On the Stability Problem in Fuzzy Banach Space

Abstract

We investigate the generalized Ulam-Hyers stability of the Cauchy functional equation and pose two open problems in fuzzy Banach space.

1. Introduction and Preliminaries

In 1940, Ulam [1] asked the first question on the stability problem. In 1941, Hyers [2] solved the problem of Ulam. This result was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Th. M. Rassias). 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 (1.1) for all 𝑥,𝑦∈𝐸, where 𝜖 and 𝑝 are constants with 𝜖>0 and 0≤𝑝<1. Then, the limit 𝐿(𝑥)=lim_𝑛→∞(1/2^𝑛)𝑓(2^𝑛𝑥) exists for all 𝑥∈𝐸 and 𝐿∶𝐸→𝐸^′ is the unique additive mapping which satisfies 𝑓(𝑥)−𝐿(𝑥)≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (1.2) for all 𝑥∈𝐸. Also, if for each 𝑥∈𝐸 the function 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ, then 𝐿 is linear.

In 1990, Th. M. Rassias [5] 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 [6] gave an affirmative solution to this question for 𝑝>1. It was shown by Gajda [6], as well as by Th. M. Rassias and Šemrl [7], that one cannot prove a Th. M. Rassias type theorem when 𝑝=1. Găvruţa [8] proved that the function 𝑓(𝑥)=𝑥ln|𝑥|, if 𝑥≠0 and 𝑓(0)=0 satisfies (1.1) with 𝜖=𝑝=1 but sup_𝑥≠0𝑓(𝑥)−𝐴(𝑥)|𝑥|≥sup_𝑛∈ℕ|𝑛ln𝑛−𝐴(𝑛)|𝑛=sup_𝑛∈ℕ|ln𝑛−𝐴(1)|=∞ (1.3) for any additive function 𝐴∶ℝ→ℝ. J. M. Rassias [9] replaced the factor ‖𝑥‖^𝑝+‖𝑦‖^𝑝 by ‖𝑥‖^𝑝1‖𝑦‖^𝑝2 for 𝑝₁,𝑝₂∈ℝ with 𝑝₁+𝑝₂≠1 (see also [10, 11]) and has obtained the following theorem.

Theorem 1.2. Let 𝑋 be a real normed linear space and 𝑌 a real complete normed linear space. Assume that 𝑓∶𝑋→𝑌 is an approximately additive mapping for which there exist constants 𝜃≥0 and 𝑝=𝑝₁+𝑝₂≠1 such that 𝑓 satisfies the inequality: 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜃‖𝑥‖^𝑝1𝑦^𝑝2 (1.4) for all 𝑥,𝑦∈𝑋. Then, there exists a unique additive mapping 𝐿∶𝑋→𝑌 satisfying 𝑓(𝑥)(𝑥≤𝜃−𝐿)|2^𝑝|‖𝑥‖−2^𝑝 (1.5) for all 𝑥∈𝑋. If, in addition, 𝑓∶𝑋→𝑌 is a mapping such that the transformation 𝑡→𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each fixed 𝑥∈𝑋, then 𝐿 is an ℝ-linear mapping.

In the case 𝑝=1, we do not have stability [12]. In 1994, a further generalization of Th. M. Rassias’ Theorem was obtained by Găvruţa [13], in which he replaced the bound 𝜖(‖𝑥‖^𝑝+‖𝑦‖^𝑝) by a general control function 𝜑(𝑥,𝑦). Isac and Th. M. Rassias [14] replaced the factor ‖𝑥‖^𝑝+‖𝑦‖^𝑝 by ‖𝑥‖^𝑝1+‖𝑦‖^𝑝2 in Theorem 1.1 and solved stability problem when 𝑝₂≤𝑝₁<1 or 1<𝑝₂≤𝑝₁, also they asked the question whether such a theorem can be proved for 𝑝₂<1<𝑝₁. Găvruţa [8] gave a negative answer to this question. Isac and Th. M. Rassias [15] applied the Ulam-Hyers-Rassias stability theory to prove fixed point theorems and study some new applications in nonlinear analysis. During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of the generalized Ulam-Hyers stability to a number of functional equations and mappings (see [16–40]). We also refer the readers to the books of Czerwik [41] and Hyers et al. [42].

