Functional Inequalities Associated with Cauchy Additive Functional Equation in Non-Archimedean Spaces

Abstract

We investigate the generalized Hyers-Ulam stability of the functional inequalities ‖𝑓((𝑥+𝑦+𝑧)/4)+𝑓((3𝑥−𝑦−4𝑧)/4)+𝑓((4𝑥+3𝑧)/4)‖≤‖2𝑓(𝑥)‖ and ‖𝑓((𝑦−𝑥)/3)+𝑓((𝑥−3𝑧)/3)+𝑓((3𝑥+3𝑧−𝑦)/3)‖≤‖𝑓(𝑥)‖ in non-Archimedean normed spaces in the spirit of the Th. M. Rassias stability approach.

1. Introduction

Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

Let (𝐺₁,⋅) be a group and let (𝐺₂,∗) be a metric group with the metric 𝑑(⋅,⋅). Given 𝜖>0, does there exist a 𝛿>0, such that if a mapping ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥⋅𝑦),ℎ(𝑥)∗ℎ(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there exists a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜖 for all 𝑥∈𝐺₁?

In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. Let 𝑓∶𝐸→𝐸^′ be a mapping between Banach spaces such that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝛿 (1.1) for all 𝑥,𝑦∈𝐸, and for some 𝛿>0. Then there exists a unique additive mapping 𝑇∶𝐸→𝐸^′ such that 𝑓(𝑥)−𝑇(𝑥)≤𝛿 (1.2) for all 𝑥∈𝐸. Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each fixed 𝑥∈𝐸, then 𝑇 is linear. In 1978, Rassias [3] proved the following theorem.

Theorem 1.1. Let 𝑓∶𝐸→𝐸^′ be a mapping from a normed vector space 𝐸 into a Banach space 𝐸^′ subject to the inequality 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖‖𝑥‖^𝑝+𝑦^𝑝 (1.3) for all 𝑥,𝑦∈𝐸, where 𝜖 and p are constants with 𝜖>0 and 𝑝<1. Then there exists a unique additive mapping 𝑇∶𝐸→𝐸^′ such that 𝑓(𝑥)−𝑇(𝑥)≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (1.4) for all 𝑥∈𝐸. If 𝑝<0 then inequality (1.3) holds for all 𝑥,𝑦≠0, and (1.4) for 𝑥≠0. Also, if the function 𝑡↦𝑓(𝑡𝑥) from ℝ into 𝐸^′ is continuous in real 𝑡 for each fixed 𝑥∈𝐸, then 𝑇 is linear.

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 stability of functional equations. The reader is referred to [5–13] for a number of results in this domain of research.

In 1994, a generalization of the Rassias theorem was obtained by Găvruţa as follows [14].

Suppose (𝐺,+) is an abelian group, 𝐸 is a Banach space, and that the so-called admissible control function 𝜑∶𝐺×𝐺→ℝ satisfies 𝜑𝑥,𝑦∶=2^∞−1_𝑛=02^−𝑛𝜑2^𝑛𝑥,2^𝑛𝑦<∞ (1.5) for all 𝑥,𝑦∈𝐺. If 𝑓∶𝐺→𝐸 is a mapping with 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜑𝑥,𝑦 (1.6) for all 𝑥,𝑦∈𝐺, then there exists a unique mapping 𝑇∶𝐺→𝐸 such that 𝑇(𝑥+𝑦)=𝑇(𝑥)+𝑇(𝑦) and ‖𝑓(𝑥)−𝑇(𝑥)‖≤𝜑(𝑥,𝑥) for all 𝑥,𝑦∈𝐺.

During the last decades, several stability problems of functional equations have been investigated by a number of mathematicians, see [15–17] and references therein for more detailed information.

By a non-Archimedean field we mean a field 𝐾 equipped with a function (valuation) |⋅| from 𝐾 into [0,1) such that |𝑟|=0 if and only if 𝑟=0, |𝑟𝑠|=|𝑟||𝑠|, and |𝑟+𝑠|≤max{|𝑟|,|𝑠|} for all 𝑟,𝑠∈𝐾. Clearly |1|=|−1|=1 and |𝑛|≤1 for all 𝑛∈ℕ.

