Hyers-Ulam Stability of Jensen Functional Inequality in p-Banach Spaces

Abstract

We prove the Hyers-Ulam stability of the following Jensen functional inequality ‖𝑓((𝑥−𝑦)/𝑛+𝑧)+𝑓((𝑦−𝑧)/𝑛+𝑥)+𝑓((𝑧−𝑥)/𝑛+𝑦)‖≤‖𝑓((𝑥+𝑦+𝑧)‖ in 𝑝-Banach spaces for any fixed nonzero integer 𝑛.

1. Introduction

The stability problem of equations originated from a question of Ulam [1] concerning the stability of group homomorphisms.

We are given a group 𝐺₁ and a metric group 𝐺₂ with metric 𝜌(⋅,⋅). Given 𝜖>0, does there exist a number 𝛿>0 such that if 𝑓∶𝐺₁→𝐺₂ satisfies 𝜌(𝑓(𝑥𝑦),𝑓(𝑥)𝑓(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then a homomorphism ℎ∶𝐺₁→𝐺₂ exists with 𝜌(𝑓(𝑥),ℎ(𝑥))<𝜖 for all 𝑥∈𝐺₁?

In 1941, Hyers [2] considered the case of approximately additive mappings between Banach spaces and proved the following result.

Suppose that 𝐸₁ and 𝐸₂ are Banach spaces and 𝑓∶𝐸₁→𝐸₂ satisfies the following condition: if there is a number 𝜖≥0 such that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖 (1.1) for all 𝑥,𝑦∈𝐸₁, then the limit ℎ(𝑥)=lim_𝑛→∞𝑓(2^𝑛𝑥)/2^𝑛 exists for all 𝑥∈𝐸₁ and there exists a unique additive mapping ℎ∶𝐸₁→𝐸₂ such that 𝑓(𝑥)−ℎ(𝑥)≤𝜖. (1.2) Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each 𝑥∈𝐸₁, then the mapping ℎ is ℝ-linear.

The method which was provided by Hyers, and which produces the additive mapping ℎ, is called a direct method. This method is the most important and most powerful tool for studying the stability of various functional equations. Hyers' theorem was generalized by Aoki [3] and Bourgin [4] for additive mappings by considering an unbounded Cauchy difference. In 1978, Rassias [5] also provided a generalization of Hyers' theorem for linear mappings which allows the Cauchy difference to be unbounded. Let 𝐸₁ and 𝐸₂ be two Banach spaces and let 𝑓∶𝐸₁→𝐸₂ be a mapping such that 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each fixed 𝑥. Assume that there exist 𝜖>0 and 0≤𝑝<1 such that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖‖𝑥‖^𝑝+𝑦^𝑝,∀𝑥,𝑦∈𝐸₁. (1.3) Then, there exists a unique ℝ-linear mapping 𝑇∶𝐸₁→𝐸₂ such that 𝑓(𝑥)−𝑇(𝑥)≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (1.4) for all 𝑥∈𝐸₁. A generalized result of Rassias' theorem was obtained by Găvruţa in [6] and Jung in [7]. In 1990, Rassias [8] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for 𝑝≥1. In 1991, Gajda [9], following the same approach as in [5], gave an affirmative solution to this question for 𝑝>1. It was shown by Gajda [9], as well as by Rassias and Šemrl [10], that one cannot prove a Rassias' type theorem when 𝑝=1. The counterexamples of Gajda [9], as well as of Rassias and Šemrl [10], have stimulated several mathematicians to invent new approximately additive or approximately linear mappings.

We recall some basic facts concerning quasinormed spaces and some preliminary results. Let 𝑋 be a real linear space. A quasinorm is a real-valued function on 𝑋 satisfying the following:(1)‖𝑥‖≥0 for all 𝑥∈𝑋 and ‖𝑥‖=0 if and only if 𝑥=0.(2)‖𝜆𝑥‖=|𝜆|‖𝑥‖ for all 𝜆∈ℝ and all 𝑥∈𝑋.(3)There is a constant 𝑀≥1 such that ‖𝑥+𝑦‖≤𝑀(‖𝑥‖+‖𝑦‖) for all 𝑥,𝑦∈𝑋.

The pair (𝑋,‖⋅‖) is called a quasinormed space if ‖⋅‖ is a quasinorm on 𝑋 [11, 12]. The smallest possible 𝑀 is called the modulus of concavity of ‖⋅‖. A quasi-Banach space is a complete quasinormed space.

A quasinorm ‖⋅‖ is called a 𝑝-norm (0<𝑝≤1) if 𝑥+𝑦^𝑝≤‖𝑥‖^𝑝+𝑦^𝑝 (1.5) for all 𝑥,𝑦∈𝑋. In this case, a quasi-Banach space is called a 𝑝-Banach space.

