Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

Abstract

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equation 𝑓(𝑥+𝑘𝑦)+𝑓(𝑥−𝑘𝑦)=𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)+(2(𝑘+1)/𝑘)𝑓(𝑘𝑦)−2(𝑘+1)𝑓(𝑦) for fixed integers 𝑘 with 𝑘≠0,±1 in fuzzy Banach spaces.

1. Introduction and Preliminaries

The stability problem of functional equations was originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms. Let (𝐺₁,⋅) be a group and let (𝐺₂,∗,𝑑) 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 a first affirmative answer to the question of Ulam for Banach spaces. Let 𝑓∶𝐸→𝐸^′ be a mapping between Banach spaces such that𝑓𝑥+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 that𝑓(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] provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. 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 (see [5–17]).

The functional equation𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (1.3) is related to a symmetric biadditive function. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic function. 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 (see [6, 18]). The biadditive function 𝐵 is given by𝐵=1𝑥,𝑦4𝑓𝑥+𝑦−𝑓𝑥−𝑦. (1.4) A Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof for functions 𝑓∶𝐴→𝐵, where 𝐴 is normed space and 𝐵 Banach space (see [19–22]). Borelli and Forti [23] generalized the stability result of quadratic functional equations as follows (cf. [24, 25]): let 𝐺 be an Abelian group, and 𝑋 a Banach space. Assume that a mapping 𝑓∶𝐺→𝑋 satisfies the functional inequality:𝑓𝑥+𝑦+𝑓𝑥−𝑦−2𝑓(𝑥)−2𝑓𝑦≤𝜑𝑥,𝑦, (1.5) for all 𝑥,𝑦∈𝐺, and 𝜑∶𝐺×𝐺→[0,∞) is a function such thatΦ𝑥,𝑦∶=^∞_𝑖=014^𝑖+1𝜑2^𝑖𝑥,2^𝑖𝑦<∞, (1.6) for all 𝑥,𝑦∈𝐺. Then there exists a unique quadratic mapping 𝑄∶𝐺→𝑋 with the property𝑓(𝑥)−𝑄(𝑥)≤Φ(𝑥,𝑥), (1.7) for all 𝑥∈𝐺.

Now, we introduce the following functional equation for fixed integers 𝑘 with 𝑘≠0,±1:𝑓+2()𝑥+𝑘𝑦+𝑓𝑥−𝑘𝑦=𝑓𝑥+𝑦+𝑓𝑥−𝑦𝑘+1𝑘𝑓𝑘𝑦−2(𝑘+1)𝑓𝑦, (1.8) with 𝑓(0)=0 in a non-Archimedean space. It is easy to see that the function 𝑓(𝑥)=𝑎𝑥+𝑏𝑥² is a solution of the functional equation (1.8), which explains why it is called additive-quadratic functional equation. For more detailed definitions of mixed type functional equations, we can refer to [26–47].

Definition 1.1 (see [48]). Let 𝑋 be a real vector space. A function 𝑁∶𝑋×ℝ→[0,1] is called a fuzzy norm on 𝑋 if for all 𝑥,𝑦∈𝑋 and all 𝑠,𝑡∈ℝ, (N1)𝑁(𝑥,𝑡)=0 for 𝑡≤0; (N2)𝑥=0 if and only if 𝑁(𝑥,𝑡)=1 for all 𝑡>0; (N3)𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) if 𝑐≠0; (N4)𝑁(𝑥+𝑦,𝑠+𝑡)≥min{𝑁(𝑥,𝑠),𝑁(𝑦,𝑡)};(N5)𝑁(𝑥,⋅) is a nondecreasing function of ℝ and lim_𝑡→∞𝑁(𝑥,𝑡)=1; (N6)for 𝑥≠0, 𝑁(𝑥,⋅) is continuous on ℝ.

The pair (𝑋,𝑁) is called a fuzzy normed vector space.

Example 1.2. Let (𝑋,‖⋅‖) be a normed linear space and 𝛼,𝛽>0. Then 𝑁(𝑥,𝑡)=𝛼𝑡𝛼𝑡+𝛽‖𝑥‖,𝑡>0,𝑥∈𝑋,0,𝑡≤0,𝑥∈𝑋, (1.9) is a fuzzy norm on 𝑋.

Definition 1.3. Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {𝑥_𝑛} in 𝑋 is said to be convergent or converge if there exists an 𝑥∈𝑋 such that lim_𝑛→∞𝑁(𝑥_𝑛−𝑥,𝑡)=1 for all 𝑡>0. In this case, 𝑥 is called the limit of the sequence {𝑥_𝑛} in 𝑋 and one denotes it by 𝑁−lim_𝑛→∞𝑥_𝑛=𝑥.

Definition 1.4. Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {𝑥_𝑛} in 𝑋 is called Cauchy if for each 𝜖>0 and each 𝑡>0 there exists an 𝑛₀∈ℕ such that for all 𝑛≥𝑛₀ and all 𝑝>0, one has 𝑁(𝑥_𝑛+𝑝−𝑥_𝑛,𝑡)>1−𝜖.

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

Example 1.5. Let 𝑁∶ℝ×ℝ→[0,1] be a fuzzy norm on ℝ defined by 𝑁()=𝑡𝑥,𝑡𝑡+|𝑥|,𝑡>0,0,𝑡≤0. (1.10) The (ℝ,𝑁) is a fuzzy Banach space. Let {𝑥_𝑛} be a Cauchy sequence in ℝ, 𝛿>0, and 𝜖=𝛿/(1+𝛿). Then there exist 𝑚∈ℕ such that for all 𝑛≥𝑚 and all 𝑝>0, one has 11+𝑥_𝑛+𝑝−𝑥_𝑛≥1−𝜖. (1.11) So |𝑥_𝑛+𝑝−𝑥_𝑛|<𝛿 for all 𝑛≥𝑚 and all 𝑝>0. Therefore {𝑥_𝑛} is a Cauchy sequence in (ℝ,|⋅|). Let 𝑥_𝑛→𝑥₀∈ℝ as 𝑛→∞. Then lim_𝑛→∞𝑁(𝑥_𝑛−𝑥₀,𝑡)=1 for all 𝑡>0.

We say that a mapping 𝑓∶𝑋→𝑌 between fuzzy normed vector spaces 𝑋 and 𝑌 is continuous at a point 𝑥∈𝑋 if for each sequence {𝑥_𝑛} converging to 𝑥₀∈𝑋, the sequence {𝑓(𝑥_𝑛)} converges to 𝑓(𝑥₀). If 𝑓∶𝑋→𝑌 is continuous at each 𝑥∈𝑋, then 𝑓∶𝑋→𝑌 is said to be continuous on 𝑋 ([49]).

