Approximate Euler-Lagrange Quadratic Mappings in Fuzzy Banach Spaces

Abstract

We consider general solution and the generalized Hyers-Ulam stability of an Euler-Lagrange quadratic functional equation 𝑓(𝑟𝑥+𝑠𝑦)+𝑟𝑠𝑓(𝑥−𝑦)=(𝑟+𝑠)[𝑟𝑓(𝑥)+𝑠𝑓(𝑦)] in fuzzy Banach spaces, where 𝑟, 𝑠 are nonzero rational numbers with 𝑟²+𝑟𝑠+𝑠²−1≠0, 𝑟+𝑠≠0.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for additive mappings on Banach spaces. Hyers's 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 Gǎvruta [5] by replacing the unbounded Cauchy difference by a general control function.

The functional equation 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥)+2𝑓𝑦 (1) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. Cholewa [6] noticed that the theorem of F. Skof is still true if the relevant domain 𝑋 is replaced by an Abelian group. Czerwik [7] proved the Hyers-Ulam stability of the quadratic functional equation. In particular, Rassias investigated the Hyers-Ulam stability for the relative Euler-Lagrange functional equation 𝑓𝑎𝑥+𝑏𝑦+𝑓𝑏𝑥−𝑎𝑦=𝑎²+𝑏²𝑓(𝑥)+𝑓𝑦 (2) in [8–10]. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [11–14]).

The theory of fuzzy space has much progressed as the theory of randomness has developed. Some mathematicians have defined fuzzy norms on a vector space from various points of view [15–19]. Following Cheng and Mordeson [20] and Bag and Samanta [15] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [21] and investigated some properties of fuzzy normed spaces [22].

We use the definition of fuzzy normed spaces given [15, 18, 23].

Definition 1 (see [15, 18, 23]). Let 𝑋 be a real vector space. A function 𝑁∶𝑋×𝐑→[0,1] is said to be 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|) for 𝑐≠0;(𝑁₄)  𝑁(𝑥+𝑦,𝑠+𝑡)≥min{𝑁(𝑥,𝑠),𝑁(𝑦,𝑡)};(𝑁₅)  𝑁(𝑥,⋅) is a nondecreasing function on 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. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [18, 24].

Definition 2 (see [15, 18, 23]). Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {𝑥_𝑛} in 𝑋 is said to be convergent or converges to 𝑥 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}, and one denotes it by 𝑁-lim_𝑛→∞𝑥_𝑛=𝑥.

Definition 3 (see [15, 18, 23]). 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 space is a Cauchy sequence. 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.

It is said that a mapping 𝑓∶𝑋→𝑌 between fuzzy normed spaces 𝑋 and 𝑌 is continuous at 𝑥₀∈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 (see [22]).

We recall the fixed point theorem from [25], which is needed in Section 4.

Theorem 4 (see [25, 26]). 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=∞ (3) 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 1996, Isac and Rassias [27] were the first to provide new application of fixed point theorems to the proof of stability theory of functional equations. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [28–30] and references therein).

Recently, Kim et al. [31] investigated the solution and the stability of the Euler-Lagrange quadratic functional equation 𝑓𝑓((𝑥)𝑦,𝑘𝑥+𝑙𝑦+𝑓𝑘𝑥−𝑙𝑦=𝑘𝑙𝑥+𝑦+𝑓𝑥−𝑦+2𝑘−𝑙)𝑘𝑓−𝑙𝑓 (4) where 𝑘, 𝑙 are non-zero rational numbers with 𝑘≠𝑙.

Najati and Jung [32] have observed the Hyers-Ulam stability of the generalized quadratic functional equation 𝑓𝑟𝑥+𝑠𝑦+𝑟𝑠𝑓𝑥−𝑦=𝑟𝑓(𝑥)+𝑠𝑓𝑦, (5) where 𝑟, 𝑠 are non-zero rational numbers with 𝑟+𝑠=1.

In this paper, we generalize the above quadratic functional equation (5) to investigate the generalized Hyers-Ulam stability of an Euler-Lagrange quadratic functional equation 𝑓𝑟𝑥+𝑠𝑦+𝑟𝑠𝑓𝑥−𝑦=(𝑟+𝑠)𝑟𝑓(𝑥)+𝑠𝑓𝑦 (6) in fuzzy Banach spaces, where 𝑟,𝑠 are non-zero rational numbers with 𝑟²+𝑟𝑠+𝑠²−1≠0, 𝑟+𝑠≠0. In particular, if 𝑟+𝑠=1 in the functional equation (6), then 𝑟²+𝑟𝑠+𝑠²−1≠0 is trivial and so (6) reduces to (5).