Given a 𝑝-norm, the formula 𝑑(𝑥,𝑦)∶=‖𝑥−𝑦‖^𝑝 gives us a translation invariant metric on 𝑋. By the Aoki-Rolewicz theorem [12], each quasinorm is equivalent to some 𝑝-norm (see also [11]). Since it is much easier to work with 𝑝-norms, henceforth, we restrict our attention mainly to 𝑝-norms. We observe that if 𝑥₁,𝑥₂,…,𝑥_𝑛 are nonnegative real numbers, then ^𝑛_𝑖=1𝑥_𝑖^𝑝≤^𝑛_𝑖=1𝑥^𝑝_𝑖, (1.6) where 0<𝑝≤1.

In 2009, Moslehian and Najati [13] introduced the Hyers-Ulam stability of the additive functional inequality: 𝑓𝑥−𝑦2+𝑧+𝑓𝑦−𝑧2+𝑥+𝑓𝑧−𝑥2+𝑦≤𝑓𝑥+𝑦+𝑧, (1.7) and then have investigated the general solution and the Hyers-Ulam stability problem for the functional inequality. The stability problems of several functional equations in quasi-normed spaces and several functional inequalities have been investigated by a number of authors and there are many interesting results concerning the stability of various functional inequalities [14–17].

In this paper, we consider a modified and general Jensen functional inequality: 𝑓𝑥−𝑦𝑛+𝑧+𝑓𝑦−𝑧𝑛+𝑥+𝑓𝑧−𝑥𝑛+𝑦≤𝑓𝑥+𝑦+𝑧 (1.8) for any fixed nonzero integer 𝑛. First of all, it is easy to see that a function 𝑓 satisfies the inequality (1.8) if and only if 𝑓(𝑥) is additive. Thus the inequality (1.8) may be called the Jensen functional inequality and the general solution of inequality (1.8) may be called the Jensen function. In the sequel, we investigate the generalized Hyers-Ulam stability of (1.8) in 𝑝-Banach spaces for any fixed nonzero integer 𝑛 by using the techniques of [14, 15].

2. Generalized Hyers-Ulam Stability

First, we present the general solution of the inequality (1.8).

Lemma 2.1. Let both 𝑋 and 𝑌 be real vector spaces. A function 𝑓∶𝑋→𝑌 satisfies (1.8) for all 𝑥,𝑦,𝑧∈𝑋 if and only if 𝑓 is additive.

Proof. Letting 𝑥=𝑦=𝑧=0 in (1.8), we have 𝑓(0)=0. Putting 𝑦=−(𝑛+1)𝑥/2 and 𝑧=(𝑛−1)𝑥/2 in (1.8), we get 𝑓𝑛²+3𝑥−𝑛2𝑛+𝑓²+3𝑥≤𝑓(02𝑛) (2.1) for all 𝑥∈𝑋. Hence 𝑓(−𝑥)=−𝑓(𝑥) for all 𝑥∈𝑋. Replacing 𝑧 by −𝑥−𝑦 in (1.8), we obtain 𝑓(1−𝑛)𝑥−(𝑛+1)𝑦𝑛()+𝑓𝑛+1𝑥+2𝑦𝑛()𝑦+𝑓−2𝑥+𝑛−1𝑛≤𝑓(0), (2.2) that is, 𝑓(1−𝑛)𝑥−(𝑛+1)𝑦+𝑓(𝑛+1)𝑥+2𝑦+𝑓−2𝑥+(𝑛−1)𝑦=0 (2.3) for all 𝑥,𝑦∈X. Putting 𝑢=(𝑛+1)𝑥+2𝑦 and 𝑣=−2𝑥+(𝑛−1)𝑦 in (2.3), we get by oddness of 𝑓, 𝑓(𝑢+𝑣)=𝑓(𝑢)+𝑓(𝑣) (2.4) for all 𝑢,𝑣∈𝑋. So 𝑓 is additive.
The proof of the converse is trivial.

From now on, assume that 𝑋 is a quasinormed space with quasinorm ‖⋅‖ and that 𝑌 is a 𝑝-Banach space with 𝑝-norm ‖⋅‖. Let 𝑀 be the modulus of concavity of ‖⋅‖ in 𝑌.

Before taking up the main subject, given a mapping 𝑓∶𝑋→𝑌, we define the difference operator 𝐷𝑓∶𝑋³→𝑌 by 𝐷𝑓𝑥,𝑦,𝑧∶=𝑓𝑥−𝑦𝑛+𝑧+𝑓𝑦−𝑧𝑛+𝑥+𝑓𝑧−𝑥𝑛+𝑦−𝑓𝑥+𝑦+𝑧 (2.5) for all 𝑥,𝑦,𝑧∈𝑋 and for any fixed nonzero integer 𝑛.

Theorem 2.2. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜑𝑥,𝑦,𝑧 (2.6) for all 𝑥,𝑦,𝑧∈𝑋 and the perturbing function 𝜑∶𝑋³→ℝ⁺ satisfies Φ𝑥,𝑦,𝑧∶=^∞_𝑖=012^𝑖𝑝𝜑2^𝑖𝑥,2^𝑖𝑦,2^𝑖𝑧^𝑝<∞ (2.7) for all 𝑥,𝑦,𝑧∈𝑋. Then, there exists a unique additive mapping ℎ∶𝑋→𝑌 defined by ℎ(𝑥)=lim_𝑘→∞(1/2^𝑘)𝑓(2^𝑘𝑥) such that 𝑓(𝑥)(𝑥≤𝑀−ℎ)2Φ𝑛(𝑛−3)𝑥𝑛²,𝑛()𝑥+3𝑛+3𝑛²,+3−2𝑛²𝑥𝑛²()𝑥+3+Φ−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,+34𝑛𝑥𝑛²+3^1/𝑝 (2.8) for all 𝑥∈𝑋.

