Stability and Superstability of Generalized (𝜃, 𝜙)-Derivations in Non-Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation

Abstract

Let 𝐴 be an algebra, and let 𝜃, 𝜙 be ring automorphisms of 𝐴. An additive mapping 𝐻∶𝐴→𝐴 is called a (𝜃,𝜙)-derivation if 𝐻(𝑥𝑦)=𝐻(𝑥)𝜃(𝑦)+𝜙(𝑥)𝐻(𝑦) for all 𝑥,𝑦∈𝐴. Moreover, an additive mapping 𝐹∶𝐴→𝐴 is said to be a generalized (𝜃,𝜙)-derivation if there exists a (𝜃,𝜙)-derivation 𝐻∶𝐴→𝐴 such that 𝐹(𝑥𝑦)=𝐹(𝑥)𝜃(𝑦)+𝜙(𝑥)𝐻(𝑦) for all 𝑥,𝑦∈𝐴. In this paper, we investigate the superstability of generalized (𝜃,𝜙)-derivations in non-Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.

1. Introduction and Preliminaries

In 1897, Hensel [1] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [2, 3].

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 [4]).

Definition 1.1. 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).

A sequence {𝑥_𝑚} in a non-Archimedean space is Cauchy’s if and only if {𝑥_𝑚+1−𝑥_𝑚} converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy’s sequence is convergent. A non-Archimedean-normed algebra is a non-Archimedean-normed space 𝐴 with a linear associative multiplication, satisfying ‖𝑥𝑦‖≤‖𝑥‖‖𝑦‖ for all 𝑥,𝑦∈𝐴. A non-Archimedean complete normed algebra is called a non-Archimedean Banach’s algebra (see [5]).

Definition 1.2. 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 generalized metric space. (𝑋,𝑑) is called complete if every 𝑑-Cauchy’s sequence in 𝑋 is d-convergent.

Definition 1.3. Let 𝐴 be a non-Archimedean algebra, and let 𝜃, 𝜙 be ring automorphisms of 𝐴. An additive mapping 𝐻∶𝐴→𝐴 is called a (𝜃,𝜙)-derivation in case 𝐻(𝑥𝑦)=𝐻(𝑥)𝜃(𝑦)+𝜙(𝑥)𝐻(𝑦) holds for all 𝑥,𝑦∈𝐴. An additive mapping 𝐹∶𝐴→𝐴 is said to be a generalized (𝜃,𝜙)-derivation if there exists a (𝜃,𝜙)-derivation 𝐻∶𝐴→𝐴 such that 𝐹𝑥𝑦=𝐹(𝑥)𝜃𝑦+𝜙(𝑥)𝐻𝑦 (1.1) for all 𝑥,𝑦∈𝐴.

We need the following fixed point theorem (see [6, 7]).

Theorem 1.4 (Non-Archimedean Alternative Contraction Principle). Suppose (𝑋,𝑑) is a non-Archimedean generalized complete metric space and Λ∶𝑋→𝑋 is a strictly contractive mapping; that is, 𝑑Λ𝑥,Λ𝑦≤𝐿𝑑𝑥,𝑦𝑥,𝑦∈𝑋 (1.2) for some 𝐿<1. If there exists a nonnegative integer 𝑘 such that 𝑑(Λ^𝑘+1𝑥,Λ^𝑘𝑥)<∞ for some 𝑥∈𝑋, then the followings are true.(a)The sequence {Λ^𝑛𝑥} converges to a fixed point 𝑥^∗ of Λ.(b)𝑥^∗ is a unique fixed point of Λ in 𝑋^∗=𝑦∈𝑋∣𝑑Λ^𝑘𝑥,𝑦<∞. (1.3)(c)If 𝑦∈𝑋^∗, then 𝑑𝑦,𝑥^∗≤𝑑Λ𝑦,𝑦. (1.4)

A functional equation (𝜉) is superstable if every approximately solution of (𝜉) is an exact solution of it.

