Abstract

We investigate the superstability of generalized derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy functional equation.

1. Introduction

A functional equation (πœ‰) is superstable if every approximately solution of (πœ‰) is an exact solution of it.

The stability of functional equations was first introduced by Ulam [1] during his talk before a Mathematical Colloquium at the University of Wisconsin in 1940.

Given a metric group 𝐺(β‹…,𝜌), a number πœ€>0, and a mapping π‘“βˆΆπΊβ†’πΊ which satisfies the inequality 𝜌(𝑓(π‘₯⋅𝑦),𝑓(π‘₯)⋅𝑓(𝑦))β‰€πœ€ for all π‘₯,𝑦 in 𝐺, does there exist an automorphism π‘Ž of 𝐺 and a constant π‘˜>0, depending only on 𝐺 such that 𝜌(π‘Ž(π‘₯),𝑓(π‘₯))β‰€π‘˜πœ€ for all π‘₯∈𝐺?

If the answer is affirmative, we would call the equation π‘Ž(π‘₯⋅𝑦)=π‘Ž(π‘₯)β‹…π‘Ž(𝑦) of automorphism is stable. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias [3] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ–(β€–π‘₯‖𝑝+‖𝑦‖𝑝),(πœ–>0,π‘βˆˆ[0,1)). In 1991, Gajda [4] answered the question for the case 𝑝>1, which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias or generalized Hyers-Ulam stability of functional equations [5, 6].

In 1992, Găvruţa [7] generalized the Th. M. Rassias Theorem as follows.

Suppose that (𝐺,+) is an ablian group, 𝑋 is a Banach space πœ‘βˆΆπΊΓ—πΊβ†’[0,∞) which satisfies 1ξ‚πœ‘(π‘₯,𝑦)=2βˆžξ“π‘›=02βˆ’π‘›πœ‘(2𝑛π‘₯,2𝑛𝑦)<∞,(1.1) for all π‘₯,π‘¦βˆˆπΊ. If π‘“βˆΆπΊβ†’π‘‹ is a mapping with ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ‘(π‘₯,𝑦),(1.2) for all π‘₯,π‘¦βˆˆπΊ, then there exists a unique mapping π‘‡βˆΆπΊβ†’π‘‹ such that 𝑇(π‘₯+𝑦)=𝑇(π‘₯)+𝑇(𝑦) and ‖𝑓(π‘₯)βˆ’π‘‡(π‘₯)β€–β‰€ξ‚πœ‘(π‘₯,π‘₯) for all π‘₯,π‘¦βˆˆπΊ.

In 1949, Bourgin [8] proved the following result, which is sometimes called the superstability of ring homomorphisms: suppose that 𝐴 and 𝐡 are Banach algebras with unit. If π‘“βˆΆπ΄β†’π΅ is a surjective mapping such that ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ–,‖𝑓(π‘₯𝑦)βˆ’π‘“(π‘₯)𝑓(𝑦)‖≀𝛿,(1.3) for some πœ–β‰₯0,𝛿β‰₯0 and for all π‘₯,π‘¦βˆˆπ΄, then 𝑓 is a ring homomorphism.

Badora [9] and Miura et al. [10] proved the Ulam-Hyers stability and the Isac and Rassias-type stability of derivations [11] (see also [12, 13]); Savadkouhi et al. [14] have contributed works regarding the stability of ternary Jordan derivations. Jung and Chang [15] investigated the stability and superstability of higher derivations on rings. Recently, Ansari-Piri and Anjidani [16] discussed the superstability of generalized derivations on Banach algebras. In this paper, we investigate the superstability of generalized derivations on non-Archimedean Banach algebras by using the fixed point methods.

2. Preliminaries

In 1897, Hensel [17] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [18, 19].

A non-Archimedean field is a field 𝕂 equipped with a function (valuation) |β‹…| from 𝕂 into [0,∞) such that |π‘Ÿ|=0 if and only if π‘Ÿ=0,|π‘Ÿπ‘ |=|π‘Ÿβ€–π‘ |, and |π‘Ÿ+𝑠|≀max{|π‘Ÿ|,|𝑠|} for all π‘Ÿ,π‘ βˆˆπ•‚ (see [20, 21]).