Th. M. Rassias [43] has obtained the following theorem and posed a problem.

Theorem 1.3. Let 𝐸₁ and 𝐸₂ be two Banach spaces, and let 𝑓∶𝐸₁→𝐸₂ be a mapping such that 𝑓(𝑡𝑥) is continuous in 𝑡 for each fixed 𝑥. Assume that there exist 𝜃≥0 and 𝑝∈[0,1) such that 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜃‖𝑥‖^𝑝+𝑦^𝑝 (1.6) for all 𝑥,𝑦∈𝑋. Let 𝑘 be a positive integer 𝑘>2. Then, there exists a unique linear mapping 𝑇∶𝐸₁→𝐸₂ such that 𝑓(𝑥)−𝑇(𝑥)≤𝑘𝜃𝑘−𝑘^𝑝‖𝑥‖^𝑝𝑠𝑘,𝑝 (1.7) for all 𝑥∈𝑋, where 𝑠1𝑘,𝑝=1+𝑘^𝑘−1_𝑚=2𝑚^𝑝. (1.8)

Th. M. Rassias Problem
What is the best possible value of 𝑘 in Theorem 1.3?
Găvruţa et al. have given a generalization of [13] and have answered to Th. M. Rassias problem [44].
In [45], J. M. Rassias et al. have investigated the generalized Ulam-Hyers “product-sum” stability of functional equations and have obtained the following theorem.

Theorem 1.4 (see [45]). Let 𝑓∶𝐸→𝐹 be a mapping which satisfies the inequality 𝑓𝑚𝑥+𝑦+𝑓𝑚𝑥−𝑦−2𝑓𝑥+𝑦−2𝑓𝑥−𝑦−2𝑚²−2𝑓(𝑥)+2𝑓𝑦_𝐹≤𝜖‖𝑥‖^𝑝_𝐸𝑦^𝑝_𝐸+‖𝑥‖_𝐸^2𝑝+𝑦_𝐸^2𝑝 (1.9)
for all 𝑥,𝑦∈𝐸 with 𝑥⟂𝑦, where 𝜖 and p are constants with 𝜖,𝑝>0 and either 𝑚>1, 𝑝<1 or 𝑚<1, 𝑝>1 with 𝑚≠0,𝑚≠±1, 𝑚≠√±2, and −1≠|𝑚|^𝑝−1<1. Then, the limit lim_𝑛→∞𝑚^−2𝑛𝑓(𝑚^𝑛𝑥) exists for all 𝑥∈𝐸 and 𝑄∶𝐸→𝐹 is the unique orthogonally Euler-Lagrange quadratic mapping such that 𝑓(𝑥)−𝑄(𝑥)_𝐹≤𝜖2𝑚²−𝑚^2𝑝‖𝑥‖_𝐸^2𝑝 (1.10) for all 𝑥∈𝐸.

Note that the mixed “product-sum” function was introduced by J. M. Rassias in 2008-2009 [46–48].

We recall some basic facts concerning fuzzy normed space.

Let 𝑋 be a real linear space. A function 𝑁∶𝑋×ℝ→[0,1] (so-called fuzzy subset) is said to be a fuzzy norm on 𝑋 if for all 𝑥,𝑦∈𝑋 and all 𝑐,𝑡∈ℝ,

(𝑁1)𝑁(𝑥,𝑐)=0 for 𝑐≤0;

(𝑁2)𝑥=0 if and only if 𝑁(𝑥,𝑐)=1 for all 𝑐>0;

(𝑁3)𝑁(𝑐𝑥,𝑡)=𝑁(𝑥,𝑡/|𝑐|) if 𝑐≠0;

(𝑁4)𝑁(𝑥+𝑦,t)≥min{𝑁(𝑥,𝑡),𝑁(𝑦,𝑡)};

(𝑁5)𝑁(𝑥,⋅) is a nondecreasing function of ℝ and lim_𝑡→∞𝑁(𝑥,𝑡)=1. (1.11)

The pair (𝑋,𝑁) is called a fuzzy normed linear space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [49–51].

