir}+{2}^{jr},{2}^{2j+2}\right\},\hfill & a=k{2}^{i},b=m{2}^{j};\hfill \end{array}$

for all $xâˆˆ\mathcal{A}$, where $i,j,k,mâ‰¥1$ are integers and $gcd\left(k,2\right)=gcd\left(m,2\right)=1$.

Proof The proof follows from Theorem 2.1, by taking

$\mathrm{Ï†}\left(x,y,z,w\right)=\mathrm{Î¸}\left({âˆ¥xâˆ¥}^{r}+{âˆ¥yâˆ¥}^{r}+{âˆ¥zâˆ¥}^{r}+{âˆ¥wâˆ¥}^{r}\right)$

for all $x,y,z,wâˆˆ\mathcal{A}$. Then we choose $L={2}^{\mathrm{â„“}\left(2\mathrm{â„“}+4âˆ’r\right)}$ and we get the desired result.â€ƒâ–¡

Corollary 2.3 Let $\mathcal{A}$, $\mathcal{B}$ be non-Archimedean Banach algebras over ${\mathbb{Q}}_{2}$, $\mathrm{â„“}âˆˆ\left\{âˆ’1,1\right\}$ be fixed and Î´, s be non-negative real numbers such that $\mathrm{â„“}s>2\mathrm{â„“}\left(\mathrm{â„“}+2\right)$. Suppose that a mapping $f:\mathcal{A}â†’\mathcal{B}$ satisfies $f\left(0\right)=0$ and

for all $x,y,z,wâˆˆ\mathcal{A}$, where a, b are positive fixed integers. Then there exists a unique quadratic homomorphism $H:\mathcal{A}â†’\mathcal{B}$ such that

for all $xâˆˆ\mathcal{A}$, where $i,j,k,mâ‰¥1$ are integers and $gcd\left(k,2\right)=gcd\left(m,2\right)=1$.

Theorem 2.4 Let $\mathrm{â„“}âˆˆ\left\{âˆ’1,1\right\}$ be fixed and let $f:\mathcal{A}â†’\mathcal{A}$ be a mapping with $f\left(0\right)=0$ if $\mathrm{â„“}=âˆ’1$, for which there exists a function $\mathrm{Ï†}:{\mathcal{A}}^{4}â†’\left[0,\mathrm{âˆž}\right)$ satisfying (2.2) and

(2.17)

for all $x,y,z,wâˆˆ\mathcal{A}$ and nonzero fixed integers a, b. Then there exists a unique quadratic derivation $\mathcal{D}:\mathcal{A}â†’\mathcal{A}$ such that

$âˆ¥f\left(x\right)âˆ’\mathcal{D}\left(x\right)âˆ¥â‰¤\frac{{L}^{\frac{1âˆ’\mathrm{â„“}}{2}}}{|4|}\mathrm{Ïˆ}\left(x\right)$
(2.18)

for all $xâˆˆ\mathcal{A}$, where $\mathrm{Ïˆ}\left(x\right)$ is defined as in Theorem 2.1.

Proof By (2.2), if $\mathrm{â„“}=1$, we obtain $\mathrm{Ï†}\left(0,0,0,0\right)=0$. Letting $x=y=z=w=0$ in (2.17), we get $f\left(0\right)â‰¤\mathrm{Ï†}\left(0,0,0,0\right)$. So $f\left(0\right)=0$ for $\mathrm{â„“}=1$.

By the same reasoning as that in the proof of Theorem 2.1, there exists a unique quadratic mapping $\mathcal{D}:\mathcal{A}â†’\mathcal{A}$ satisfying (2.18). The mapping $\mathcal{D}:\mathcal{A}â†’\mathcal{A}$ is given by $\mathcal{D}\left(x\right)={lim}_{nâ†’\mathrm{âˆž}}{T}^{n}f\left(x\right)={lim}_{nâ†’\mathrm{âˆž}}\frac{1}{{4}^{\mathrm{â„“}n}}f\left({2}^{\mathrm{â„“}n}x\right)$ for all $xâˆˆ\mathcal{A}$. It follows from (2.17) that

for all $z,wâˆˆ\mathcal{A}$. Therefore $\mathcal{D}:\mathcal{A}â†’\mathcal{A}$ is a quadratic derivation satisfying (2.18).â€ƒâ–¡

Very recently, J. M. Rassias [43] considered the Cauchy difference controlled by the product and sum of powers of norms, that is, $\mathrm{Î¸}\left\{{âˆ¥xâˆ¥}^{p}{âˆ¥yâˆ¥}^{p}+\left({âˆ¥xâˆ¥}^{2p}+{âˆ¥yâˆ¥}^{2p}\right)\right\}$.

Corollary 2.5 Let $\mathcal{A}$ be a non-Archimedean Banach algebra over ${\mathbb{Q}}_{2}$, $\mathrm{â„“}âˆˆ\left\{âˆ’1,1\right\}$ be fixed and Îµ, p, q be non-negative real numbers such that $\mathrm{â„“}\left(p+q\right)>2\mathrm{â„“}\left(\mathrm{â„“}+2\right)$. Suppose that a mapping $f:\mathcal{A}â†’\mathcal{A}$ satisfies

for all $x,y,z,wâˆˆ\mathcal{A}$, where a, b are positive fixed integers. Then there exists a unique quadratic derivation $\mathcal{D}:\mathcal{A}â†’\mathcal{A}$ such that

for all $xâˆˆ\mathcal{A}$, where $i,j,k,mâ‰¥1$ are integers and $gcd\left(k,2\right)=gcd\left(m,2\right)=1$.

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Correspondence to Madjid Eshaghi Gordji or Choonkil Park.

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Eshaghi Gordji, M., Khodaei, H., Khodabakhsh, R. et al. Fixed points and quadratic equations connected with homomorphisms and derivations on non-Archimedean algebras. Adv Differ Equ 2012, 128 (2012). https://doi.org/10.1186/1687-1847-2012-128

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### Keywords

• fixed point approach
• non-Archimedean algebra