Definition 2.1. Let 𝑋 be a vector space over a scalar field 𝕂 with a non-Archimedean nontrivial valuation |β‹…|. A function β€–β‹…β€–βˆΆπ‘‹β†’β„ is a non-Archimedean norm (valuation) if it satisfies the following conditions:(NA1)β€–π‘₯β€–=0 if and only if π‘₯=0,(NA2)β€–π‘Ÿπ‘₯β€–=|π‘Ÿ|β€–π‘₯β€– for all π‘Ÿβˆˆπ•‚ and π‘₯βˆˆπ‘‹,(NA3)β€–π‘₯+𝑦‖≀max{β€–π‘₯β€–,‖𝑦‖} for all π‘₯,π‘¦βˆˆπ‘‹ (the strong triangle inequality).
A sequence {π‘₯π‘š} in a non-Archimedean space is Cauchy if and only if {π‘₯π‘š+1βˆ’π‘₯π‘š} converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy sequence is convergent. A non-Archimedean normed algebra is a non-Archimedean normed space 𝐴 with a linear associative multiplication, satisfying β€–π‘₯𝑦‖≀‖π‘₯‖‖𝑦‖ for all π‘₯,π‘¦βˆˆπ΄. A non-Archimedean complete normed algebra is called a non-Archimedean Banach algebra (see [22]).

Example 2.2. Let 𝑝 be a prime number. For any nonzero rational number π‘₯=(π‘Ž/𝑏)𝑝𝑛π‘₯ such that π‘Ž and 𝑏 are integers not divisible by 𝑝, define the 𝑝–adic absolute value |π‘₯|π‘βˆΆ=π‘βˆ’π‘›π‘₯. Then, |β‹…| is a non-Archimedean norm on β„š. The completion of β„š with respect to |β‹…| is denoted by β„šπ‘ which is called the 𝑝-adic number field.

Definition 2.3. Let 𝑋 be a nonempty set and π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞] satisfy the following properties: (D1)𝑑(π‘₯,𝑦)=0 if and only if π‘₯=𝑦,(D2)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯) (symmetry),(D3)𝑑(π‘₯,𝑧)≀max{𝑑(π‘₯,𝑦),𝑑(𝑦,𝑧)} (strong triangle in equality),for all π‘₯,𝑦,π‘§βˆˆπ‘‹. Then, (𝑋,𝑑) is called a non-Archimedean generalized metric space. (𝑋,𝑑) is called complete if every 𝑑-Cauchy sequence in 𝑋 is 𝑑-convergent.

Definition 2.4. Let 𝐴 be a non-Archimedean algebra. An additive mapping π·βˆΆπ΄β†’π΄ is said to be a ring derivation if 𝐷(π‘₯𝑦)=𝐷(π‘₯)𝑦+π‘₯𝐷(𝑦) for all π‘₯,π‘¦βˆˆπ΄. An additive mapping π»βˆΆπ΄β†’π΄ is said to be a generalized ring derivation if there exists a ring derivation π·βˆΆπ΄β†’π΄ such that 𝐻(π‘₯𝑦)=π‘₯𝐻(𝑦)+𝐷(π‘₯)𝑦,(2.1) for all π‘₯,π‘¦βˆˆπ΄.
We need the following fixed point theorem (see [23, 24]).

Theorem 2.5 (non-Archimedean alternative Contraction Principle). Suppose that (𝑋,𝑑) is a non-Archimedean generalized complete metric space and Ξ›βˆΆπ‘‹β†’π‘‹ is a strictly contractive mapping; that is, 𝑑(Ξ›π‘₯,Λ𝑦)≀𝐿𝑑(π‘₯,𝑦),(π‘₯,π‘¦βˆˆπ‘‹),(2.2) for some 𝐿<1. If there exists a nonnegative integer π‘˜ such that 𝑑(Ξ›π‘˜+1π‘₯,Ξ›π‘˜π‘₯)<∞ for some π‘₯βˆˆπ‘‹, then the followings are true:
(a) the sequence {Λ𝑛π‘₯} converges to a fixed point π‘₯βˆ— of Ξ›,
(b) π‘₯βˆ— is a unique fixed point of Ξ› in π‘‹βˆ—=ξ€½ξ€·Ξ›π‘¦βˆˆπ‘‹βˆ£π‘‘π‘˜ξ€Έξ€Ύπ‘₯,𝑦<∞,(2.3)
(c) if π‘¦βˆˆπ‘‹βˆ—, then 𝑑𝑦,π‘₯βˆ—ξ€Έβ‰€π‘‘(Λ𝑦,𝑦).(2.4)