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:(i)‖𝑥‖=0 if and only if 𝑥=0;(ii)‖𝑟𝑥‖=|𝑟|‖𝑥‖(𝑟∈𝕂,𝑥∈𝑋); (iii)the strong triangle inequality (ultrametric), namely, 𝑥+𝑦≤max‖𝑥‖,𝑦𝑥,𝑦∈𝑋. (1.7)

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), (1.8) 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 we mean one in which every Cauchy sequence is convergent (see [18–22]).

Gilányi [23] and Rätz [24] showed that if 𝑓 satisfies the functional inequality 2𝑓(𝑥)+2𝑓𝑦−𝑓𝑥𝑦⁻¹≤𝑓𝑥𝑦, (1.9) then 𝑓 satisfies the Jordan-von Neumann functional equation 2𝑓(𝑥)+2𝑓𝑦=𝑓𝑥𝑦+𝑓𝑥𝑦⁻¹. (1.10) Gilányi [23] and Fechner [25] proved the generalized Hyers-Ulam stability of the functional inequality (1.3).

Cho and Kim [26] proved the generalized Hyers-Ulam stability of the following functional inequalities: 𝑓𝑥−𝑦2−𝑧+𝑓𝑦+2𝑓(𝑧)≤𝑓𝑥+𝑦2,𝑓(𝑥)𝑦(𝑧≤+𝑧+𝜑𝑥,𝑦,𝑧+𝑓+2𝑓)2𝑓𝑥+𝑦2+𝑧+𝜑𝑥,𝑦,𝑧, (1.11) which are associated with Jordan-von Neumann-type Cauchy-Jensen additive functional equations.

Now, we consider the following functional inequality: 𝑓𝑥+𝑦+𝑧4+𝑓3𝑥−𝑦−4𝑧4+𝑓4𝑥+3𝑧4≤(𝑥𝑓2𝑓),𝑦−𝑥3+𝑓𝑥−3𝑧3+𝑓3𝑥+3𝑧−𝑦3≤𝑓(𝑥), (1.12)(1.13) which is associated with Cauchy additive functional equation.

The purpose of this paper is to prove that if 𝑓 satisfies the inequalities (1.12) and (1.13), which satisfies certain conditions, then 𝑓 is Cauchy additive, and thus we prove the generalized Hyers-Ulam stability of the functional inequalities (1.12) and (1.13) in non-Archimedean normed spaces.

2. Stability of Functional Inequality (1.12)

In this section, we prove the generalized Hyers-Ulam stability of the functional inequality (1.12). Throughout this section, we assume that 𝑆 is an additive semigroup and 𝑋 is a complete non-Archimedean space.

We need the following lemma in the main results.

Lemma 2.1. Let 𝑓∶𝑆→𝑋 be a mapping such that 𝑓𝑥+𝑦+𝑧4+𝑓3𝑥−𝑦−4𝑧4+𝑓4𝑥+3𝑧4≤2𝑓(𝑥) (2.1) for all 𝑥,𝑦,𝑧∈𝑆. If |3|>|2|, then the mapping 𝑓 is Cauchy additive.

Proof. Letting 𝑥=𝑦=𝑧=0 in (2.1), we get |3|‖𝑓(0)‖≤|2|‖𝑓(0)‖. So, 𝑓(0)=0. Letting 𝑥=𝑧=0 and replacing 𝑦 by 4𝑦 in (2.1), we get ‖𝑓(𝑦)+𝑓(−𝑦)‖≤|2|‖𝑓(0)‖ for all 𝑦∈𝑆. So, 𝑓(−𝑦)=−𝑓(𝑦) for all 𝑦∈𝑆. Setting 𝑥=0 in (2.1), we obtain 𝑓𝑦+𝑧4+𝑓−𝑦−4𝑧4+𝑓3𝑧4≤|2|𝑓(0). (2.2) So, 𝑓𝑦+𝑧4+𝑓−𝑦−4𝑧4+𝑓3𝑧4=0 (2.3) for all 𝑦,𝑧∈𝑆. Replacing 𝑦 by 2𝑧 in (2.3), we get 2𝑓3𝑧4=𝑓3𝑧2 (2.4) for all 𝑧∈𝑆. Using (2.4), we obtain 𝑓(2𝑧)=2𝑓(𝑧) and 𝑓(4𝑧)=4𝑓(𝑧) for all 𝑧∈𝑆. Letting 𝑥=0, 𝑤₁=(𝑦+𝑧)/4 and 𝑤₂=(𝑦−4𝑧)/4, in (2.1) we get 𝑓𝑤₁+𝑓𝑤₂=𝑓𝑤₁+𝑤₂ (2.5) for all 𝑤₁,𝑤₂∈𝑆. Hence, 𝑓 is additive.

