# Nearly Quadraticn-Derivations on Non-Archimedean Banach Algebras

Authors: Madjid Eshaghi Gordji; Badrkhan Alizadeh; Young Whan Lee; Gwang Hui Kim
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

Letn>1 be an integer, let A be an algebra, and X be an A-module. A quadratic function D:A→X is called a quadratic n-derivation if D(∏i=1nai)=D(a1)a22⋯an2+a12D(a2)a32⋯an2+⋯+a12a22⋯an−12D(an) for all a1,...,an∈A. We investigate the Hyers-Ulam stability of quadratic n-derivations from non-Archimedean Banach algebras into non-Archimedean Banach modules by using the Banach fixed point theorem.


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## Body

## 1. Introduction

A functional equation(ξ) is stable if any function g satisfying the equation (ξ) approximately is near to a true solution of (ξ).The stability of functional equations was first introduced by Ulam [1] in 1964. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Th. M. Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences ∥f(x+y)-f(x)-f(y)∥≤ϵ(∥x∥p+∥y∥p),(ϵ>0,p∈[0,1)). In 1994, a generalization of Th. M. Rassias theorem was obtained by Gǎvruţa [4], who replaced the bound ϵ(∥x∥p+∥y∥p) by a general control function φ(x,y) (see also [5–7]).Every solution of the following functional equation(1.1)f(x+y)+f(x-y)=2f(x)+2f(y)is said to be a quadratic function [8]. It is well known that a mapping f between real vector spaces is quadratic mapping if and only if there exists a unique symmetric biadditive mapping B1 such that f(x)=B1(x,x) for all x. The biadditive mapping B1 is given by B1(x,y)=(1/4)(f(x+y)-f(x-y)).The stability problem of the quadratic functional equation was proved by Skof [9] for mappings f:A→B, where A is a normed space and B is a Banach space (see also [10, 11]). Let A be an algebra and let X be a A-bimodule. A quadratic function D:A→X is called a quadratic n-derivation if(1.2)D(∏i=1nai)=D(a1)a22⋯an2+a12D(a2)a32⋯an2+⋯+a12a22⋯an-12D(an)for alla1,…,an∈A. Recently, Gordji and Ghobadipour [12] introduced the quadratic derivations on Banach algebras. Indeed, they investigated the Hyers-Ulam-Aoki-Rassias stability and Ulam-Gavruta-Rassias type stability of quadratic derivations on Banach algebras.More recently, Gordji et al. [13] investigated the Hyers-Ulam stability and the superstability of higher ring derivations on non-Archimedean Banach algebras (see also [12–32]). In this paper we investigate the Hyers-Ulam stability of quadratic n-derivations from non-Archimedean Banach algebras into non-Archimedean Banach modules by using the weighted space method (see [33]).