Let (𝑋,𝑁) be a fuzzy normed space and let {𝑥_𝑛} be a sequence in 𝑋. Then, {𝑥_𝑛} is said to be convergent if there exists 𝑥∈𝑋 such that lim_𝑛→∞𝑁(𝑥_𝑛−𝑥,𝑡)=1 for all 𝑡>0. In that case, 𝑥 is called the limit of the sequence {𝑥_𝑛} and we denote it by lim_𝑛→∞𝑥_𝑛=𝑥.

A sequence {𝑥_𝑛} in a fuzzy normed space (𝑋,𝑁) is called Cauchy if, for each 𝜖>0 and 𝛿>0, one can find some 𝑛₀ such that 𝑁𝑥_𝑚−𝑥_𝑛,𝛿>1−𝜖 (1.12)

for all 𝑛,𝑚≥𝑛₀.

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

Stability of Cauchy, Jensen, quadratic, and cubic function equation in fuzzy normed spaces have first been investigated in [50–53].

In this paper, we give a generalization of the results from [13] and pose two open problems in fuzzy Banach space. For convenience, we use the following abbreviation for a given mapping 𝑓: 𝐷𝑓𝑥,𝑦=∶𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦. (1.13)

2. Stability of the Cauchy Functional Equation

Hereafter, unless otherwise stated, we will assume that 𝑋 is real vector space, (𝑌,𝑁) is a complete fuzzy norm space and 𝑘 is a fixed integer greater than 1.

Theorem 2.1. Let (𝑍,𝑁^′) be a fuzzy normed space and 𝜑∶𝑋×𝑋→𝑍 be a mapping such that, 𝜑(𝑘𝑥,𝑘𝑦)=𝛼𝜑(𝑥,𝑦) for some 𝛼 with 0<𝛼<𝑘. Suppose that 𝑓∶𝑋→𝑌 be mapping such that 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′𝜑𝑥,𝑦,𝑡 (2.1) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡. Then, there is a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 such that 𝑇_𝑘(𝑥)=lim_𝑛→∞𝑓(𝑘^𝑛𝑥)/𝑘^𝑛 and 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑀_𝑘(𝑥,(𝑘−𝛼)𝑡), (2.2) where 𝑀_𝑘(𝑥,𝑡)∶=min{𝑁^′(𝜑(𝑥,𝑖𝑥),𝑡)∶1≤𝑖<𝑘}.