Theorem 2.2. Let 𝜑∶𝑆×𝑆×𝑆→ℝ⁺∪{0} be a function such that lim_𝑛→∞𝜑2max^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|2|^𝑛,𝜑−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧|2|^𝑛=0 (2.6) for all 𝑥,𝑦,𝑧∈𝑆 and let the limit 𝜑(𝑧)∶=lim_𝑛→∞𝜑maxmax0,2.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,𝜑0,−2.2^𝑖𝑧,−2^𝑖𝑧|2|^𝑖;0≤𝑖<𝑛 (2.7) exists for all 𝑧∈𝑆. Suppose that 𝑓∶𝑆→𝑋 with 𝑓(0)=0 is a mapping satisfying 𝑓𝑥+𝑦+𝑧4+𝑓3𝑥−𝑦−4𝑧4+𝑓4𝑥+3𝑧4≤|2|𝑓(𝑥)+𝜑𝑥,𝑦,𝑧 (2.8) for all 𝑥,𝑦,𝑧∈𝑆. Then there exists an additive mapping ℎ∶𝑆→𝑋 such that 𝑓(𝑧)−𝑓(−𝑧)2(𝑧≤1−ℎ)|2|²𝜑(𝑧) (2.9) for all 𝑧∈𝑆. Moreover, if lim_𝑘→∞lim_𝑛→∞𝜑maxmax0,2.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,𝜑0,−2.2^𝑖𝑧,−2^𝑖𝑧|2|^𝑖;𝑘≤𝑖<𝑛+𝑘=0 (2.10) then ℎ is the unique additive mapping satisfying (2.9).