3. Non-Archimedean Superstability of Generalized Derivations

Hereafter, we will assume that 𝐴 is a non-Archimedean Banach algebra with unit over a non-Archimedean field 𝕂.

Theorem 3.1. Let πœ‘βˆΆπ΄Γ—π΄β†’[0,∞) be a function. Suppose that 𝑓,π‘”βˆΆπ΄β†’π΄ are mappings such that 𝑔 is additive and ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ‘(π‘₯,𝑦),(3.1)‖𝑓(π‘₯𝑦)βˆ’π‘₯𝑓(𝑦)βˆ’π‘”(π‘₯)π‘¦β€–β‰€πœ‘(π‘₯,𝑦),(3.2) for all π‘₯,π‘¦βˆˆπ΄. If there exists a natural number π‘˜βˆˆπ•‚ and 0<𝐿<1, ||π‘˜||βˆ’1||π‘˜||πœ‘(π‘˜π‘₯,π‘˜π‘¦),βˆ’1||π‘˜||πœ‘(π‘˜π‘₯,𝑦),βˆ’1πœ‘(π‘₯,π‘˜π‘¦)β‰€πΏπœ‘(π‘₯,𝑦),(3.3) for all π‘₯,π‘¦βˆˆπ΄. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.

Proof. By induction on 𝑖, we prove that ‖𝑓(𝑖π‘₯)βˆ’π‘–π‘“(π‘₯)‖≀max{πœ‘(0,0),πœ‘(π‘₯,π‘₯),πœ‘(2π‘₯,π‘₯),…,πœ‘((π‘–βˆ’1)π‘₯,π‘₯)},(3.4) for all π‘₯∈𝐴 and 𝑖β‰₯2. Let π‘₯=𝑦 in (3.1). Then, ‖𝑓(2π‘₯)βˆ’2𝑓(π‘₯)‖≀max{πœ‘(0,0),πœ‘(π‘₯,π‘₯)},π‘›βˆˆβ„•0,π‘₯∈𝐴.(3.5) This proves (3.4) for 𝑖=2. Let (3.4) holds for 𝑖=1,2,…,𝑗. Replacing π‘₯ by 𝑗π‘₯ and 𝑦 by π‘₯ in (3.1) for each π‘›βˆˆβ„•0, and for all π‘₯∈𝐴, we get ‖𝑓((𝑗+1)π‘₯)βˆ’π‘“(𝑗π‘₯)βˆ’π‘“(π‘₯)‖≀max{πœ‘(0,0),πœ‘(𝑗π‘₯,π‘₯)}.(3.6) Since 𝑓((𝑗+1)π‘₯)βˆ’π‘“(𝑗π‘₯)βˆ’π‘“(π‘₯)=𝑓((𝑗+1)π‘₯)βˆ’(𝑗+1)𝑓(π‘₯)+(𝑗+1)𝑓(π‘₯)βˆ’π‘“(𝑗π‘₯)βˆ’π‘“(π‘₯)=𝑓((𝑗+1)π‘₯)βˆ’(𝑗+1)𝑓(π‘₯)+𝑗𝑓(π‘₯)βˆ’π‘“(𝑗π‘₯),(3.7) for all π‘₯∈𝐴, it follows from induction hypothesis and (3.6) that {β€–}‖𝑓((𝑗+1)π‘₯)βˆ’(𝑗+1)𝑓(π‘₯)‖≀max‖𝑓((𝑗+1)π‘₯)βˆ’π‘“(𝑗π‘₯)βˆ’π‘“(π‘₯)β€–,‖𝑗𝑓(π‘₯)βˆ’π‘“(𝑗π‘₯)≀max{πœ‘(0,0),πœ‘(π‘₯,π‘₯),πœ‘(2π‘₯,π‘₯),…,πœ‘((𝑗)π‘₯,π‘₯)},(3.8) for all π‘₯∈𝐴. This proves (3.4) for all 𝑖β‰₯2. In particular, ‖𝑓(π‘˜π‘₯)βˆ’π‘˜π‘“(π‘₯)β€–β‰€πœ“(π‘₯),(3.9) for all π‘₯∈𝐴 where πœ“(π‘₯)=max{πœ‘(0,0),πœ‘(π‘₯,π‘₯),πœ‘(2π‘₯,π‘₯),…,πœ‘((π‘˜βˆ’1)π‘₯,π‘₯)}(π‘₯∈𝐴).(3.10)