Proof. By induction on 𝑘, we show that 𝑁𝑓(𝑘𝑥)−𝑘𝑓(𝑥),𝑡≥𝑀_𝑘(𝑥,𝑡)∶=min𝑁^′𝜑(𝑥,𝑖𝑥),𝑡∶1≤𝑖<𝑘 (2.3) for all 𝑥∈𝑋 and all positive real number 𝑡. Letting 𝑦=𝑥 in (2.1), we get 𝑁𝑓(2𝑥)−2𝑓(𝑥),𝑡≥𝑁^′𝜑(𝑥,𝑥),𝑡. (2.4) So we get (2.3) for 𝑘=2.
Assume that (2.3) holds for 𝑘 with 𝑘>2. Letting 𝑦=𝑘𝑥 in (2.1), we get 𝑁𝑓((𝑘+1)𝑥)−𝑓(𝑥)−𝑓(𝑘𝑥),𝑡≥𝑁^′𝜑(𝑥,𝑘𝑥),𝑡. (2.5) for all 𝑥∈𝑋. By using (2.3) and (2.5), we get (2.3) for 𝑘+1 and this completes the induction argument. Replacing 𝑥 by 𝑘^𝑛𝑥 in (2.3), we get 𝑁𝑓𝑘^𝑛+1𝑥−𝑘𝑓𝑘^𝑛𝑥,𝑡≥𝑀_𝑘𝑘^𝑛𝑥,𝑡. (2.6) Thus 𝑁𝑓𝑘^𝑛+1𝑥𝑘^𝑛+1−𝑓𝑘^𝑛𝑥𝑘^𝑛,𝑡𝑘^𝑛+1≥𝑀_𝑘𝑡𝑥,𝛼^𝑛 (2.7) for all 𝑥∈𝑋 and all positive real number 𝑡. Hence, 𝑁1𝑘^𝑛+1𝑓𝑘^𝑛+1𝑥−1𝑘^𝑚𝑓𝑘^𝑚𝑥,^𝑛_𝑖=𝑚𝛼^𝑖𝑘^𝑖+1𝑡≥𝑁^𝑛_𝑖=𝑚1𝑘^𝑖+1𝑓𝑘^𝑖+1𝑥−1𝑘^𝑖𝑓𝑘^𝑖𝑥,^𝑛_𝑖=𝑚𝛼^𝑖𝑘^𝑖+1𝑡≥min^𝑛_𝑖=𝑚𝑁1𝑘^𝑖+1𝑓𝑘^𝑖+1𝑥−1𝑘^𝑖𝑓𝑘^𝑖𝑥,𝛼^𝑖𝑘^𝑖+1𝑡≥𝑀_𝑘(𝑥,𝑡). (2.8) Let 𝜖>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑀_𝑘(𝑥,𝑡)=1, there is some 𝑡₀>0 such that 𝑀_𝑘(𝑥,𝑡₀)>1−𝜖. Since ∑^∞_𝑛=0(𝛼^𝑛/𝑘^𝑛)𝑡₀<∞, there is some 𝑛₀∈𝑁 such that ∑^𝑛_𝑖=𝑚(𝛼^𝑖/𝑘^𝑖)𝑡₀<𝑘𝛿 for all 𝑛>𝑚≥𝑛₀. It follows that 𝑁1𝑘^𝑛+1𝑓𝑘^𝑛+1𝑥−1𝑘^𝑚𝑓𝑘^𝑚𝑥1,𝛿≥𝑁𝑘^𝑛+1𝑓𝑘^𝑛+1𝑥−1𝑘^𝑚𝑓𝑘^𝑚𝑥,^𝑛_𝑖=𝑚𝛼^𝑖𝑘^𝑖+1𝑡₀≥𝑀_𝑘𝑥,𝑡₀>1−𝜖 (2.9)