Proof. Putting 𝑥=0 and 𝑦=2𝑧 in (2.8), we get 2𝑓3𝑧4+𝑓−3𝑧2≤𝜑(0,2𝑧,𝑧) (2.11) for all 𝑧∈𝑆. Replacing 𝑧 by 4𝑧/3 in (2.11), we obtain 2𝑓(𝑧)+𝑓(−2𝑧)≤𝜑0,8𝑧3,4𝑧3 (2.12) for all 𝑧∈𝑆. Replacing 𝑧 by −𝑧 in (2.12), we get 2𝑓(−𝑧)+𝑓(2𝑧)≤𝜑0,−8𝑧3,−4𝑧3 (2.13) for all 𝑧∈𝑆. Let 𝑔(𝑧)∶=(𝑓(𝑧)−𝑓(−𝑧))/2. It follows from (2.12) and (2.13) that 𝑔()(𝑧≤12𝑧−2𝑔)|2|max𝜑0,8𝑧3,4𝑧3,𝜑0,−8𝑧3,−4𝑧3 (2.14) for all 𝑧∈𝑆. Replacing 𝑧 by 2^𝑛−1𝑧 in (2.14), we get 𝑔2^𝑛𝑧2^𝑛−𝑔2^𝑛−1𝑧2^𝑛−1≤1|2|^𝑛+1max𝜑0,8.2^𝑛−1𝑧3,4.2^𝑛−1𝑧3,𝜑0,−8.2^𝑛−1𝑧3,−4.2^𝑛−1𝑧3 (2.15) for all 𝑧∈𝑆. It follows from (2.6) and (2.15) that the sequence {𝑔(2^𝑛𝑧)/2^𝑛} is Cauchy. Since 𝑋 is complete, we conclude that {𝑔(2^𝑛𝑧)/2^𝑛} is convergent. Set ℎ(𝑧)∶=lim_𝑛→∞(𝑔(2^𝑛𝑧)/2^𝑛) for all 𝑧∈𝑆. Using induction one can show that 𝑔2^𝑛𝑧2^𝑛(𝑧≤1−𝑔)|2|²𝜑maxmax0,8.2^𝑘𝑧/3,4.2^𝑘𝑧/3|2|^𝑘,𝜑0,−8.2^𝑘𝑧/3,−4.2^𝑘𝑧/3|2|^𝑘;0≤𝑘<𝑛 (2.16) for all 𝑧∈𝑆 and 𝑛∈ℕ. By taking 𝑛 to approach infinity in (2.16) and using (2.7) one obtains (2.9).
It follows from (2.8) that ℎ𝑥+𝑦+𝑧4+ℎ3𝑥−𝑦−4𝑧4+ℎ4𝑥+3𝑧4=lim_𝑛→∞1|2|^𝑛𝑔2^𝑛𝑥+𝑦+𝑧4+𝑔2^𝑛3𝑥−𝑦−4𝑧4+𝑔2^𝑛4𝑥+3𝑧4=lim_𝑛→∞1|2|^𝑛+1𝑓2^𝑛𝑥+𝑦+𝑧4+𝑓2^𝑛3𝑥−𝑦−4𝑧4+𝑓2^𝑛(4𝑥+3𝑧)4−𝑓2^𝑛−𝑥−𝑦−𝑧4−𝑓2^𝑛−3𝑥+𝑦+4𝑧4−𝑓2^𝑛(−4𝑥−3𝑧)4≤1|2|^𝑛𝑓2^𝑛𝑧−𝑓−2^𝑛𝑧+lim_𝑛→∞1|2|𝜑2max^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|2|^𝑛,𝜑−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧|2|^𝑛=|2|‖ℎ(𝑧)‖ (2.17) for all 𝑥,𝑦,𝑧∈𝑆. So, ℎ𝑥+𝑦+𝑧4+ℎ3𝑥−𝑦−4𝑧4+ℎ4𝑥+3𝑧4≤|2|‖ℎ(𝑧)‖ (2.18) for all 𝑥,𝑦,𝑧∈𝑆. By Lemma 2.1, the mapping ℎ∶𝑆→𝑋 is additive.
Now, let 𝑇∶𝑆→𝑋 be another additive mapping satisfying (2.9). Then we have ‖ℎ(𝑧)(𝑧=1−𝑇)‖|2|^𝑘ℎ2^𝑘𝑧−𝑇2^𝑘𝑧≤1|2|^𝑘max𝑇2^𝑘𝑧−𝑔2^𝑘𝑧,𝑔2^𝑘𝑧−ℎ2^𝑘𝑧≤1|2|²lim_𝑘→∞lim_𝑛→∞𝜑maxmax0,2.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,𝜑0,−2.2^𝑖𝑧,−2^𝑖𝑧|2|^𝑖;𝑘≤𝑖<𝑖+𝑘=0 (2.19) for all 𝑧∈𝑆. Therefore ℎ=𝑇. This completes the proof of the uniqueness of ℎ.

Corollary 2.3. Let 𝑝>1 and 𝜃 be positive real numbers, and let 𝑓∶𝑆→𝑋 be a mapping satisfying 𝑓𝑥+𝑦+𝑧4+𝑓3𝑥−𝑦−4𝑧4+𝑓4𝑥+3𝑧4≤|2|𝑓(𝑥)+𝜃‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝 (2.20) for all 𝑥,𝑦,𝑧∈𝑆. If |2|<1 then there exists a unique additive mapping ℎ∶𝑆→𝑋 such that 𝑓(𝑧)−𝑓(−𝑧)2−ℎ(𝑧)≤2𝜃|2|²‖𝑧‖^𝑝 (2.21) for all 𝑧∈𝑆.

