Generalized Hyers-Ulam Stability of Quadratic Functional Inequality

Abstract

We establish the general solution of the functional inequality ‖𝑓(𝑥−𝑦)+𝑓(𝑦−𝑧)+𝑓(𝑥−𝑧)−3𝑓(𝑥)−3𝑓(𝑦)−3𝑓(𝑧)‖≤‖𝑓(𝑥+𝑦+𝑧)‖ and then investigate the generalized Hyers-Ulam stability of this inequality in Banach spaces and in non-Archimedean Banach spaces.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms.

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

Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces by using, so called, direct method. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias’ theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Rassias’ approach.

Let 𝐸₁ and 𝐸₂ be vector spaces. A mapping 𝑓∶𝐸₁→𝐸₂ is called quadratic if and only if it is a solution of the quadratic functional equation: 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (1) for all 𝑥,𝑦∈𝑋. It is well known that a function 𝑓 between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function 𝐵 such that 𝑓(𝑥)=𝐵(𝑥,𝑥) for all 𝑥, where the mapping 𝐵 is given by 𝐵(𝑥,𝑦)=(1/4)[𝑓(𝑥+𝑦)−𝑓(𝑥−𝑦)] ([6, 7]). The Hyers-Ulam stability of the quadratic functional equation (1) was first proved by Skof [8] for functions 𝑓∶𝐸₁→𝐸₂, where 𝐸₁ is a normed space and 𝐸₂ is a Banach space. Cholewa [9] noticed that Skof’s theorem is also valid if 𝐸₁ is replaced by an abelian group. In 1992, Czerwik [10] proved the generalized Hyers-Ulam stability of quadratic functional equation (1) in the spirit of Rassias approach by using direct method [11].

The generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by Cădariu and Radu [12, 13] by using fixed point method (see also [14, 15]).

Gilányi [16] and Rätz [17] proved that for a function 𝑓∶𝐺→𝐸 mapping from an abelian group 𝐺 divisible by 2 into an inner product space 𝐸, the functional inequality 2𝑓(𝑥)+2𝑓𝑦−𝑓𝑥𝑦⁻¹≤𝑓𝑥𝑦𝑥,𝑦∈𝐺 (2) implies the Jordan-von Neumann functional equation 2𝑓(𝑥)+2𝑓𝑦=𝑓𝑥𝑦+𝑓𝑥𝑦⁻¹𝑥,𝑦∈𝐺. (3)

Fechner [18] and Gilányi [19] have proved the generalized Hyers-Ulam stability of the functional inequality (2). Park et al. [20] have investigated the generalized Hyers-Ulam stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, and Kim et al. [21] have proved the generalized Hyers-Ulam stability of Jensen functional inequality in 𝑝-Banach spaces. The stability problems of several functional equations and inequalities have been extensively investigated by a number of authors and there are many interesting results concerning the stability of various functional equations and inequalities [6, 22].

In 2001, Bae and Kim [23] investigated the Hyers-Ulam stability of the quadratic functional equation 𝑓()(𝑥)𝑦(𝑧)𝑥+𝑦+𝑧+𝑓𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧=3𝑓+3𝑓+3𝑓 (4) which is equivalent to the original quadratic functional equation (1). Now, let us turn our attention to investigate the generalized stability problem of the following quadratic functional inequality: 𝑓()(𝑥)𝑦(𝑧≤𝑓.𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧−3𝑓−3𝑓−3𝑓)𝑥+𝑦+𝑧 (5)

In this paper, we make an attempt to establish the generalized Hyers-Ulam stability of a new quadratic functional inequality (5) by using fixed point method and direct method. In Section 2, we establish the general solution of the functional inequality (5). And then we prove the generalized Hyers-Ulam stability of the functional inequality (5) in Banach spaces by using fixed point method. In Section 3, we verify the generalized Hyers-Ulam stability of the functional inequality (5) in Banach spaces by using direct method. In Section 4, we investigate the generalized Hyers-Ulam stability of the functional inequality (5) in non-Archimedean Banach spaces by using fixed point method. In Section 5, we prove the generalized Hyers-Ulam stability of the functional inequality (5) in non-Archimedean Banach spaces by using direct method.