for all 𝑥∈𝑋 and all nonnegative integers 𝑛 and 𝑚 with 𝑛>𝑚≥𝑛₀. Therefore, the sequence {(1/𝑘^𝑛)𝑓(𝑘^𝑛𝑥)} is a Cauchy sequence in (𝑌,𝑁) for all 𝑥∈𝑋. Since (𝑌,𝑁) is complete, the sequence {(1/𝑘^𝑛)𝑓(𝑘^𝑛𝑥)} converges in 𝑌 for all 𝑥∈𝑋. So one can define the mapping 𝑇_𝑘∶𝑋→𝑌 by 𝑇_𝑘(𝑥)∶=lim_𝑛→∞1𝑘^𝑛𝑓𝑘^𝑛𝑥 (2.10) for all 𝑥∈𝑋. Now, we show that 𝑇_𝑘 is an additive mapping. It follows from (2.1) and (2.10) that 𝑁𝐷𝑇_𝑘𝑥,𝑦,𝑡=lim_𝑛→∞𝑁𝐷𝑓𝑘^𝑛𝑥,𝑘^𝑛𝑦𝑘^𝑛,𝑡≥lim_𝑛→∞𝑁^′𝜑𝑘^𝑛𝑥,𝑘^𝑛𝑦𝑘^𝑛,𝑡=lim_𝑛→∞𝑁^′𝜑,𝑘𝑥,𝑦^𝑛𝛼^𝑛𝑡=1 (2.11)
for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡. Therefore, the mapping 𝑇_𝑘 is additive. Moreover, if we put 𝑚=0 in (2.8), we observe that 𝑁1𝑘^𝑛+1𝑓𝑘^𝑛+1𝑥−𝑓(𝑥),^𝑛_𝑖=0𝛼^𝑖𝑘^𝑖+1𝑡≥𝑀_𝑘(𝑥,𝑡). (2.12) Therefore, 𝑁1𝑘^𝑛+1𝑓𝑘^𝑛+1𝑥−𝑓(𝑥),𝑡≥𝑀_𝑘𝑡𝑥,∑^𝑛_𝑖=0𝛼^𝑖/𝑘^𝑖+1. (2.13) It follows from (2.13), for large enough 𝑛, that 𝑁𝑇_𝑘(𝑥)(𝑥)𝑁𝑓𝑘−𝑓,𝑡≥min^𝑛+1𝑥𝑘^𝑛+1−𝑓(𝑥),𝑡,𝑁𝑇_𝑘(𝑥)−𝑓𝑘^𝑛+1𝑥𝑘^𝑛+1,𝑡≥𝑀_𝑘𝑡𝑥,∑^𝑛_𝑖=0𝛼^𝑖/𝑘^𝑖+1≥𝑀_𝑘(𝑥,(𝑘−𝛼)𝑡). (2.14) Now, we show that 𝑇_𝑘 is unique. Let 𝑇^′ be another additive mapping from 𝑋 into 𝑌, which satisfies the required inequality. Then, for each 𝑥∈𝑋 and 𝑡>0, we have 𝑁𝑇_𝑘(𝑥)−𝑇^′(𝑥),𝑡≥min𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡,𝑁𝑓(𝑥)−𝑇^′(𝑥),𝑡≥𝑀_𝑘(𝑥,(𝑘−𝛼)𝑡). (2.15) So, 𝑁𝑇_𝑘(𝑥)−𝑇^′(𝑥)𝑇,𝑡=𝑁^𝑘𝑘^𝑛𝑥𝑘^𝑛−𝑇^′𝑘^𝑛𝑥𝑘^𝑛𝑇,𝑡=𝑁_𝑘𝑘^𝑛𝑥−𝑇^′𝑘^𝑛𝑥,𝑘^𝑛𝑡≥𝑀_𝑘𝑘^𝑛𝑥,(𝑘−𝛼)𝑘^𝑛𝑡≥𝑀_𝑘()𝑘𝑥,𝑘−𝛼^𝑛𝛼^𝑛𝑡. (2.16) Hence, the right-hand side of the above inequality tends to 1 as 𝑛→∞. It follows that 𝑇_𝑘(𝑥)=𝑇^′(𝑥) for all 𝑥∈𝑋.