Definition 1.6. Let 𝑋 be a set. A function 𝑑∶𝑋×𝑋→[0,∞] is called a generalized metric on 𝑋 if 𝑑 satisfies the following conditions:(1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦 for all 𝑥,𝑦∈𝑋; (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.

Theorem 1.7. 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 all 𝑥∈𝑋, either 𝑑𝐽^𝑛𝑥,𝐽^𝑛+1𝑥=∞, (1.12) for all nonnegative integers 𝑛, or there exists a positive integer 𝑛₀ such that(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞ for all 𝑛₀≥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.

We have the following theorem from [42], which investigates the solution of (1.8).

Theorem 1.8. A function 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies (1.8) for all 𝑥,𝑦∈𝑋 if and only if there exist functions 𝐴∶𝑋→𝑌 and 𝑄∶𝑋×𝑋→𝑌, such that 𝑓(𝑥)=𝐴(𝑥)+𝑄(𝑥,𝑥) for all 𝑥∈𝑋, where the function 𝑄 is symmetric biadditive and 𝐴 is additive.

2. A Fixed Point Method

Using the fixed point methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.8) in fuzzy Banach spaces. Throughout this paper, assume that 𝑋 is a vector space and that (𝑌,𝑁) is a fuzzy Banach space.

Theorem 2.1. Let 𝜑∶𝑋²→[0,∞) be a mapping such that there exists an 𝛼<1 with 𝜑≤|𝑘|𝑥𝑥,𝑦𝛼𝜑𝑘,𝑦𝑘, (2.1) for all 𝑥,𝑦∈𝑋. Let 𝑓∶𝑋→𝑌 be an odd function satisfying 𝑓(0)=0 and 𝑁𝑓𝑘𝑘−2()𝑥+𝑦+𝑓𝑥−𝑦−𝑓𝑘𝑥+𝑦−𝑓𝑘𝑥−𝑦𝑘+1𝑘𝑓()𝑓𝑦≥𝑡𝑘𝑦+2𝑘+1,𝑡,𝑡+𝜑𝑥,𝑦 (2.2) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then 𝐴(𝑥)∶=𝑁−lim_𝑛→∞(𝑓(𝑘^𝑛𝑥)/𝑘^𝑛) exists for all 𝑥∈𝑋 and defines a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑁𝑓(𝑥)(𝑥)≥|−||𝛼)𝑡−𝐴,𝑡(|2𝑘+22𝑘+2(|2𝑘+2|−|2𝑘+2|𝛼)𝑡+𝜑(0,𝑥), (2.3) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Note that 𝑓(0)=0 and 𝑓(−𝑥)=−𝑓(𝑥) for all 𝑥∈𝑋 since 𝑓 is an odd function. Putting 𝑥=0 in (2.2), we get 𝑁𝑓𝑘𝑦𝑘𝑦,𝑡−𝑓||≥𝑡2𝑘+2𝑡+𝜑0,𝑦, (2.4) for all 𝑦∈𝑋 and all 𝑡>0. Replacing 𝑦 by 𝑥 in (2.4), we have 𝑁𝑓(𝑘𝑥)𝑘(𝑥),𝑡−𝑓||≥𝑡2𝑘+2𝑡+𝜑(0,𝑥), (2.5) for all 𝑥∈𝑋 and all 𝑡>0. Consider the set 𝑆∶={ℎ∶𝑋→𝑌;ℎ(0)=0} and introduce the generalized metric on 𝑆: 𝑑𝑔,ℎ=inf_{()𝜇∈0,+∞}𝑁𝑔(𝑥)(𝑥)≥𝑡−ℎ,𝜇𝑡𝑡+𝜑(0,𝑥),∀𝑥∈𝑋, (2.6) where, as usual, inf𝜙=+∞. It is easy to show that (𝑆,𝑑) is complete (see [50]). We consider the mapping 𝐽∶(𝑆,𝑑)→(𝑆,𝑑) as follows: (𝑥)1𝐽𝑔∶=𝑘𝑔(𝑘𝑥), (2.7) for all 𝑥∈𝑋. Let 𝑔,ℎ∈𝑆 be given such that 𝑑(𝑔,ℎ)=𝛽. Then 𝑁𝑔(𝑥)(𝑥)≥𝑡−ℎ,𝛽𝑡𝑡+𝜑(0,𝑥), (2.8) for all 𝑥∈𝑋 and all 𝑡>0. Hence 𝑁(𝑥)(𝑥)1𝐽𝑔−𝐽ℎ,𝛼𝛽𝑡=𝑁𝑘𝑔()−1𝑘𝑥𝑘ℎ()𝑔()(),|𝑘|≥|𝑘|𝑘𝑥,𝛼𝛽𝑡=𝑁𝑘𝑥−ℎ𝑘𝑥𝛼𝛽𝑡𝛼𝑡|𝑘|()≥|𝑘|𝛼𝑡+𝜑0,𝑥𝛼𝑡|𝑘||𝑘|()=𝑡𝛼𝑡+𝛼𝜑0,𝑥(),𝑡+𝜑0,𝑥 (2.9) for all 𝑥∈𝑋 and all 𝑡>0. So 𝑑(𝑔,ℎ)=𝛽 implies that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝛽. This means that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝑑(𝑔,ℎ) for all 𝑔,ℎ∈𝑆. It follows from (2.5) that 𝑑≤1𝑓,𝐽𝑓|2𝑘+2|. (2.10) By Theorem 1.7, there exists a mapping 𝐴∶𝑋→𝑌 satisfying the following.
(1) 𝐴 is a fixed point of 𝐽, that is, 𝑘𝐴(𝑥)=𝐴(𝑘𝑥), (2.11) 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 a unique mapping satisfying (2.11) such that there exists a 𝜇∈(0,∞) satisfying 𝑁𝑓(𝑥)(𝑥)≥𝑡−𝐴,𝜇𝑡𝑡+𝜑(0,𝑥), (2.12) for all 𝑥∈𝑋.
(2) 𝑑(𝐽^𝑛𝑓,𝐴)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞(𝑓(𝑘^𝑛𝑥)/𝑘^𝑛)=𝐴(𝑥), for all 𝑥∈𝑋.
(3) 𝑑(𝑓,𝐴)≤(1/(1−𝛼))𝑑(𝑓,𝐽𝑓), which implies the inequality 𝑑≤1𝑓,𝐴|2𝑘+2|−|2𝑘+2|𝛼. (2.13) This implies that the inequality (2.3) holds.
It follows from (2.1) and (2.2) that 𝑁𝑓𝑘^𝑛𝑥+𝑘𝑦𝑘^𝑛+𝑓𝑘^𝑛𝑥−𝑘𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥+𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥−𝑦𝑘^𝑛−2(𝑘+1)𝑘𝑓𝑘^𝑛+1𝑦𝑘^𝑛()𝑓𝑘+2𝑘+1^𝑛𝑦𝑘^𝑛,𝑡𝑘^𝑛≥𝑡𝑡+𝜑𝑘^𝑛𝑥,𝑘^𝑛𝑦, (2.14) for all 𝑥,𝑦∈𝑋, all 𝑡>0, and all 𝑛∈ℕ. So 𝑁𝑓𝑘^𝑛𝑥+𝑘𝑦𝑘^𝑛+𝑓𝑘^𝑛𝑥−𝑘𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥+𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥−𝑦𝑘^𝑛−2(𝑘+1)𝑘𝑓𝑘^𝑛+1𝑦𝑘^𝑛()𝑓𝑘+2𝑘+1^𝑛𝑦𝑘^𝑛≥|𝑘|,𝑡_𝑛𝑡|𝑘|^𝑛𝑡+|𝑘|^𝑛𝛼^𝑛𝜑,𝑥,𝑦 (2.15) for all 𝑥,𝑦∈𝑋, all 𝑡>0, and all 𝑛∈ℕ. Since lim_𝑛→∞(|𝑘|^𝑛𝑡/(|𝑘|^𝑛𝑡+|𝑘|^𝑛𝛼^𝑛𝜑(𝑥,𝑦)))=1 for all 𝑥,𝑦∈𝑋 and all 𝑡>0, we obtain that 𝑁𝐴𝑘𝑘−2()𝑥+𝑦+𝐴𝑥−𝑦−𝐴𝑘𝑥+𝑦−𝐴𝑘𝑥−𝑦𝑘+1𝑘𝐴()𝐴𝑦𝑘𝑦+2𝑘+1,𝑡=1, (2.16) for all 𝑥,𝑦,𝑧∈𝑋 and all 𝑡>0. Hence the mapping 𝐴∶𝑋→𝑌 is additive, as desired.

