Non-Archimedean Hyers-Ulam Stability of an Additive-Quadratic Mapping

Abstract

Using fixed point method and direct method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation 𝑟²𝑓((𝑥+𝑦+𝑧)/𝑟)+𝑟²𝑓((𝑥−𝑦+𝑧)/𝑟)+𝑟²𝑓((𝑥+𝑦−𝑧)/𝑟)+𝑟²𝑓((−𝑥+𝑦+𝑧)/𝑟)=4𝑓(𝑥)+4𝑓(𝑦)+4𝑓(𝑧), where 𝑟 is a positive real number, in non-Archimedean normed spaces.

1. Introduction and Preliminaries

A classical question in the theory of functional equations is the following: “When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?” If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940. In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Th. M. Rassias [3] proved a generalization of Hyers’ theorem for linear mappings. Furthermore, in 1994, a generalization of the Th. M. Rassias’ theorem was obtained by Găvruţa [4] by replacing the bound 𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜙(𝑥,𝑦).

In 1983, the Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings 𝑓∶𝑋→𝑌, where 𝑋 is a normed space and 𝑌 is a Banach space. In 1984, Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain 𝑋 is replaced by an Abelian group and, in 2002, Czerwik [7] proved the Hyers-Ulam stability of the quadratic functional equation. The reader is referred to [2–4, 6–48] and references therein for detailed information on stability of functional equations.

In 1897, Hensel [24] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [16, 26–28, 37]).

Definition 1.1. By a non-Archimedean field, one means a field 𝕂 equipped with a function (valuation) |⋅|∶𝕂→[0,∞) such that for all 𝑟, 𝑠∈𝕂, the following conditions hold:(1)|𝑟|=0 if and only if 𝑟=0;(2)|𝑟𝑠|=|𝑟‖𝑠|;(3)|𝑟+𝑠|≤max{|𝑟|,|𝑠|}.

Definition 1.2. Let 𝑋 be a vector space over a scalar field 𝕂 with a non-Archimedean nontrivial valuation |⋅|. A function ‖⋅‖∶𝑋→𝑅 is a non-Archimedean norm (valuation) if it satisfies the following conditions:(1)‖𝑥‖=0 if and only if 𝑥=0;(2)‖𝑟𝑥‖=|𝑟‖|𝑥‖(𝑟∈𝕂,𝑥∈𝑋);(3)the strong triangle inequality (ultrametric), namely,𝑥+𝑦≤max‖𝑥‖,𝑦,𝑥,𝑦∈𝑋. (1.1) Then (𝑋,‖⋅‖) is called a non-Archimedean space, due to the fact that

𝑥_𝑛−𝑥_𝑚≤max𝑥_𝑗+1−𝑥_𝑗∶𝑚≤𝑗≤𝑛−1(𝑛>𝑚). (1.2)

Definition 1.3. A sequence {𝑥_𝑛} is Cauchy if and only if {𝑥_𝑛+1−𝑥_𝑛} converges to zero in a non-Archimedean space. By a complete non-Archimedean space, one means one in which every Cauchy sequence is convergent.

Definition 1.4. Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies(1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(3)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋.
One recalls a fundamental result in fixed point theory.

Theorem 1.5 (see [7, 17]). Let (𝑋,𝑑) be a complete generalized metric space and let 𝐽∶𝑋→𝑋 be a strictly contractive mapping with Lipschitz constant 𝛼<1. Then for each given element 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=∞ (1.3) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞,forall𝑛≥𝑛₀;(2)the sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽;(3)𝑦^∗ is the unique fixed point of 𝐽 in the set 𝑌={𝑦∈𝑋∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞};(4)𝑑(𝑦,𝑦^∗)≤(1/(1−𝛼))𝑑(𝑦,𝐽𝑦)forall𝑦∈𝑌.

In 1998, D. H. Hyers, G. Isac and Th. M. Rassias [25] provided applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [15, 39, 40, 42]).

This paper is organized as follows. In Section 2, using fixed point method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation:𝑟²𝑓𝑥+𝑦+𝑧𝑟+𝑟²𝑓𝑥−𝑦+𝑧𝑟+𝑟²𝑓𝑥+𝑦−𝑧𝑟+𝑟²𝑓−𝑥+𝑦+𝑧𝑟=4𝑓(𝑥)+4𝑓𝑦+4𝑓(𝑧), (1.4) where 𝑥, 𝑦, 𝑧∈𝑋, in non-Archimedean normed space. In Section 3, using direct method, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.4) in non-Archimedean normed spaces.

2. Stability of the Functional Equation (1.4): A Fixed Point Approach

In this section, we deal with the stability problem for the quadratic functional equation (1.4).