Proof. Defining 𝜑∶𝑆×𝑆×𝑆→𝑋 by 𝜑(𝑥,𝑦,𝑧)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝+‖𝑧‖^𝑝) we have lim_𝑛→∞𝜑2max^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|2|^𝑛,𝜑−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧|2|^𝑛=lim_𝑛→∞𝜃|2|^𝑛𝑝|2|^𝑛‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝𝜑(𝑧)=0,∶=lim_𝑛→∞𝜑maxmax0,2.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,𝜑0,−2.2^𝑖𝑧,−2^𝑖𝑧|2|^𝑖;0≤𝑖<𝑛=lim_𝑛→∞|2|max^(𝑖+1)𝑝+|2|^𝑖𝑝|2|^𝑖𝜃‖𝑧‖^𝑝;0≤𝑖<𝑛≤2𝜃‖𝑧‖^𝑝,lim_𝑘→∞lim_𝑛→∞𝜑maxmax0,2.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,𝜑0,−2.2^𝑖𝑧,−2^𝑖𝑧|2|^𝑖;𝑘≤𝑖<𝑛+𝑘≤lim_𝑘→∞|2|^𝑘𝑝𝜃‖𝑧‖^𝑝=0 (2.22) for all 𝑧∈𝑆.
Applying Theorem 2.2, we conclude the required result.

Corollary 2.4. Let 𝜓∶ℝ⁺∪{0}→ℝ⁺∪{0} be a function satisfying 𝜓(|2|𝑟)≤𝜓(|2|)𝜓(𝑟)(𝑟≥0),𝜓(|2|)<|2|. (2.23) Let 𝜃>0, let 𝑆 be a normed space and let 𝑓∶𝑆→𝑋 fulfill the inequality 𝑓𝑥+𝑦+𝑧4+𝑓3𝑥−𝑦−4𝑧4+𝑓4𝑥+3𝑧4≤|2|𝑓(𝑥)+𝜃𝜓(‖𝑥‖)+𝜓𝑦+𝜓(‖𝑧‖) (2.24) for all 𝑥,𝑦,𝑧∈𝑆. Then there exists a unique additive mapping ℎ∶𝑆→𝑋 such that 𝑓(𝑧)−𝑓(−𝑧)2−ℎ(𝑧)≤2𝜃|2|²𝜓(‖𝑧‖) (2.25) for all 𝑧∈𝑆.

Proof. Defining 𝜑∶𝑆×𝑆×𝑆→𝑋 by 𝜑(𝑥,𝑦,𝑧)∶=𝜃[𝜓(‖𝑥‖)+𝜓(‖𝑦‖)+𝜓(‖𝑧‖)] we have lim_𝑛→∞𝜑2max^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|2|^𝑛,𝜑−2^𝑛𝑥,−2^𝑛𝑦,−2^𝑛𝑧|2|^𝑛≤𝜃lim_𝑛→∞𝜓2max(||)|2|^𝑛𝜑𝜑(𝑧)𝑥,𝑦,𝑧,𝜑−𝑥,−𝑦,−𝑧=0,∶=lim_𝑛→∞𝜑maxmax0,2.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,𝜑0,−2.2^𝑖𝑧,−2^𝑖𝑧|2|^𝑖𝑧,;0≤𝑖<𝑛=2𝜃𝜓(‖‖)lim_𝑘→∞lim_𝑛→∞𝜑maxmax0,2.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,𝜑0,−2.2^𝑖𝑧,−2^𝑖𝑧|2|^𝑖;𝑘≤𝑖<𝑛+𝑘≤lim_𝑘→∞𝜓(|2|)|2|^𝑘𝜓(‖𝑧‖)=0 (2.26) for all 𝑧∈𝑆.
Applying Theorem 2.2, we conclude the required result.

Theorem 2.5. Let 𝜑∶𝑆×𝑆×𝑆→ℝ⁺∪{0} be a function such that lim_𝑛→∞max|2|^𝑛𝜑2^−𝑛𝑥,2^−𝑛𝑦,2^−𝑛,|2|^𝑛𝜑−2^−𝑛𝑥,−2^−𝑛𝑦,−2^−𝑛𝑧=0 (2.27) for all 𝑥,𝑦,𝑧∈𝑆 and let the limit 𝜑(𝑧)∶=lim_𝑛→∞maxmax|2|^𝑛𝜑0,2.2^−𝑖𝑧,2^−𝑖𝑧,|2|^𝑛𝜑0,−2.2^−𝑖𝑧,−2^−𝑖𝑧;0≤𝑖<𝑛 (2.28) exist for all 𝑧∈𝑆. Suppose that 𝑓∶𝑆→𝑋 with 𝑓(0)=0 is a mapping satisfying 𝑓𝑥+𝑦+𝑧4+𝑓3𝑥−𝑦−4𝑧4+𝑓4𝑥+3𝑧4≤|2|𝑓(𝑥)+𝜑𝑥,𝑦,𝑧 (2.29) for all 𝑥,y,𝑧∈𝑆. Then there exists an additive mapping ℎ∶𝑆→𝑋 such that 𝑓(𝑧)−𝑓(−𝑧)2(𝑧≤1−ℎ)|2|𝜑(𝑧) (2.30) for all 𝑧∈𝑆. Moreover, if lim_𝑘→∞lim_𝑛→∞maxmax|2|^𝑖𝜑0,2.2^−𝑖𝑧,2^−𝑖𝑧,|2|^𝑖𝜑0,−2.2^−𝑖𝑧,−2^−𝑖𝑧;𝑘≤𝑖<𝑛+𝑘=0 (2.31) then ℎ is the unique additive mapping satisfying (2.30).