Corollary 2.2. Let 𝜃≥0 and let 𝑟 be a real positive number with 𝑟<1. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying 𝑁𝑓𝑘𝑘−2()𝑥+𝑦+𝑓𝑥−𝑦−𝑓𝑘𝑥+𝑦−𝑓𝑘𝑥−𝑦𝑘+1𝑘𝑓()𝑓𝑦≥𝑡𝑘𝑦+2𝑘+1,𝑡𝑡+𝜃‖𝑥‖^𝑟+𝑦^𝑟, (2.17) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then the limit 𝐴(𝑥)∶=𝑁−lim_𝑛→∞(𝑓(𝑘^𝑛𝑥)/𝑘^𝑛) exists for each 𝑥∈𝑋 and defines a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑁𝑓(𝑥)(𝑥)≥|𝑘|−|𝑘|−𝐴,𝑡2𝑘+2||^𝑟𝑡|2𝑘+2|(|𝑘|−|𝑘|^𝑟)𝑡+|𝑘|𝜃‖𝑥‖^𝑟, (2.18) for all 𝑥∈𝑋 and all 𝑡>0.

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

Theorem 2.3. Let 𝜑∶𝑋²→[0,∞) be a mapping such that there exists an 𝛼<1 with 𝜑𝑥𝑘,𝑦𝑘≤𝛼|𝑘|𝜑𝑥,𝑦, (2.19) for all 𝑥,𝑦∈𝑋. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying 𝑓(0)=0 and (2.2). Then the limit 𝐴(𝑥)∶=𝑁−lim_𝑛→∞𝑘^𝑛𝑓(𝑥/𝑘^𝑛) exists for all 𝑥∈𝑋 and defines a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑁𝑓(𝑥)(𝑥)≥|−||𝛼)𝑡−𝐴,𝑡(|2𝑘+22𝑘+2(|2𝑘+2|−|2𝑘+2|𝛼)𝑡+𝛼𝜑(0,𝑥), (2.20) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. Let (𝑆,𝑑) be the generalized metric space defined as in the proof of Theorem 2.1.
Consider the mapping 𝐽∶𝑆→𝑆 by (𝑥)𝑥𝐽𝑔∶=𝑘𝑔𝑘, (2.21) for all 𝑔∈𝑆. Let 𝑔,ℎ∈𝑆 be given such that 𝑑(𝑔,ℎ)=𝛽. Then 𝑁𝑔(𝑥)(𝑥)≥𝑡−ℎ,𝛽𝑡𝑡+𝜑(0,𝑥), (2.22) for all 𝑥∈𝑋 and all 𝑡>0. Hence 𝑁(𝑥)(𝑥)𝑥𝐽𝑔−𝐽ℎ,𝛼𝛽𝑡=𝑁𝑘𝑔𝑘𝑥−𝑘ℎ𝑘𝑔𝑥,𝛼𝛽𝑡=𝑁𝑘𝑥−ℎ𝑘,𝛼𝛽𝑡|𝑘|≥(𝛼𝑡/|𝑘|)|𝑘|()≥𝑡𝛼𝑡/+𝜑0,𝑥/𝑘(),𝑡+𝜑0,𝑥 (2.23) for all 𝑥∈𝑋 and all 𝑡>0. So 𝑑(𝑔,ℎ)=𝛽 implies that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝛽. This means that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝑑(𝑔,ℎ) for all 𝑔,ℎ∈𝑆. It follows from (2.5) that 𝑁𝑥𝑘𝑓𝑘−𝑓(𝑥),𝑘𝑡||≥𝑡2𝑘+2()≥𝑡𝑡+𝜑0,𝑥/𝑘𝑡+(𝛼/|𝑘|)𝜑(0,𝑥), (2.24) for all 𝑥∈𝑋 and all 𝑡>0. Therefore 𝑁𝑥𝑘𝑓𝑘−𝑓(𝑥),𝛼𝑡||≥𝑡2𝑘+2𝑡+𝜑(0,𝑥). (2.25) So 𝑑(𝑓,𝐽𝑓)≤𝛼. By Theorem 1.7, there exists a mapping 𝐴∶𝑋→𝑌 satisfying the following.
(1) 𝐴 is a fixed point of 𝐽, that is, 𝐴𝑥𝑘=1𝑘𝐴(𝑥), (2.26) for all 𝑥∈𝑋. The mapping 𝐴 is a unique fixed point of 𝐽 in the set Ω={ℎ∈𝑆∶𝑑(𝑔,ℎ)<∞}. This implies that 𝐴 is a unique mapping satisfying (2.26) such that there exists 𝜇∈(0,∞) satisfying 𝑁𝑓(𝑥)(𝑥)≥𝑡−𝐴,𝜇𝑡𝑡+𝜑(0,𝑥), (2.27) for all 𝑥∈𝑋 and 𝑡>0.
(2)𝑑(𝐽^𝑛𝑓,𝐴)→0 as 𝑛→∞. This implies the equality 𝑁−lim_𝑛→∞𝑘^𝑛𝑓(𝑥/𝑘^𝑛)=𝐴(𝑥) for all 𝑥∈𝑋.
(3) 𝑑(𝑓,𝐴)≤𝑑(𝑓,𝐽𝑓)/(1−𝐿) with 𝑓∈Ω, which implies the inequality 𝑑≤𝛼𝑓,𝐴|2𝑘+2|−|2𝑘+2|𝛼. (2.28) This implies that the inequality (2.20) holds.
The rest of proof is similar to the proof of Theorem 2.1.