## 2. Preliminaries

Let us recall that a non-Archimedean field is a field𝕂 equipped with a function (valuation) |·| from 𝕂 into [0,∞) such that |r|=0 if and only if r=0,|rs|=|r||s|, and |r+s|≤max{|r|,|s|} for all r,s∈𝕂. An example of a non-Archimedean valuation is the mapping |·| taking everything but 0 into 1 and |0|=0. This valuation is called trivial (see [34]).Definition 2.1.
LetX be a vector space over a scalar field 𝕂 with a non-Archimedean non-trivial valuation |·|. A function ∥·∥:X→ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions:(NA1)
∥x∥=0 if and only if x=0;(NA2)
∥rx∥=|r|∥x∥ for all r∈𝕂 and x∈X;(NA3)
∥x+y∥≤max{∥x∥,∥y∥} for all x,y∈X (the strong triangle inequality).In 1897, Hensel [35] introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications. The most important examples of non-Archimedean spaces are p-adic numbers. Let p be a prime number. For any nonzero rational number x=(a/b)pnx such that a and b are integers not divisible by p, define the p-adic absolute value |x|p:=p-nx. Then |·|p is a non-Archimedean norm on ℚ. The completion of ℚ with respect to |·|p is denoted by ℚp which is called the p-adic number field.Definition 2.2.
LetX be a nonempty set and let d:X×X→[0,∞) satisfy the following properties:(D1)
d(x,y)=0 if and only if x=y,(D2)
d(x,y)=d(y,x) (symmetry),(D3)
d(x,z)≤max{d(x,y),d(y,z)} (strong triangle inequality),for all x,y,z∈X. Then (X,d) is called a non-Archimedean metric space. (X,d) is called a non-Archimedean complete metric space if every d-Cauchy sequence in X is d-convergent.Theorem 2.3 (Non-Archimedean Banach Contraction Principle).
Let(X,d) be a non-Archimedean complete metric space and let T:X→X be a contraction; that is, there exists α∈[0,1) such that
(2.1)d(Tx,Ty)≤αd(x,y),∀x,y∈X.
Then there exists a unique element a∈X such that Ta=a. Moreover, a=limn→∞Tnx, and
(2.2)d(a,x)≤d(x,Tx),∀x∈X.Proof.
A similar argument as Archimedean case can be applied to show thatT has a unique element a∈X such that Ta=a and a=limn→∞Tnx. It follows from strong triangle inequality that for all x∈X and for each n∈ℕ, we have
(2.3)d(Tnx,x)≤max⁡{d(T(x),x),…,d(Tn(x),Tn-1(x))}≤max⁡{d(T(x),x),…,αn-1d(T(x),(x))}=d(T(x),x).