2. Stability of (5) by Fixed Point Method

In [24], Rassias introduced the following equality: ^𝑛_𝑖,𝑗=1𝑥_𝑖−𝑥_𝑗²=2𝑛^𝑛_𝑖=1𝑥_𝑖²,^𝑛_𝑖=1𝑥_𝑖=0 (6) for a fixed integer 𝑛≥3. Let 𝑉, 𝑊 be real vector space. It has been proved that if a mapping 𝑓∶𝑉→𝑊 satisfies ^𝑛_𝑖,𝑗=1𝑓𝑥_𝑖−𝑥_𝑗=2𝑛^𝑛_𝑖=1𝑓𝑥_𝑖, (7) for all 𝑥₁,…,𝑥_𝑛∈𝑉 with ∑^𝑛_𝑖=1𝑥_𝑖=0, then the mapping 𝑓 is realized as the sum of an additive mapping and a quadratic mapping [25]. Park [25] and Jang et al. [26] have proved the generalized Hyers-Ulam stability of the functional equation (7). In particular, if 𝑛=3 and even function 𝑓 satisfies (7), then it is easy to see that 𝑓 satisfies the equation 𝑓(𝑥)𝑦𝑥−𝑦+𝑓2𝑥+𝑦+𝑓𝑥+2𝑦=3𝑓+3𝑓+3𝑓𝑥+𝑦 (8) for all 𝑥,𝑦∈𝑋. Thus, we first consider the general solution of functional equation (8) to verify the general solution of functional inequality (5).

Lemma 1. Let both 𝑋 and 𝑌 be vector spaces. A function 𝑓∶𝑋→𝑌 satisfies (8) if and only if 𝑓 is quadratic.

Proof. If we put 𝑥=𝑦=0 in (8), then we have 𝑓(0)=0. Letting 𝑦=0 in (8), we have 𝑓(2𝑥)=4𝑓(𝑥) for all 𝑥∈𝑋. Similarly, we can easily show that 𝑓(3𝑥)=9𝑓(𝑥) and 𝑓(5𝑥)=25𝑓(𝑥) for all 𝑥∈𝑋. Replacing 𝑦 by −𝑥 in (8), we get 𝑓(−𝑥)=𝑓(𝑥) for all 𝑥∈𝑋. Substituting 𝑥+𝑦 for 𝑦 in (8), we obtain 𝑓𝑦(𝑥)+𝑓3𝑥+𝑦+𝑓3𝑥+2𝑦=3𝑓+3𝑓𝑥+𝑦+3𝑓2𝑥+𝑦 (9) for all 𝑥,𝑦∈𝑋. Switching 𝑥 with 𝑦 in (9) yields 𝑓(𝑥)𝑦+𝑓𝑥+3𝑦+𝑓2𝑥+3𝑦=3𝑓+3𝑓𝑥+𝑦+3𝑓𝑥+2𝑦 (10) for all 𝑥,𝑦∈𝑋. Adding (9) to (10) and using (8), we arrive at 𝑓(𝑥)𝑦3𝑥+2𝑦+𝑓2𝑥+3𝑦+𝑓3𝑥+𝑦+𝑓𝑥+3𝑦=11𝑓+11𝑓+15𝑓𝑥+𝑦−3𝑓𝑥−𝑦 (11) for all 𝑥,𝑦∈𝑋. Letting 𝑦∶=2𝑦 in (8), we obtain 𝑓(𝑥)𝑦𝑥−2𝑦+4𝑓𝑥+𝑦+𝑓𝑥+4𝑦=3𝑓+12𝑓+3𝑓𝑥+2𝑦 (12) for all 𝑥,𝑦∈𝑋. Switching 𝑥 with 𝑦 in (12) and using the evenness of 𝑓, one gets 𝑓(𝑥)𝑦2𝑥−𝑦+4𝑓𝑥+𝑦+𝑓4𝑥+𝑦=12𝑓+3𝑓+3𝑓2𝑥+𝑦 (13) for all 𝑥,𝑦∈𝑋. Adding (12) to (13) and using (8), we have 𝑓(𝑥)𝑦𝑥+4𝑦+𝑓4𝑥+𝑦=21𝑓+21𝑓+2𝑓𝑥+𝑦−6𝑓𝑥−𝑦 (14) for all 𝑥,𝑦∈𝑋. Putting 𝑥∶=𝑥−𝑦 and 𝑦∶=2𝑥+3𝑦 in (8), we get 𝑓𝑥+4𝑦+𝑓4𝑥+𝑦=3𝑓𝑥−𝑦+3𝑓2𝑥+3𝑦+3𝑓3𝑥+2𝑦−25𝑓𝑥+𝑦 (15) for all 𝑥,𝑦∈𝑋. From (14) and (15), it follows that 𝑓(𝑥)𝑦3𝑥+2𝑦+𝑓2𝑥+3𝑦=9𝑓𝑥+𝑦+7𝑓+7𝑓−3𝑓𝑥−𝑦 (16) for all 𝑥,𝑦∈𝑋. By (11) and (16), we obtain 𝑓3𝑥+𝑦+𝑓𝑥+3𝑦=6𝑓𝑥+𝑦+4𝑓(𝑥)+4𝑓𝑦 (17) for all 𝑥,𝑦∈𝑋. Letting 𝑥∶=(𝑥+𝑦)/2 and 𝑦∶=(𝑥−𝑦)/2 in (17), we have 𝑓2𝑥+𝑦+𝑓2𝑥−𝑦=6𝑓(𝑥)+𝑓𝑥+𝑦+𝑓𝑥−𝑦 (18) for all 𝑥,𝑦∈𝑋. Interchanging 𝑥 for 𝑦 in (18) and using the evenness of 𝑓, one gets 𝑓𝑥+2𝑦+𝑓𝑥−2𝑦=6𝑓𝑦+𝑓𝑥+𝑦+𝑓𝑥−𝑦 (19) for all 𝑥,𝑦∈𝑋. Substituting 2𝑦 for 𝑦 in (18), we get by virtue of (19) (𝑥)(𝑥)𝑦,4𝑓𝑥+𝑦+4𝑓𝑥−𝑦=6𝑓+𝑓𝑥+2𝑦+𝑓𝑥−2𝑦=6𝑓+6𝑓+𝑓𝑥+𝑦+𝑓𝑥−𝑦 (20) which yields 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (21) for all 𝑥,𝑦∈𝑋. Thus, the mapping 𝑓 is quadratic.
The proof of the converse is trivial.