Corollary 2.4. Let 𝜃≥0 and let 𝑟 be a real number with 𝑟>1. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an odd mapping satisfying (2.17). Then 𝐴(𝑥)∶=𝑁−lim_𝑛→∞𝑘^𝑛𝑓(𝑥/𝑘^𝑛) exists for each 𝑥∈𝑋 and defines a unique additive mapping 𝐴∶𝑋→𝑌 such that 𝑁𝑓(𝑥)(𝑥)≥|𝑘|−𝐴,𝑡2𝑘+2||^𝑟−|𝑘|𝑡|2𝑘+2|(|𝑘|^𝑟−|𝑘|)𝑡+|𝑘|𝜃‖𝑥‖^𝑟, (2.29) for all 𝑥∈𝑋 and all 𝑡>0.

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

Theorem 2.5. Let 𝜑∶𝑋²→[0,∞) be a function such that there exists an 𝛼<1 with 𝜑𝑥,𝑦≤𝑘²𝑥𝛼𝜑𝑘,𝑦𝑘, (2.30) for all 𝑥,𝑦∈𝑋. Let 𝑓∶𝑋→𝑌 be an even mapping with 𝑓(0)=0 and satisfying (2.2). Then 𝑄(𝑥)∶=𝑁−lim_𝑛→∞(𝑓(𝑘^𝑛𝑥)/𝑘^2𝑛) exists for all 𝑥∈𝑋 and defines a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑁𝑓(𝑥)(𝑥)≥(2|𝑘||𝑘|𝛼)𝑡−𝑄,𝑡−2(2|𝑘|−2|𝑘|𝛼)𝑡+𝜑(0,𝑥), (2.31) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. Replacing 𝑥 by 𝑘𝑥 in (2.2), we get 𝑁𝑓𝑘𝑘−2()𝑥+𝑦+𝑓𝑥−𝑦−𝑓𝑘𝑥+𝑦−𝑓𝑘𝑥−𝑦𝑘+1𝑘𝑓()𝑓𝑦≥𝑡𝑘𝑦+2𝑘+1,𝑡,𝑡+𝜑𝑘𝑥,𝑦 (2.32) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Putting 𝑥=0 and replacing 𝑦 by 𝑥 in (2.32), we have 𝑁𝑓(𝑘𝑥)𝑘(𝑥),𝑡−𝑘𝑓2≥𝑡𝑡+𝜑(0,𝑥), (2.33) for all 𝑥∈𝑋 and all 𝑡>0. By (2.33), (𝑁3), and (𝑁4), we get 𝑁𝑓(𝑘𝑥)𝑘²(𝑥),𝑡−𝑓2|𝑘|≥𝑡𝑡+𝜑(0,𝑥), (2.34) for all 𝑥∈𝑋 and all 𝑡>0. Consider the set 𝑆^∗∶={ℎ∶𝑋→𝑌;ℎ(0)=0} and introduce the generalized metric on 𝑆^∗: 𝑑𝑔,ℎ=inf_{𝜇∈(0,+∞)}𝑁𝑔(𝑥)(𝑥)≥𝑡−ℎ,𝜇𝑡𝑡+𝜑(0,𝑥),∀𝑥∈𝑋, (2.35) where, as usual, inf𝜙=+∞. It is easy to show that (𝑆^∗,𝑑) is complete (see [50]). Now we consider the linear mapping 𝐽∶(𝑆^∗,𝑑)→(𝑆^∗,𝑑) such that (𝑥)1𝐽𝑔∶=𝑘²𝑔(𝑘𝑥), (2.36) for all 𝑥∈𝑋. Proceeding as in the proof of Theorem 2.1, we obtain that 𝑑(𝑔,ℎ)=𝛽 implies that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝛽. This means that 𝑑(𝐽𝑔,𝐽ℎ)≤𝛼𝑑(𝑔,ℎ) for all 𝑔,ℎ∈𝑆. It follows from (2.34) that 𝑑≤1𝑓,𝐽𝑓2|𝑘|. (2.37) By Theorem 1.7, there exists a mapping 𝑄∶𝑋→𝑌 such that one has the folowing.
(1) 𝑄 is a fixed point of 𝐽, that is, 𝑘²𝑄(𝑥)=𝑄(𝑘𝑥), (2.38) for all 𝑥∈𝑋. The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀={𝑔∈𝑆^∗∶𝑑(ℎ,𝑔)<∞}. This implies that 𝑄 is a unique mapping satisfying (2.38) such that there exists a 𝜇∈(0,∞) satisfying 𝑁(𝑓(𝑥)−𝑄(𝑥),𝜇𝑡)≥𝑡/(𝑡+𝜑(0,𝑥)) for all 𝑥∈𝑋.
(2)𝑑(𝐽^𝑛𝑓,𝑄)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞(𝑓(𝑘^𝑛𝑥)/𝑘^2𝑛)=𝑄(𝑥) for all 𝑥∈𝑋.
(3)𝑑(𝑓,𝑄)≤(1/(1−𝛼))𝑑(𝑓,𝐽𝑓), which implies the inequality 𝑑(𝑓,𝑄)≤1/(2|𝑘|−2|𝑘|𝛼). This implies that the inequality (2.31) holds.
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.6. Let 𝜃≥0 and let 𝑟 be a real positive number with 𝑟<1. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an even mapping with 𝑓(0)=0 and satisfying (2.17). Then the limit 𝑄(𝑥)∶=𝑁−lim_𝑛→∞(𝑓(𝑘^𝑛𝑥)/𝑘^2𝑛) exists for each 𝑥∈𝑋 and defines a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥2𝑘²−2𝑘^2𝑟𝑡2𝑘²−2𝑘^2𝑟𝑡+|𝑘|𝜃‖𝑥‖^𝑟, (2.39) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. The proof follows from Theorem 2.5 by taking 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑟+‖𝑦‖^𝑟) for all 𝑥,𝑦∈𝑋. Then we can choose 𝛼=𝑘^2𝑟−2 and we get the desired result.

