Nearly Quadratic n-Derivations on Non-Archimedean Banach Algebras

Abstract

Let 𝑛>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(∏^𝑛_𝑖=1𝑎_𝑖)=𝐷(𝑎₁)𝑎²₂⋯𝑎²_𝑛+𝑎²₁𝐷(𝑎₂)𝑎²₃⋯𝑎²_𝑛+⋯+𝑎²₁𝑎²₂⋯𝑎²_𝑛−1𝐷(𝑎_𝑛) for all 𝑎₁,...,𝑎_𝑛∈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.

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 equation𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (1.1)

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 if𝐷^𝑛_𝑖=1𝑎_𝑖=𝐷𝑎₁𝑎²₂⋯𝑎²_𝑛+𝑎²₁𝐷𝑎₂𝑎²₃⋯𝑎²_𝑛+⋯+𝑎²₁𝑎²₂⋯𝑎²_𝑛−1𝐷𝑎_𝑛 (1.2)

for all 𝑎₁,…,𝑎_𝑛∈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 field 𝕂 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. Let 𝑋 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)𝑝^𝑛𝑥 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 ℚ_𝑝 which is called the 𝑝-adic number field.

Definition 2.2. Let 𝑋 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 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋. (2.1) Then there exists a unique element 𝑎∈𝑋 such that 𝑇𝑎=𝑎. Moreover, 𝑎=lim_𝑛→∞𝑇^𝑛𝑥, and 𝑑(𝑎,𝑥)≤𝑑(𝑥,𝑇𝑥),∀𝑥∈𝑋. (2.2)

Proof. A similar argument as Archimedean case can be applied to show that 𝑇 has a unique element 𝑎∈𝑋 such that 𝑇𝑎=𝑎 and 𝑎=lim_𝑛→∞𝑇^𝑛𝑥. It follows from strong triangle inequality that for all 𝑥∈𝑋 and for each 𝑛∈ℕ, we have 𝑑𝑇^𝑛𝑥,𝑥≤max𝑑(𝑇(𝑥),𝑥),…,𝑑𝑇^𝑛(𝑥),𝑇^𝑛−1(𝑥𝑑(𝑇(𝑥)))≤max,𝑥,…,𝛼^𝑛−1𝑑(𝑇(𝑥),(𝑥(𝑇(𝑥)).))=𝑑,𝑥 (2.3)

3. Main Results

In this section 𝐴 denotes a non-Archimedean Banach algebra over a non-Archimedean field 𝕂 and 𝑋 is a non-Archimedean Banach 𝐴-module.

Theorem 3.1. Let 𝜑∶𝐴×𝐴→[0,∞),𝜓∶𝐴×⋯×𝐴→[0,∞) be functions. Let 𝑓∶𝐴→𝑋 be a given mapping such that 𝑓(0)=0, 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦≤𝜑𝑥,𝑦 (3.1) and that 𝑓^𝑛_𝑖=1𝑥_𝑖−𝑓𝑥₁𝑥²₂⋯𝑥²_𝑛−𝑥²₁𝑓𝑥₂𝑥²₃⋯𝑥²_𝑛−⋯−𝑥²₁⋯𝑥²_𝑛−1𝑓𝑥_𝑛≤𝜓𝑥₁,…,𝑥_𝑛 (3.2) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴. Suppose that there exist a natural number 𝑘∈𝕂 and 𝐿,𝐾∈(0,1), such that |𝑘|²𝜑𝑘⁻¹𝑥,𝑘⁻¹𝑦≤𝐿𝜑𝑥,𝑦,|𝑘|²𝜓𝑘⁻¹𝑥₁,…,𝑘⁻¹𝑥_𝑛≤𝐾𝜓𝑥₁,…,𝑥_𝑛 (3.3) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴. Then there exists a unique quadratic 𝑛-derivation ℎ from 𝐴 into 𝑋 such that 𝑓(𝑥)(𝑥≤(𝑥)−ℎ)𝐿Φ|𝑘|² (3.4) for all 𝑥∈𝐴, where Φ(𝑥)=max𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),…,𝜑((𝑘−1)𝑥,𝑥)(𝑥∈𝐴). (3.5)