Proof. Replacing 𝑧 by −𝑥−𝑦 in (2.6), we obtain 𝑓(1−𝑛)𝑥−(𝑛+1)𝑦𝑛()+𝑓𝑛+1𝑥+2𝑦𝑛()𝑦+𝑓−2𝑥+𝑛−1𝑛≤𝜑𝑥,𝑦,−𝑥−𝑦 (2.9) for all 𝑥,𝑦∈𝑋. Letting 𝑥=(𝑛−3)𝑥/(𝑛²+3) and 𝑦=(𝑛+3)𝑥/(𝑛²+3) in (2.9), we get 𝑓−2𝑥𝑛𝑥+2𝑓𝑛()𝑥≤𝜑𝑛−3𝑛²,()𝑥+3𝑛+3𝑛²,+3−2𝑛𝑥𝑛²+3 (2.10) for all 𝑥∈𝑋. Putting 𝑥=−(𝑛+1)𝑧/2 and 𝑦=(𝑛−1)𝑧/2 in (2.6), we have 𝑓−𝑛²+3𝑧𝑛2𝑛+𝑓²+3𝑧−()𝑧2𝑛≤𝜑𝑛+12,(𝑛−1)𝑧2,𝑧 (2.11) for all 𝑧∈𝑋. Replacing 𝑧 by 4𝑥/(𝑛²+3) in (2.11), we obtain 𝑓−2𝑥𝑛+𝑓2𝑥𝑛()𝑥≤𝜑−2𝑛+1𝑛²,2()𝑥+3𝑛−1𝑛²,+34𝑥𝑛²+3 (2.12) for all 𝑥∈𝑋. It follows from (2.10) and (2.12) that 𝑓2𝑥𝑛−2𝑓2𝑥𝑛≤𝑀𝑓−2𝑥𝑛𝑥+2𝑓𝑛+𝑓−2𝑥𝑛+𝑓2𝑥𝑛𝜑()𝑥≤𝑀𝑛−3𝑛²,()𝑥+3𝑛+3𝑛²,+3−2𝑛𝑥𝑛²()𝑥+3+𝜑−2𝑛+1𝑛²,2()𝑥+3𝑛−1𝑛²,+34𝑥𝑛²+3 (2.13) for all 𝑥∈𝑋. If we replace 𝑥 by 𝑛𝑥 in (2.13), then we get that 𝑓()(𝑥𝜑𝑛()𝑥2𝑥−2𝑓)≤𝑀𝑛−3𝑛²,𝑛()𝑥+3𝑛+3𝑛²,+3−2𝑛²𝑥𝑛²()𝑥+3+𝜑−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,+34𝑛𝑥𝑛².+3 (2.14)
It follows from (2.14) that 𝑓2^𝑙𝑥2^𝑙−𝑓2^𝑚𝑥2^𝑚𝑝≤^𝑚−1_𝑖=𝑙12^𝑖𝑓2^𝑖𝑥−12^𝑖+1𝑓2^𝑖+1𝑥^𝑝=^𝑚−1_𝑖=𝑙12^𝑖𝑝𝑓2^𝑖𝑥−12𝑓2^𝑖+1𝑥^𝑝≤𝑀^𝑝2^{𝑝𝑚−1𝑖=𝑙}12^𝑖𝑝𝜑𝑛(𝑛−3)2^𝑖𝑥𝑛²,𝑛()2+3𝑛+3^𝑖𝑥𝑛²,+3−2𝑛²2^𝑖𝑥𝑛²+3^𝑝()2+𝜑−2𝑛𝑛+1^𝑖𝑥𝑛²,()2+32𝑛𝑛−1^𝑖𝑥𝑛²,()2+34𝑛^𝑖𝑥𝑛²+3^𝑝 (2.15) for all nonnegative integers 𝑚 and 𝑙 with 𝑚>𝑙≥0 and 𝑥∈𝑋. Since the right-hand side of (2.15) tends to zero as 𝑙→∞, by the convergence of the series (2.7), we obtain that the sequence {𝑓(2^𝑚𝑥)/2^𝑚} is Cauchy for all 𝑥∈𝑋. Because of the fact that 𝑌 is complete, it follows that the sequence {𝑓(2^𝑚𝑥)/2^𝑚} converges in 𝑌. Therefore, we can define a mapping ℎ∶𝑋→𝑌 as ℎ(𝑥)=lim_𝑚→∞𝑓2^𝑚𝑥2^𝑚,𝑥∈𝑋. (2.16) Moreover, letting 𝑙=0 and taking 𝑚→∞ in (2.15), we get 𝑓(𝑥)(𝑥≤𝑀−ℎ)2Φ𝑛(𝑛−3)𝑥𝑛²,𝑛()𝑥+3𝑛+3𝑛²,+3−2𝑛²𝑥𝑛²()𝑥+3+Φ−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,+34𝑛𝑥𝑛²+3^1/𝑝 (2.17) for all 𝑥∈𝑋.
It follows from (2.6) and (2.7) that ℎ𝑥−𝑦𝑛+𝑧+ℎ𝑦−𝑧𝑛+𝑥+ℎ𝑧−𝑥𝑛+𝑦^𝑝=lim_𝑚→∞12^𝑚𝑓2^𝑚𝑥−𝑦𝑛+𝑧+𝑓2^𝑚𝑦−𝑧𝑛+𝑥+𝑓2^𝑚𝑧−𝑥𝑛+𝑦^𝑝≤lim_𝑚→∞12^𝑚𝑓2^𝑚𝑥+𝑦+𝑧^𝑝+12^𝑚𝑝𝜑2^𝑚𝑥,2^𝑚𝑦,2^𝑚𝑧^𝑝=ℎ𝑥+𝑦+𝑧^𝑝 (2.18) for all 𝑥,𝑦,𝑧∈𝑋. So the mapping ℎ is additive.
Next, let ℎ′∶𝑋→𝑌 be another additive mapping satisfying (2.8). Then, we have ℎ(𝑥)−ℎ^′(𝑥)^𝑝=12^𝑘ℎ2^𝑘𝑥−12^𝑘ℎ^′2^𝑘𝑥^𝑝≤12^𝑘𝑝ℎ2^𝑘𝑥−𝑓2^𝑘𝑥^𝑝+𝑓2^𝑘𝑥−ℎ^′2^𝑘𝑥^𝑝≤^∞_𝑖=02𝑀^𝑝2^{(𝑖+𝑘+1)𝑝}𝜑𝑛(𝑛−3)2^𝑖+𝑘𝑥𝑛²,𝑛()2+3𝑛+3^𝑖+𝑘𝑥𝑛²,+3−2𝑛²2^𝑖+𝑘𝑥𝑛²+3^𝑝()2+𝜑−2𝑛𝑛+1^𝑖+𝑘𝑥𝑛²,()2+32𝑛𝑛−1^𝑖+𝑘𝑥𝑛²,()2+34𝑛^𝑖+𝑘𝑥𝑛²+3^𝑝=^∞_𝑖=𝑘2𝑀^𝑝2^(𝑖+1)𝑝𝜑𝑛(𝑛−3)2^𝑖𝑥𝑛²,𝑛()2+3𝑛+3^𝑖𝑥𝑛²,+3−2𝑛²2^𝑖𝑥𝑛²+3^𝑝()2+𝜑−2𝑛𝑛+1^𝑖𝑥𝑛²,()2+32𝑛𝑛−1^𝑖𝑥𝑛²,()2+34𝑛^𝑖𝑥𝑛²+3^𝑝 (2.19) for all 𝑘∈ℕ and all 𝑥∈𝑋. Taking the limit as 𝑘→∞, we conclude that ℎ(𝑥)=ℎ^′(𝑥) (2.20) for all 𝑥∈𝑋. This completes the proof.