The stability of functional equations was first introduced by Ulam [8] during his talk before a mathematical colloquium at the University of Wisconsin in 1940. In 1941, Hyers [9] gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias [10] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy’s differences ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑝),(𝜖>0,𝑝∈[0,1)). Moreover, John Rassias [11–13] investigated the stability of some functional equations when the control function is the product of powers of norms. In 1991, Gajda [14] answered the question for the case 𝑝>1, which was raised by Rassias. This new concept is known as the Hyers-Ulam-Rassias or the generalized Hyers-Ulam stability of functional equations ([11–13, 15–35]).

In 1992, Gǎvruţa [36] generalized the Rassias theorem as follows.

Suppose (𝐺,+) is an ablian group, 𝑋 is a Banach space, 𝜑∶𝐺×𝐺→[0,∞) satisfies𝜑=1𝑥,𝑦2^∞_𝑛=02^−𝑛𝜑2^𝑛𝑥,2^𝑛𝑦<∞, (1.5)

for all 𝑥,𝑦∈𝐺. If 𝑓∶𝐺→𝑋 is a mapping with𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜑𝑥,𝑦, (1.6)

for all 𝑥,𝑦∈𝐺, then there exists a unique mapping 𝑇∶𝐺→𝑋 such that 𝑇(𝑥+𝑦)=𝑇(𝑥)+𝑇(𝑦) and ‖𝑓(𝑥)−𝑇(𝑥)‖≤𝜑(𝑥,𝑥) for all 𝑥,𝑦∈𝐺.

In 1949, Bourgin [37] proved the following result, which is sometimes called the superstability of ring homomorphisms: suppose that 𝐴 and 𝐵 are Banach algebras with unit. If 𝑓∶𝐴→𝐵 is a surjective mapping such that 𝑓(𝑥)𝑦𝑓(𝑥)𝑓𝑦𝑥+𝑦−𝑓−𝑓≤𝜖,𝑥𝑦−𝑓≤𝛿, (1.7) for some 𝜖≥0, 𝛿≥0 and for all 𝑥,𝑦∈𝐴, then 𝑓 is a ring homomorphism.

The first superstability result concerning derivations between operator algebras was obtained by Šemrl in [38]. Badora [39] proved the superstability of the functional equation 𝑓(𝑥𝑦)=𝑥𝑓(𝑦)+𝑓(𝑥)𝑦, where 𝑓 is a mapping on normed algebra 𝐴 with unit. Ansari-Piri and Anjidani [40] discussed the superstability of generalized derivations on Banach’s algebras. Recently, Eshaghi Gordji et al. [41] investigated the stability and superstability of higher ring derivations on non-Archimedean Banach’s algebras (see also [42]). In this paper, we investigate the superstability of generalized (𝜃,𝜙)-derivations on non-Archimedean Banach algebras by using the fixed point methods.

2. Non-Archimedean Superstability of Generalized (𝜃,𝜙)-Derivations

In this paper, we assume that 𝐴 is a non-Archimedean Banach’s algebra, with unit over a non-Archimedean field 𝕂, and 𝜃, 𝜙 are ring automorphisms of 𝐴.

Theorem 2.1. Let 𝜑,𝜓∶𝐴×𝐴→[0,∞) be functions. Suppose that 𝑓∶𝐴→𝐴 is a mapping such that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜑𝑥,𝑦,𝑓𝑥𝑦−𝑓(𝑥)𝜃𝑦−𝜙(𝑥)𝑔𝑦≤𝜓𝑥,𝑦, (2.1)(2.2) for all 𝑥,𝑦∈𝐴. If there exist constants 𝐾,𝐿<1 and a natural number 𝑘∈𝕂, |𝑘|⁻¹𝜑𝑘𝑥,𝑘𝑦≤𝐿𝜑𝑥,𝑦,|𝑘|⁻¹𝜓𝑘𝑥,𝑦,|𝑘|⁻¹𝜓𝑥,𝑘𝑦≤𝐾𝜓𝑥,𝑦, (2.3) for all 𝑥,𝑦∈𝐴, then 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