Theorem 2.1. Let X be a non-Archimedean normed space and 𝑌 a complete non-Archimedean space. Let 𝜑∶𝑋³→[0,∞) be a function such that there exists an 𝛼<1 with 1|2|𝜑2𝑥,2𝑦,2𝑧≤𝛼𝜑𝑥,𝑦,𝑧 (2.1) for all 𝑥, 𝑦, 𝑧∈𝑋. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying 𝑟²𝑓𝑥+𝑦+𝑧𝑟+𝑟²𝑓𝑥−𝑦+𝑧𝑟+𝑟²𝑓𝑥+𝑦−𝑧𝑟+𝑟²𝑓−𝑥+𝑦+𝑧𝑟−4𝑓(𝑥)−4𝑓𝑦−4𝑓(𝑧)_Y≤𝜑𝑥,𝑦,𝑧 (2.2) for all 𝑥, 𝑦, 𝑧∈𝑋. Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)_𝑌≤1𝜑()1−𝛼max2𝑥,0,0|8|,𝜑(𝑥,𝑥,0)|8| (2.3) for all 𝑥∈𝑋.

Proof. Note that 𝑓(0)=0 and 𝑓(−𝑥)=−𝑓(𝑥) for all 𝑥∈𝑋 since 𝑓 is an odd mapping. Putting 𝑦=𝑧=0 in (2.2) and replacing 𝑥 by 2𝑥, we get 𝑟²𝑓2𝑥𝑟−2𝑓(2𝑥)_𝑌≤1|2|𝜑(2𝑥,0,0) (2.4) for all 𝑥∈𝑋. Putting 𝑦=𝑥 and 𝑧=0 in (2.2), we have 𝑟²𝑓2𝑥𝑟−4𝑓(𝑥)_𝑌≤1|2|𝜑(𝑥,𝑥,0) (2.5) for all 𝑥∈𝑋. By (2.4) and (2.5), we get 𝑓(2𝑥)2−𝑓(𝑥)_𝑌≤1|4|max𝑟²𝑓2𝑥𝑟−2𝑓(2𝑥)_𝑌,𝑟²𝑓2𝑥𝑟−4𝑓(𝑥)_𝑌≤1|8|max𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0). (2.6) Consider the set 𝑆∶={ℎ∶𝑋→𝑌} and introduce the generalized metric on 𝑆: 𝑑𝑔,ℎ=inf𝜇∈(0,+∞)∶𝑔(𝑥)−ℎ(𝑥)_𝑌≤𝜇max𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0),∀𝑥∈𝑋, (2.7) where, as usual, inf𝜙=+∞. It is easy to show that (𝑆,𝑑) is complete (see [31]). Now we consider the linear mapping 𝐽∶𝑆→𝑆 such that (𝑥)1𝐽𝑔∶=2𝑔(2𝑥) (2.8) for all 𝑥∈𝑋.
Let 𝑔, ℎ∈𝑆 be given such that 𝑑(𝑔,ℎ)=𝜆. Then𝑔(𝑥)−ℎ(𝑥)_𝑌≤𝜆max𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0) (2.9) for all 𝑥∈𝑋. Hence 𝐽𝑔(𝑥)−𝐽ℎ(𝑥)_𝑌=12𝑔()−12𝑥2ℎ(2𝑥)_𝑌=1|2|𝑔(2𝑥)−ℎ(2𝑥)_𝑌≤𝜆max𝜑(4𝑥,0,0),𝜑(2𝑥,2𝑥,0)|2|≤𝛼𝜆max𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0) (2.10) for all 𝑥∈𝑋. So 𝑑(𝑔,ℎ)=𝜆 implies that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝜆. This means that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝑑(𝑔,ℎ) for all 𝑔, ℎ∈𝑆. It follows from (2.6) that 𝑑≤1𝑓,𝐽𝑓|8|. (2.11) By Theorem 1.5, there exists a mapping 𝐴∶𝑋→𝑌 satisfying the following.(1)𝐴 is a fixed point of 𝐽, that is,2𝐴(𝑥)=𝐴(2𝑥) (2.12) for all 𝑥∈𝑋. The mapping 𝐴 is a unique fixed point of 𝐽 in the set 𝑀={𝑔∈𝑆∶𝑑(ℎ,𝑔)<∞}. This implies that 𝐴 is the unique mapping satisfying (2.12) such that there exists a 𝜇∈(0,∞) satisfying ‖𝑓(𝑥)−𝐴(𝑥)‖_𝑌≤𝜇max{𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0)} for all 𝑥∈𝑋.(2)𝑑(𝐽^𝑛𝑓,𝐴)→0 as 𝑛→∞. This implies the equalitylim_𝑛→∞𝑓2^𝑛𝑥2^𝑛=𝐴(𝑥) (2.13) for all 𝑥∈𝑋.(3)𝑑(𝑓,𝐴)≤(1/(1−𝛼))𝑑(𝑓,𝐽𝑓), which implies the inequality 𝑑(𝑓,𝐴)≤1/(|8|(1−𝛼)). This implies that the inequalities (2.3) holds.
It follows from (2.1) and (2.2) that𝑟²𝐴𝑥+𝑦+𝑧𝑟+𝑟²𝐴𝑥−𝑦+𝑧𝑟+𝑟²𝐴𝑥+𝑦−𝑧𝑟+𝑟²𝐴−𝑥+𝑦+𝑧𝑟−4𝐴(𝑥)−4𝐴(𝑦)−4𝐴(𝑧)_𝑌=lim_𝑛→∞1|2|^𝑛𝑟²𝑓2^𝑛𝑥+𝑦+𝑧𝑟+𝑟²𝑓2^𝑛𝑥−𝑦+𝑧𝑟+𝑟²𝑓2^𝑛𝑥+𝑦−𝑧𝑟+𝑟²𝑓2^𝑛−𝑥+𝑦+𝑧𝑟−4𝑓2^𝑛𝑥−4𝑓2^𝑛𝑦−4𝑓2^𝑛𝑧_𝑌≤lim_𝑛→∞1|2|^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧≤lim_𝑛→∞𝛼^𝑛𝜑𝑥,𝑦,𝑧=0 (2.14) for all 𝑥, 𝑦, 𝑧∈𝑋. So 𝑟²𝐴𝑥+𝑦+𝑧𝑟+𝑟²𝐴𝑥−𝑦+𝑧𝑟+𝑟²𝐴𝑥+𝑦−𝑧𝑟+𝑟²𝐴−𝑥+𝑦+𝑧𝑠=4𝐴(𝑥)+4𝐴𝑦+4𝐴(𝑧) (2.15) for all 𝑥, 𝑦, 𝑧∈𝑋. Hence 𝐴∶𝑋→𝑌 satisfying (1.4).
It follows from (2.1) and (2.6) that‖𝐴(2𝑥)−2𝐴(𝑥)‖_𝑌=lim_𝑛→∞𝑓2^𝑛+1𝑥2^𝑛−2𝑓2^𝑛𝑥2^𝑛_𝑌≤lim_𝑛→∞|2||8||2|^𝑛max𝜑2^𝑛+1𝑥,0,0,𝜑2^𝑛𝑥,2^𝑛𝑥,0≤lim_𝑛→∞1|4|𝛼^𝑛max𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0)=0 (2.16) for all 𝑥∈𝑋. So 𝐴(2𝑥)=2𝐴(𝑥) for all 𝑥∈𝑋. Hence 𝐴∶𝑋→𝑌 is additive and we get the desired result.