Theorem 2.2. Let (𝑍,𝑁^′) be a fuzzy normed space and, Φ∶𝑋×𝑋→𝑍 be a mapping such that Φ(𝑘⁻¹𝑥,𝑘⁻¹𝑦)=𝛼⁻¹Φ(𝑥,𝑦) for some 𝛼 with 𝛼>𝑘. Suppose that 𝑓∶𝑋→𝑌 be mapping such that 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′Φ𝑥,𝑦,𝑡 (2.17) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡. Then, there is a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 such that 𝑇_𝑘(𝑥)=lim_𝑛→∞𝑘^𝑛𝑓(𝑥/𝑘^𝑛) and 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑀_𝑘(𝑥,(𝛼−𝑘)𝑡), (2.18) where 𝑀_𝑘(𝑥,𝑡)∶=min{𝑁^′(Φ(𝑥,𝑖𝑥),𝑡)∶1≤𝑖<𝑘}.

Proof. Similarly to the proof of Theorem 2.1, we have 𝑁𝑓(𝑘𝑥)−𝑘𝑓(𝑥),𝑡≥𝑀_𝑘(𝑥,𝑡) (2.19) for all 𝑥∈𝑋 and all positive real number 𝑡. Replacing 𝑥 by 𝑥/𝑘^𝑛+1 in (2.19), we get 𝑁𝑓𝑥𝑘^𝑛𝑥−𝑘𝑓𝑘^𝑛+1,𝑡≥𝑀_𝑘𝑥𝑘^𝑛+1,𝑡. (2.20) Thus, 𝑁𝑘^𝑛𝑓𝑥𝑘^𝑛−𝑘^𝑛+1𝑓𝑥𝑘^𝑛+1,𝑘^𝑛𝑡≥𝑀_𝑘𝑥,𝛼^𝑛+1𝑡 (2.21) for all 𝑥∈𝑋 and all positive real number 𝑡. Hence, 𝑁𝑘^𝑛+1𝑓𝑥𝑘^𝑛+1−𝑘^𝑚𝑓𝑥𝑘^𝑚,^𝑛_𝑖=𝑚𝑘^𝑖𝛼^𝑖+1𝑡≥𝑁^𝑛_𝑖=𝑚𝑘^𝑖+1𝑓𝑥𝑘^𝑖+1−𝑘^𝑖𝑓𝑥𝑘^𝑖,^𝑛_𝑖=𝑚𝑘^𝑖𝛼^𝑖+1𝑡≥min^𝑛_𝑖=𝑚𝑁𝑘^𝑖+1𝑓𝑥𝑘^𝑖+1−𝑘^𝑖𝑓𝑥𝑘^𝑖,𝑘^𝑖𝛼^𝑖+1𝑡≥𝑀_𝑘(𝑥,𝑡). (2.22)
Let 𝜖>0 and 𝛿>0 be given. Since lim_𝑡→∞𝑀_𝑘(𝑥,𝑡)=1, there is some 𝑡₀>0 such that 𝑀_𝑘(𝑥,𝑡₀)>1−𝜖. Since ∑^∞_𝑛=0(𝑘^𝑛/𝛼^𝑛)𝑡₀<∞, there is some 𝑛₀∈𝑁 such that ∑^𝑛_𝑖=𝑚(𝑘^𝑖/𝛼^𝑖)𝑡₀<𝛼𝛿 for all 𝑛>𝑚≥𝑛₀. It follows from (2.22) that 𝑁𝑘^𝑛+1𝑓𝑥𝑘^𝑛+1−𝑘^𝑚𝑓𝑥𝑘^𝑚,𝛿≥𝑁𝑘^𝑛+1𝑓𝑥𝑘^𝑛+1−𝑘^𝑚𝑓𝑥𝑘^𝑚,^𝑛_𝑖=𝑚𝑘^𝑖𝛼^𝑖+1𝑡₀≥𝑀_𝑘𝑥,𝑡₀>1−𝜖 (2.23) for all 𝑥∈𝑋 and all nonnegative integers 𝑛 and 𝑚 with 𝑛>𝑚≥𝑛₀. Therefore, the sequence {𝑘^𝑛𝑓(𝑥/𝑘^𝑛)} is a Cauchy sequence in (𝑌,𝑁) for all 𝑥∈𝑋. Since (𝑌,𝑁) is complete, the sequence {𝑘^𝑛𝑓(𝑥/𝑘^𝑛)} converges in 𝑌 for all 𝑥∈𝑋. So one can define the mapping 𝑇_𝑘∶𝑋→𝑌 by 𝑇_𝑘(𝑥)∶=lim_𝑛→∞𝑘^𝑛𝑓𝑥𝑘^𝑛 (2.24) for all 𝑥∈𝑋. The rest of the proof is similar to the proof of Theorem 2.1

Theorem 2.3. Let 𝑋 be a normed space, let (𝑍,𝑁^′) be a fuzzy normed space, and let 𝜓∶[0,∞)→[0,∞) be a function such that
(1)𝜓(𝑡𝑠)=𝜓(𝑡)𝜓(𝑠),
(2)𝜓(𝑡)<𝑡 for all 𝑡>1.
Suppose that a mapping 𝑓∶𝑋→𝑌 satisfies the inequality: 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′𝜓(‖𝑥‖)+𝜓𝑦𝑧₀,𝑡 (2.25) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡, where 𝑧₀ is a fixed vector of 𝑍. Then, there exists a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 satisfying 𝑇_𝑘(𝑥)∶=lim_𝑛→∞(𝑓(𝑘^𝑛𝑥)/𝑘^𝑛) and 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑁^′𝜓(‖𝑥‖)𝑧₀,𝑘−𝜓(𝑘)𝜎_𝑘𝜓𝑡 (2.26) for all 𝑥∈𝑋, where 𝜎_𝑘(𝜓)=max{1+𝜓(𝑖)∶1≤𝑖<𝑘}. Moreover, 𝑇_𝑘=𝑇₂ for all 𝑘≥2.