Now, we present the general solution of the functional inequality (5) by using Lemma 1.

Lemma 2. Let both 𝑋 and 𝑌 be vector spaces. A mapping 𝑓∶𝑋→𝑌 satisfies the functional inequality (5) for all 𝑥,𝑦,𝑧∈𝑋 if and only if 𝑓 is quadratic.

Proof. Let 𝑓 satisfy the functional inequality (5). If we replace 𝑥,𝑦,and𝑧 in (5) by 0, then we have 𝑓(0)=0. Replacing 𝑧 by −𝑥−𝑦 in (5), we obtain 𝑓(𝑥)𝑦𝑥−𝑦+𝑓𝑥+2𝑦+𝑓2𝑥+𝑦−3𝑓−3𝑓−3𝑓−𝑥−𝑦=0 (22) for all 𝑥,𝑦∈𝑋. Letting 𝑦∶=−𝑥 and 𝑧∶=0 in (5), we get 𝑓(2𝑥)−2𝑓(𝑥)−2𝑓(−𝑥)=0 (23) for all 𝑥∈𝑋. Putting 𝑦∶=0 in (22) yields 𝑓(2𝑥)−𝑓(𝑥)−3𝑓(−𝑥)=0 (24) for all 𝑥∈𝑋. From (23) and (24), we have 𝑓(−𝑥)=𝑓(𝑥) for all 𝑥∈𝑋. Thus, it follows from (22) that 𝑓(𝑥)𝑦𝑥−𝑦+𝑓𝑥+2𝑦+𝑓2𝑥+𝑦=3𝑓+3𝑓+3𝑓𝑥+𝑦 (25) for all 𝑥,𝑦∈𝑋. So 𝑓 is quadratic by Lemma 1.
The proof of the converse is trivial.

Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies the following:(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.

Before taking up the main subject, we recall the fixed point theorem from [14].

Theorem 3 (see [14]). Let (𝑋,𝑑) be a complete generalized metric space and let 𝐽∶𝑋→𝑋 be a strictly contractive mapping with Lipschitz constant 𝐿<1. Then for each given element 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=∞ (26) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞, for all 𝑛≥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−𝐿))𝑑(𝑦,𝐽𝑦) for all 𝑦∈𝑌.

In this part, let 𝑋 be a normed space and 𝑌 a Banach space.

Theorem 4. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional inequality 𝑓()(𝑥)𝑦(𝑧≤𝑓𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧−3𝑓−3𝑓−3𝑓)𝑥+𝑦+𝑧+𝜑𝑥,𝑦,𝑧 (27) for all 𝑥,𝑦,𝑧∈𝑋 and that there exists a constant 𝐿 with 0<𝐿<1 for which the perturbing function 𝜑∶𝑋³→[0,∞) satisfies 𝜑,𝜑𝑥2𝑥,2𝑦,2𝑧≤4𝐿𝜑𝑥,𝑦,𝑧2,𝑦2,𝑧2≤𝐿4𝜑𝑥,𝑦,𝑧,𝑟𝑒𝑠𝑝. (28) for all 𝑥,𝑦,𝑧∈𝑋. Then, there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 given by 𝑄(𝑥)=lim_𝑚→∞(1/4^𝑚)𝑓(2^𝑚𝑥) (𝑄(𝑥)=lim_𝑚→∞4^𝑚𝑓(𝑥/2^𝑚), resp.) such that 𝑓(𝑥)(𝑥≤1−𝑄)()(,𝑓(𝑥)(𝑥≤𝐿4−4𝐿3𝜑𝑥,−𝑥,0+2𝜑𝑥,0,−𝑥)−𝑄)()(4−4𝐿3𝜑𝑥,−𝑥,0+2𝜑𝑥,0,−𝑥),𝑟𝑒𝑠𝑝. (29) for all 𝑥∈𝑋.