Proof. By induction on 𝑖, we prove that for each 𝑛∈ℕ₀, for all 𝑥∈𝐴 and 𝑖≥2, 𝑓(𝑖𝑥)−𝑖𝑓(𝑥)≤max𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),…,𝜑((𝑖−1)𝑥,𝑥). (2.4) Let 𝑥=𝑦 in (2.1), then 𝑓(2𝑥)−2𝑓(𝑥)≤max𝜑(0,0),𝜑(𝑥,𝑥),𝑛∈ℕ₀,𝑥∈𝐴. (2.5) This proves (2.4) for 𝑖=2. Let (2.4) hold for 𝑖=1,2,…,𝐽. Replacing 𝑥 by 𝑗𝑥 and 𝑦 by 𝑥 in (2.1) for each 𝑛∈ℕ₀, and for all 𝑥∈𝐴, we get 𝑓𝑗+1𝑥−𝑓𝑗𝑥−𝑓(𝑥)≤max𝜑(0,0),𝜑𝑗𝑥,𝑥. (2.6) Since 𝑓𝑥(𝑥)𝑥−𝑓(𝑥)+𝑓(𝑥)(𝑥)𝑥−𝑓(𝑥)(𝑥),𝑗+1−𝑓𝑗𝑥−𝑓=𝑓𝑗+1𝑗+1𝑗+1−𝑓𝑗𝑥−𝑓=𝑓𝑗+1𝑗+1+𝑗𝑓−𝑓𝑗𝑥 (2.7) for all 𝑥∈𝐴, it follows from induction hypothesis and (2.6) that, for all 𝑥∈𝐴, 𝑓𝑥−𝑓(𝑥𝑓𝑥(𝑥,(𝑥)𝜑()()()𝑗.𝑗+1𝑗+1)≤max𝑗+1−𝑓𝑗𝑥−𝑓)𝑗𝑓−𝑓𝑗𝑥≤max0,0,𝜑𝑥,𝑥,𝜑2𝑥,𝑥,…,𝜑𝑥,𝑥 (2.8) This proves (2.4) for all 𝑖≥2. In particular, for all 𝑥∈𝐴, 𝑓(𝑘𝑥)−𝑘𝑓(𝑥)≤Φ(𝑥), (2.9) where Φ(𝑥)=max𝜑(0,0),𝜑(𝑥,𝑥),𝜑(2𝑥,𝑥),…,𝜑((𝑘−1)𝑥,𝑥)(𝑥∈𝐴). (2.10)
Let us define a set 𝑋 of all functions 𝑟∶𝐴→𝐴 by 𝑋={𝑟∶𝐴⟶𝐴} (2.11) and introduce 𝑑 on 𝑋 as follows: 𝑑(𝑟,𝑠)=inf{𝛼>0∶‖𝑟(𝑥)−𝑠(𝑥)‖≤𝛼Φ(𝑥)∀𝑥∈𝐴}. (2.12) It is easy to see that 𝑑 defines a generalized complete metric on 𝑋. Define 𝐽∶𝑋→𝑋 by 𝐽(𝑟)(𝑥)=𝑘⁻¹𝑟(𝑘𝑥). Then 𝐽 is strictly contractive on 𝑋, in fact if ‖𝑟(𝑥)−𝑠(𝑥)‖≤𝛼Φ(𝑥)(𝑥∈𝐴), (2.13) then, by (2.3), ‖𝐽(𝑟)(𝑥)−𝐽(𝑠)(𝑥)‖=|𝑘|⁻¹‖𝑟(𝑘𝑥)−𝑠(𝑘𝑥)‖≤𝛼|𝑘|⁻¹Φ(𝑘𝑥)≤𝐿𝛼Φ(𝑥)(𝑥∈𝐴). (2.14) It follows that 𝑑(𝐽(𝑟),𝐽(𝑠))≤𝐿𝑑(𝑟,𝑠)𝑔,ℎ∈𝑋. (2.15) Hence, 𝐽 is strictly contractive mapping with the Lipschitz constant 𝐿. By (2.9), 𝐽𝑓(𝑥)−𝑓(𝑥)=𝑘⁻¹𝑓()(𝑥,|𝑘|𝑘𝑥−𝑓)₋₁𝑓(𝑘𝑥)−𝑘𝑓(𝑥)≤|𝑘|⁻¹Φ(𝑥)(𝑥∈𝐴). (2.16) This means that 𝑑(𝐽(𝑓),𝑓)≤1/|𝑘|. By Theorem 1.4, 𝐽 has a unique fixed point ℎ∶𝐴→𝐴 in the set 𝑈=𝑟∈𝑋∶𝑑𝑟,𝐽𝑓<∞, (2.17) and, for each 𝑥∈𝐴, ℎ(𝑥)=lim_𝑚→∞𝐽^𝑚𝑓(𝑥)=lim𝑘^−𝑚𝑓𝑘^𝑚𝑥. (2.18)