Proof. By induction on 𝑖, one can show that for all 𝑥∈𝐴 and 𝑖≥2, 𝑓(𝑖𝑥)−𝑖²𝑓(𝑥)≤max𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),…,𝜑((𝑖−1)𝑥,𝑥). (3.6) Let 𝑥=𝑦 in (3.1). Then 𝑓(2𝑥)−2²𝑓(𝑥)≤max𝜑(0,0),𝜑(𝑥,𝑥)(𝑥∈𝐴). (3.7) 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 𝑓𝑗+1𝑥+𝑓𝑗−1𝑥−2𝑓𝑗𝑥−2𝑓(𝑥)≤max𝜑(0,0),𝜑𝑗𝑥,𝑥 (3.8) for all 𝑥∈𝐴. Since 𝑓𝑗+1𝑥+𝑓𝑗−1𝑥−2𝑓𝑗𝑥−2𝑓(𝑥)=𝑓𝑗+1𝑥−𝑗+1²𝑓(𝑥)+𝑓𝑗−1𝑥−𝑗−1²𝑓(𝑥)−2𝑓𝑗𝑥−𝑗²𝑓(𝑥) (3.9) for all 𝑥∈𝐴, it follows from induction hypothesis and (3.8) that for all 𝑥∈𝐴, 𝑓𝑗+1𝑥−𝑗+1²𝑓(𝑥𝑓𝑥𝑥(𝑥,𝑓𝑥−)≤max𝑗+1+𝑓𝑗−1−2𝑓𝑗𝑥−2𝑓)𝑗−1𝑗−1²𝑓(𝑥),|2|𝑗²𝑓(𝑥)𝜑()()()𝑗.−𝑓𝑗𝑥≤max0,0,𝜑𝑥,𝑥,𝜑2𝑥,𝑥,…,𝜑𝑥,𝑥 (3.10) This proves (3.6) for all 𝑖≥2. In particular 𝑓(𝑘𝑥)−𝑘²𝑓(𝑥)≤Φ(𝑥)(𝑥∈𝐴). (3.11) Replacing 𝑥 by 𝑘⁻¹𝑥 in (3.11), we get 𝑓(𝑥)−𝑘²𝑓𝑘⁻¹𝑥≤Φ𝑘⁻¹𝑥≤𝐿|𝑘|²Φ(𝑥) (3.12) for all 𝑥∈𝐴. Let Ω be the set of all functions 𝑢∶𝐴→𝑋. We define the metric 𝑑 on Ω as follows: 𝑑(𝑢,𝑣)=sup_𝑥∈𝐴𝐷(𝑥), (3.13) 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 ‖𝑢(𝑥)−𝑣(𝑥)‖≤𝛼Φ(𝑥)(𝑥∈𝐴), (3.14) then by (3.3), ‖𝐽(𝑢)(𝑥)−𝐽(𝑣)(𝑥)‖=|𝑘|²𝑢𝑘⁻¹𝑥−𝑣𝑘⁻¹𝑥≤𝛼|𝑘|²Φ𝑘⁻¹𝑥≤𝐿𝛼Φ(𝑥),(𝑥∈𝐴). (3.15) It follows that 