If we put 𝜑(𝑥,𝑦,𝑧)∶=𝜃(‖𝑥‖^𝑟1‖𝑦‖^𝑟2‖𝑧‖^𝑟3) and 𝜑(𝑥,𝑦,𝑧)∶=𝜃₁‖𝑥‖^𝑟1+𝜃₂‖𝑦‖^𝑟2+𝜃₃‖𝑧‖^𝑟3 in the following corollaries, respectively, then we lead to the desired results.

Corollary 2.3. Let 𝑟_𝑖>0 for 𝑖=1,2,3 with ∑³_𝑖=1𝑟_𝑖<1 and 𝜃≥0. If a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the following functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜃‖𝑥‖^𝑟1𝑦^𝑟2‖𝑧‖^𝑟3 (2.21) for all 𝑥,𝑦,𝑧∈𝑋, then there exists a unique additive mapping ℎ∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤‖𝑥‖−ℎ)𝑀𝜃^𝑟_𝑝√2^𝑝−2^𝑟𝑝𝑛(𝑛−3)𝑛²+3^𝑟1𝑝𝑛(𝑛+3)𝑛²+3^𝑟2𝑝2𝑛²𝑛²+3^𝑟3𝑝+2𝑛(𝑛+1)𝑛²+1^𝑟1𝑝2𝑛(𝑛−3)𝑛²+3^𝑟2𝑝4𝑛𝑛²+3^{𝑟3𝑝1/𝑝} (2.22) for all 𝑥∈𝑋, where 𝑟=∑³_𝑖=1𝑟_𝑖.