Therefore, each𝑥,𝑦∈𝐴, ℎ𝑥+𝑦−ℎ(𝑥)−ℎ𝑦=lim_𝑚→∞|𝑘|^−𝑚𝑓𝑘^𝑚𝑥+𝑦−𝑓𝑘^𝑚𝑥−𝑓𝑘^𝑚𝑦≤lim_𝑚→∞|𝑘|^−𝑚max𝜑(0,0),𝜑𝑘^𝑛𝑥,𝑘^𝑛𝑦≤lim_𝑚→∞𝐿^𝑚𝜑𝑥,𝑦=0. (2.19) This shows that ℎ is additive.
Replacing 𝑦 by 𝑘^𝑛𝑦 in (2.2), we get 𝑓𝑘^𝑛𝑥𝑦−𝑓(𝑥)𝜃𝑘^𝑛𝑦−𝜙(𝑥)𝑔𝑘^𝑛𝑦≤𝜓𝑥,𝑘^𝑛𝑦, (2.20) and so 𝑓𝑘^𝑛𝑥𝑦𝑘^𝑛(𝑥)𝜃𝑦(𝑥)𝑔𝑘−𝑓−𝜙^𝑛𝑦𝑘^𝑛≤1|𝑘|^𝑛𝜓𝑥,𝑘^𝑛𝑦≤𝐾^𝑛𝜓𝑥,𝑦, (2.21) for all 𝑥,𝑦∈𝐴 and each 𝑛∈ℕ. By taking 𝑛→∞, we have ℎ𝑥𝑦=𝑓(𝑥)𝜃𝑦+lim_𝑛→∞𝜙(𝑥)𝑔𝑘^𝑛𝑦𝑘^𝑛, (2.22) for all 𝑥,𝑦∈𝐴.
Fix 𝑚∈ℕ. By (2.22), we have 𝑓𝑘^𝑚𝑥𝜃𝑦=ℎ𝑘^𝑚𝑥𝑦−lim_𝑛→∞𝜙𝑘^𝑚𝑥𝑔𝑘^𝑛𝑦𝑘^𝑛=𝑓(𝑥)𝜃𝑘^𝑚𝑦+lim_𝑛→∞𝜙(𝑥)𝑔𝑘^𝑛𝑘^𝑚𝑥𝑘^𝑛−𝑘^𝑚lim_𝑛→∞𝜙(𝑥)𝑔𝑘^𝑛𝑥𝑘^𝑛=𝑘^𝑚𝑓(𝑥)𝜃𝑦+𝑘^𝑚lim_𝑛→∞𝜙(𝑥)𝑔𝑘^𝑛+𝑚𝑥𝑘^𝑛+𝑚−𝑘^𝑚lim_𝑛→∞𝜙(𝑥)𝑔𝑘^𝑛𝑥𝑘^𝑛=𝑘^𝑚𝑓(𝑥)𝜃𝑦, (2.23) for all 𝑥,𝑦∈𝐴. Then 𝑓(𝑥)𝜃(𝑦)=(𝑓(𝑘^𝑚𝑥)/𝑘^𝑚)𝜃(𝑦) for all 𝑥,𝑦∈𝐴 and each 𝑚∈ℕ, and so, by taking 𝑚→∞, we have 𝑓(𝑥)𝜃(𝑦)=ℎ(𝑥)𝜃(𝑥). Now we obtain ℎ=𝑓, since 𝐴 is with unit. Replacing 𝑥 by 𝑘^𝑛𝑥 in (2.2), we obtain 𝑓𝑘^𝑛𝑥𝑦−𝑓𝑘^𝑛𝑥𝜃𝑦−𝜙𝑘^𝑛𝑥𝑔𝑦≤𝜓𝑘^𝑛𝑥,𝑦, (2.24) and; hence, 𝑓𝑘^𝑛𝑥𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥𝑘^𝑛𝜃𝑦(𝑥)𝑔𝑦≤1−𝜙|𝑘|^𝑛𝜓𝑘^𝑛𝑥,𝑦≤𝐾^𝑛𝜓𝑥,𝑦, (2.25) for all 𝑥,𝑦∈𝐴 and each 𝑛∈ℕ. Sending 𝑛 to infinite, we have 𝑓𝑥𝑦=𝑓(𝑥)𝜃𝑦+𝜙(𝑥)𝑔𝑦. (2.26) By (2.26), we get 𝜙(𝑧)𝑔(𝑧)𝜃()𝜃𝑦()𝑔𝑦(𝑧)𝜃=𝑓(𝑧)𝜃(𝑥)(𝑧)𝑔(𝑥𝜃𝑦()𝑔𝑦(𝑧)𝜃(𝑧𝑔(𝑥)𝜃𝑦(𝑥)𝑔𝑦,𝑥𝑦=𝑓𝑧𝑥𝑦−𝑓𝑥𝑦=𝑓𝑧𝑥+𝜙𝑧𝑥−𝑓𝑥𝑦+𝜙)+𝜙𝑧𝑥−𝑓𝑥𝑦=𝜙)+𝜙 (2.27) for all 𝑥,𝑦,𝑧∈𝐴. Therefore, we have 𝑔(𝑥𝑦)=𝑔(𝑥)𝜃(𝑦)+𝜙(𝑥)𝑔(𝑦).
Since 𝑓(𝑥𝑦)=𝑓(𝑥)𝜃(𝑦)+𝜙(𝑥)𝑔(𝑦),  𝑓 is additive, and 𝐴 is with unit, 𝑔 is additive.