Proof. It follows from (2.14) that 𝑧2𝑔2(𝑧≤1−𝑔)|2|max𝜑0,4𝑧3,2𝑧3,𝜑0,−4𝑧3,−2𝑧3 (2.32) for all 𝑧∈𝑆. Hence, 2^𝑛𝑔2^−𝑛𝑧−2^(𝑛+1)𝑔2^{−(𝑛+1)𝑧}≤|2|^𝑛max𝜑0,4.2^−𝑛𝑧3,2.2^−𝑛𝑧3,𝜑0,−4.2^−𝑛𝑧3,−2.2^−𝑛𝑧3 (2.33) for all 𝑧∈𝑆. It follows from (2.27) and (2.33) that the sequence {2^𝑛𝑔(2^−𝑛𝑧)} is a Cauchy sequence for all 𝑧∈𝑆. Since 𝑋 is complete, the sequence {2^𝑛𝑔(2^−𝑛𝑧)} converges. So, one can define the mapping ℎ∶𝑆→𝑋 by ℎ(𝑧)∶=lim_𝑛→∞{2^𝑛𝑔(2^−𝑛𝑧)} for all 𝑧∈𝑆.
The rest of the proof is similar to the proof of Theorem 2.2.

Remark 2.6. We can obtain similar results to Corollary 2.3 for 𝑝<1 and Corollary 2.4.

3. Stability of Functional Inequality (1.13)

We prove the generalized Hyers-Ulam stability of the functional inequality (1.13). Throughout this section, we assume that 𝑆 is an additive semigroup and 𝑋 is a complete non-Archimedean space.

We need the following lemma in the main results.

Lemma 3.1. Let 𝑓∶𝑆→𝑋 be a mapping such that 𝑓𝑦−𝑥3+𝑓𝑥−3𝑧3+𝑓3𝑥+3𝑧−𝑦3≤𝑓(𝑥) (3.1) for all 𝑥,𝑦,𝑧∈𝑆. If 𝑓(0)=0, then the mapping 𝑓 is Cauchy additive.

Proof. Letting 𝑥=𝑦=0 in (3.1), we get 𝑓(−𝑧)+𝑓(𝑧)≤𝑓(0)=0 (3.2) for all 𝑧∈𝑆. Hence, 𝑓(−𝑧)=−𝑓(𝑧) for all 𝑧∈𝑆. Letting 𝑥=0 and 𝑦=6𝑧 in (3.1), we get 𝑓(2𝑧)−2𝑓(𝑧)≤𝑓(0)=0 (3.3) for all 𝑧∈𝑆. Hence, 𝑓(2𝑧)=2𝑓(𝑧) (3.4) for all 𝑧∈𝑆. Letting 𝑥=0 and 𝑦=9𝑧 in (3.1), we get 𝑓(3𝑧)−𝑓(𝑧)−2𝑓(𝑧)≤𝑓(0)=0 (3.5) for all 𝑧∈𝑆. Hence, 𝑓(3𝑧)=3𝑓(𝑧) (3.6) for all 𝑧∈𝑆. Letting 𝑥=0 in (3.1), we get 𝑓𝑦3()𝑦+𝑓−𝑧+𝑓𝑧−3≤𝑓(0)=0 (3.7) for all 𝑥,𝑦,𝑧∈𝑆. So, 𝑓𝑦3()𝑦+𝑓−𝑧+𝑓𝑧−3=0 (3.8) for all 𝑥,𝑦,𝑧∈𝑆. Let 𝑡₁=𝑧−𝑦/3 and 𝑡₂=𝑦/3 in (3.8). Then 𝑓𝑡₂−𝑓𝑡₁+𝑡₂+𝑓𝑡₁=0 (3.9) for all 𝑡₁,𝑡₂∈𝑆. So, 𝑓 is additive.