Corollary 2.4. Let 0<𝑟_𝑖<1 and 𝜃_𝑖≥0 for 𝑖=1,2,3. If a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the following functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜃₁‖𝑥‖^𝑟1+𝜃₂𝑦^𝑟2+𝜃₃‖𝑧‖^𝑟3 (2.23) for all 𝑥,𝑦,𝑧∈𝑋, then there exists a unique additive mapping ℎ∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥𝑛()−ℎ)≤𝑀𝑛−3𝑛²+3^𝑟1𝑝+2𝑛(𝑛+1)𝑛²+3^𝑟1𝑝𝜃^𝑝₁‖𝑥‖^𝑟1𝑝2^𝑝−2^𝑟1𝑝+𝑛(𝑛+3)𝑛²+3^𝑟2𝑝+2𝑛(𝑛−1)𝑛²+3^𝑟2𝑝𝜃^𝑝₂‖𝑥‖^𝑟2𝑝2^𝑝−2^𝑟2𝑝+2𝑛²𝑛²+3^𝑟3𝑝+4𝑛𝑛²+3^𝑟3𝑝𝜃^𝑝₃‖𝑥‖^𝑟3𝑝2^𝑝−2^{𝑟3𝑝1/𝑝} (2.24) for all 𝑥∈𝑋.

Theorem 2.5. Suppose that a mapping 𝑓∶𝑋→𝑌 satisfies the functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜑𝑥,𝑦,𝑧 (2.25) for all 𝑥,𝑦,𝑧∈𝑋, and the perturbing function 𝜑∶𝑋³→ℝ⁺ satisfies Φ𝑥,𝑦,𝑧∶=^∞_𝑖=02^𝑖𝑝𝜑𝑥2^𝑖+1,𝑦2^𝑖+1𝑦,𝑧2^𝑖+1𝑝<∞ (2.26) for all 𝑥,𝑦,𝑧∈𝑋. Then, there exists a unique additive mapping ℎ∶𝑋→𝑌 defined by ℎ(𝑥)=lim_𝑘→∞2^𝑘𝑓(𝑥/2^𝑘) such that 𝑓(𝑥)(𝑥Φ𝑛()𝑥−ℎ)≤𝑀𝑛−3𝑛²,𝑛()𝑥+3𝑛+3𝑛²,+3−2𝑛²𝑥𝑛²()𝑥+3+Φ−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,+34𝑛𝑥𝑛²+3^1/𝑝 (2.27) for all 𝑥∈𝑋.

Proof. We note that 𝑓(0)=0 since 𝜑(0,0,0)=0 by the convergence of (2.26). Now, if we replace 𝑥 by 𝑥/2 in (2.14), 𝑓(𝑥)𝑥−2𝑓2𝜑𝑛()𝑥≤𝑀𝑛−32𝑛²,𝑛()𝑥+3𝑛+32𝑛²,+3−𝑛²𝑥𝑛²()𝑥+3+𝜑−𝑛𝑛+1𝑛²,𝑛()𝑥+3𝑛−1𝑛²,+32𝑛𝑥𝑛²+3 (2.28) for all 𝑥∈𝑋. Then, it follows from the last inequality that 𝑓(𝑥)−2^𝑚𝑓𝑥2^𝑚𝑝≤𝑀^𝑝𝑚−1_𝑖=02^𝑖𝑝𝜑𝑛(𝑛−3)𝑥2^𝑖+1𝑛²,𝑛()𝑥+3𝑛+32^𝑖+1𝑛²,+3−2𝑛²𝑥2^𝑖+1𝑛²+3^𝑝()𝑥+𝜑−2𝑛𝑛+12^𝑖+𝑖𝑛²,()𝑥+32𝑛𝑛−12^𝑖+1𝑛²,+34𝑛𝑥2^𝑖+1𝑛²+3^𝑝 (2.29) for all nonnegative integer 𝑚 and all 𝑥∈𝑋. The remaining proof is similar to the corresponding part of Theorem 2.2. This completes the proof.

If we put 𝜑(𝑥,𝑦,𝑧)∶=𝜃(‖𝑥‖^𝑟1‖𝑦‖^𝑟2‖𝑧‖^𝑟3) and 𝜑(𝑥,𝑦,𝑧)∶=𝜃₁‖𝑥‖^𝑟1+𝜃₂‖𝑦‖^𝑟2+𝜃₃‖𝑧‖^𝑟3 in the following corollaries, respectively, then we lead to the desired results.

Corollary 2.6. Let 𝑟_𝑖>0 for 𝑖=1,2,3 with ∑³_𝑖=1𝑟_𝑖>1 and 𝜃≥0. If a mapping 𝑓∶𝑋→𝑌 satisfies the following functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜃‖𝑥‖^𝑟1𝑦^𝑟2‖𝑧‖^𝑟3 (2.30) for all 𝑥,𝑦,𝑧∈𝑋, then there exists a unique additive mapping ℎ∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤‖𝑥‖−ℎ)𝑀𝜃^𝑟_𝑝√2^𝑟𝑝−2^𝑝𝑛(𝑛−3)𝑛²+3^𝑟1𝑝𝑛(𝑛+3)𝑛²+3^𝑟2𝑝2𝑛²𝑛²+3^𝑟3p+2𝑛(𝑛+1)𝑛²+1^𝑟1𝑝2𝑛(𝑛−3)𝑛²+3^𝑟2𝑝4𝑛𝑛²+3^{𝑟3𝑝1/𝑝} (2.31) for all 𝑥∈𝑋, where 𝑟=∑³_𝑖=1𝑟_𝑖.