2. General Solution of (6)

Lemma 5 (see [31]). A mapping 𝑓∶𝑋→𝑌 between linear spaces satisfies the functional equation 𝑓𝑓((𝑥)𝑦,𝑘𝑥+𝑙𝑦+𝑓𝑘𝑥−𝑙𝑦=𝑘𝑙𝑥+𝑦+𝑓𝑥−𝑦+2𝑘−𝑙)𝑘𝑓−𝑙𝑓 (7) where 𝑘, 𝑙 are non-zero rational numbers with 𝑘≠𝑙 if and only if 𝑓 is quadratic.

Lemma 6. Let 𝑋 and 𝑌 be vector spaces and 𝑓∶𝑋→𝑌 an odd function satisfying (6). Then 𝑓≡0.

Proof. Putting 𝑥=0 (resp., 𝑦=0) in (6), we get 𝑓𝑠𝑦=𝑠(𝑠+2𝑟)𝑓𝑦,𝑓(𝑟𝑥)=𝑟²𝑓(𝑥) (8) for all 𝑥,𝑦∈𝑋. Replacing 𝑦 by −𝑦 in (6) and adding the obtained functional equation to (6), we get 𝑓()𝑓(𝑥)𝑓𝑟𝑥+𝑠𝑦+𝑓𝑟𝑥−𝑠𝑦=2𝑟𝑟+𝑠−𝑟𝑠𝑥+𝑦+𝑓𝑥−𝑦 (9) for all 𝑥,𝑦∈𝑋. Replacing 𝑦 by 𝑟𝑦 in (9) and using (8), we get ()𝑓(𝑥)𝑓𝑟𝑓𝑥+𝑠𝑦+𝑟𝑓𝑥−𝑠𝑦=2𝑟+𝑠−𝑠𝑥+𝑟𝑦+𝑓𝑥−𝑟𝑦 (10) for all 𝑥,𝑦∈𝑋. Again if we replace 𝑥 by 𝑠𝑥 in (10) and use (8), we get 𝑟(𝑓()𝑓(𝑥)−𝑓2𝑟+𝑠)𝑥+𝑦+𝑓𝑥−𝑦=2𝑟+𝑠)(2𝑟+𝑠𝑠𝑥+𝑟𝑦+𝑓𝑠𝑥−𝑟𝑦 (11) for all 𝑥,𝑦∈𝑋. Exchanging 𝑥 for 𝑦 in (6) and using the oddness of 𝑓, we have 𝑓𝑠𝑥+𝑟𝑦=(𝑟+𝑠)𝑟𝑓𝑦+𝑠𝑓(𝑥)+𝑟𝑠𝑓𝑥−𝑦 (12) for all 𝑥,𝑦∈𝑋. Replacing 𝑦 by −𝑦 in (12) and adding the obtained functional equation to (12), we get 𝑓()𝑓(𝑥)𝑓𝑠𝑥+𝑟𝑦+𝑓𝑠𝑥−𝑟𝑦=2𝑠𝑟+𝑠+𝑟𝑠𝑥+𝑦+𝑓𝑥−𝑦 (13) for all 𝑥,𝑦∈𝑋. So it follows from (11) and (13) that 𝑓𝑥+𝑦+𝑓𝑥−𝑦=2𝑓(𝑥) (14) for all 𝑥,𝑦∈𝑋. It easily follows from (14) that 𝑓 is additive; that is, 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) for all 𝑥,𝑦∈𝑋. Since 𝑟 is a rational number, 𝑓(𝑟𝑥)=𝑟𝑓(𝑥) for all 𝑥∈𝑋. Therefore, it follows from (8) that 𝑟(𝑟−1)𝑓(𝑥)=0 for all 𝑥∈𝑋. Since 𝑟, 𝑠 are nonzero, we infer that 𝑓≡0 if 𝑟≠1.
If 𝑟=1, then 𝑠≠0,−1, and thus we see easily that 𝑓≡0 by the similar argument above.

Lemma 7. Let 𝑋 and 𝑌 be vector spaces and 𝑓∶𝑋→𝑌 an even function satisfying (6). Then 𝑓 is quadratic.