Corollary 2.2. Let 𝜃 be a positive real number and 𝑞 a real number with 0<𝑞<1. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying 𝑟²𝑓𝑥+𝑦+𝑧𝑟+𝑟²𝑓𝑥−𝑦+𝑧𝑟+𝑟²𝑓𝑥+𝑦−𝑧𝑟+𝑟²𝑓−𝑥+𝑦+𝑧𝑟−4𝑓(𝑥)−4𝑓𝑦−4𝑓(𝑧)_Y≤𝜃‖𝑥‖^𝑞+𝑦^𝑞+‖𝑧‖^𝑞 (2.17) for all 𝑥, 𝑦, 𝑧∈𝑋. Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)_𝑌≤2|2|^𝑞𝜃‖𝑥‖^𝑞|2|^𝑞+3−|2|⁴ (2.18) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.1 by taking 𝜑(𝑥,𝑦,𝑧)=𝜃(‖𝑥‖^𝑞+‖𝑦‖^𝑞+‖𝑧‖^𝑞) for all 𝑥, 𝑦, 𝑧∈𝑋. Then we can choose 𝛼=|2|^1−𝑞 and we get the desired result.

Theorem 2.3. Let 𝑋 be a non-Archimedean normed space and 𝑌 a complete non-Archimedean space. Let 𝜑∶𝑋³→[0,∞) be a function such that there exists an 𝛼<1 with |2|𝜑𝑥2,𝑦2,𝑧2≤𝛼𝜑𝑥,𝑦,𝑧 (2.19) for all 𝑥, 𝑦, 𝑧∈𝑋. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (2.2). Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)_𝑌≤𝛼𝜑()1−𝛼max2𝑥,0,0|8|,𝜑(𝑥,𝑥,0)|8| (2.20) for all 𝑥∈𝑋.

Proof. Let (𝑆,𝑑) be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝐽𝑔(𝑥)∶=2𝑔(𝑥/2) for all 𝑥∈𝑋.
Replacing 𝑥 by 𝑥/2 in (2.6) and using (2.19), we have𝑓(𝑥)𝑥−2𝑓2_𝑌≤1|4|𝜑()𝑥max𝑥,0,0,𝜑2,𝑥2≤𝛼,0|8|max𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0). (2.21) So 𝑑(𝑓,𝐽𝑓)≤𝛼/|8|.
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.4. Let 𝜃 be a positive real number and 𝑞 a real number with 𝑞>1. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (2.17). Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)_𝑌≤2|2|^𝑞𝜃‖𝑥‖^𝑞|2|⁴−|2|^𝑞+3 (2.22) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.3 by taking 𝜑(𝑥,𝑦,𝑧)=𝜃(‖𝑥‖^𝑞+‖𝑦‖^𝑞+‖𝑧‖^𝑞) for all 𝑥, 𝑦, 𝑧∈𝑋Then we can choose 𝛼=|2|^𝑞−1 and we get the desired result.

Theorem 2.5. Let 𝑋 be a non-Archimedean normed space and 𝑌 a complete non-Archimedean space. Let 𝜑∶𝑋³→[0,∞) be a function such that there exists an 𝛼<1 with 1|4|𝜑2𝑥,2𝑦,2𝑧≤𝛼𝜑𝑥,𝑦,𝑧 (2.23) for all 𝑥, 𝑦, 𝑧∈𝑋. Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (2.2). Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)_𝑌≤1|4)𝜑()|(1−𝛼max2𝑥,0,0|4|,𝜑(𝑥,𝑥,0)|2| (2.24) for all 𝑥∈𝑋.