Theorem 2.7. Let 𝜑∶𝑋²→[0,∞) be a function such that there exists an 𝛼<1 with 𝜑𝑥𝑘,𝑦𝑘≤𝛼𝑘²𝜑𝑥,𝑦, (2.40) for all 𝑥,𝑦∈𝑋. Let 𝑓∶𝑋→𝑌 be an even mapping with 𝑓(0)=0 and satisfying (2.2). Then the limit 𝑄(𝑥)∶=𝑁−lim_𝑛→∞𝑘^2𝑛𝑓(𝑥/𝑘^𝑛) exists for all 𝑥∈𝑋 and defines a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑁𝑓(𝑥)(𝑥)≥(2|𝑘||𝑘|𝛼)𝑡−𝑄,𝑡−2(2|𝑘|−2|𝑘|𝛼)𝑡+𝛼𝜑(0,𝑥), (2.41) for all 𝑥∈𝑋 and 𝑡>0.

Proof. Let (𝑆^∗,𝑑) be the generalized metric space defined as in the proof of Theorem 2.5. It follows from (2.34) that 𝑁𝑘²𝑓𝑥𝑘(𝑥),|𝑘|𝑡−𝑓2≥𝑡()≥𝑡𝑡+𝜑0,𝑥/𝑘𝑡+𝛼/𝑘²𝜑(0,𝑥), (2.42) for all 𝑥∈𝑋 and 𝑡>0. So 𝑁𝑓(𝑥)−𝑘²𝑓𝑥𝑘,𝛼𝑡2|𝑘|≥𝑡𝑡+𝜑(0,𝑥). (2.43) The rest of the proof is similar to the proofs of Theorems 2.1 and 2.3.

Corollary 2.8. Let 𝜃≥0 and let 𝑟 be a real number with 𝑟>1. Let 𝑋 be a normed vector space with norm ‖⋅‖. Let 𝑓∶𝑋→𝑌 be an even mapping with 𝑓(0)=0 and satisfying (2.17). Then 𝑄(𝑥)∶=𝑁−lim_𝑛→∞𝑘^2𝑛𝑓(𝑥/𝑘^𝑛) exists for each 𝑥∈𝑋 and defines a unique quadratic mapping 𝑄∶𝑋→𝑌 such that 𝑁𝑓(𝑥)(𝑥)≥2|𝑘|−𝑄,𝑡^2𝑟+1−2|𝑘|³𝑡2|𝑘|^2𝑟+1−2|𝑘|³𝑡+𝑘²𝜃‖𝑥‖^𝑟, (2.44) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. It follows from Theorem 2.7 by taking 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑟+‖𝑦‖^𝑟) for all 𝑥,𝑦∈𝑋. Then we can choose 𝛼=𝑘^2−2𝑟 and we get the desired result.

3. Direct Method

In this section, using direct method, we prove the Hyers-Ulam stability of functional equation (1.8) in fuzzy Banach spaces. Throughout this section, we assume that 𝑋 is a linear space, (𝑌,𝑁) is a fuzzy Banach space, and (𝑍,𝑁^′) is a fuzzy normed space. Moreover, we assume that 𝑁(𝑥,⋅) is a left continuous function on ℝ.