Proof. Putting 𝑥=𝑦=0 in (6), we get 𝑓(0)=0 since 𝑟²+𝑟𝑠+𝑠²−1≠0. Replacing 𝑥 by 𝑥+𝑦 in (6), we obtain 𝑓𝑟𝑥+(𝑟+𝑠)𝑦=(𝑟+𝑠)𝑟𝑓𝑥+𝑦+𝑠𝑓𝑦−𝑟𝑠𝑓(𝑥) (15) for all 𝑥,𝑦∈𝑋. Replacing 𝑦 by −𝑦 in (15) and using the evenness of 𝑓, we get 𝑓𝑟𝑥−(𝑟+𝑠)𝑦=(𝑟+𝑠)𝑟𝑓𝑥−𝑦+𝑠𝑓𝑦−𝑟𝑠𝑓(𝑥) (16) for all 𝑥,𝑦∈𝑋. Adding (15) and (16), we get 𝑓()𝑦()𝑦(𝑓(𝑥)−()𝑓𝑦𝑟𝑥+𝑟+𝑠+𝑓𝑟𝑥−𝑟+𝑠=𝑟𝑟+𝑠)𝑥+𝑦+𝑓𝑥−𝑦−2𝑠𝑟𝑓𝑟+𝑠 (17) for all 𝑥,𝑦∈𝑋. Thus (17) can be rewritten by 𝑓𝑓((𝑥)𝑦,𝑘𝑥+𝑙𝑦+𝑓𝑘𝑥−𝑙𝑦=𝑘𝑙𝑥+𝑦+𝑓𝑥−𝑦+2𝑘−𝑙)𝑘𝑓−𝑙𝑓 (18) where 𝑘∶=𝑟, 𝑙∶=𝑟+𝑠 for all 𝑥,𝑦∈𝑋. Therefore, it follows from Lemma 5 that 𝑓 is quadratic.

Theorem 8. Let 𝑓∶𝑋→𝑌 be a function between vector spaces 𝑋 and 𝑌. Then 𝑓 satisfies (6) if and only if 𝑓 is quadratic.

Proof. Let 𝑓_𝑜 and 𝑓_𝑒 be the odd and the even parts of 𝑓. Suppose that 𝑓 satisfies (6). It is clear that 𝑓_𝑜 and 𝑓_𝑒 satisfy (6). By Lemmas 6 and 7, 𝑓_𝑜≡0 and 𝑓_𝑒 is quadratic. Since 𝑓=𝑓_𝑜+𝑓_𝑒, we conclude that 𝑓 is quadratic.
Conversely, if a mapping 𝑓 is quadratic, then it is easy to see that 𝑓 satisfies (6).

3. Stability of (6) by Direct Method

Throughout this paper, we assume that 𝑋 is a linear space, (𝑌,𝑁) is a fuzzy Banach space, and (𝑍,𝑁^′) is a fuzzy normed space.

For notational convenience, given a mapping 𝑓∶𝑋→𝑌, we define a difference operator 𝐷_𝑟𝑠𝑓∶𝑋²→𝑌 of (6) by 𝐷_𝑟𝑠𝑓−((𝑥)𝑦𝑥,𝑦∶=𝑓𝑟𝑥+𝑠𝑦+𝑟𝑠𝑓𝑥−𝑦𝑟+𝑠)𝑟𝑓+𝑠𝑓 (19) for all 𝑥,𝑦∈𝑋.

Theorem 9. Assume that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑁𝐷_𝑟𝑠𝑓𝑥,𝑦,𝑡≥𝑁^′𝜑𝑥,𝑦,𝑡, (20) and 𝜑∶𝑋²→𝑍 is a mapping for which there is a constant 𝑐∈𝐑 satisfying 0<|𝑐|<(𝑟+𝑠)² such that 𝑁^′𝜑(𝑟+𝑠)𝑥,(𝑟+𝑠)𝑦,𝑡≥𝑁^′𝑐𝜑𝑥,𝑦,𝑡 (21) for all 𝑥∈𝑋 and all 𝑡>0. Then one can find a unique Euler-Lagrange quadratic mapping 𝑄∶𝑋→𝑌 satisfying the equation 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜑(𝑥,𝑥)(𝑟+𝑠)²−|𝑐|,𝑡,𝑡>0, (22) for all 𝑥∈𝑋.