Proof. Putting 𝑦=𝑥 and 𝑧=0 in (2.2), we have 𝑟²𝑓2𝑥𝑟−4𝑓(𝑥)_𝑌≤1|2|𝜑(𝑥,𝑥,0) (2.25) for all 𝑥∈𝑋.
Substituting 𝑦=𝑧=0 and then replacing 𝑥 by 2𝑥 in (2.2), we obtain𝑟²𝑓2𝑥𝑟−𝑓(2𝑥)_𝑌≤1|4|𝜑(2𝑥,0,0). (2.26) By (2.25) and (2.26), we get 𝑓(2𝑥)4−𝑓(𝑥)_𝑌=1|4|𝑟²𝑓2𝑥𝑟−4𝑓(𝑥)−𝑟²𝑓2𝑥𝑟+𝑓(2𝑥)_𝑌≤1|4|max𝑟²𝑓2𝑥𝑟−4𝑓(𝑥)_𝑌,𝑟²𝑓2𝑥𝑟−𝑓(2𝑥)_𝑌≤1|4|𝜑()max2𝑥,0,0|4|,𝜑(𝑥,𝑥,0)|2|. (2.27) Consider the set 𝑆^∗={𝑔∶𝑋→𝑌;𝑔(0)=0} and the generalized metric 𝑑^∗ in 𝑆^∗ defined by 𝑑𝑔,ℎ=inf𝜇∈(0,+∞)∶𝑔(𝑥)−ℎ(𝑥)_𝑌𝜑()≤𝜇max2𝑥,0,0|4|,𝜑(𝑥,𝑥,0)|2|,∀𝑥∈𝑋, (2.28) where, as usual, inf𝜙=+∞. It is easy to show that (𝑆^∗,𝑑^∗) is complete (see [31]).
Now we consider the linear mapping 𝐽∶(𝑆^∗,𝑑^∗)→(𝑆^∗,𝑑^∗) such that 𝐽𝑔(𝑥)∶=(1/4)𝑔(2𝑥) for all 𝑥∈𝑋.
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.6. Let 𝜃 be a positive real number and 𝑞 a real number with 𝑞>1. Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (2.17). Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)−Q(𝑥)_𝑌≤1|4|−|4|^𝑞‖𝑥‖max2𝜃^𝑞|2|,|2|^𝑞𝜃‖𝑥‖^𝑞|4| (2.29) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.5 by taking 𝜑(𝑥,𝑦,𝑧)=𝜃(‖𝑥‖^𝑞+‖𝑦‖^𝑞+‖𝑧‖^𝑞) for all 𝑥, 𝑦, 𝑧∈𝑋. Then we can choose 𝛼=|4|^𝑞−1 and we get the desired result.

Theorem 2.7. Let 𝑋 be a non-Archimedean normed space and 𝑌 a complete non-Archimedean space. Let 𝜑∶𝑋³→[0,∞) be a function such that there exists an 𝛼<1 with |4|𝜑𝑥2,𝑦2,𝑧2≤𝛼𝜑𝑥,𝑦,𝑧 (2.30) for all 𝑥, 𝑦, 𝑧∈𝑋. Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (2.2). Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)_𝑌≤𝛼|4)𝜑()|(1−𝛼max2𝑥,0,0|4|,𝜑(𝑥,𝑥,0)|2| (2.31) for all 𝑥∈𝑋.

Proof. It follows from (2.27) that 𝑓(𝑥)𝑥−4𝑓2_𝑌1≤max|4|𝜑(),1𝑥,0,0|2|𝜑𝑥2,𝑥2≤𝛼,0|4|𝜑()max2𝑥,0,0|4|,𝜑(𝑥,𝑥,0)|2|. (2.32)
The rest of the proof is similar to the proof of Theorems 2.1 and 2.5.

Corollary 2.8. Let 𝜃 be a positive real number and 𝑞 a real number with 0<𝑞<1. Let 𝑓∶𝑋→𝑌 be an even mapping satisfying 𝑓(0)=0 and (2.17). Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)−𝑄(𝑥)_𝑌≤|4|^1−𝑞|4|−|4|^2−𝑞‖𝑥‖max2𝜃^𝑞|2|,|2|^𝑞𝜃‖𝑥‖^𝑞|4| (2.33) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.7 by taking 𝜑(𝑥,𝑦,𝑧)=𝜃(‖𝑥‖^𝑞+‖𝑦‖^𝑞+‖𝑧‖^𝑞) for all 𝑥, 𝑦, 𝑧∈𝑋. Then we can choose 𝛼=|4|^1−𝑞 and we get the desired result.

Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 and (1.4). Let 𝑓_𝑒(𝑥)∶=(𝑓(𝑥)+𝑓(−𝑥))/2 and 𝑓_𝑜(𝑥)=(𝑓(𝑥)−𝑓(−𝑥))/2. Then 𝑓_𝑒 is an even mapping satisfying (1.4) and 𝑓_𝑜 is an odd mapping satisfying (1.4) such that 𝑓(𝑥)=𝑓_𝑒(𝑥)+𝑓_𝑜(𝑥). So we obtain the following.