Theorem 3.1. Assume that a mapping 𝑓∶𝑋→𝑌 is an odd mapping with 𝑓(0)=0 satisfying the inequality 𝑁𝑓𝑘𝑘−2()𝑥+𝑦+𝑓𝑥−𝑦−𝑓𝑘𝑥+𝑦−𝑓𝑘𝑥−𝑦𝑘+1𝑘𝑓𝑘𝑦+2(𝑘+1)𝑓𝑦,𝑡≥𝑁^′𝜑𝑥,𝑦,𝑡, (3.1) for all 𝑥,𝑦∈𝑋, 𝑡>0, and 𝜑∶𝑋²→𝑍 is a mapping for which there is a constant 𝑟∈ℝ satisfying 0<|𝑟|<1/|𝑘| such that 𝑁^′𝜑𝑥𝑘,𝑦𝑘,𝑡≥𝑁^′𝜑,𝑡𝑥,𝑦|𝑟|, (3.2) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 satisfying (1.8) and the inequality 𝑁𝑓(𝑥)−𝐴(𝑥),𝑡≥𝑁^′𝜑(),||𝑡0,𝑥2𝑘+2|(1−𝑘𝑟|)|𝑟|, (3.3) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. It follows from (3.2) that 𝑁^′𝜑𝑥𝑘^𝑗,𝑦𝑘^𝑗,𝑡≥𝑁^′𝜑,𝑡𝑥,𝑦|𝑟|^𝑗, (3.4) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Putting 𝑥=0 in (3.1) and then replacing 𝑦 by 𝑥/𝑘, we get 𝑁𝑥𝑘𝑓𝑘(𝑥),|𝑘|𝑡−𝑓|2𝑘+2|≥𝑁^′𝜑𝑥0,𝑘,𝑡, (3.5) for all 𝑥∈𝑋 and all 𝑡>0. Replacing 𝑥 by 𝑥/𝑘^𝑗 in (3.5), we have 𝑁𝑘^𝑗+1𝑓𝑥𝑘^𝑗+1−𝑘^𝑗𝑓𝑥𝑘^𝑗,|𝑘|^𝑗+1𝑡|2𝑘+2|≥𝑁^′𝜑𝑥0,𝑘^𝑗+1,𝑡≥𝑁^′𝜑(),𝑡0,𝑥|𝑟|^𝑗+1, (3.6) for all 𝑥∈𝑋, all 𝑡>0, and all integer 𝑗≥0. So 𝑁𝑓(𝑥)−𝑘^𝑛𝑓𝑥𝑘^𝑛,^𝑛−1_𝑗=0|𝑘|^𝑗+1|𝑟|^𝑗+1𝑡|2𝑘+2|=𝑁^𝑛−1_𝑗=0𝑘^𝑗+1𝑓𝑥𝑘^𝑗+1−𝑘^𝑗𝑓𝑥𝑘^𝑗,^𝑛−1_𝑗=0|𝑘|^𝑗+1|𝑟|^𝑗+1𝑡|2𝑘+2|≥min_{0≤𝑗≤𝑛−1}𝑁𝑘^𝑗+1𝑓𝑥𝑘^𝑗+1−𝑘^𝑗𝑓𝑥𝑘^𝑗,|𝑘|^𝑗+1|𝑟|^𝑗+1𝑡|2𝑘+2|≥min_{0≤𝑗≤𝑛−1}𝑁^′𝜑(0,𝑥),𝑡=𝑁^′𝜑(0,𝑥),𝑡, (3.7) which yields 𝑁𝑘^𝑛+𝑝𝑓𝑥𝑘^𝑛+𝑝−𝑘^𝑝𝑓𝑥𝑘^𝑝,^𝑛−1_𝑗=0|𝑘|^𝑗+𝑝+1|𝑟|^𝑗+1𝑡|2𝑘+2|≥𝑁^′𝜑𝑥0,2^𝑝,𝑡≥𝑁^′𝜑(),𝑡0,𝑥|𝑟|^𝑝, (3.8) for all 𝑥∈𝑋, 𝑡>0, and all integers 𝑛>0, 𝑝≥0. So 𝑁𝑘^𝑛+𝑝𝑓𝑥𝑘^𝑛+𝑝−𝑘^𝑝𝑓𝑥𝑘^𝑝,^𝑛−1_𝑗=0|𝑘|^𝑗+𝑝+1|𝑟|^𝑗+𝑝+1𝑡|2𝑘+2|≥𝑁^′𝜑(0,𝑥),𝑡, (3.9) for all 𝑥∈𝑋, 𝑡>0, and any integers 𝑛>0, 𝑝≥0. Hence one can obtain 𝑁𝑘^𝑛+𝑝𝑓𝑥𝑘^𝑛+𝑝−𝑘^𝑝𝑓𝑥𝑘^𝑝,𝑡≥𝑁^′𝜑(),𝑡0,𝑥∑^𝑛−1_𝑗=0|𝑘|^𝑗+𝑝+1|𝑟|^𝑗+𝑝+1/|2𝑘+2|, (3.10) for all 𝑥∈𝑋, 𝑡>0, and any integers 𝑛>0, 𝑝≥0. Since the series ∑^+∞_𝑗=0𝑘^𝑗|𝑟|^𝑗 is a convergent series, we see by taking the limit 𝑝→∞ in the last inequality that the sequence {𝑘^𝑛𝑓(𝑥/𝑘^𝑛)} is a Cauchy sequence in the fuzzy Banach space (𝑌,𝑁) and so it converges in 𝑌. Therefore a mapping 𝐴∶𝑋→𝑌 defined by 𝐴(𝑥)∶=𝑁−lim_𝑛→∞𝑘^𝑛𝑓(𝑥/𝑘^𝑛) is well defined for all 𝑥∈𝑋. This means that lim_𝑛→∞𝑁𝐴(𝑥)−𝑘^𝑛𝑓𝑥𝑘^𝑛,𝑡=1, (3.11) for all 𝑥∈𝑋 and all 𝑡>0. In addition, it follows from (3.10) that 𝑁𝑓(𝑥)−𝑘^𝑛𝑓𝑥𝑘^𝑛,𝑡≥𝑁^′𝜑(),𝑡0,𝑥∑^𝑛−1_𝑗=0|𝑘|^𝑗+1|𝑟|^𝑗+1/|2𝑘+2|, (3.12) for all 𝑥∈𝑋 and all 𝑡>0. So 𝑁𝑓(𝑥)−𝐴(𝑥),𝑡≥min𝑁𝑓(𝑥)−𝑘^𝑛𝑓𝑥𝑘^𝑛,(1−𝜖)𝑡,𝑁𝐴(𝑥)−𝑘^𝑛𝑓𝑥𝑘^𝑛,𝜖𝑡≥𝑁^′𝜑(0,𝑥),𝜖𝑡∑^𝑛−1_𝑗=0|𝑘|^𝑗+1|𝑟|^𝑗+1/|2𝑘+2|≥𝑁^′𝜑(),||𝑘𝑟0,𝑥2𝑘+2|(1−|||)𝜖𝑡||,𝑘𝑟 (3.13) for sufficiently large 𝑛 and for all 𝑥∈𝑋, 𝑡>0, and 𝜖 with 0<𝜖<1. Since 𝜀 is arbitrary and 𝑁^′ is left continuous, we obtain 𝑁𝑓(𝑥)−𝐴(𝑥),𝑡≥𝑁^′𝜑(),||𝑘𝑟𝑡0,𝑥2𝑘+2|(1−|||)|𝑘𝑟|, (3.14) for all 𝑥∈𝑋 and 𝑡>0. It follows from (3.1) that 𝑁𝑓𝑘^𝑛𝑥+𝑘𝑦𝑘^𝑛+𝑓𝑘^𝑛𝑥−𝑘𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥+𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥−𝑦𝑘^𝑛−2(𝑘+1)𝑘𝑓𝑘^𝑛+1𝑦𝑘^𝑛()𝑓𝑘+2𝑘+1^𝑛𝑦𝑘^𝑛,𝑡≥𝑁^′𝜑𝑘^𝑛𝑥,𝑘^𝑛𝑦,𝑡|𝑘|^𝑛≥𝑁^′𝜑,𝑡𝑥,𝑦|𝑟|^𝑛|𝑘|^𝑛⟶1as𝑛⟶+∞, (3.15) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Therefore, we obtain in view of (3.11) 𝑁𝐴𝑘𝑘−2()𝑥+𝑦+𝐴𝑥−𝑦−𝐴𝑘𝑥+𝑦−𝐴𝑘𝑥−𝑦𝑘+1𝑘𝐴()𝐴𝑦𝑁𝐴𝑘𝑘−2()𝑘𝑦+2𝑘+1,𝑡≥min𝑥+𝑦+𝐴𝑥−𝑦−𝐴𝑘𝑥+𝑦−𝐴𝑘𝑥−𝑦𝑘+1𝑘𝐴()𝐴𝑦−𝑓𝑘𝑘𝑦+2𝑘+1^𝑛𝑥+𝑘𝑦𝑘^𝑛+𝑓𝑘^𝑛𝑥−𝑘𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥+𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥−𝑦𝑘^𝑛−2(𝑘+1)𝑘𝑓𝑘^𝑛+1𝑦𝑘^𝑛()𝑓𝑘+2𝑘+1^𝑛𝑦𝑘^𝑛,𝑡2,𝑁𝑓𝑘^𝑛𝑥+𝑘𝑦𝑘^𝑛+𝑓𝑘^𝑛𝑥−𝑘𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥+𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥−𝑦𝑘^𝑛−2(𝑘+1)𝑘𝑓𝑘^𝑛+1𝑦𝑘^𝑛()𝑓𝑘+2𝑘+1^𝑛𝑦𝑘^𝑛,𝑡2𝑓𝑘=𝑁^𝑛𝑥+𝑘𝑦𝑘^𝑛+𝑓𝑘^𝑛𝑥−𝑘𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥+𝑦𝑘^𝑛−𝑓𝑘^𝑛𝑥−𝑦𝑘^𝑛−2(𝑘+1)𝑘𝑓𝑘^𝑛+1𝑦𝑘^𝑛()𝑓𝑘+2𝑘+1^𝑛𝑦𝑘^𝑛,𝑡2forsufficientlylarge𝑛≥𝑁^′𝜑,𝑡𝑥,𝑦2|𝑘|^𝑛|𝑟|^𝑛⟶1as𝑛⟶+∞, (3.16) for all 𝑥,𝑦∈𝑋 and all 𝑡>0, which implies that 𝐴𝑘𝑘+2()𝑥+𝑦+𝐴𝑥−𝑦=𝐴𝑘𝑥+𝑦+𝐴𝑘𝑥−𝑦𝑘+1𝑘𝐴𝑘𝑦−2(𝑘+1)𝐴𝑦. (3.17) Hence the mapping 𝐴∶𝑋→𝑌 is additive, as desired.
To prove the uniqueness, let there be another mapping 𝐿∶𝑋→𝑌 which satisfies the inequality (3.3). Since 𝐿(𝑘^𝑛𝑥)=𝑘^𝑛𝐿(𝑥) for all 𝑥∈𝑋, we have 𝑁(𝐴(𝑥)−𝐿(𝑥),𝑡)=𝑁𝑘^𝑛𝐴𝑥𝑘^𝑛−𝑘^𝑛𝐿𝑥𝑘^𝑛𝑁𝑘,𝑡≥min^𝑛𝐴𝑥𝑘^𝑛−𝑘^𝑛𝑓𝑥𝑘^𝑛,𝑡2,𝑁𝑘^𝑛𝑓𝑥𝑘^𝑛−𝑘^𝑛𝐿𝑥𝑘^𝑛,𝑡2≥𝑁^′𝜑𝑥0,𝑘^𝑛,|2𝑘+2|(1−|𝑘||𝑟|)𝑡2|𝑘|^𝑛+1|𝑟|≥𝑁^′𝜑(),||𝑘𝑟𝑡0,𝑥2𝑘+2|(1−|||)2|𝑘|^𝑛+1|𝑟|^𝑛+1⟶1as𝑛⟶∞, (3.18) for all 𝑡>0. Therefore 𝐴(𝑥)=𝐿(𝑥) for all 𝑥∈𝑋. This completes the proof.