Proof. We observe from (21) that 𝑁^′𝜑(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦,𝑡≥𝑁^′𝑐^𝑛𝜑𝑥,𝑦,𝑡=𝑁^′𝜑,𝑡𝑥,𝑦|𝑐|^𝑛𝑁,𝑡>0,_′𝜑(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦,|𝑐|^𝑛𝑡≥𝑁^′𝜑𝑥,𝑦,𝑡,𝑡>0, (23) for all 𝑥,𝑦∈𝑋. Putting 𝑦∶=𝑥 in (20), we obtain 𝑁𝑓((𝑟+𝑠)𝑥)−(𝑟+𝑠)²𝑓(𝑥),𝑡≥𝑁^′𝜑(),𝑓(𝑥)−𝑓)𝑥)𝑥,𝑥,𝑡or𝑁((𝑟+𝑠(𝑟+𝑠)²,𝑡(𝑟+𝑠)²≥𝑁^′𝜑(𝑥,𝑥),𝑡 (24) for all 𝑥∈𝑋. Therefore it follows from (23), (24) that 𝑁𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛−𝑓(𝑟+𝑠)^𝑛+1𝑥(𝑟+𝑠)^2(𝑛+1),|𝑐|^𝑛𝑡(𝑟+𝑠)^2(𝑛+1)≥𝑁^′𝜑(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑥,|𝑐|^𝑛𝑡≥𝑁^′𝜑(𝑥,𝑥),𝑡 (25) for all 𝑥∈𝑋 and any integer 𝑛≥0. So 𝑁𝑓(𝑥)−𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛,^𝑛−1_𝑖=0|𝑐|^𝑖𝑡(𝑟+𝑠)^2(𝑖+1)=𝑁^𝑛−1_𝑖=0𝑓(𝑟+𝑠)^𝑖𝑥(𝑟+𝑠)^2𝑖−𝑓(𝑟+𝑠)^𝑖+1𝑥(𝑟+𝑠)^2(𝑖+1)_{𝑛−1𝑖=0},|𝑐|^𝑖𝑡(𝑟+𝑠)^2(𝑖+1)≥min_{0≤𝑖≤𝑛−1}𝑁𝑓(𝑟+𝑠)^𝑖𝑥(𝑟+𝑠)^2𝑖−𝑓(𝑟+𝑠)^𝑖+1𝑥(𝑟+𝑠)^2(𝑖+1),|𝑐|^𝑖𝑡(𝑟+𝑠)^2(𝑖+1)≥𝑁^′𝜑(𝑥,𝑥),𝑡,𝑡>0, (26) which yields 𝑁𝑓(𝑟+𝑠)^𝑚𝑥(𝑟+𝑠)^2𝑚−𝑓(𝑟+𝑠)^𝑚+𝑝𝑥(𝑟+𝑠)^2(𝑚+𝑝),^{𝑚+𝑝−1}_𝑖=𝑚|𝑐|^𝑖𝑡(𝑟+𝑠)^2(𝑖+1)=𝑁^{𝑚+𝑝−1}_𝑖=𝑚𝑓(𝑟+𝑠)^𝑖𝑥(𝑟+𝑠)^2𝑖−𝑓(𝑟+𝑠)^𝑖+1𝑥(𝑟+𝑠)^2(𝑖+1)_{𝑚+𝑝−1𝑖=𝑚},|𝑐|^𝑖𝑡(𝑟+𝑠)^2(𝑖+1)≥min_{𝑚≤𝑖≤𝑚+𝑝−1}𝑁𝑓(𝑟+𝑠)^𝑖𝑥(𝑟+𝑠)^2𝑖−𝑓(𝑟+𝑠)^𝑖+1𝑥(𝑟+𝑠)^2(𝑖+1),|𝑐|^𝑖𝑡(𝑟+𝑠)^2(𝑖+1)≥𝑁^′𝜑(𝑥,𝑥),𝑡,𝑡>0, (27) for all 𝑥∈𝑋 and any integers 𝑝>0, 𝑚≥0. Hence one obtains 𝑁𝑓(𝑟+𝑠)^𝑚𝑥(𝑟+𝑠)^2𝑚−𝑓(𝑟+𝑠)^𝑚+𝑝𝑥(𝑟+𝑠)^2(𝑚+𝑝),𝑡≥𝑁^′𝜑(),𝑡𝑥,𝑥∑^{𝑚+𝑝−1}_𝑖=𝑚|𝑐|^𝑖/(𝑟+𝑠)^2(𝑖+1) (28) for all 𝑥∈𝑋 and any integers 𝑝>0, 𝑚≥0, 𝑡>0. Since ∑^{𝑚+𝑝−1}_𝑖=𝑚(|𝑐|^𝑖/(𝑟+𝑠)^2𝑖) is convergent series, we see by taking the limit 𝑚→∞ in the last inequality that a sequence {𝑓((𝑟+𝑠)^𝑛𝑥)/(𝑟+𝑠)^2𝑛} is Cauchy in the fuzzy Banach space (𝑌,𝑁) and so it converges in 𝑌. Therefore a mapping 𝑄∶𝑋→𝑌 defined by 𝑄(𝑥)∶=𝑁−lim_𝑛→∞𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛 (29) is well defined for all 𝑥∈𝑋. It means that lim_𝑛→∞𝑁(𝑓((𝑟+𝑠)^𝑛𝑥)/(𝑟+𝑠)^2𝑛−𝑄(𝑥),𝑡)=1, 𝑡>0, for all 𝑥∈𝑋. In addition, we see from (26) that 𝑁𝑓(𝑥)−𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛,𝑡≥𝑁^′𝜑(),𝑡𝑥,𝑥∑^𝑛−1_𝑖=0|𝑐|^𝑖/(𝑟+𝑠)^2(𝑖+1) (30) and so, for any 𝜀>0, 