Let 𝑋 be the set of all functions π‘ŸβˆΆπ΄β†’π΄. We define π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞] as follows:𝑑(π‘Ÿ,𝑠)=inf{𝛼>0βˆΆβ€–π‘Ÿ(π‘₯)βˆ’π‘ (π‘₯)β€–β‰€π›Όπœ“(π‘₯)βˆ€π‘₯∈𝐴}.(3.11) It is easy to see that 𝑑 defines a generalized complete metric on 𝑋. Define π½βˆΆπ‘‹β†’π‘‹ by 𝐽(π‘Ÿ)(π‘₯)=π‘˜βˆ’1π‘Ÿ(π‘˜π‘₯). Then, 𝐽 is strictly contractive on 𝑋, in fact, if β€–π‘Ÿ(π‘₯)βˆ’π‘ (π‘₯)β€–β‰€π›Όπœ“(π‘₯),(π‘₯∈𝐴),(3.12) then by (3.3), ||π‘˜||‖𝐽(π‘Ÿ)(π‘₯)βˆ’π½(𝑠)(π‘₯)β€–=βˆ’1||π‘˜||β€–π‘Ÿ(π‘˜π‘₯)βˆ’π‘ (π‘˜π‘₯)β€–β‰€π›Όβˆ’1πœ“(π‘˜π‘₯)β‰€πΏπ›Όπœ“(π‘₯),(π‘₯∈𝐴).(3.13) It follows that 𝑑(𝐽(π‘Ÿ),𝐽(𝑠))≀𝐿𝑑(π‘Ÿ,𝑠)(π‘Ÿ,π‘ βˆˆπ‘‹).(3.14) Hence, 𝐽 is a strictly contractive mapping with Lipschitz constant 𝐿. By (3.9), β€–β€–β€–π‘˜(𝐽𝑓)(π‘₯)βˆ’π‘“(π‘₯)β€–=βˆ’1𝑓‖‖,||π‘˜||(π‘˜π‘₯)βˆ’π‘“(π‘₯)βˆ’1||π‘˜||‖𝑓(π‘˜π‘₯)βˆ’π‘˜π‘“(π‘₯)β€–β‰€βˆ’1πœ“(π‘₯)(π‘₯∈𝐴).(3.15) This means that 𝑑(𝐽(𝑓),𝑓)≀1/|π‘˜|. By Theorem 2.5, 𝐽 has a unique fixed point β„ŽβˆΆπ΄β†’π΄ in the set π‘ˆ={π‘Ÿβˆˆπ‘‹βˆΆπ‘‘(π‘Ÿ,𝐽(𝑓))<∞},(3.16) and for each π‘₯∈𝐴, β„Ž(π‘₯)=limπ‘šβ†’βˆžπ½π‘š(𝑓(π‘₯))=limπ‘˜βˆ’π‘šπ‘“(π‘˜π‘šπ‘₯).(3.17)
Therefore,β€–β„Ž(π‘₯+𝑦)βˆ’β„Ž(π‘₯)βˆ’β„Ž(𝑦)β€–=limπ‘šβ†’βˆž||π‘˜||βˆ’π‘šβ€–π‘“(π‘˜π‘š(π‘₯+𝑦))βˆ’π‘“(π‘˜π‘šπ‘₯)βˆ’π‘“(π‘˜π‘šπ‘¦)‖≀limπ‘šβ†’βˆž||π‘˜||βˆ’π‘šmax{πœ‘(0,0),πœ‘(π‘˜π‘›π‘₯,π‘˜π‘›π‘¦)}≀limπ‘šβ†’βˆžπΏπ‘šπœ‘(π‘₯,𝑦)=0,(3.18) for all π‘₯,π‘¦βˆˆπ΄. This shows that β„Ž is additive.