Proof. Replacing 𝑧 by −𝑥−𝑦 in (27), we obtain 𝑓(𝑥)𝑦𝑥−𝑦+𝑓𝑥+2𝑦+𝑓2𝑥+𝑦−3𝑓−3𝑓−3𝑓−𝑥−𝑦≤𝜑𝑥,𝑦,−𝑥−𝑦 (30) for all 𝑥,𝑦∈𝑋. Letting 𝑦=−𝑥 and 𝑧=0 in (27), we get 𝑓(2𝑥)−2𝑓(𝑥)−2𝑓(−𝑥)≤𝜑(𝑥,−𝑥,0) (31) for all 𝑥∈𝑋. Putting 𝑦=0 in (30) yields 𝑓(2𝑥)−𝑓(𝑥)−3𝑓(−𝑥)≤𝜑(𝑥,0,−𝑥) (32) for all 𝑥∈𝑋. From (31) and (32), it follows that 𝑓(2𝑥)−4𝑓(𝑥)≤3𝜑(𝑥,−𝑥,0)+2𝜑(𝑥,0,−𝑥)∶≡Φ(𝑥) (33) and so 𝑓(𝑥)−14𝑓(≤12𝑥)4Φ(𝑥) (34) for all 𝑥∈𝑋.
Consider the set of mappings 𝑆∶={ℎ∶𝑋⟶𝑌∣ℎ(0)=0} (35) and introduce the generalized metric on 𝑆: 𝑑[)∶𝑔(𝑥)(𝑥(𝑥),𝑔,ℎ=inf𝜇∈0,∞−ℎ)≤𝜇Φ,∀𝑥∈𝑋 (36) where, as usual, inf∅=+∞. It is easy to show that (𝑆,𝑑) is a complete generalized metric space (see the proof of Theorem 3.1 of [27]).
Now we consider a linear mapping 𝐽∶𝑆→𝑆 such that (𝑥)1𝐽𝑔∶=4𝑔(2𝑥) (37) for all 𝑥∈𝑋. Then it is well-known that 𝐽 is a strictly contractive mapping with Lipschitz constant 𝐿, and it follows from (34) that 𝑑(𝑓,𝐽𝑓)≤1/4. By Theorem 3, there exists a mapping 𝑄∶𝑋→𝑌 satisfying the following.(1)𝑄 is a fixed point of 𝐽, that is, 𝑄(2𝑥)=4𝑄(𝑥) for all 𝑥∈𝑋. The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}. This implies that 𝑄 is a unique fixed point of 𝐽 such that there exists a 𝜇∈(0,∞) satisfying 𝑓(𝑥)−𝑄(𝑥)≤𝜇Φ(𝑥), (38) for all 𝑥∈𝑋.(2)𝑑(𝐽^𝑚𝑓,𝑄)→0 as 𝑚→∞. This implies the equality 𝑄(𝑥)=lim_𝑚→∞(1/4^𝑚)𝑓(2^𝑚𝑥) for all 𝑥∈𝑋.(3)𝑑(𝑓,𝑄)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓)≤1/(4−4𝐿), which implies that the inequality (29) holds.
Now, we show that the mapping 𝑄 is quadratic. It follows from (27) and (28) that 𝑄()(𝑥)𝑦(𝑧−𝑄𝑥−𝑦+𝑄𝑦−𝑧+𝑄𝑥−𝑧−3𝑄−3𝑄−3𝑄)𝑥+𝑦+𝑧≤lim_𝑚→∞14^𝑚𝜑2^𝑚𝑥,2^𝑚𝑦,2^𝑚𝑧≤lim_𝑚→∞𝐿^𝑚𝜑𝑥,𝑦,𝑧=0 (39) for all 𝑥,𝑦,𝑧∈𝑋. Thus, it follows from Lemma 2 that the mapping 𝑄∶𝑋→𝑌 is quadratic, as desired.

We obtain the following corollary concerning the stability for approximate mappings controlled by a sum of powers of norms.