Corollary 2.7. Let 𝑟_𝑖>1 and 𝜃_𝑖≥0 for 𝑖=1,2,3. If a mapping 𝑓∶𝑋→𝑌 satisfies the following functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜃₁‖𝑥‖^𝑟1+𝜃₂𝑦^𝑟2+𝜃₃‖𝑧‖^𝑟3 (2.32) for all 𝑥,𝑦,𝑧∈𝑋, then there exists a unique additive mapping ℎ∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥𝑛()−ℎ)≤𝑀𝑛−3𝑛²+3^𝑟1𝑝+2𝑛(𝑛+1)𝑛²+3^𝑟1𝑝𝜃^𝑝₁‖𝑥‖^𝑟1𝑝2^{𝑟1𝑝−2𝑝}+𝑛(𝑛+3)𝑛²+3^𝑟2𝑝+2𝑛(𝑛−1)𝑛²+3^𝑟2𝑝𝜃^𝑝₂‖𝑥‖^𝑟2𝑝2^𝑟2𝑝−2^𝑝+2𝑛²𝑛²+3^𝑟3𝑝+4𝑛𝑛²+3^𝑟3𝑝𝜃^𝑝₃‖𝑥‖^𝑟3𝑝2^𝑟3𝑝−2^𝑝1/𝑝 (2.33) for all 𝑥∈𝑋.

The following is a simple example that the additive functional inequality 𝐷𝑓(𝑥,𝑦,𝑧)≤𝜃(‖𝑥‖+‖𝑦‖+‖𝑧‖) is not stable for the singular case 𝑟₁,𝑟₂,𝑟₃=1 in Corollaries 2.4 and 2.7.

Example 2.8. Fix 𝜃≥0 and put 𝜇∶=𝜃/8. Let 𝜙∶ℝ→ℝ be defined by 𝜙(𝑥)=[),(),(],𝜇for𝑥∈1,∞𝜇𝑥for𝑥∈−1,1−𝜇for𝑥∈−∞,−1 (2.34) and define 𝑓∶ℝ→ℝ by 𝑓(𝑥)=^∞_𝑖=0𝜙2^𝑖𝑥2^𝑖,∀𝑥∈ℝ, (2.35) which can be found in [9]. It follows from the same argument as in the example of [9] that 𝑓 satisfies the functional inequality 𝑓𝑥−𝑦𝑛+𝑧+𝑓𝑦−𝑧𝑛+𝑥+𝑓𝑧−𝑥𝑛−𝑓|𝑥|+𝑦+|𝑧|+𝑦𝑥+𝑦+𝑧≤8𝜇 (2.36) for all 𝑥,𝑦,𝑧∈ℝ. In fact, if 𝑥=𝑦=𝑧=0, then (2.36) is trivially fulfilled. Next, if 0<|𝑥|+|𝑦|+|𝑧|<1, then there exists an 𝑁∈ℕ such that 12^𝑁≤|𝑥|+𝑦+|𝑧|<12^𝑁−1, (2.37) which implies that 2^𝑖𝑥−𝑦𝑛+𝑧,2^𝑖𝑦−𝑧𝑛+𝑥,2^𝑖𝑧−𝑥𝑛+𝑦,2^𝑖∈(),{}.𝑥+𝑦+𝑧−1,1∀𝑖∈0,…,𝑁−1 (2.38) Thus, we see that 𝜙2^𝑖𝑥−𝑦𝑛+𝑧+𝜙2^𝑖𝑦−𝑧𝑛+𝑥+𝜙2^𝑖𝑧−𝑥𝑛+𝑦−𝜙2^𝑖𝑥+𝑦+𝑧=0 (2.39) for all 𝑖∈{0,…,𝑁−1}. As a result, we infer that 𝑓𝑥−𝑦/𝑛+𝑧+𝑓𝑦−𝑧/𝑛+𝑥+𝑓((𝑧−𝑥)/𝑛)+𝑦−𝑓𝑥+𝑦+𝑧|𝑥|+𝑦+|𝑧|≤^∞_𝑖=𝑁𝜙2^𝑖𝑥−𝑦/𝑛+𝑧+𝜙2^𝑖𝑦−𝑧/𝑛+𝑥+𝜙2^𝑖((𝑧−𝑥)/𝑛)+𝑦−𝜙2^𝑖𝑥+𝑦+𝑧2^𝑖|𝑥|+𝑦+|𝑧|≤8𝜇 (2.40) for all 𝑥,𝑦,𝑧∈ℝ. Finally, if |𝑥|+|𝑦|+|𝑧|≥1, then one has by use of boundedness of 𝑓𝑓𝑥−𝑦/𝑛+𝑧+𝑓𝑦−𝑧/𝑛+𝑥+𝑓((𝑧−𝑥)/𝑛)+𝑦−𝑓𝑥+𝑦+𝑧|𝑥|+𝑦+|𝑧|≤8𝜇 (2.41) for all 𝑥,𝑦,𝑧∈ℝ. Therefore, 𝑓 satisfies the functional inequality (2.36) and so 𝐷𝑓𝑥,𝑦,𝑧≤8𝜇|𝑥|+𝑦+|𝑧| (2.42) for all 𝑥,𝑦,𝑧∈ℝ. However, there do not exist an additive function 𝑇∶ℝ→ℝ and a constant 𝑐>0 such that 𝑓(𝑥)−𝑇(𝑥)≤𝑐|𝑥|∀𝑥∈ℝ. (2.43)