Corollary 3.2. Let 𝑋 be a normed space and let (ℝ,𝑁^′) be a fuzzy Banach space. Assume that there exist real numbers 𝜃≥0 and 𝑝>1 such that an odd mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the following inequality: 𝑁𝑓𝑘𝑘−2()𝑥+𝑦+𝑓𝑥−𝑦−𝑓𝑘𝑥+𝑦−𝑓𝑘𝑥−𝑦𝑘+1𝑘𝑓𝑘𝑦+2(𝑘+1)𝑓𝑦,𝑡≥𝑁^′𝜃‖𝑥‖^𝑝+𝑦^𝑝,𝑡, (3.19) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then there is a unique additive mapping 𝐴∶𝑋→𝑌 satisfying (1.8) and the inequality 𝑁𝑓(𝑥)−𝐴(𝑥),𝑡≥𝑁^′𝜃‖𝑥‖^𝑝||,|𝑘|2𝑘+2^𝑝−|𝑘||𝑘|𝑡. (3.20)

Proof. Let 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) and |𝑟|=|𝑘|^−𝑝. Applying Theorem 3.1, we get desired results.

Theorem 3.3. Let 𝑓∶𝑋→𝑌 be an odd mapping with 𝑓(0)=0 satisfying the inequality (3.1) and let 𝜑∶𝑋²→𝑍 be a mapping for which there exists a constant 𝑟∈ℝ satisfying 0<|𝑟|<|𝑘| such that 𝑁^′𝜑𝑥,𝑦,|𝑟|𝑡≥𝑁^′𝜑𝑥𝑘,𝑦𝑘,𝑡, (3.21) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 satisfying (1.8) and the following inequality: 𝑁𝑓(𝑥)−𝐴(𝑥),𝑡≥𝑁^′𝜑(),|𝑘|−|𝑟𝑡0,𝑥2𝑘+2|(||)|𝑘|, (3.22) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. It follows from (3.5) that 𝑁𝑓(𝑘𝑥)𝑘(𝑥),𝑡−𝑓|2𝑘+2|≥𝑁^′𝜑(0,𝑥),𝑡, (3.23) for all 𝑥∈𝑋 and all 𝑡>0. Replacing 𝑥 by 𝑘^𝑛𝑥 in (3.41), we obtain 𝑁𝑓𝑘^𝑛+1𝑥𝑘^𝑛+1−𝑓𝑘^𝑛𝑥𝑘^𝑛,𝑡|2𝑘+2|𝑘^𝑛≥𝑁^′𝜑0,𝑘^𝑛𝑥,𝑡≥𝑁^′𝜑(),𝑡0,𝑥|𝑟|^𝑛. (3.24) So 𝑁𝑓𝑘^𝑛+1𝑥𝑘^𝑛+1−𝑓𝑘^𝑛𝑥𝑘^𝑛,|𝑟|^𝑛𝑡|2𝑘+2||𝑘|^𝑛≥𝑁^′𝜑(0,𝑥),𝑡, (3.25) for all 𝑥∈𝑋 and all 𝑡>0. Proceeding as in the proof of Theorem 3.1, we obtain that 𝑁𝑓(𝑥)−𝑓𝑘^𝑛𝑥𝑘^𝑛,^𝑛−1_𝑗=0|𝑟|^𝑗𝑡|2𝑘+2||𝑘|^𝑗≥𝑁^′𝜑(0,𝑥),𝑡, (3.26) for all 𝑥∈𝑋, all 𝑡>0, and any integer 𝑛>0. So 𝑁𝑓(𝑥)−𝑓𝑘^𝑛𝑥𝑘^𝑛,𝑡≥𝑁^′𝜑(),𝑡0,𝑥∑^𝑛−1_𝑗=0|𝑟|^𝑗/|2𝑘+2||𝑘|^𝑗. (3.27) The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.4. Let 𝑋 be a normed space and let (ℝ,𝑁^′) be a fuzzy Banach space. Assume that there exist real numbers 𝜃≥0 and 0<𝑝<1 such that an odd mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies (3.19). Then there exists a unique additive mapping 𝐴∶𝑋→𝑌 satisfying (1.8) and the inequality 𝑁𝑓(𝑥)−𝐴(𝑥),𝑡≥𝑁^′𝜑(),|||𝑘|−|𝑘|0,𝑥2𝑘+2^𝑝𝑡|𝑘|. (3.28)

Proof. Let 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) and |𝑟|=|𝑘|^𝑝. Applying Theorem 3.3, we get the desired results.