Corollary 5. Let 𝜃≥0 be a real number and 𝑝 a positive real number with 𝑝≠2. If a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑓()(𝑥)𝑦(𝑧≤𝑓‖𝑥‖𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧−3𝑓−3𝑓−3𝑓)𝑥+𝑦+𝑧+𝜃^𝑝+𝑦^𝑝+‖𝑧‖^𝑝 (40) for all 𝑥,𝑦,𝑧∈𝑋, then we can find a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying the inequality 𝑓(𝑥)−𝑄(𝑥)≤10𝜃4−2^𝑝‖𝑥‖^𝑝 (41) for all 𝑥∈𝑋.

3. Stability of (5) by Direct Method

We now investigate stability problem of the quadratic functional inequality (5) with perturbed control function 𝜑. In this section, let 𝑋 be a normed space and 𝑌 a Banach space.

Theorem 6. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional inequality (27) and that the perturbing function 𝜑∶𝑋³→[0,∞) satisfies ^∞_𝑖=014^𝑖𝜑2^𝑖𝑥,2^𝑖𝑦,2^𝑖𝑧<∞,_∞𝑖=14^𝑖𝜑𝑥2^𝑖,𝑦2^𝑖,𝑧2^𝑖<∞,𝑟𝑒𝑠𝑝. (42) for all 𝑥,𝑦,𝑧∈𝑋. Then, there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 defined by 𝑄(𝑥)=lim_𝑚→∞(1/4^𝑚)𝑓(2^𝑚𝑥) (𝑄(𝑥)=lim_𝑚→∞4^𝑚𝑓(𝑥/2^𝑚), resp.) such that 𝑓(𝑥)(𝑥≤1−𝑄)4^∞_𝑖=014^𝑖3𝜑2^𝑖𝑥,−2^𝑖𝑥,0+2𝜑2^𝑖𝑥,0,−2^𝑖𝑥,𝑓(𝑥)(𝑥≤1−𝑄)4^∞_𝑖=14^𝑖𝑥3𝜑2^𝑖,−𝑥2^𝑖𝑥,0+2𝜑2^𝑖,0,−𝑥2^𝑖,𝑟𝑒𝑠𝑝. (43) for all 𝑥∈𝑋.

Proof. It follows from (33) that 𝑓(𝑥)−14𝑓(≤12𝑥)43𝜑(𝑥,−𝑥,0)+2𝜑(𝑥,0,−𝑥) (44) for all 𝑥∈𝑋. Therefore, we prove from inequality (44) that for any integers 𝑚 and 𝑙 with 𝑚>𝑙≥0𝑓2^𝑙𝑥4^𝑙−𝑓2^𝑚𝑥4^𝑚≤^𝑚−1_𝑖=𝑙14^𝑖𝑓2^𝑖𝑥−14𝑓2^𝑖+1𝑥≤14^𝑚−1_𝑖=𝑙14^𝑖3𝜑2^𝑖𝑥,−2^𝑖𝑥,0+2𝜑2^𝑖𝑥,0,−2^𝑖𝑥 (45) for all 𝑥∈𝑋. Since the right-hand side of (45) tends to zero as 𝑙→∞, we obtain that the sequence {𝑓(2^𝑚𝑥)/4^𝑚} is Cauchy for all 𝑥∈𝑋. Because of the fact that 𝑌 is complete, it follows that the sequence {𝑓(2^𝑚𝑥)/4^𝑚} converges in 𝑌. Therefore, we can define a mapping 𝑄∶𝑋→𝑌 as 𝑄(𝑥)=lim_𝑚→∞𝑓2^𝑚𝑥4^𝑚,𝑥∈𝑋. (46)
Moreover, letting 𝑙=0 and taking 𝑚→∞ in (45), we get the desired inequality (43).
It follows from (27) and (42) that 𝑄()(𝑥)𝑦(𝑧−𝑄𝑥−𝑦+𝑄𝑦−𝑧+𝑄𝑥−𝑧−3𝑄−3𝑄−3𝑄)𝑥+𝑦+𝑧≤lim_𝑚→∞14^𝑚𝜑2^𝑚𝑥,2^𝑚𝑦,2^𝑚𝑧=0 (47) for all 𝑥,𝑦,𝑧∈𝑋. So the mapping 𝑄 is quadratic.
Next, let 𝑄^′∶𝑋→𝑌 be another quadratic mapping satisfying (43). Then, we have 𝑄(𝑥)−𝑄^′(𝑥=1)4^𝑘𝑄2^𝑘𝑥−14^𝑘𝑄^′2^𝑘𝑥≤14^𝑘𝑄2^𝑘𝑥−𝑓2^𝑘𝑥+𝑓2^𝑘𝑥−𝑄^′2^𝑘𝑥≤12^∞_𝑖=014^𝑖+𝑘3𝜑2^𝑖+𝑘𝑥,−2^𝑖+𝑘2𝑥,0+2𝜑^𝑖+𝑘𝑥,0,−2^𝑖+𝑘𝑥=12^∞_𝑖=𝑘14^𝑖3𝜑2^𝑖𝑥,−2^𝑖𝑥,0+2𝜑2^𝑖𝑥,0,−2^𝑖𝑥 (48) for all 𝑘∈ℕ and all 𝑥∈𝑋. Taking the limit as 𝑘→∞, we conclude that 𝑄(𝑥)=𝑄^′(𝑥) for all 𝑥∈𝑋. This completes the proof.