## 3. Main Results

In this sectionA denotes a non-Archimedean Banach algebra over a non-Archimedean field 𝕂 and X is a non-Archimedean Banach A-module.Theorem 3.1.
Letφ:A×A→[0,∞),ψ:A×⋯×A→[0,∞) be functions. Let f:A→X be a given mapping such that f(0)=0,
(3.1)‖f(x+y)+f(x-y)-2f(x)-2f(y)‖≤φ(x,y)
and that
(3.2)‖f(∏i=1nxi)-f(x1)x22⋯xn2-x12f(x2)x32⋯xn2-⋯-x12⋯xn-12f(xn)‖≤ψ(x1,…,xn)
for all x1,…,xn,x,y∈A. Suppose that there exist a natural number k∈𝕂 and L,K∈(0,1), such that
(3.3)|k|2φ(k-1x,k-1y)≤Lφ(x,y),|k|2ψ(k-1x1,…,k-1xn)≤Kψ(x1,…,xn)
for all x1,…,xn,x,y∈A. Then there exists a unique quadratic n-derivation h from A into X such that
(3.4)‖f(x)-h(x)‖≤LΦ(x)|k|2
for all x∈A, where
(3.5)Φ(x)=max⁡{φ(0,0),φ(x,x),φ(2x,x),…,φ((k-1)x,x)}(x∈A).Proof.
By induction oni, one can show that for all x∈A and i≥2,
(3.6)‖f(ix)-i2f(x)‖≤max⁡{φ(0,0),φ(x,x),φ(2x,x),…,φ((i-1)x,x)}.
Let x=y in (3.1). Then
(3.7)‖f(2x)-22f(x)‖≤max⁡{φ(0,0),φ(x,x)}(x∈A).
This proves (3.6) for i=2. Let (3.6) hold for i=1,2,…,j. Replacing x by jx and y by x in (3.1) for all x∈A, we get
(3.8)‖f((j+1)x)+f((j-1)x)-2f(jx)-2f(x)‖≤max⁡{φ(0,0),φ(jx,x)}
for all x∈A. Since
(3.9)f((j+1)x)+f((j-1)x)-2f(jx)-2f(x)=f((j+1)x)-(j+1)2f(x)+f((j-1)x)-(j-1)2f(x)-2[f(jx)-j2f(x)]
for all x∈A, it follows from induction hypothesis and (3.8) that for all x∈A,
(3.10)‖f((j+1)x)-(j+1)2f(x)‖≤max⁡{(j-1)2‖f((j+1)x)+f((j-1)x)-2f(jx)-2f(x)‖,‖f((j-1)x)-(j-1)2f(x)‖,|2|‖j2f(x)-f(jx)‖}≤max⁡{φ(0,0),φ(x,x),φ(2x,x),…,φ((j)x,x)}.
This proves (3.6) for all i≥2. In particular
(3.11)‖f(kx)-k2f(x)‖≤Φ(x)(x∈A).
Replacing x by k-1x in (3.11), we get
(3.12)‖f(x)-k2f(k-1x)‖≤Φ(k-1x)≤L|k|2Φ(x)
for all x∈A. Let Ω be the set of all functions u:A→X. We define the metric d on Ω as follows:
(3.13)d(u,v)=sup⁡x∈AD(x),
where D(x)=(∥u(x)-v(x)∥)/Φ(x) if Φ(x)≠0 and D(x)=∥u(x)-v(x)∥ if Φ(x)=0. One has the operator J:Ω→Ω by J(u)(x)=k2u(k-1x). Then J is strictly contractive on Ω; in fact, if
(3.14)‖u(x)-v(x)‖≤αΦ(x)(x∈A),
then by (3.3),
(3.15)‖J(u)(x)-J(v)(x)‖=|k|2‖u(k-1x)-v(k-1x)‖≤α|k|2Φ(k-1x)≤LαΦ(x),(x∈A).
It follows that
(3.16)d(J(u),J(v))≤Ld(u,v)(u,v∈Ω).
Hence J is a contractive with Lipschitz constant L. By Theorem 2.3, J has a unique fixed point h:A→X and
(3.17)h(x)=lim⁡m→∞Jm(f(x))=lim⁡k2mf(k-mx)
for all x∈A.
Therefore(3.18)‖h(x+y)+h(x-y)-2h(x)-2h(y)‖=lim⁡m→∞|k|2m‖f(k-m(x+y))+f(k-m(x-y))-2f(k-mx)-2f(k-my)‖≤lim⁡m→∞|k|2mφ(k-mx,k-my)≤lim⁡m→∞Lmφ(x,y)=0
for all x,y∈A. This shows that h is quadratic. It follows from Theorem 2.3 that
(3.19)d(f,h)≤d(J(f),f),
that is,
(3.20)‖f(x)-h(x)‖≤LΦ(x)|k|2(x∈A).Replacing xi by k-mxi,i=1,…,n in (3.2), we get
(3.21)‖f(∏i=1nk-mnxi)-f(k-mx1)k-2m(n-1)x22⋯xn2-k-2m(n-1)x12f(k-mx2)x32⋯k-2m(n-1)xn2-⋯-x12⋯xn-12f(k-mxn)∏i=1nk-mnxi‖≤ψ(k-mx1,…,k-mxn),