𝑁𝑓(𝑥)(𝑥)𝑁𝑓(𝑥)−𝑓()−𝑄,𝑡≥min𝑟+𝑠^𝑛𝑥(𝑟+𝑠)^2𝑛,()𝑡,𝑁𝑓()1−𝜀𝑟+𝑠^𝑛𝑥(𝑟+𝑠)^2𝑛−𝑄(𝑥),𝜀𝑡≥𝑁^′𝜑(),()𝑡𝑥,𝑥1−𝜀∑^𝑛−1_𝑖=0|𝑐|^𝑖/(𝑟+𝑠)^2(𝑖+1)≥𝑁^′𝜑(𝑥,𝑥),(1−𝜀)(𝑟+𝑠)²−|𝑐|𝑡,0<𝜀<1, (31) for sufficiently large 𝑛 and for all 𝑥∈𝑋 and all 𝑡>0. Since 𝜀 is arbitrary and 𝑁^′ is left continuous, we obtain 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜑(𝑥,𝑥),(𝑟+𝑠)²−|𝑐|𝑡,𝑡>0, (32) for all 𝑥∈𝑋, which yields the approximation (22).
In addition, it is clear from (20) and (𝑁₅) that the following relation 𝑁𝐷_𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦(𝑟+𝑠)^2𝑛,𝑡≥𝑁^′𝜑(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦,(𝑟+𝑠)^2𝑛𝑡≥𝑁^′𝜑,()𝑥,𝑦𝑟+𝑠^2𝑛|𝑐|^𝑛𝑡⟶1as𝑛⟶∞ (33) holds for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Therefore, we obtain by use of lim_𝑛→∞𝑁𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛−𝑄(𝑥),𝑡=1(𝑡>0) (34) that 𝑁𝐷_𝑟𝑠𝑄𝑁𝐷𝑥,𝑦,𝑡≥min_𝑟𝑠𝑄−𝐷𝑥,𝑦^𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦(𝑟+𝑠)^2𝑛,𝑡2,𝑁𝐷_𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦(𝑟+𝑠)^2𝑛,𝑡2𝐷=𝑁^𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦𝑟^2𝑛,𝑡2,forsufficientlylarge𝑛≥𝑁^′𝜑,()𝑥,𝑦𝑟+𝑠^2𝑛2|𝑐|^𝑛𝑡,𝑡>0⟶1as𝑛⟶∞ (35) which implies 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 by (𝑁₂). Thus we find that 𝑄 is an Euler-Lagrange quadratic mapping satisfying (6) and (22) near the approximate quadratic mapping 𝑓∶𝑋→𝑌.
To prove the aforementioned uniqueness, we assume now that there is another quadratic mapping 𝑄^′∶𝑋→𝑌 which satisfies (22). Then one establishes by using the equality 𝑄^′((𝑟+𝑠)^𝑛𝑥)=(𝑟+𝑠)^2𝑛𝑄(𝑥) and (22) that 𝑁𝑄(𝑥)−𝑄^′(𝑥)𝑄(),𝑡=𝑁𝑟+𝑠^𝑛𝑥(𝑟+𝑠)^2𝑛−𝑄^′(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛𝑁𝑄(),𝑡≥min𝑟+𝑠^𝑛𝑥(𝑟+𝑠)^2𝑛−𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛,𝑡2,𝑁𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛−𝑄^′(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛,𝑡2≥𝑁^′𝜑(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)²−|𝑐|(𝑟+𝑠)^2𝑛𝑡2≥𝑁^′𝜑(),()𝑥,𝑥𝑟+𝑠²−|𝑐|(𝑟+𝑠)^2𝑛𝑡2|𝑐|^𝑛,𝑡>0,∀𝑛∈ℕ, (36) which tends to 1 as 𝑛→∞ by (𝑁₅). Therefore one obtains 𝑄(𝑥)=𝑄^′(𝑥) for all 𝑥∈𝑋, completing the proof of uniqueness.