Theorem 2.9. Let 𝑋 be a non-Archimedean normed space and 𝑌 a complete non-Archimedean space. Let 𝜑∶𝑋³→[0,∞) be a function such that there exists an 𝛼<1 with 1|4|𝜑2𝑥,2𝑦,2𝑧≤𝛼𝜑𝑥,𝑦,𝑧 (2.34) for all 𝑥, 𝑦, 𝑧∈𝑋. Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 and (2.2). Then there exist an additive mapping 𝐴∶𝑋→𝑌 and a quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)−𝐴(𝑥)−𝑄(𝑥)_𝑌𝑓(𝑥)()≤max−𝑓−𝑥2−𝐴(𝑥)_𝑌,𝑓(𝑥)+𝑓(−𝑥)2−𝑄(𝑥)_𝑌1≤max𝜑()1−𝛼max2𝑥,0,0|8|,𝜑(𝑥,𝑥,0)|8|,1|4)𝜑()|(1−𝛼max2𝑥,0,0|4|,𝜑(𝑥,𝑥,0)|2| (2.35) for all 𝑥∈𝑋.

3. Stability of the Functional Equation (1.4): A Direct Method

In this section, using direct method, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.4) in non-Archimedean space.

Theorem 3.1. Let 𝐺 be an additive semigroup and 𝑋 a non-Archimedean Banach space. Assume that 𝜁∶𝐺³→[0,+∞) is a function such that lim_𝑛→∞|2|^𝑛𝜑𝑥2^𝑛,𝑦2^𝑛,𝑧2^𝑛=0 (3.1) for all 𝑥, 𝑦, 𝑧∈𝐺. Suppose that, for any 𝑥∈𝐺, the limit Ω(𝑥)=lim_𝑛→∞max|2|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1,0;0≤𝑘<𝑛 (3.2) exists and 𝑓∶𝐺→𝑋 is an odd mapping satisfying 𝑟²𝑓𝑥+𝑦+𝑧𝑟+𝑟²𝑓𝑥−𝑦+𝑧𝑟+𝑟²𝑓𝑥+𝑦−𝑧𝑟+𝑟²𝑓−𝑥+𝑦+𝑧𝑟−4𝑓(𝑥)−4𝑓𝑦−4𝑓(𝑧)_X≤𝜁𝑥,𝑦,𝑧. (3.3) Then the limit 𝐴(𝑥)∶=lim_𝑛→∞2^𝑛𝑓𝑥2^𝑛 (3.4) exists for all 𝑥∈G and defines an additive mapping 𝐴∶𝐺→𝑋 such that 𝑓(𝑥)(𝑥≤1−𝐴)|4|Ω(𝑥). (3.5) Moreover, if lim_𝑗→∞lim_𝑛→∞max|2|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1,0;𝑗≤𝑘<𝑛+𝑗=0, (3.6) then 𝐴 is the unique additive mapping satisfying (3.5).