The proof of the following theorem is similar to that in Theorem 2.1; hence, it is omitted.

Theorem 2.2. Let 𝜑,𝜓∶𝐴×𝐴→[0,∞) be functions. Suppose that 𝑓∶𝐴→𝐴 and 𝑔∶𝐴→𝐴 are mappings such that 𝑓(𝑥)𝑦,𝑓𝑦(𝑥)𝑦,𝑥+𝑦−𝑓−𝑓≤𝜑𝑥,𝑦𝑥𝑦−𝑥𝑓−𝑔≤𝜓𝑥,𝑦 (2.28) for all 𝑥,𝑦∈𝐴. If there exists constants 𝐾,𝐿<1 and a natural number 𝑘∈𝕂, |𝑘|𝜑𝑘⁻¹𝑥,𝑘⁻¹𝑦≤𝐿𝜑𝑥,𝑦,|𝑘|𝜓𝑘⁻¹𝑥,𝑦,|𝑘|𝜓𝑥,𝑘⁻¹𝑦≤𝐾𝜓𝑥,𝑦, (2.29) for all 𝑥,𝑦∈𝐴, then 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

In the following corollaries ℚ_𝑝 is the field of 𝑝-adic numbers.

Corollary 2.3. Let 𝐴 be a non-Archimedean Banach algebra over ℚ_𝑝, 𝜀>0, and let 𝑝₁,𝑝₂∈(1,∞). Suppose that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝1𝑦^𝑝2,𝑓𝑥𝑦−𝑥𝑓𝑦−𝑔(𝑥)𝑦≤𝜀‖𝑥‖^𝑝1𝑦^𝑝2, (2.30) for all 𝑥,𝑦∈𝐴. Then 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