Replacing π‘₯ by π‘˜π‘›π‘₯ in (3.2) to get‖𝑓(π‘˜π‘›π‘₯𝑦)βˆ’π‘˜π‘›π‘₯𝑓(𝑦)βˆ’π‘”(π‘˜π‘›π‘₯)π‘¦β€–β‰€πœ‘(π‘˜π‘›π‘₯,𝑦),(3.19) and so ‖‖‖𝑓(π‘˜π‘›π‘₯𝑦)π‘˜π‘›βˆ’π‘₯𝑓(𝑦)βˆ’π‘”(π‘˜π‘›π‘₯)π‘˜π‘›π‘¦β€–β€–β€–β‰€1||π‘˜||π‘›πœ‘(π‘˜π‘›π‘₯,𝑦)β‰€πΏπ‘›πœ‘(π‘₯,𝑦),(3.20) for all π‘₯,π‘¦βˆˆπ΄ and all π‘›βˆˆβ„•. By taking π‘›β†’βˆž, we have β„Ž(π‘₯𝑦)=π‘₯𝑓(𝑦)+limπ‘›β†’βˆžπ‘”(π‘˜π‘›π‘₯)π‘˜π‘›π‘¦,(3.21) for all π‘₯,π‘¦βˆˆπ΄.
Fix π‘šβˆˆβ„•. By (3.21), we haveπ‘₯𝑓(π‘˜π‘šπ‘¦)=β„Ž(π‘˜π‘šπ‘₯𝑦)βˆ’limπ‘›β†’βˆžξ‚΅π‘”(π‘˜π‘›π‘₯)π‘˜π‘›(π‘˜π‘šξ‚Άπ‘¦)=π‘˜π‘šπ‘₯𝑓(𝑦)+limπ‘›β†’βˆžξ‚΅π‘”(π‘˜π‘›π‘˜π‘šπ‘₯)π‘˜π‘›π‘¦ξ‚Άβˆ’π‘˜π‘šlimπ‘›β†’βˆžξ‚΅π‘”(π‘˜π‘›π‘₯)π‘˜π‘›π‘¦ξ‚Ά=π‘˜π‘šπ‘₯𝑓(𝑦)+π‘˜π‘šlimπ‘›β†’βˆžξƒ©π‘”ξ€·π‘˜π‘›+π‘šπ‘₯ξ€Έπ‘˜π‘›+π‘šπ‘¦ξƒͺβˆ’π‘˜π‘šlimπ‘›β†’βˆžξ‚΅π‘”(π‘˜π‘›π‘₯)π‘˜π‘›π‘¦ξ‚Ά=π‘˜π‘šπ‘₯𝑓(𝑦),(3.22) for all π‘₯,π‘¦βˆˆπ΄. Then, π‘₯𝑓(𝑦)=π‘₯(𝑓(π‘˜π‘šπ‘¦)/π‘˜π‘š) for all π‘₯,π‘¦βˆˆπ΄ and each π‘šβˆˆβ„•, and so by taking π‘šβ†’βˆž, we have π‘₯𝑓(𝑦)=π‘₯β„Ž(𝑦). Now, we obtain β„Ž=𝑓, since 𝐴 is with unit. Replacing 𝑦 by π‘˜π‘›π‘¦ in (3.2), we obtain ‖𝑓(π‘˜π‘›(π‘₯𝑦))βˆ’π‘₯𝑓(π‘˜π‘›π‘¦)βˆ’π‘˜π‘›π‘”(π‘₯)π‘¦β€–β‰€πœ‘(π‘₯,π‘˜π‘›π‘¦),(3.23) and hence, ‖‖‖𝑓(π‘˜π‘›π‘₯𝑦)π‘˜π‘›βˆ’π‘₯𝑓(π‘˜π‘›π‘¦)π‘˜π‘›β€–β€–β€–β‰€1βˆ’π‘”(π‘₯)𝑦||π‘˜||π‘›πœ‘(π‘₯,π‘˜π‘›π‘¦)β‰€πΏπ‘›πœ‘(π‘₯,𝑦),(3.24) for all π‘₯,π‘¦βˆˆπ΄ and each π‘›βˆˆβ„•. Letting 𝑛 tends to infinite, we have 𝑓(π‘₯𝑦)=π‘₯𝑓(𝑦)+𝑔(π‘₯)𝑦.(3.25) Now, we show that 𝑔 is a ring derivation. By (3.25), we get 𝑔(π‘₯𝑦)𝑧=𝑓(π‘₯𝑦𝑧)βˆ’π‘₯𝑦𝑓(𝑧)=π‘₯𝑓(𝑦𝑧)+𝑔(π‘₯)π‘¦π‘§βˆ’π‘₯𝑦𝑓(𝑧)=(π‘₯𝑔(𝑦)+𝑔(π‘₯)𝑦)𝑧,(3.26) for all π‘₯,𝑦,π‘§βˆˆπ΄. Therefore, we have 𝑔(π‘₯𝑦)=π‘₯𝑔(𝑦)+𝑔(π‘₯)𝑦.