Proof. By (2.21), we know that 𝑓(𝑥)𝑥−2𝑓2_𝑋≤1|4|𝜑()𝑥max𝑥,0,0,𝜑2,𝑥2,0 (3.7) for all 𝑥∈𝐺. Replacing 𝑥 by 𝑥/2^𝑛 in (3.7), we obtain 2^𝑛𝑓𝑥2^𝑛−2^𝑛+1𝑓𝑥2^𝑛+1_𝑋≤|2|^𝑛|4|𝜑𝑥max2^𝑛𝑥,0,0,𝜑2^𝑛+1,𝑥2^𝑛+1,0. (3.8) Thus, it follows from (3.1) and (3.8) that the sequence {2^𝑛𝑓(𝑥/2^𝑛)}_𝑛≥1 is a Cauchy sequence. Since 𝑋 is complete, it follows that {2^𝑛𝑓(𝑥/2^𝑛)}_𝑛≥1 is convergent. Set 𝐴(𝑥)∶=lim_𝑛→∞2^𝑛𝑓𝑥2^𝑛. (3.9) By induction on 𝑛, one can show that 2^𝑛𝑓𝑥2^𝑛−𝑓(𝑥)_𝑋≤1|4|max|2|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1,0;0≤𝑘<𝑛 (3.10) for all 𝑛≥1 and 𝑥∈𝐺. Indeed, (3.10) holds for 𝑛=1 by (3.7). Now, if (3.10) holds for 𝑛, then by (3.8), we have 2^𝑛+1𝑓𝑥2^𝑛+1(𝑥=2−𝑓)_𝑛+1𝑓𝑥2^𝑛+1±2^𝑛𝑓𝑥2^𝑛(𝑥2−𝑓)≤max^𝑛+1𝑓𝑥2^𝑛+1−2^𝑛𝑓𝑥2^𝑛,2^𝑛𝑓𝑥2^𝑛(𝑥≤1−𝑓)|4|max|2|^𝑛𝜑𝑥max2^𝑛𝑥,0,0,𝜑2^𝑛+1,𝑥2^𝑛+1,2|,0max|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1=1,0;0≤𝑘<𝑛|4|max|2|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1,0;0≤𝑘<𝑛+1. (3.11) By taking 𝑛→∞ in (3.10) and using (3.2), one obtains (3.5). By (3.1) and (3.3), we get 𝑟²𝐴𝑥+𝑦+𝑧𝑟+𝑟²𝐴𝑥−𝑦+𝑧𝑟+𝑟²𝐴𝑥+𝑦−𝑧𝑟+𝑟²𝐴−𝑥+𝑦+𝑧𝑟−4𝐴(𝑥)−4𝐴𝑦−4𝐴(𝑧)_𝑋=lim_𝑛→∞|2|^𝑛𝑟²𝑓𝑥+𝑦+𝑧2^𝑛𝑟+𝑟²𝑓𝑥−𝑦+𝑧2^𝑛𝑟+𝑟²𝑓𝑥+𝑦−𝑧2^𝑛𝑟+𝑟²𝑓−𝑥+𝑦+𝑧2^𝑛𝑟𝑥−4𝑓2^𝑛𝑦−4𝑓2^𝑛𝑧−4𝑓2^𝑛_𝑋≤lim_𝑛→∞|2|^𝑛𝜑𝑥2^𝑛,𝑦2^𝑛,𝑧2^𝑛=0 (3.12) for all 𝑥, 𝑦, 𝑧∈𝑋. Therefore, the mapping 𝐴∶𝐺→𝑋 satisfies (1.4).
To prove the uniqueness property of 𝐴, let 𝐿 be another mapping satisfying (3.5). Then we have‖𝐴(𝑥)−𝐿(𝑥)‖_𝑋=lim_𝑗→∞|2|^𝑗𝐴𝑥2^𝑗𝑥−𝐿2^𝑗_𝑋≤lim_𝑗→∞|2|^𝑗𝐴𝑥max2^𝑗𝑥−𝑓2^𝑗_𝑋,𝑓𝑥2^𝑗𝑥−𝐿2^𝑗_𝑋≤lim_𝑗→∞lim_𝑛→∞max|2|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1,0;𝑗≤𝑘<𝑛+𝑗=0 (3.13) for all 𝑥∈𝐺. Therefore, 𝐴=𝐿. This completes the proof.

Corollary 3.2. Let 𝜉∶[0,∞)→[0,∞) be a function satisfying 𝜉|2|⁻¹𝑡≤𝜉|2|⁻¹𝜉(𝑡),𝜉|2|⁻¹<|2|⁻¹ (3.14) for all 𝑡≥0. Assume that 𝜅>0 and 𝑓∶𝐺→𝑋 is a mapping with 𝑓(0)=0 such that 𝑟²𝑓𝑥+𝑦+𝑧𝑟+𝑟²𝑓𝑥−𝑦+𝑧𝑟+𝑟²𝑓𝑥+𝑦−𝑧𝑟+𝑟²𝑓−𝑥+𝑦+𝑧𝑟−4𝑓(𝑥)−4𝑓𝑦−4𝑓(𝑧)_X≤𝜅𝜉(‖𝑥‖)+𝜉𝑦+𝜉(‖𝑧‖) (3.15) for all 𝑥, 𝑦, 𝑧∈𝐺. Then there exists a unique additive mapping 𝐴∶𝐺→𝑋 such that 𝑓(𝑥)−𝐴(𝑥)_𝑋≤1|4|𝑥,2max𝜅𝜁(‖‖)|2|𝜅𝜁(‖𝑥‖). (3.16)

Proof. One can define 𝜁∶𝐺³→[0,∞) by 𝜁(𝑥,𝑦,𝑧)∶=𝜅(𝜉(‖𝑥‖)+𝜉(‖𝑦‖)+𝜉(‖𝑧‖)). Then we have lim_𝑛→∞|2|^𝑛𝜁𝑥2^𝑛,𝑦2^𝑛,𝑧2^𝑛≤lim_𝑛→∞|2|𝜉|2|^−1𝑛𝜁𝑥,𝑦,𝑧=0 (3.17) for all 𝑥, 𝑦, 𝑧∈𝐺. The last equality comes from the fact that |2|𝜉(|2|⁻¹)<1. On the other hand, it follows that Ω(𝑥)=lim_𝑛→∞max|2|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1𝜑()𝑥,0;0≤𝑘<𝑛≤max𝑥,0,0,𝜑2,𝑥2𝑥,2,0=max𝜅𝜁(‖‖)|2|𝜅𝜁(‖𝑥‖) (3.18) exists for all 𝑥∈𝐺. Also, we have lim_𝑗→∞lim_𝑛→∞max|2|^𝑘𝜑𝑥max2^𝑘𝑥,0,0,𝜑2^𝑘+1,𝑥2^𝑘+1,0;𝑗≤𝑘<𝑛+𝑗=lim_𝑗→∞|2|^𝑗𝜑𝑥max2^𝑗𝑥,0,0,𝜑2^𝑗+1,𝑥2^𝑗+1,0=0. (3.19) Thus, applying Theorem 3.1, we have the conclusion. This completes the proof.