Remark 2.9. The stability problem on the singular case 𝑟=1 in Corollaries 2.3 and 2.6 is not easy and it remains with us unsolved for providing a counterexample on the singular case 𝑟=1.

3. Alternative Generalized Hyers-Ulam Stability of (1.8)

From now on, we investigate the generalized Hyers-Ulam stability of the functional inequality (1.8) using the contractive property of perturbing term of the inequality (1.8).

Theorem 3.1. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜑𝑥,𝑦,𝑧 (3.1) for all 𝑥,𝑦,𝑧∈𝑋 and there exists a constant 𝐿 with 0<𝐿<1 for which the perturbing function 𝜑∶𝑋³→ℝ⁺ satisfies 𝜑2𝑥,2𝑦,2𝑧≤2𝐿𝜑𝑥,𝑦,𝑧 (3.2) for all 𝑥,𝑦,𝑧∈𝑋. Then, there exists a unique additive mapping ℎ∶𝑋→𝑌 given by ℎ(𝑥)=lim_𝑘→∞(1/2^𝑘)𝑓(2^𝑘𝑥) such that 𝑓(𝑥)(𝑥≤𝑀−ℎ)2√^𝑝1−𝐿^𝑝𝜑𝑛(𝑛−3)𝑥𝑛²,𝑛()𝑥+3𝑛+3𝑛²,+3−2𝑛²𝑥𝑛²+3^𝑝()𝑥+𝜑−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,+34𝑛𝑥𝑛²+3^𝑝1/𝑝 (3.3) for all 𝑥∈𝑋.

Proof. It follows from (2.15) and (3.2) that 𝑓2^𝑙𝑥2^𝑙−𝑓2^𝑚𝑥2^𝑚𝑝≤𝑀^𝑝2^{𝑝𝑚−1𝑖=𝑙}12^𝑖𝑝𝜑𝑛(𝑛−3)2^𝑖𝑥𝑛²,𝑛()2+3𝑛+3^𝑖𝑥𝑛²,+3−2𝑛²2^𝑖𝑥𝑛²+3^𝑝()2+𝜑−2𝑛𝑛+1^𝑖𝑥𝑛²,()2+32𝑛𝑛−1^𝑖𝑥𝑛²,()2+34𝑛^𝑖𝑥𝑛²+3^𝑝≤𝑀^𝑝2^{𝑝𝑚−1𝑖=𝑙}𝐿^𝑖𝑝𝜑𝑛(𝑛−3)𝑥𝑛²,𝑛()𝑥+3𝑛+3𝑛²,+3−2𝑛²𝑥𝑛²+3^𝑝()𝑥+𝜑−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,()𝑥+34𝑛𝑛²+3^𝑝 (3.4) for all nonnegative integers 𝑚 and 𝑙 with 𝑚>𝑙≥0 and 𝑥∈𝑋. Since the sequence {𝑓(2^𝑚𝑥)/2^𝑚} is Cauchy for all 𝑥∈𝑋, we can define a mapping ℎ∶𝑋→𝑌 by ℎ(𝑥)=lim_𝑚→∞𝑓2^𝑚𝑥2^𝑚,𝑥∈𝑋. (3.5) Moreover, letting 𝑙=0 and 𝑚→∞ in the last inequality yields the approximation (3.3).
The remaining proof is similar to the corresponding part of Theorem 2.2. This completes the proof.

Corollary 3.2. Let 𝜉∶[0,∞)→[0,∞) be a nontrivial function satisfying 𝜉(2𝑡)≤𝜉(2)𝜉(𝑡),(𝑡≥0),0<𝜉(2)<2. (3.6) If 𝑓∶𝑋→𝑌 with 𝑓(0)=0 is a mapping satisfying the following functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜃𝜉(‖𝑥‖)+𝜉𝑦+𝜉(‖𝑧‖) (3.7) for all 𝑥,𝑦,𝑧∈𝑋 and for some 𝜃≥0, then there exists a unique additive mapping ℎ∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤−ℎ)𝑀𝜃_𝑝2^𝑝−𝜉(2)^𝑝𝜉𝑛(𝑛−3)𝑛²𝑥+3‖‖^𝑝𝑛()+𝜉𝑛+3𝑛²𝑥+3‖‖^𝑝+𝜉2𝑛²𝑛²𝑥+3‖‖^𝑝()+𝜉2𝑛𝑛+1𝑛²𝑥‖+3‖^𝑝()+𝜉2𝑛𝑛−1𝑛²𝑥‖+3‖^𝑝+𝜉4𝑛𝑛²𝑥‖+3‖^𝑝1/𝑝 (3.8) for all 𝑥∈𝑋.