We remark that, if 𝑟+𝑠=1 in Theorem 9, then 𝑁^′(𝜑(𝑥,𝑦),𝑡)≥𝑁^′(𝜑(𝑥,𝑦),𝑡/|𝑐|^𝑛)→1 as 𝑛→∞, and so 𝜑(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋. Hence 𝐷_𝑟𝑠𝑓(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋 and 𝑓 is itself a quadratic mapping.

Theorem 10. Assume that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑁𝐷_𝑟𝑠𝑓𝑥,𝑦,𝑡≥𝑁^′𝜑𝑥,𝑦,𝑡 (37) and 𝜑∶𝑋²→𝑍 is a mapping for which there is a constant 𝑐∈ℝ satisfying |𝑐|>(𝑟+𝑠)² such that 𝑁^′𝜑𝑥(),𝑦𝑟+𝑠(𝑟+𝑠),𝑡≥𝑁^′1𝑐𝜑𝑥,𝑦,𝑡,𝑡>0, (38) for all 𝑥∈𝑋 and all 𝑡>0. Then one can find a unique Euler-Lagrange quadratic mapping 𝑄∶𝑋→𝑌 satisfying the equation 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜑(𝑥,𝑥)|𝑐|−(𝑟+𝑠)²,𝑡,𝑡>0, (39) for all 𝑥∈𝑋.

Proof. It follows from (24) and (38) that 𝑁𝑓(𝑥)−(𝑟+𝑠)²𝑓𝑥(),𝑡𝑟+𝑠|𝑐|≥𝑁^′𝜑(𝑥,𝑥),𝑡,𝑡>0 (40) for all 𝑥∈𝑋. Therefore it follows that 𝑁𝑓(𝑥)−(𝑟+𝑠)^2𝑛𝑓𝑥(𝑟+𝑠)^𝑛,^𝑛−1_𝑖=0(𝑟+𝑠)^2𝑖|𝑐|^𝑖+1𝑡≥𝑁^′𝜑(𝑥,𝑥),𝑡,𝑡>0, (41) for all 𝑥∈𝑋 and any integer 𝑛>0. Thus we see from the last inequality that 𝑁𝑓(𝑥)−(𝑟+𝑠)^2𝑛𝑓𝑥(𝑟+𝑠)^𝑛,𝑡≥𝑁^′𝜑(),𝑡𝑥,𝑥∑^𝑛−1_𝑖=0(𝑟+𝑠)^2𝑖/|𝑐|^𝑖+1≥𝑁^′𝜑(𝑥,𝑥),|𝑐|−(𝑟+𝑠)²𝑡,𝑡>0. (42)
The remaining assertion goes through by the similar way to the corresponding part of Theorem 9.

We also observe that, if 𝑟+𝑠=1 in Theorem 10, then 𝑁^′(𝜑(𝑥,𝑦),𝑡)≥𝑁^′(𝜑(𝑥,𝑦),|𝑐|^𝑛𝑡)→1 as 𝑛→∞, and so 𝜑(𝑥,𝑦)=0 for all 𝑥,𝑦∈𝑋. Hence 𝐷_𝑟𝑠𝑓=0 and 𝑓 is itself a quadratic mapping.