Theorem 3.3. Let 𝐺 be an additive semigroup and 𝑋 a non-Archimedean Banach space. Assume that 𝜁∶𝐺³→[0,+∞) is a function such that lim_𝑛→∞𝜁2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|2|^𝑛=0 (3.20) for all 𝑥, 𝑦, 𝑧∈𝐺. Suppose that, for any 𝑥∈𝐺, the limit Ω(𝑥)=lim_𝑛→∞𝜑2maxmax^𝑘+1𝑥,0,0,𝜑2^𝑘𝑥,2^𝑘𝑥,0|2|^𝑘;0≤𝑘<𝑛 (3.21) exists and 𝑓∶𝐺→𝑋 is an odd mapping satisfying (3.3). Then the limit 𝐴(𝑥)∶=lim_𝑛→∞(𝑓(2^𝑛𝑥)/2^𝑛) exists for all 𝑥∈𝐺 and 𝑓(𝑥)−𝐴(𝑥)_𝑋≤1|8|Ω(𝑥) (3.22) for all 𝑥∈𝐺. Moreover, if lim_𝑗→∞lim_𝑛→∞𝜑2maxmax^𝑘+1𝑥,0,0,𝜑2^𝑘𝑥,2^𝑘𝑥,0|2|^𝑘;𝑗≤𝑘<𝑛+𝑗=0, (3.23) then 𝐴 is the unique mapping satisfying (3.22).

Proof. By (2.6), we get 𝑓(2𝑥)2−𝑓(𝑥)_𝑋≤max𝜑(2𝑥,0,0),𝜑(𝑥,𝑥,0)|8| (3.24) for all 𝑥∈𝐺. Replacing 𝑥 by 2^𝑛𝑥 in (3.24), we obtain 𝑓2^𝑛+1𝑥2^𝑛+1−𝑓2^𝑛𝑥2^𝑛_𝑋≤max𝜑2^𝑛+1𝑥,0,0,𝜑2^𝑛𝑥,2^𝑛𝑥,0|2|^𝑛+3. (3.25) Thus it follows from (3.20) and (3.25) that the sequence {𝑓(2^𝑛𝑥)/2^𝑛}_𝑛≥1 is convergent. Set 𝐴(𝑥)∶=lim_𝑛→∞𝑓2^𝑛𝑥2^𝑛. (3.26) On the other hand, it follows from (3.25) that 𝑓2^𝑝𝑥2^𝑞−𝑓2^𝑞𝑥2^𝑞_𝑋=^𝑞−1_𝑘=𝑝𝑓2^𝑘+1𝑥2^𝑘+1−𝑓2^𝑘𝑥2^𝑘_𝑋𝑓2≤max^𝑘+1𝑥2^𝑘+1−𝑓2^𝑘𝑥2^𝑘_𝑋≤1;𝑝≤𝑘<𝑞−1|8|𝜑2maxmax^𝑘+1𝑥,0,0,𝜑2^𝑘𝑥,2^𝑘𝑥,0|2|^𝑘;𝑝≤𝑘<𝑞 (3.27) for all 𝑥∈𝐺 and 𝑝, 𝑞≥0 with 𝑞>𝑝≥0. Letting 𝑝=0, taking 𝑞→∞ in the last inequality, and using (3.21), we obtain (3.22).
The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof.

Theorem 3.4. Let 𝐺 be an additive semigroup and 𝑋 a non-Archimedean Banach space. Assume that 𝜁∶𝐺³→[0,+∞) is a function such that lim_𝑛→∞|4|^𝑛𝜑𝑥2^𝑛,𝑦2^𝑛,𝑧2^𝑛=0 (3.28) for all 𝑥, 𝑦, 𝑧∈𝐺. Suppose that, for any 𝑥∈𝐺, the limit Θ(𝑥)=lim_𝑛→∞max|4|^𝑘1max|4|𝜑𝑥2^𝑘,1,0,0|2|𝜑𝑥2^𝑘+1,𝑥2^𝑘+1,0;0≤𝑘<𝑛 (3.29) exists and 𝑓∶𝐺→𝑋 is an even mapping satisfying 𝑓(0)=0 and (3.3). Then the limit 𝐴(𝑥)∶=lim_𝑛→∞4^𝑛𝑓(𝑥/2^𝑛) exists for all 𝑥∈𝐺 and defines a quadratic mapping 𝑄∶𝐺→𝑋 such that 𝑓(𝑥)−𝑄(𝑥)_𝑋≤Θ(𝑥). (3.30) Moreover, if lim_𝑗→∞lim_𝑛→∞max|4|^𝑘1max|4|𝜑𝑥2^𝑘,1,0,0|2|𝜑𝑥2^𝑘+1,𝑥2^𝑘+1,0;𝑗≤𝑘<𝑛+𝑗=0, (3.31) then 𝑄 is the unique additive mapping satisfying (3.30).