Proof. Let 𝜑(𝑥,𝑦)=𝜓(𝑥,𝑦)=𝜀(‖𝑥‖^𝑝1‖𝑦‖^𝑝2) for all 𝑥,𝑦∈𝐴; then 𝑝⁻¹𝜑𝑝𝑥,𝑝𝑦=𝑝^{𝑝1+𝑝2−1}𝜀‖𝑥‖^𝑝1𝑦^𝑝2,𝑝⁻¹𝜑𝑝𝑥,𝑦=𝑝^𝑝1−1𝜀‖𝑥‖^𝑝1𝑦^𝑝2,𝑝⁻¹𝜑𝑥,𝑝𝑦=𝑝^𝑝2−1𝜀‖𝑥‖^𝑝1𝑦^𝑝2. (2.31) Put 𝐿=𝐾=max𝑝^𝑝1−1,𝑝^𝑝2−1,𝑝^{𝑝1+𝑝2−1}=max𝑝^1−𝑝1,𝑝^1−𝑝2,𝑝^{1−𝑝1−𝑝2}. (2.32) So, by Theorem 2.1, 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

Corollary 2.4. Let 𝐴 be a non-Archimedean Banach algebra over ℚ_𝑝, 𝜀>0, and let 𝑝₁,𝑝₂,𝑝₁+𝑝₂∈(−∞,1). Suppose that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝1𝑦^𝑝2,𝑓𝑥𝑦−𝑥𝑓𝑦−𝑔(𝑥)𝑦≤𝜀‖𝑥‖^𝑝1𝑦^𝑝2, (2.33) for all 𝑥,𝑦∈𝐴. Then 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

Proof. Let 𝜑(𝑥,𝑦)=𝜓(𝑥,𝑦)=𝜀(‖𝑥‖^𝑝1‖𝑦‖^𝑝2) for all 𝑥,𝑦∈𝐴, then 𝑝𝜑𝑝⁻¹𝑥,𝑝⁻¹𝑦=𝑝^{1−𝑝1−𝑝2}𝜀‖𝑥‖^𝑝1𝑦^𝑝2,𝑝𝜑𝑝⁻¹𝑥,𝑦=𝑝^1−𝑝1𝜀‖𝑥‖^𝑝1𝑦^𝑝2,𝑝𝜑𝑥,𝑝⁻¹𝑦=𝑝^1−𝑝2𝜀‖𝑥‖^𝑝1𝑦^𝑝2. (2.34) Put 𝐿=𝐾=max𝑝^1−𝑝1,𝑝^1−𝑝2,𝑝^{1−𝑝1−𝑝2}=max𝑝^𝑝1−1,𝑝^𝑝2−1,𝑝^{𝑝1+𝑝2−1}. (2.35) So, by Theorem 2.2, 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

Similarly, we can obtain the following results.

Corollary 2.5. Let 𝐴 be a non-Archimedean Banach’s algebra over ℚ_𝑝, 𝜀>0, 𝛿>0, and let 𝑝₁,𝑝₂∈(1,∞). Suppose that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝1+𝑦^𝑝2,𝑓𝑥𝑦−𝑥𝑓𝑦−𝑔(𝑥)𝑦≤𝛿‖𝑥‖^𝑝1𝑦^𝑝2, (2.36) for all 𝑥,𝑦∈𝐴. Then 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

Corollary 2.6. Let 𝐴 be a non-Archimedean Banach’s algebra over ℚ_𝑝, 𝜀>0, 𝛿>0, and let 𝑝₁,𝑝₂∈(1,∞). Suppose that 𝑓(𝑥)𝑦,𝑓𝑦(𝑥)𝑦‖𝑥‖max𝑥+𝑦−𝑓−𝑓𝑥𝑦−𝑥𝑓−𝑔≤𝜀min^𝑝1+𝑦^𝑝2,‖𝑥‖^𝑝1𝑦^𝑝2, (2.37) for all 𝑥,𝑦∈𝐴. Then 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

Corollary 2.7. Let 𝐴 be a non-Archimedean Banach’s algebra over ℚ_𝑝, 𝜀>0,𝛿>0, and let 𝑝₁,𝑝₂,𝑝₁+𝑝₂∈(−∞,1). Suppose that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜀‖𝑥‖^𝑝1+𝑦^𝑝2,𝑓𝑥𝑦−𝑥𝑓𝑦−𝑔(𝑥)𝑦≤𝛿‖𝑥‖^𝑝1𝑦^𝑝2, (2.38) for all 𝑥,𝑦∈𝐴. Then 𝑓 is a generalized (𝜃,𝜙)-derivation and 𝑔 is a (𝜃,𝜙)-derivation.

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 number: 2011-0005197).