𝑑(𝐽(𝑢),𝐽(𝑣))≤𝐿𝑑(𝑢,𝑣)(𝑢,𝑣∈Ω). (3.16) Hence 𝐽 is a contractive with Lipschitz constant 𝐿. By Theorem 2.3, 𝐽 has a unique fixed point ℎ∶𝐴→𝑋 and ℎ(𝑥)=lim_𝑚→∞𝐽^𝑚𝑓(𝑥)=lim𝑘^2𝑚𝑓𝑘^−𝑚𝑥 (3.17) for all 𝑥∈𝐴.
Therefore ℎ𝑥+𝑦+ℎ𝑥−𝑦−2ℎ(𝑥)−2ℎ𝑦=lim_𝑚→∞|𝑘|^2𝑚𝑓𝑘^−𝑚𝑥+𝑦+𝑓𝑘^−𝑚𝑥−𝑦−2𝑓𝑘^−𝑚𝑥−2𝑓𝑘^−𝑚𝑦≤lim_𝑚→∞|𝑘|^2𝑚𝜑𝑘^−𝑚𝑥,𝑘^−𝑚𝑦≤lim_𝑚→∞𝐿^𝑚𝜑𝑥,𝑦=0 (3.18) for all 𝑥,𝑦∈𝐴. This shows that ℎ is quadratic. It follows from Theorem 2.3 that 𝑑𝑓,ℎ≤𝑑𝐽𝑓,𝑓, (3.19) that is, 𝑓(𝑥)(𝑥≤(𝑥)−ℎ)𝐿Φ|𝑘|²(𝑥∈𝐴). (3.20)Replacing 𝑥_𝑖 by 𝑘^−𝑚𝑥_𝑖,𝑖=1,…,𝑛 in (3.2), we get 𝑓^𝑛_𝑖=1𝑘^−𝑚𝑛𝑥_𝑖−𝑓𝑘^−𝑚𝑥₁𝑘^{−2𝑚(𝑛−1)}𝑥²₂⋯𝑥²_𝑛−𝑘^{−2𝑚(𝑛−1)}𝑥²₁𝑓𝑘^−𝑚𝑥₂𝑥²₃⋯𝑘^{−2𝑚(𝑛−1)}𝑥²_𝑛−⋯−𝑥²₁⋯𝑥²_𝑛−1𝑓𝑘^−𝑚𝑥_𝑛≤𝜓𝑘^−𝑚𝑥₁,…,𝑘^−𝑚𝑥_𝑛, (3.21) and so |𝑘|^2𝑚𝑛𝑓^𝑛_𝑖=1𝑘^−𝑚𝑛𝑥_𝑖−𝑓𝑘^−𝑚𝑥₁𝑘^{−2𝑚(𝑛−1)}𝑥²₂⋯𝑥²_𝑛−𝑘^{−2𝑚(𝑛−1)}𝑥²₁𝑓𝑘^−𝑚𝑥₂𝑥²₃⋯𝑥²_𝑛−⋯−𝑘^{−2𝑚(𝑛−1)}𝑥²₁⋯𝑥²_𝑛−1𝑓𝑘^−𝑚𝑥_𝑛=2^2𝑚𝑛𝑓^𝑛_𝑖=1𝑘^−𝑚𝑛𝑥_𝑖−𝑘^2𝑚𝑓𝑘^−𝑚𝑥₁𝑥²₂⋯𝑥²_𝑛−𝑥²₁𝑘^2𝑚𝑓𝑘^−𝑚𝑥₂𝑥²₃⋯𝑥²_𝑛−⋯−𝑥²₁⋯𝑥²_𝑛−1𝑘^2𝑚𝑓𝑘^−𝑚𝑥_𝑛≤|𝑘|^2𝑚𝑛𝜓𝑘^−𝑚𝑥₁,…,𝑘^−𝑚𝑥_𝑛≤|𝑘|^2𝑚𝑛𝐾^𝑚|𝑘|^2𝑚𝜓𝑥₁,…,𝑥_𝑛 (3.22) for all 𝑥₁,…,𝑥_𝑛∈𝐴 and each 𝑚∈ℕ. By taking 𝑚→∞, we have ℎ^𝑛_𝑖=1𝑥_𝑖=ℎ𝑥₁𝑥²₂⋯𝑥²_𝑛+𝑥²₁ℎ𝑥₂𝑥²₃⋯𝑥²_𝑛+⋯−𝑥²₁⋯𝑥²_𝑛−1ℎ𝑥_𝑛 (3.23) for all 𝑥₁,…,𝑥_𝑛∈𝐴.