The proof of following theorem is similar to that in Theorem 3.1, hence it is omitted.

Theorem 3.2. Let πœ‘βˆΆπ΄Γ—π΄β†’[0,∞) be a function. Suppose that 𝑓,π‘”βˆΆπ΄β†’π΄ are mappings such that 𝑔 is additive and ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ‘(π‘₯,𝑦),‖𝑓(π‘₯𝑦)βˆ’π‘₯𝑓(𝑦)βˆ’π‘”(π‘₯)π‘¦β€–β‰€πœ‘(π‘₯,𝑦),(3.27) for all π‘₯,π‘¦βˆˆπ΄. If there exists a natural number π‘˜βˆˆπ•‚ and 0<𝐿<1, ||π‘˜||πœ‘ξ€·π‘˜βˆ’1π‘₯,π‘˜βˆ’1𝑦,||π‘˜||πœ‘ξ€·π‘˜βˆ’1ξ€Έ,||π‘˜||πœ‘ξ€·π‘₯,𝑦π‘₯,π‘˜βˆ’1π‘¦ξ€Έβ‰€πΏπœ‘(π‘₯,𝑦),(3.28) for all π‘₯,π‘¦βˆˆπ΄. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.

The following results are immediate corollaries of Theorems 3.1 and 3.2 and Example 2.3.

Corollary 3.3. Let 𝐴 be a non-Archimedean Banach algebra over β„šπ‘, πœ€>0, and 𝑝1,𝑝2∈(1,∞). Suppose that 𝑓,π‘”βˆΆπ΄β†’π΄ are mappings such that 𝑔 is additive and ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ€β€–π‘₯‖𝑝1‖𝑦‖𝑝2ξ€Έ,(‖𝑓π‘₯y)βˆ’π‘₯𝑓(𝑦)βˆ’π‘”(π‘₯)π‘¦β€–β‰€πœ€β€–π‘₯‖𝑝1‖𝑦‖𝑝2ξ€Έ,(3.29) for all π‘₯,π‘¦βˆˆπ΄. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.

Corollary 3.4. Let 𝐴 be a non-Archimedean Banach algebra over β„šπ‘, πœ€>0 and 𝑝1,𝑝2,𝑝1+𝑝2∈(βˆ’βˆž,1). Suppose that 𝑓,π‘”βˆΆπ΄β†’π΄ are mappings such that 𝑔 is additive and ‖𝑓(π‘₯+𝑦)βˆ’π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœ€β€–π‘₯‖𝑝1‖𝑦‖𝑝2ξ€Έ,(‖𝑓π‘₯𝑦)βˆ’π‘₯𝑓(𝑦)βˆ’π‘”(π‘₯)π‘¦β€–β‰€πœ€β€–π‘₯‖𝑝1‖𝑦‖𝑝2ξ€Έ,(3.30) for all π‘₯,π‘¦βˆˆπ΄. Then, 𝑓 is a generalized ring derivation and 𝑔 is a ring derivation.