We obtain the following corollary concerning the stability for approximate mappings controlled by a sum of powers of norms.

Corollary 7. Let 𝜃_𝑖≥0 be a real number and 𝑝_𝑖 a positive real number with 𝑝_𝑖<2 or 𝑝_𝑖>2 for all 𝑖=1,2,3. If a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑓()(𝑥)𝑦(𝑧≤𝑓𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧−3𝑓−3𝑓−3𝑓)𝑥+𝑦+𝑧+𝜃₁‖𝑥‖^𝑝1+𝜃₂𝑦^𝑝2+𝜃₃‖𝑧‖^𝑝3 (49) for all 𝑥,𝑦,𝑧∈𝑋, then we can find a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying the inequality 𝑓(𝑥)−𝑄(𝑥)≤5𝜃₁4−2^𝑝1‖𝑥‖^𝑝1+3𝜃₂4−2^𝑝2‖𝑥‖^𝑝2+2𝜃₃4−2^𝑝3‖𝑥‖^𝑝3 (50) for all 𝑥∈𝑋.

4. Stability of (5) in Non-Archimedean Spaces by Fixed Point Method

A non-Archimedean valuation in a field 𝕂 is a function |⋅|: 𝕂→[0,∞) with the following:(i)|𝑟|=0 if and only if 𝑟=0;(ii)|𝑟𝑠|=|𝑟||𝑠| for all 𝑟,𝑠∈𝕂;(iii)|𝑟+𝑠|≤max{|𝑟|,|𝑠|} for all 𝑟,𝑠∈𝕂.

Any field endowed with a non-Archimedean valuation is said to be a non-Archimedean field; in any such field we have |1|=|−1|=1 and |𝑛|≤1 for all 𝑛∈ℕ.

Definition 8. Let X be a linear space over a field 𝕂 with a non-Archimedean nontrivial valuation |⋅|. A function ‖⋅‖∶𝑋→[0,∞) is said to be a non-Archimedean norm if it satisfies the following conditions:(i)‖𝑥‖=0 if and only if 𝑥=0;(ii)‖𝑟𝑥‖=|𝑟|‖𝑥‖, for all 𝑟∈𝕂 and 𝑥∈𝑋;(iii)‖𝑥+𝑦‖≤max{‖𝑥‖,‖𝑦‖}, for all 𝑥,𝑦∈𝑋.Then (𝑋,‖⋅‖) is called a non-Archimedean normed space.

Definition 9. Let {𝑥_𝑛} be a sequence in a non-Archimedean normed space 𝑋.(1) A sequence {𝑥_𝑛}^∞_𝑛=1 in a non-Archimedean space is a Cauchy sequence if the sequence {𝑥_𝑛+1−𝑥_𝑛}^∞_𝑛=1 converges to zero.(2) The sequence {𝑥_𝑛} is said to be convergent if, for any 𝜀>0, there are a positive integer 𝑁 and 𝑥∈𝑋 such that 𝑥_𝑛−𝑥<𝜀,∀𝑛≥𝑁. (51) Then the point 𝑥∈𝑋 is called the limit of the sequence {𝑥_𝑛}, which is denoted by lim_𝑛→∞𝑥_𝑛=𝑥.(3) If every Cauchy sequence in 𝑋 converges, then the non-Archimedean normed space 𝑋 is called a non-Archimedean Banach space.

In 2007, Moslehian and Rassias [28] proved the generalized Hyers-Ulam stability of the Cauchy and quadratic functional equations in non-Archimedean normed spaces. Some papers [29, 30] on the stability of various functional equations in non-Archimedean normed spaces have been published after their stability results.

In this section, assume that 𝑋 is a non-Archimedean normed space and that 𝑌 is a non-Archimedean Banach space. Now, we are going to investigate the stability of the functional inequality (5) in non-Archimedean Banach space by using fixed point method.