In the following corollaries we will assume that 𝐴 is a non-Archimedean Banach algebra over 𝕂=ℚ_𝑝 the field of 𝑝-adic numbers, where 𝑝>2 is a prime number.

Corollary 3.2. Let 𝑟<1 and let 𝜀 be 𝛿 be positive real numbers. Suppose that 𝑓∶𝐴→𝑋 is a mapping such that 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑦‖≤𝜀‖𝑥‖^𝑟‖𝑦^𝑟,𝑓^𝑛_𝑖=1𝑥_𝑖−𝑓𝑥₁𝑥²₂⋯𝑥²_𝑛−𝑥²₁𝑓𝑥₂𝑥²₃⋯𝑥²_𝑛−⋯−𝑥²₁⋯𝑥²_𝑛−1𝑓𝑥_𝑛≤𝛿max𝑥₁^r,…,𝑥_𝑛^r (3.24) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴. Then there exists a unique quadratic 𝑛-derivation ℎ from 𝐴 into 𝑋 such that 𝑓(𝑥)−ℎ(𝑥)≤𝜀𝑝^2𝑟‖𝑥‖^2𝑟 (3.25) for all 𝑥∈𝐴.

Proof. By (3.24), 𝑓(0)=0. Let 𝜑(𝑥,𝑦)=𝜀‖𝑥‖^𝑟‖𝑦‖^𝑟 and 𝜓(𝑥₁,…,𝑥_𝑛}=𝛿max{‖𝑥₁‖^𝑟,…,‖𝑥_𝑛‖^𝑟} for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴. Then 𝑝²𝜑𝑝⁻¹𝑥,𝑝⁻¹𝑦=𝑝^2𝑟−2𝜑𝑥,𝑦,𝑝²𝜓𝑝⁻¹𝑥₁,…,𝑝⁻¹𝑥_𝑛=𝑝^𝑟−2𝜓𝑥₁,…,𝑥_𝑛 (3.26) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴.
Moreover, Φ(𝑥)=max𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),…,𝜑𝑝−1𝑥,𝑥=𝜖‖𝑥‖^2𝑟(𝑥∈𝐴). (3.27) Put 𝐿=𝑝^2𝑟−2 and 𝐾=𝑝^𝑟−2 in Theorem 3.1. Then there exists a unique quadratic 𝑛-derivation ℎ from 𝐴 into 𝑋 such that 𝑓(𝑥)−ℎ(𝑥)≤𝜀𝑝^2𝑟‖𝑥‖^2𝑟 (3.28) for all 𝑥∈𝐴.

Similarly, we can prove the following result.

Corollary 3.3. Let 𝑟<2 and let 𝜀 be 𝛿 be positive real numbers. Suppose that 𝑓∶𝐴→𝑋 is a mapping such that 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑦≤𝜀max‖𝑥‖^𝑟,𝑦^𝑟,𝑓^𝑛_𝑖=1𝑥_𝑖−𝑓𝑥₁𝑥²₂⋯𝑥²_𝑛−𝑥²₁𝑓𝑥₂𝑥²₃⋯𝑥²_𝑛−⋯−𝑥²₁⋯𝑥²_𝑛−1𝑓𝑥_𝑛≤𝛿max𝑥₁^𝑟,…,𝑥_𝑛^𝑟 (3.29) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴. Then there exists a unique quadratic 𝑛-derivation ℎ from 𝐴 into 𝑋 such that 𝑓(𝑥)−ℎ(𝑥)≤𝜀𝑝^𝑟‖𝑥‖^𝑟 (3.30) for all 𝑥∈𝐴.

Remark 3.4. We can use similar arguments to obtain corollaries like Corollaries 3.2 and 3.3, when 𝑟>1 and 𝑟>2.

By using the same technique of proving Theorem 3.1, we can prove the following result.

Remark 3.5. Let 𝜑∶𝐴×𝐴→[0,∞),𝜓∶𝐴×⋯×𝐴→[0,∞) be functions. Let 𝑓∶𝐴→𝑋 be a given mapping such that 𝑓(0)=0, 𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦≤𝜑𝑥,𝑦 (3.31) and that 𝑓^𝑛_𝑖=1𝑥_𝑖−𝑓𝑥₁𝑥²₂⋯𝑥²_𝑛−𝑥²₁𝑓𝑥₂𝑥²₃⋯𝑥²_𝑛−⋯−𝑥²₁⋯𝑥²_𝑛−1𝑓𝑥_𝑛≤𝜓𝑥₁,…,𝑥_𝑛 (3.32) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴. Suppose that there exist a natural number 𝑘∈𝕂 and 𝐿,𝐾∈(0,1), such that 𝜑𝑘𝑥,𝑦≤|𝑘|²𝐿𝜑𝑥,𝑦,𝜓𝑘𝑥₁,…,𝑘𝑥_𝑛≤|𝑘|²𝐾𝜓𝑥₁,…,𝑥_𝑛 (3.33) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝐴. Then there exists a unique quadratic 𝑛-derivation 𝑑 from 𝐴 into 𝑋 such that 𝑓(𝑥)−𝑑(𝑥)≤|𝑘|²𝐿Φ(𝑥) (3.34) for all 𝑥∈𝐴, where Φ(𝑥)𝜑()()𝑥=max0,0,𝜑𝑥,𝑥,𝜑2𝑥,𝑥,…,𝜑())𝑘−1,𝑥(𝑥∈𝐴. (3.35)

Acknowledgment

The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021253).