Proof. Let 𝜑𝑥,𝑦=𝜓(‖𝑥‖)+𝜓‖𝑦‖𝑧₀ (2.27) for all 𝑥,𝑦∈𝑋. So, 𝜑𝑘𝑥,𝑘𝑦=𝜓(𝑘)𝜑𝑥,𝑦. (2.28) where 𝜓(𝑘)<𝑘. By using Theorem 2.1, we can get (2.26). Now, we show that 𝑇_𝑘=𝑇₂. It follows from (1) that 𝜓(𝑘^𝑛)=(𝜓(𝑘))^𝑛. Replacing 𝑥 by 2^𝑛𝑥 in (2.26), we get 𝑁𝑇_𝑘2^𝑛𝑥−𝑓2^𝑛𝑥,𝑡≥𝑁^′𝜓2^𝑛𝑥𝑧₀,𝑘−𝜓(𝑘)𝜎_𝑘𝜓𝑡 (2.29) for all 𝑥∈𝑋. So we have 𝑁𝑇_𝑘(𝑥)−𝑓2^𝑛𝑥2^𝑛,𝑡≥𝑁^′𝜓(‖𝑥‖)𝑧₀,𝑘−𝜓(𝑘)𝜎_𝑘𝜓𝜓(2^𝑛)2^𝑛𝑡 (2.30) Using (2) and passing the limit 𝑛→∞ in (2.30), we get 𝑇_𝑘=𝑇₂.

Theorem 2.4. Let 𝑋 be a normed space, let (𝑍,𝑁^′) be a fuzzy normed space, and let 𝜓∶[0,∞)→[0,∞) be a function such that
(1)𝜓(𝑡𝑠)=𝜓(𝑡)𝜓(𝑠),
(2)𝜓(𝑡)>𝑡 for all 𝑡>1.
Suppose that a mapping 𝑓∶𝑋→𝑌 satisfies the inequality: 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′𝜓(‖𝑥‖)+𝜓𝑦𝑧₀,𝑡 (2.31) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡, where 𝑧₀ is a fixed vector of 𝑍. Then, there exists a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 satisfying 𝑇_𝑘(𝑥)∶=lim_𝑛→∞𝑘^𝑛𝑓(𝑥/𝑘^𝑛) and 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑁^′𝜓(‖𝑥‖)𝑧₀,𝜓(𝑘)−𝑘𝜎_𝑘𝜓𝑡 (2.32) for all 𝑥∈𝑋, where 𝜎_𝑘𝜓=max1+𝜓(𝑖)∶1≤𝑖<𝑘. (2.33)
Moreover, 𝑇_𝑘=𝑇₂   for all 𝑘≥2.

Proof. Let Φ𝑥,𝑦=𝜓(‖𝑥‖)+𝜓𝑦𝑧₀ (2.34) for all 𝑥,𝑦∈𝑋. So, we have Φ𝑘⁻¹𝑥,𝑘⁻¹𝑦=𝜓𝑘⁻¹Φ𝑥,𝑦, (2.35) where 𝜓(𝑘⁻¹)=𝜓(𝑘)⁻¹<𝑘⁻¹. It follows from (1) that 𝜓(𝑘^−𝑛)=(𝜓(𝑘))^−𝑛. By using Theorem 2.2, we can get (2.32). Now, we show that 𝑇_𝑘=𝑇₂. Replacing 𝑥 by 𝑥/2^𝑛 in (2.32), we get 𝑁𝑇_𝑘𝑥2^𝑛𝑥−𝑓2^𝑛,𝑡≥𝑁^′𝜓𝑥2^𝑛𝑧₀,𝜓(𝑘)−𝑘𝜎_𝑘𝜓𝑡. (2.36) for all 𝑥∈𝑋. So we have 𝑁𝑇_𝑘(𝑥)−2^𝑛𝑓𝑥2^𝑛,𝑡≥𝑁^′𝜓(‖𝑥‖)𝑧₀,𝜓(𝑘)−𝑘2^𝑛𝜎_𝑘𝜓𝜓(2^−𝑛)𝑡. (2.37) Using (2) and passing the limit 𝑛→∞ in (2.37), we get 𝑇_𝑘=𝑇₂.