Theorem 3.2. Let 𝜑∶𝑆×𝑆×𝑆→[0,∞) be a function such that lim_𝑛→∞𝜑2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧|2|^𝑛=0 (3.10) for all 𝑥,𝑦,𝑧∈𝑆 and let the limit 𝜑(z)∶=lim_𝑛→∞𝜑maxmax0,6.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,|2|𝜑0,3.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖;0≤𝑖<𝑛 (3.11) exist for all 𝑧∈𝑆. Suppose that 𝑓∶𝑆→𝑋 with 𝑓(0)=0 is a mapping satisfying 𝑓𝑦−𝑥3+𝑓𝑥−3𝑧3+𝑓3𝑥+3𝑧−𝑦3≤𝑓(𝑥)+𝜑𝑥,𝑦,𝑧 (3.12) for all 𝑥,𝑦,𝑧∈𝑆. Then there exists an additive mapping 𝑇∶𝑆→𝑋 such that 𝑓(𝑧)(𝑧≤1−𝑇)|2|𝜑(𝑧) (3.13) for all 𝑧∈𝑆. Moreover, if lim_𝑘→∞lim_𝑛→∞𝜑maxmax0,6.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,|2|𝜑0,3.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖;𝑘≤𝑖<𝑛+𝑘=0 (3.14) then 𝑇 is the unique additive mapping satisfying (3.13).

Proof. Letting 𝑥=0 and 𝑦=6𝑧 in (3.12), we get 𝑓(2𝑧)+2𝑓(−𝑧)≤𝜑(0,6𝑧,𝑧) (3.15) for all 𝑧∈𝑆. Putting 𝑥=0 and 𝑦=3𝑧 in (3.12), we get 2𝑓(𝑧)+2𝑓(−𝑧)≤|2|𝜑(0,3𝑧,𝑧)≤𝜑(0,3𝑧,𝑧) (3.16) for all 𝑧∈𝑆. It follows from (3.15) and (3.16) that 𝑓(2𝑧)−2𝑓(𝑧)≤max𝜑(0,6𝑧,𝑧),𝜑(0,3𝑧,𝑧) (3.17) for all 𝑧∈𝑆. Replacing 𝑧 by 2^𝑛−1𝑧 in (3.17), we get 𝑓2^𝑛𝑧2^𝑛−𝑓2^𝑛−1𝑧2^𝑛−1≤1|2|^𝑛max𝜑0,6.2^𝑛−1𝑧,2^𝑛−1𝑧,𝜑0,3.2^𝑛−1𝑧,2^𝑛−1𝑧 (3.18) for all 𝑧∈𝑆. It follows from (3.10) and (3.18) that the sequence {𝑓(2^𝑛𝑧)/2^𝑛} is Cauchy. Since 𝑋 is complete, we conclude that {𝑓(2^𝑛𝑧)/2^𝑛} is convergent. Set 𝑇(𝑧)∶=lim_𝑛→∞(𝑓(2^𝑛𝑧)/2^𝑛) for all 𝑧∈𝑆. Using induction one can show that 𝑓2^𝑛𝑧2^𝑛(𝑧≤1−𝑓)|2|𝜑maxmax0,6.2^𝑘𝑧,2^𝑘𝑧|2|^𝑘,|2|𝜑0,3.2^𝑘𝑧,2^𝑘𝑧|2|^𝑘;0≤𝑘<𝑛 (3.19) for all 𝑧∈𝑆 and 𝑛∈ℕ. By taking 𝑛 to approach infinity in (3.19) and using (3.11) one obtains (3.13).
Replacing 𝑥,  𝑦, and 𝑧 by 2^𝑛𝑥,2^𝑛𝑦, and 2^𝑛𝑧, respectively, in (3.12) we get 𝑓2^𝑛𝑦−𝑥3.2^𝑛2+𝑓^𝑛(𝑥−3𝑧)3.2^𝑛2+𝑓^𝑛3𝑥+3𝑧−𝑦3.2^𝑛≤𝑓2^𝑛𝑥2^𝑛+1|2|^𝑛𝜑2^𝑛𝑥,2^𝑛𝑦,2^𝑛𝑧 (3.20) for all 𝑥,𝑦,𝑧∈𝑆. Taking the limit as 𝑛→inf𝑡𝑦 and using (3.10) we get 𝑇𝑦−𝑥3+𝑇𝑥−3𝑧3+𝑇3𝑥+3𝑧−𝑦3≤‖𝑇(𝑥)‖ (3.21) for all 𝑥,𝑦,𝑧∈𝑆. By Lemma 3.1, the mapping 𝑇∶𝑆→𝑋 is additive.
If 𝑇^′ is another additive mapping satisfying (3.13), then 𝑇(𝑧)−𝑇^′(𝑧=1)|2|^𝑘𝑇2^𝑘𝑧−𝑇^′2^𝑘𝑧≤1|2|^𝑘max𝑇2^𝑘𝑧−𝑓2^𝑘𝑧,𝑓2^𝑘𝑧−𝑇^′2^𝑘𝑧≤1|2|lim_𝑘→∞lim_𝑛→∞𝜑maxmax0,6.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖,|2|𝜑0,3.2^𝑖𝑧,2^𝑖𝑧|2|^𝑖;𝑘≤𝑖<𝑖+𝑘=0 (3.22) for all 𝑧∈𝑆. Therefore 𝑇^′=𝑇. This proves the uniqueness of 𝑇. Hence, the mapping 𝑇∶𝑆→𝑋 is a unique additive mapping satisfying (3.13).