Theorem 10. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional inequality (27) and that there exists a constant 𝐿 with 0<𝐿<1 for which the perturbing function 𝜑∶𝑋³→[0,∞) satisfies 𝜑≤|4|,𝜑𝑥2𝑥,2𝑦,2𝑧𝐿𝜑𝑥,𝑦,𝑧2,𝑦2,𝑧2≤𝐿|4|𝜑𝑥,𝑦,𝑧,𝑟𝑒𝑠𝑝. (52) for all 𝑥,𝑦,𝑧∈𝑋. Then, there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 given by 𝑄(𝑥)=lim_𝑚→∞(1/4^𝑚)𝑓(2^𝑚𝑥) (𝑄(𝑥)=lim_𝑚→∞4^𝑚𝑓(𝑥/2^𝑚), resp.) such that 𝑓(𝑥)(𝑥≤1−𝑄)|4)Φ(𝑥),𝑓(𝑥)(𝑥≤𝐿|(1−𝐿−𝑄)|4)Φ(𝑥)|(1−𝐿,𝑟𝑒𝑠𝑝. (53) for all 𝑥∈𝑋, where Φ(𝑥)∶=max{|3|𝜑(𝑥,−𝑥,0),|2|𝜑(𝑥,0,−𝑥)} for all 𝑥∈𝑋.

Proof. From (31) and (32), we get by using the non-Archimedean norm 𝑓()(𝑥3𝑓()(𝑥)(,|2𝑓()(𝑥)(3|𝜑(),|2|𝜑((𝑥)2𝑥−4𝑓)≤max||2𝑥−2𝑓−2𝑓−𝑥)|2𝑥−𝑓−3𝑓−𝑥)≤max|𝑥,−𝑥,0𝑥,0,−𝑥)∶≡Φ (54)
and so 𝑓(𝑥)−14𝑓(≤12𝑥)|4|Φ(𝑥) (55) for all 𝑥∈𝑋. Applying the similar argument to the corresponding proof of Theorem 4 on the complete generalized metric space (𝑆,𝑑), we get the desired result.

We obtain the following corollary concerning the stability for approximate mappings controlled by a sum of powers of norms.

Corollary 11. Let 𝜃≥0 be a real number and 𝑝 a positive real number with 𝑝≠2. If a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑓()(𝑥)𝑦(𝑧≤𝑓‖𝑥‖𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧−3𝑓−3𝑓−3𝑓)𝑥+𝑦+𝑧+𝜃^𝑝+𝑦^𝑝+‖𝑧‖^𝑝 (56) for all 𝑥,𝑦,𝑧∈𝑋, then we can find a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying the inequality 𝑓(𝑥)(𝑥≤3|,|2−𝑄)max{||}|2|²−|2|^𝑝2𝜃‖𝑥‖^𝑝 (57) for all 𝑥∈𝑋.

5. Stability of (5) in Non-Archimedean Spaces by Direct Method

Now, we are going to investigate the stability of the functional inequality (5) in non-Archimedean Banach space by direct method. In this section, assume that 𝑋 is a non-Archimedean normed space and that 𝑌 is a non-Archimedean Banach space.

Theorem 12. Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the functional inequality (27) and that 𝜑∶𝑋³→[0,∞) is a function such that lim_𝑚→∞1|4|^𝑚𝜑2^𝑚𝑥,2^𝑚𝑦,2^𝑚𝑧=0,lim_𝑚→∞|4|^𝑚𝜑𝑥2^𝑚,𝑦2^𝑚,𝑧2^𝑚=0,𝑟𝑒𝑠𝑝. (58) for all 𝑥,𝑦,𝑧∈𝑋 and 𝜑(𝑥)=lim_𝑚→∞1max|4|^𝑘Φ2^𝑘𝑥𝜑(𝑥)∶0≤𝑘<𝑚=lim_𝑚→∞max|4|^𝑘Φ𝑥2^𝑘∶1≤𝑘<𝑚+1,𝑟𝑒𝑠𝑝. (59) exists for all 𝑥∈𝑋, where Φ(𝑥)∶=max|3|𝜑(𝑥,−𝑥,0),|2|𝜑(𝑥,0,−𝑥) (60) for all 𝑥∈𝑋. Then there exists a quadratic mapping 𝑄∶𝑋→𝑌 defined by 𝑄(𝑥)=lim_𝑚→∞(1/4^𝑚)𝑓(2^𝑚𝑥), (𝑄(𝑥)=lim_𝑚→∞4^𝑚𝑓(𝑥/2^𝑚), resp.) such that 𝑓(𝑥)(𝑥≤1−𝑄)|4|𝜑(𝑥) (61) for all 𝑥∈𝑋. Moreover, if lim_𝑙→∞lim_𝑚→∞1max|4|^𝑘Φ2^𝑘𝑥∶𝑙≤𝑘<𝑚+𝑙=0,lim_𝑙→∞lim_𝑚→∞max|4|^𝑘Φ𝑥2^𝑘∶𝑙+1≤𝑘<𝑚+𝑙+1=0,𝑟𝑒𝑠𝑝. (62) for all 𝑥∈𝑋, then 𝑄 is a unique quadratic mapping satisfying (61).