Proof. It follows from (2.27) that 𝑓(𝑥)𝑥−4𝑓2_𝑋1≤max|4|𝜑(),1𝑥,0,0|2|𝜑𝑥2,𝑥2,0. (3.32) Replacing 𝑥 by 𝑥/2^𝑛 in (3.32), we have 4^𝑛𝑓𝑥2^𝑛−4^𝑛+1𝑓𝑥2^𝑛+1_𝑋≤|4|^𝑛1max|4|𝜑𝑥2^𝑛,1,0,0|2|𝜑𝑥2^𝑛+1,𝑥2^𝑛+1,0. (3.33) It follows from (3.28) and (3.32) that the sequence {4^𝑛𝑓(𝑥/2^𝑛)}_𝑛≥1 is Cauchy sequence.
The rest of the proof is similar to the proof of Theorem 3.1.

Similarly, we can obtain the following. We will omit the proof.

Theorem 3.5. Let 𝐺 be an additive semigroup and 𝑋 a non-Archimedean Banach space. Assume that 𝜁∶𝐺³→[0,+∞) is a function such that lim_𝑛→∞𝜁2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|4|^𝑛=0 (3.34) for all 𝑥, 𝑦, 𝑧∈𝐺. Suppose that, for any 𝑥∈𝐺, the limit Θ(𝑥)=lim_𝑛→∞1max|4|^𝑘1max|4|𝜑2^𝑘+1,1𝑥,0,0|2|𝜑2^𝑘𝑥,2^𝑘𝑥,0;0≤𝑘<𝑛 (3.35) exists and 𝑓∶𝐺→𝑋 is an even mapping satisfying 𝑓(0)=0 and (3.3). Then the limit 𝑄(𝑥)∶=lim_𝑛→∞(𝑓(2^𝑛𝑥)/4^𝑛) exists for all 𝑥∈𝐺 and 𝑓(𝑥)−𝑄(𝑥)_𝑋≤1|4|Θ(𝑥) (3.36) for all 𝑥∈𝐺. Moreover, if lim_𝑗→∞lim_𝑛→∞1max|4|^𝑘1max|4|𝜑2^𝑘+1,1𝑥,0,0|2|𝜑2^𝑘𝑥,2^𝑘𝑥,0;𝑗≤𝑘<𝑛+𝑗=0, (3.37) then 𝑄 is the unique mapping satisfying (3.36).

Let 𝑓∶𝑋→𝑌 be a mapping satisfying 𝑓(0)=0 and (1.4). Let 𝑓_𝑒(𝑥)∶=(𝑓(𝑥)+𝑓(−𝑥))/2 and 𝑓_𝑜(𝑥)=(𝑓(𝑥)−𝑓(−𝑥))/2. Then 𝑓_𝑒 is an even mapping satisfying (1.4) and 𝑓_𝑜 is an odd mapping satisfying (1.4) such that 𝑓(𝑥)=𝑓_𝑒(𝑥)+𝑓_𝑜(𝑥). So we obtain the following.

Theorem 3.6. Let 𝐺 be an additive semigroup and 𝑋 a non-Archimedean Banach space. Assume that 𝜁∶𝐺³→[0,+∞) is a function such that lim_𝑛→∞𝜁2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|4|^𝑛=0 (3.38) for all 𝑥, 𝑦, 𝑧∈𝐺. Suppose that the limits Ω(𝑥)=lim_𝑛→∞𝜑2maxmax^𝑘+1𝑥,0,0,𝜑2^𝑘𝑥,2^𝑘𝑥,0|2|^𝑘,Θ(𝑥);0≤𝑘<𝑛=lim_𝑛→∞𝜑2maxmax^𝑘+1𝑥,0,0,𝜑2^𝑘𝑥,2^𝑘𝑥,0|4|^𝑘;0≤𝑘<𝑛 (3.39) exist for all 𝑥∈𝐺 and 𝑓∶𝐺→𝑋 is a mapping satisfying 𝑓(0)=0 and (3.3). Then there exist an additive mapping 𝐴∶𝐺→𝑋 and a quadratic mapping 𝑄∶𝐺→𝑋 such that 𝑓(𝑥)−𝐴(𝑥)−𝑄(𝑥)_𝑋𝑓(𝑥)()≤max+𝑓−𝑥2−𝑄(𝑥)_𝑋,𝑓(𝑥)−𝑓(−𝑥)2−𝐴(𝑥)_𝑋1≤max|8|Ω(x),1|4|Θ(𝑥) (3.40) for all 𝑥∈𝐺. Moreover, if lim_𝑗→∞lim_𝑛→∞𝜑2maxmax^𝑘+1𝑥,0,0,𝜑2^𝑘𝑥,2^𝑘𝑥,0|2|^𝑘;𝑗≤𝑘<𝑛+𝑗=0,lim_𝑗→∞lim_𝑛→∞1max|4|^𝑘1max|4|𝜑2^𝑘+1,1𝑥,0,0|2|𝜑2^𝑘𝑥,2^𝑘𝑥,0;𝑗≤𝑘<𝑛+𝑗=0, (3.41) then 𝐴 and 𝑄 are the unique mappings satisfying (3.40).

4. Conclusion

We linked here three different disciplines, namely, the non-Archimedean normed spaces, functional equations, and fixed point theory. We established the generalized Hyers-Ulam stability of the functional equation (1.4) in non-Archimedean normed spaces.