Theorem 2.5. Let 𝑋 be a normed space, let 𝑝 be a nonnegative real number such that 𝑝≠1, and let 𝐻∶[0,∞)×[0,∞)→[0,∞) be a homogeneous function of degree 𝑝. Suppose that (𝑍,𝑁^′) be a fuzzy normed space and let 𝑓∶𝑋→𝑌 be mapping such that 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′𝐻‖𝑥‖,𝑦𝑧₀,𝑡 (2.38) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡, where 𝑧₀ is a fixed vector of 𝑍. Then, there exists a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 such that 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑀_𝑘𝑥,𝑘^𝑝−𝑘𝑡, (2.39) where 𝑀_𝑘(𝑥,𝑡)∶=min{𝑁^′(‖𝑥‖^𝑝𝐻(1,𝑖)𝑧₀,𝑡)∶1≤𝑖<𝑘}.

Proof. The proof follows from Theorems 2.1 and 2.2.

For the particular cases 𝐻(𝑥,𝑦)=𝜃(𝑥^𝑝+𝑦^𝑝),𝐻(𝑥,𝑦)=𝑥^𝑟𝑦^𝑠,𝐻(𝑥,𝑦)=𝑥^𝑟𝑦^𝑠+𝑥^𝑟+𝑠+𝑦^𝑟+𝑠(𝑟+𝑠=𝑝), and 𝐻(𝑥,𝑦)=min{𝑥^𝑝,𝑦^𝑝}, we have the following corollaries.

Corollary 2.6. Let 𝑋 be a normed space, let 𝑝 be a nonnegative real number such that 𝑝≠1. Suppose that (𝑍,𝑁′) be a fuzzy normed space and 𝑓∶𝑋→𝑌 be mapping such that 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′‖𝑥‖^𝑝+𝑦^𝑝𝜃,𝑡 (2.40) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡, where 𝜃 is a fixed vector of 𝑍. Then, there exists a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 such that 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑁^′‖𝑥‖^𝑝𝑘𝜃,^𝑝−𝑘1+(𝑘−1)^𝑝𝑡. (2.41)

Corollary 2.7. Let 𝑋 be a normed space, r,𝑠 be non-negative real numbers such that 𝑝∶=𝑟+𝑠≠1. Suppose that (𝑍,𝑁^′) be a fuzzy normed space and 𝑓∶𝑋→𝑌 be mapping such that 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′‖𝑥‖^𝑟𝑦^𝑠𝜃,𝑡 (2.42) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡, where 𝜃 is a fixed vector of 𝑍. Then there exists a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 such that 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑁^′‖𝑥‖^𝑝𝑘𝜃,^𝑝−𝑘(𝑘−1)^𝑠𝑡. (2.43)

Corollary 2.8. Let 𝑋 be a normed space, and let 𝑟,𝑠 be nonnegative real numbers such that 𝑝∶=𝑟+𝑠≠1. Suppose that (𝑍,𝑁^′) be a fuzzy normed space and let 𝑓∶𝑋→𝑌 be mapping such that 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′𝜃‖𝑥‖^𝑟𝑦^𝑠+𝜃‖𝑥‖^𝑟+𝑠+𝜃𝑦^𝑟+𝑠,𝑡 (2.44) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡, where 𝜃 is a fixed vector of 𝑍. Then, there exists a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 such that 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑁^′‖𝑥‖^𝑝𝑘𝜃,^𝑝−𝑘(𝑘−1)^𝑠+(𝑘−1)^𝑝𝑡.+1 (2.45)

Corollary 2.9. Let 𝑋 be a normed space, let 𝑝 be a nonnegative real number such that 𝑝≠1. Suppose that (𝑍,𝑁^′) be a fuzzy normed space and let 𝑓∶𝑋→𝑌 be mapping such that 𝑁𝐷𝑓𝑥,𝑦,𝑡≥𝑁^′min‖𝑥‖^𝑝,𝑦^𝑝𝜃,𝑡 (2.46) for all 𝑥,𝑦∈𝑋 and all positive real number 𝑡, where 𝜃 is a fixed vector of 𝑍. Then, there exists a unique additive mapping 𝑇_𝑘∶𝑋→𝑌 such that 𝑁𝑇_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝑁^′‖𝑥‖^𝑝𝜃,𝑘^𝑝−𝑘𝑡. (2.47)

Problem 1. Whether Theorem 2.5 and/or such Corollaries can be proved for 𝑝=1?

Problem 2. What is the best possible value of 𝑘 in Corollaries 2.6 and 2.7?

Acknowledgment

G. H. Kim 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. 2011-0005197).