Proof. Letting 𝜑(𝑥,𝑦,𝑧)=𝜃{𝜉(‖𝑥‖)+𝜉(‖𝑦‖)+𝜉(‖𝑧‖)} and applying Theorem 3.1 with 𝐿∶=𝜉(2)/2, we obtain the desired result.

Theorem 3.3. Suppose that a mapping 𝑓∶𝑋→𝑌 satisfies the functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜑𝑥,𝑦,𝑧 (3.9) for all 𝑥,𝑦,𝑧∈𝑋 and there exists a constant 𝐿 with 0<𝐿<1 for which the perturbing function 𝜑∶𝑋³→ℝ⁺ satisfies 𝜑𝑥2,𝑦2,𝑧2≤𝐿2𝜑𝑥,𝑦,𝑧 (3.10) for all 𝑥,𝑦,𝑧∈𝑋. Then, there exists a unique additive mapping ℎ∶𝑋→𝑌 defined by ℎ(𝑥)=lim_𝑘→∞2^𝑘𝑓(𝑥/2^𝑘) such that 𝑓(𝑥)−ℎ(𝑥)≤𝑀𝐿2√^𝑝1−𝐿^𝑝𝜑𝑛(𝑛−3)𝑥𝑛²,𝑛()𝑥+3𝑛+3𝑛²,+3−2𝑛²𝑥𝑛²+3^𝑝()𝑥+𝜑−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,+34𝑛𝑥𝑛²+3^𝑝1/𝑝 (3.11) for all 𝑥∈𝑋.

Proof. We observe that 𝑓(0)=0 because 𝜑(0,0,0)=0, which follows from the condition 𝜑(0,0,0)≤𝐿/2𝜑(0,0,0). It follows from (2.29) and (3.10) that 𝑓(𝑥)−2^𝑚𝑓𝑥2^𝑚𝑝≤𝑀^p𝑚−1_𝑖=02^𝑖𝑝𝜑𝑛(𝑛−3)𝑥2^𝑖+1𝑛²,𝑛()𝑥+3𝑛+32^𝑖+1𝑛²,+3−2𝑛²𝑥2^𝑖+1𝑛²+3^𝑝()𝑥+𝜑−2𝑛𝑛+12^𝑖+𝑖𝑛²,()𝑥+32𝑛𝑛−12^𝑖+1𝑛²,+34𝑛𝑥2^𝑖+1𝑛²+3^𝑝≤𝑀^𝑝2^{𝑝𝑚−1𝑖=0}𝐿^(𝑖+1)𝑝𝜑𝑛(𝑛−3)𝑥𝑛²,𝑛()𝑥+3𝑛+2𝑛²,+3−2𝑛²𝑥𝑛²+3^𝑝()𝑥+𝜑−2𝑛𝑛+1𝑛²,()𝑥+32𝑛𝑛−1𝑛²,+34𝑛𝑥𝑛²+3^𝑝 (3.12) for all nonnegative integer 𝑚 and all 𝑥∈𝑋.
The remaining proof is similar to the corresponding part of Theorem 2.2. This completes the proof.

Corollary 3.4. Let 𝜉∶[0,∞)→[0,∞) be a nontrivial function satisfying 𝜉𝑡21≤𝜉2𝜉(𝑡),()1𝑡≥0,0<𝜉2<12. (3.13) If 𝑓∶𝑋→𝑌 is a mapping satisfying the following functional inequality 𝐷𝑓𝑥,𝑦,𝑧≤𝜃𝜉(‖𝑥‖)+𝜉𝑦+𝜉(‖𝑧‖) (3.14) for all 𝑥,𝑦,𝑧∈𝑋 and for some 𝜃≥0, then there exists a unique additive mapping ℎ∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤()−ℎ)𝑀𝜃𝜉1/2_𝑝1−2^𝑝𝜉(1/2)^𝑝𝜉𝑛(𝑛−3)𝑛²𝑥+3‖‖^𝑝𝑛()+𝜉𝑛+3𝑛²𝑥+3‖‖^𝑝+𝜉2𝑛²𝑛²𝑥+3‖‖^𝑝()+𝜉2𝑛𝑛+1𝑛²𝑥+3‖‖^𝑝()+𝜉2𝑛𝑛−1𝑛²𝑥+3‖‖^𝑝+𝜉4𝑛𝑛²𝑥+3‖‖^𝑝1/𝑝 (3.15) for all 𝑥∈𝑋.

Proof. Letting 𝜑(𝑥,𝑦,𝑧)=𝜃{𝜉(‖𝑥‖)+𝜉(‖𝑦‖)+𝜉(‖𝑧‖)} and applying Theorem 3.3 with 𝐿∶=2𝜉(1/2), we lead to the approximation.

Acknowledgment

This study was supported by the Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (No. 2012R1A1A2008139).