Corollary 3.3. Let 𝑝>1 and 𝜃 be positive real numbers, and let 𝑓∶𝑆→𝑋 be a mapping satisfying 𝑓𝑦−𝑥3+𝑓𝑥−3𝑧3+𝑓3𝑥+3𝑧−𝑦3≤𝑓(𝑥)+𝜃‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝 (3.23) for all 𝑥,𝑦,𝑧∈𝑆. If |2|<1 then there exists a unique additive mapping 𝑇∶𝑆→𝑋 such that 𝑓(𝑧)−𝑇(𝑧)≤2𝜃|2|‖𝑧‖^𝑝 (3.24) for all 𝑧∈𝑆.

Proof. Letting 𝜑(𝑥,𝑦,𝑧)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝+‖𝑧‖^𝑝) in Theorem 3.2, we obtain the result.

Corollary 3.4. Let 𝜓∶[0,∞)→[0,∞) be a function satisfying 𝜓(|2|𝑟)≤𝜓(|2|)𝜓(𝑟)(𝑟≥0),𝜓(|2|)<|2|. (3.25) Let 𝜃>0, let 𝑆 be a normed space, and let 𝑓∶𝑆→𝑋 fulfill the inequality 𝑓𝑦−𝑥3+𝑓𝑥−3𝑧3+𝑓3𝑥+3𝑧−𝑦3≤𝑓(𝑥)+𝜃𝜓(‖𝑥‖)+𝜓𝑦+𝜓(‖𝑧‖) (3.26) for all 𝑥,𝑦,𝑧∈𝑆. Then there exists a unique additive mapping 𝑇∶𝑆→𝑋 such that 𝑓(𝑧)−𝑇(𝑧)≤2𝜃|2|𝜓(‖𝑧‖) (3.27) for all 𝑧∈𝑆.

Proof. If we define 𝜑(𝑥,𝑦,𝑧)∶=𝜃[𝜓(‖𝑥‖)+𝜓(‖𝑦‖)+𝜓(‖𝑧‖)] in Theorem 3.2, then we get the result.

Remark 3.5. We can formulate a similar statement to Theorem 3.2 in which we can define the sequence 𝑇(𝑧)∶=lim_𝑛→∞2^𝑛𝑓(𝑧/2^𝑛) under suitable conditions on the function 𝜑 and 𝜓 and then obtain similar results to Corollary 3.3 for 𝑝<1 and Corollary 3.4.