Corollary 11. Let 𝑋 be a normed space and (𝐑,𝑁^′) a fuzzy normed space. Assume that there exist real numbers 𝜃₁,𝜃₂≥0 and 𝑝 is real number such that either 𝑝<2 or 𝑝>2. If a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑁𝐷_𝑟𝑠𝑓𝑥,𝑦,𝑡≥𝑁^′𝜃₁‖𝑥‖^𝑝+𝜃₂𝑦^𝑝,𝑡 (43) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then one can find a unique Euler-Lagrange quadratic mapping 𝑄∶𝑋→𝑌 satisfying the equation 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 and the inequality 𝑁𝑓(𝑥)(𝑥)≤𝑁−𝑄,𝑡_′𝜃₁+𝜃₂‖𝑥‖^𝑝(𝑟+𝑠)²−|𝑟+𝑠|^𝑝,||||𝑁,𝑡𝑖𝑓𝑝<2,𝑟+𝑠>1,𝑝>2,𝑟+𝑠<1_′𝜃₁+𝜃₂‖𝑥‖^𝑝|𝑟+𝑠|^𝑝−(𝑟+𝑠)²,||||,𝑡𝑖𝑓𝑝>2,𝑟+𝑠>1,𝑝<2,𝑟+𝑠<1 (44) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. Taking 𝜑(𝑥,𝑦)=𝜃₁‖𝑥‖^𝑝+𝜃₂‖𝑦‖^𝑝 and applying Theorems 9 and 10, we obtain the desired approximation, respectively.

Corollary 12. Assume that, for 𝑟+𝑠≠1, there exists a real number 𝜃≥0 such that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑁𝐷_𝑟𝑠𝑓𝑥,𝑦,𝑡≥𝑁^′(𝜃,𝑡) (45) for all 𝑥,𝑦∈𝑋 and all 𝑡>0. Then one can find a unique Euler-Lagrange quadratic mapping 𝑄∶𝑋→𝑌 satisfying the equation 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜃(𝑟+𝑠)²−1,𝑡 (46) for all 𝑥∈𝑋 and all 𝑡>0.

We remark that, if 𝜃=0, then 𝑁(𝐷_𝑟𝑠𝑓(𝑥,𝑦),𝑡)≥𝑁^′(0,𝑡)=1, and so 𝐷_𝑟𝑠𝑓(𝑥,𝑦)=0. Thus we get that 𝑓=𝑄 is itself a quadratic mapping.

4. Stability of (6) by Fixed Point Method

Now, in the next theorem, we are going to consider a stability problem concerning the stability of (6) by using a fixed point theorem of the alternative for contraction mappings on generalized complete metric spaces due to Margolis and Diaz [25].

Theorem 13. Assume that there exists constant 𝑐∈𝐑 with |𝑐|≠1 and 𝑞>0 satisfying 0<|𝑐|^1/𝑞<(𝑟+𝑠)² such that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑁𝐷_𝑟𝑠𝑓𝑥,𝑦,𝑡₁+𝑡₂≥min𝑁^′𝜑(𝑥),𝑡^𝑞₁,𝑁^′𝜑𝑦,𝑡^𝑞₂ (47) for all 𝑥,𝑦∈𝑋, 𝑡_𝑖>0  (𝑖=1,2), and 𝜑∶𝑋→𝑍 is a mapping satisfying 𝑁^′𝜑((𝑟+𝑠)𝑥),𝑡≥𝑁^′𝑐𝜑(𝑥),𝑡 (48) for all 𝑥∈𝑋 and all 𝑡>0. Then there exists a unique Euler-Lagrange quadratic mapping 𝑄∶𝑋→𝑌 satisfying the equation 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′2^𝑞𝜑(𝑥)(𝑟+𝑠)²−|𝑐|^1/𝑞𝑞,𝑡^𝑞 (49) for all 𝑥∈𝑋 and all 𝑡>0.