Theorem 3.5. Let 𝑓∶𝑋→𝑌 be an even mapping with 𝑓(0)=0 satisfying the inequality (3.1) and let 𝜑∶𝑋²→𝑍 be a mapping for which there exists a constant 𝑟∈ℝ such that 0<|𝑟|<1/𝑘² and that 𝑁^′𝜑𝑥𝑘,𝑦𝑘,𝑡≥𝑁^′𝜑,𝑡𝑥,𝑦|𝑟|, (3.29) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (1.8) and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜑(),2𝑘0,𝑥1−²𝑟𝑡|𝑘𝑟|, (3.30) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. Replacing 𝑥 by 𝑘𝑥 in (3.1), we get 𝑁𝑓𝑘𝑘−2()𝑥+𝑦+𝑓𝑥−𝑦−𝑓𝑘𝑥+𝑦−𝑓𝑘𝑥−𝑦𝑘+1𝑘𝑓𝑘𝑦+2(𝑘+1)𝑓𝑦,𝑡≥𝑁^′𝜑𝑘𝑥,𝑦,𝑡, (3.31) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Putting 𝑥=0 and replacing 𝑦 by 𝑥 in (3.31), we have 𝑁𝑓(𝑘𝑥)𝑘²(𝑥),𝑡−𝑓|2𝑘|≥𝑁^′𝜑(0,𝑥),𝑡, (3.32) for all 𝑥∈𝑋 and all 𝑡>0. Replacing 𝑥 by 𝑥/𝑘 in (3.32), we find 𝑁𝑘²𝑓𝑥𝑘(𝑥),|𝑘|𝑡−𝑓2≥𝑁^′𝜑𝑥0,𝑘,𝑡, (3.33) for all 𝑥∈𝑋 and all 𝑡>0. Also, replacing 𝑥 by 𝑥/𝑘^𝑛 in (3.33), we obtain 𝑁𝑘^2𝑛+2𝑓𝑥𝑘^𝑛−𝑘^2𝑛𝑓𝑥𝑘^𝑛,|𝑘|^2𝑛+1𝑡2≥𝑁^′𝜑𝑥0,𝑘^𝑛+1,𝑡≥𝑁^′𝜑(),𝑡0,𝑥|𝑟|^𝑛+1. (3.34) So 𝑁𝑘^2𝑛+2𝑓𝑥𝑘^𝑛−𝑘^2𝑛𝑓𝑥𝑘^𝑛,|𝑘|^2𝑛+1|𝑟|^𝑛+1𝑡2≥𝑁^′𝜑(0,𝑥),𝑡, (3.35) for all 𝑥∈𝑋 and all 𝑡>0. Proceeding as in the proof of Theorem 3.1, we obtain that 𝑁𝑓(𝑥)−𝑘^2𝑛𝑓𝑥𝑘^𝑛,^𝑛−1_𝑗=0|𝑘|^2𝑗+1|𝑟|^𝑗+1𝑡2≥𝑁^′𝜑(0,𝑥),𝑡, (3.36) for all 𝑥∈𝑋, all 𝑡>0, and any integer 𝑛>0. So 𝑁𝑓(𝑥)−𝑘^2𝑛𝑓𝑥𝑘^𝑛,𝑡≥𝑁^′𝜑(),𝑡0,𝑥∑^𝑛−1_𝑗=0|𝑘|^2𝑗+1|𝑟|^𝑗+1𝑡/2. (3.37) The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.6. Let 𝑋 be a normed space and let (ℝ,𝑁^′) be a fuzzy Banach space. Assume that there exist real numbers 𝜃≥0 and 𝑝>1 such that an even mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality (3.19). Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (1.8) and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜃‖𝑥‖^𝑝,2𝑘^2𝑝−𝑘²𝑡|𝑘|. (3.38)

Proof. Let 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) and |𝑟|=|𝑘|^−2𝑝. Applying Theorem 3.5, we get the desired results.

Theorem 3.7. Assume that an even mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality (3.1) and 𝜑∶𝑋²→𝑍 is a mapping for which there is a constant 𝑟∈ℝ satisfying 0<|𝑟|<𝑘² such that 𝑁^′𝜑𝑥,𝑦,|𝑟|𝑡≥𝑁^′𝜑𝑥𝑘,𝑦𝑘,𝑡, (3.39) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (1.8) and the following inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜑(),2𝑘0,𝑥²−|𝑟|𝑡|𝑘|, (3.40) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. It follows from (3.32) that 𝑁𝑓(𝑘𝑥)𝑘²(𝑥),𝑡−𝑓|2𝑘|≥𝑁^′𝜑(0,𝑥),𝑡, (3.41) for all 𝑥∈𝑋 and all 𝑡>0. Replacing 𝑥 by 𝑘^𝑛𝑥 in (3.41), we obtain 𝑁𝑓𝑘^𝑛+1𝑥𝑘^2𝑛+2−𝑓𝑘^𝑛𝑥𝑘^2𝑛,𝑡2|𝑘|^2𝑛+1≥𝑁^′𝜑0,𝑘^𝑛𝑥,𝑡≥𝑁^′𝜑(),𝑡0,𝑥|𝑟|^𝑛, (3.42) for all 𝑥∈𝑋 and all 𝑡>0. So 𝑁𝑓𝑘^𝑛+1𝑥𝑘^2𝑛+2−𝑓𝑘^𝑛𝑥𝑘^2𝑛,|𝑟|^𝑛𝑡2|𝑘|^2𝑛+1≥𝑁^′𝜑(0,𝑥),𝑡, (3.43) for all 𝑥∈𝑋 and all 𝑡>0. So 𝑁𝑓(𝑥)−𝑓𝑘^𝑛𝑥𝑘^2𝑛,𝑡≥𝑁^′𝜑(),𝑡0,𝑥∑^𝑛−1_𝑗=0|𝑟|^𝑗𝑡/2|𝑘|^2𝑗+1. (3.44) The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.8. Let 𝑋 be a normed space and let (ℝ,𝑁^′) be a fuzzy Banach space. Assume that there exist real numbers 𝜃≥0 and 0<𝑝<1 such that an even mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies (3.19). Then there is a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (1.8) and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜑(),2𝑘0,𝑥²−𝑘^2𝑝𝑡|𝑘|, (3.45) for all 𝑥∈𝑋, all 𝑡>0.

Proof. Let 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) and |𝑟|=𝑘^2𝑝. Applying Theorem 3.7, we get the desired results.

Acknowledgment

The fifth 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 no. 2010-0010243).