Proof. Replacing 𝑥 by 2^𝑚−1𝑥 and dividing by |4|^𝑚−1 in (55), we have 𝑓2^𝑚−1𝑥4^𝑚−1−𝑓2^𝑚𝑥4^𝑚≤1|4|1|4|^𝑚−1Φ2^𝑚−1𝑥 (63) for all 𝑥∈𝑋. It follows from (58) and (63) that the sequence {𝑓(2^𝑚𝑥)/4^𝑚} is Cauchy for all 𝑥∈𝑋, and the sequence {𝑓(2^𝑚𝑥)/4^𝑚} converges in the non-Archimedean Banach space 𝑌. Therefore, we can define a mapping 𝑄∶𝑋→𝑌 as 𝑄(𝑥)=lim_𝑚→∞𝑓2^𝑚𝑥4^𝑚,𝑥∈𝑋. (64) Applying the similar argument to the corresponding proof of Theorem 6, we get the required result.

Corollary 13. Let  𝜌∶[0,∞)→[0,∞) be a function satisfying (i)  𝜌(|2|𝑡)≤𝜌(|2|)𝜌(𝑡) for all 𝑡≥0 and (ii) 𝜌(|2|)<|2|².
Suppose that 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑓()(𝑥)𝑦(𝑧≤𝑓𝜌𝑥𝑦𝑧𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧−3𝑓−3𝑓−3𝑓)𝑥+𝑦+𝑧+𝜀(‖‖)+𝜌+𝜌(‖‖) (65) for all 𝑥,𝑦,𝑧∈𝑋 and for some 𝜀>0. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥≤1−𝑄)|4|max{|3|,|2|}2𝜀𝜌(‖𝑥‖) (66) for all 𝑥∈𝑋.

Proof. Letting 𝜑(𝑥,𝑦,𝑧)=𝜀{𝜌(‖𝑥‖)+𝜌(‖𝑦‖)+𝜌(‖𝑧‖)}, we obtain lim_𝑚→∞1|4|^𝑚𝜑2^𝑚𝑥,2^𝑚𝑦,2^𝑚𝑧≤𝜌(|2|)|4|^𝑚𝜑𝑥,𝑦,𝑧=0 (67) for all 𝑥,𝑦,𝑧∈𝑋. It follows from (60) that Φ(𝑥)3|𝜑(),|2|𝜑(3|𝑥,|2|𝑥3|,|2𝑥∶=max|𝑥,−𝑥,0𝑥,0,−𝑥)=max|2𝜀𝜌(‖‖)2𝜀𝜌(‖‖)=max{||}2𝜀𝜌(‖‖),𝑥∈𝑋. (68) By direct calculation, 𝜑(𝑥)=lim_𝑚→∞1max|4|^𝑘Φ2^𝑘𝑥∶0≤𝑘<𝑚=Φ(𝑥) (69) exists and lim_𝑙→∞lim_𝑚→∞1max|4|^𝑘Φ2^𝑘𝑥∶𝑙≤𝑘<𝑚+𝑙=lim_𝑙→∞1|4|^𝑙Φ2^𝑙𝑥=0 (70) holds for all 𝑥∈𝑋. Applying Theorem 12, we conclude that 𝑓(𝑥)(𝑥≤1−𝑄)|4|𝜑(𝑥)=1|4|Φ(𝑥)=1|4|max{|3|,|2|}2𝜀𝜌(‖𝑥‖) (71) for all 𝑥∈𝑋.

Corollary 14. Let 𝜌∶[0,∞)→[0,∞) be a function satisfying (i) 𝜌(𝑡/|2|)≤𝜌(1/|2|)𝜌(𝑡) for all 𝑡≥0 and (ii) 𝜌(1/|2|)<|2|⁻².
Suppose that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑓()(𝑥)𝑦(𝑧≤𝑓𝜌𝑥𝑦𝑧𝑥−𝑦+𝑓𝑦−𝑧+𝑓𝑥−𝑧−3𝑓−3𝑓−3𝑓)𝑥+𝑦+𝑧+𝜀(‖‖)+𝜌+𝜌(‖‖) (72) for all 𝑥,𝑦,𝑧∈𝑋 and for some 𝜀>0. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑓(𝑥)(𝑥3|,|2𝑥𝜌1−𝑄)≤max{||}2𝜀𝜌(‖‖)|2|≤1|4|max{|3|,|2|}2𝜀𝜌(‖𝑥‖) (73) for all 𝑥∈𝑋.

Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2012R1A1A2008139).