and so
(3.22)|k|2mn‖f(∏i=1nk-mnxi)-f(k-mx1)k-2m(n-1)x22⋯xn2-k-2m(n-1)x12f(k-mx2)x32⋯xn2-⋯-k-2m(n-1)x12⋯xn-12f(k-mxn)∏i=1nk-mnxi‖=‖22mnf(∏i=1nk-mnxi)-k2mf(k-mx1)x22⋯xn2-x12k2mf(k-mx2)x32⋯xn2-⋯-x12⋯xn-12k2mf(k-mxn)∏i=1nk-mnxi‖≤|k|2mnψ(k-mx1,…,k-mxn)≤|k|2mnKm|k|2mψ(x1,…,xn)
for all x1,…,xn∈A and each m∈ℕ. By taking m→∞, we have
(3.23)h(∏i=1nxi)=h(x1)x22⋯xn2+x12h(x2)x32⋯xn2+⋯-x12⋯xn-12h(xn)
for all x1,…,xn∈A.In the following corollaries we will assume thatA is a non-Archimedean Banach algebra over 𝕂=ℚp the field of p-adic numbers, where p>2 is a prime number.Corollary 3.2.
Letr<1 and let ε be δ be positive real numbers. Suppose that f:A→X is a mapping such that
(3.24)‖f(x+y)+f(x-y)-2f(x)-2(y)∥≤ε∥x∥r∥y‖r,‖f(∏i=1nxi)-f(x1)x22⋯xn2-x12f(x2)x32⋯xn2-⋯-x12⋯xn-12f(xn)‖≤δmax⁡{‖x1‖r,…,‖xn‖r}
for all x1,…,xn,x,y∈A. Then there exists a unique quadratic n-derivation h from A into X such that
(3.25)‖f(x)-h(x)‖≤εp2r‖x‖2r
for all x∈A.Proof.
By (3.24), f(0)=0. Let φ(x,y)=ε∥x∥r∥y∥r and ψ(x1,…,xn}=δmax{∥x1∥r,…,∥xn∥r} for all x1,…,xn,x,y∈A. Then
(3.26)|p|2φ(p-1x,p-1y)=p2r-2φ(x,y),|p|2ψ(p-1x1,…,p-1xn}=pr-2ψ(x1,…,xn}
for all x1,…,xn,x,y∈A.
Moreover,(3.27)Φ(x)=max⁡{φ(0,0),φ(x,x),φ(2x,x),…,φ((p-1)x,x)}=ϵ‖x‖2r(x∈A).
Put L=p2r-2 and K=pr-2 in Theorem 3.1. Then there exists a unique quadratic n-derivation h from A into X such that
(3.28)‖f(x)-h(x)‖≤εp2r‖x‖2r
for all x∈A.Similarly, we can prove the following result.Corollary 3.3.
Letr<2 and let ε be δ be positive real numbers. Suppose that f:A→X is a mapping such that
(3.29)‖f(x+y)+f(x-y)-2f(x)-2(y)‖≤εmax⁡{‖x‖r,‖y‖r},‖f(∏i=1nxi)-f(x1)x22⋯xn2-x12f(x2)x32⋯xn2-⋯-x12⋯xn-12f(xn)‖≤δmax⁡{‖x1‖r,…,‖xn‖r}
for all x1,…,xn,x,y∈A. Then there exists a unique quadratic n-derivation h from A into X such that
(3.30)‖f(x)-h(x)‖≤εpr‖x‖r
for all x∈A.Remark 3.4.
We can use similar arguments to obtain corollaries like Corollaries3.2 and 3.3, when r>1 and r>2.By using the same technique of proving Theorem3.1, we can prove the following result.Remark 3.5.
Letφ:A×A→[0,∞),ψ:A×⋯×A→[0,∞) be functions. Let f:A→X be a given mapping such that f(0)=0,
(3.31)‖f(x+y)+f(x-y)-2f(x)-2f(y)‖≤φ(x,y)
and that
(3.32)‖f(∏i=1nxi)-f(x1)x22⋯xn2-x12f(x2)x32⋯xn2-⋯-x12⋯xn-12f(xn)‖≤ψ(x1,…,xn)
for all x1,…,xn,x,y∈A. Suppose that there exist a natural number k∈𝕂 and L,K∈(0,1), such that
(3.33)φ(kx,y)≤|k|2Lφ(x,y),ψ(kx1,…,kxn)≤|k|2Kψ(x1,…,xn)
for all x1,…,xn,x,y∈A. Then there exists a unique quadratic n-derivation d from A into X such that
(3.34)‖f(x)-d(x)‖≤|k|2LΦ(x)
for all x∈A, where
(3.35)Φ(x)=max⁡{φ(0,0),φ(x,x),φ(x2,x),…,φ(x(k-1),x)}(x∈A).



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