Proof. We consider the set of functions Ω∶=𝑔∶𝑋⟶𝑌∣𝑔(0)=0 (50) and define a generalized metric on Ω as follows: 𝑑_Ω𝑔,ℎ∶=inf𝐾∈(0,∞)∶𝑁𝑔(𝑥)−ℎ(𝑥),𝐾𝑡≥𝑁^′𝜑(𝑥),𝑡^𝑞,∀𝑥∈𝑋,∀𝑡>0. (51) Then one can easily see that (Ω,𝑑_Ω) is a complete generalized metric space [33, 34].
Now, we define an operator 𝐽∶Ω→Ω as (𝑥)=𝑔)𝑥)𝐽𝑔((𝑟+𝑠(𝑟+𝑠)² (52) for all 𝑔∈Ω, 𝑥∈𝑋.
We first prove that 𝐽 is strictly contractive on Ω. For any 𝑔,ℎ∈Ω, let 𝜀∈[0,∞) be any constant with 𝑑_Ω(𝑔,ℎ)≤𝜀. Then we deduce from the use of (48) and the definition of 𝑑_Ω(𝑔,ℎ) that 𝑁𝑔(𝑥)−ℎ(𝑥),𝜀𝑡≥𝑁^′𝜑(𝑥),𝑡^𝑞𝑔)𝑥),∀𝑥∈𝑋,𝑡>0⟹𝑁((𝑟+𝑠(𝑟+𝑠)²−ℎ((𝑟+𝑠)𝑥)(𝑟+𝑠)²,|𝑐|^1/𝑞𝜀𝑡(𝑟+𝑠)²≥𝑁^′𝜑((𝑟+𝑠)𝑥),|𝑐|𝑡^𝑞(𝑥)(𝑥),|𝑐|⟹𝑁𝐽𝑔−𝐽ℎ^1/𝑞𝜀𝑡(𝑟+𝑠)²≥𝑁^′𝜑(𝑥),𝑡^𝑞,∀𝑥∈𝑋,𝑡>0,⟹𝑑_Ω≤|𝑐|𝐽𝑔,𝐽ℎ^1/𝑞𝜀(𝑟+𝑠)². (53) Since 𝜀 is arbitrary constant with 𝑑_Ω(𝑔,ℎ)≤𝜀, we see that, for any 𝑔,ℎ∈Ω, 𝑑_Ω≤|𝑐|𝐽𝑔,𝐽ℎ^1/𝑞(𝑟+𝑠)²𝑑_Ω𝑔,ℎ, (54) which implies 𝐽 is strictly contractive with constant |𝑐|^1/𝑞/(𝑟+𝑠)²<1 on Ω.
We now want to show that 𝑑(𝑓,𝐽𝑓)<∞. If we put 𝑦∶=𝑥, 𝑡_𝑖∶=𝑡  (𝑖=1,2) in (47), then we arrive at 𝑁𝑓(𝑥)−𝑓((𝑟+𝑠)𝑥)(𝑟+𝑠)²,2𝑡(𝑟+𝑠)²≥𝑁^′𝜑(𝑥),𝑡^𝑞, (55) which yields 𝑑_Ω(𝑓,𝐽𝑓)≤2/(𝑟+𝑠)² and so 𝑑_Ω(𝐽^𝑛𝑓,𝐽^𝑛+1𝑓)≤𝑑_Ω(𝑓,𝐽𝑓)≤2/(𝑟+𝑠)² for all 𝑛∈𝐍.
Using the fixed point theorem of the alternative for contractions on generalized complete metric spaces due to Margolis and Diaz [25], we see the following (i), (ii), and (iii).
(i) There is a mapping 𝑄∶𝑋→𝑌 with 𝑄(0)=0 such that 𝑑_Ω≤1𝑓,𝑄1−|𝑐|^1/𝑞/(𝑟+𝑠)²𝑑_Ω≤2𝑓,𝐽𝑓(𝑟+𝑠)²−|𝑐|^1/𝑞 (56) and 𝑄 is a fixed point of the operator 𝐽; that is, (1/(𝑟+𝑠)²)𝑄((𝑟+𝑠)𝑥)=𝐽𝑄(𝑥)=𝑄(𝑥) for all 𝑥∈𝑋. Thus we can get 𝑁𝑓(𝑥)−𝑄(𝑥),2𝑡(𝑟+𝑠)²−|𝑐|^1/𝑞≥𝑁^′𝜑(𝑥),𝑡^𝑞,𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′𝜑(𝑥),(𝑟+𝑠)²−|𝑐|^1/𝑞𝑞2^𝑞𝑡^𝑞 (57) for all 𝑡>0 and all 𝑥∈𝑋.
(ii) Consider 𝑑_Ω(𝐽^𝑛𝑓,𝑄)→0 as 𝑛→∞. Thus we obtain 𝑁𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛(𝑥)𝑓()−𝑄,𝑡=𝑁𝑟+𝑠^𝑛𝑥−𝑄(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^2𝑛𝑡≥𝑁^′2^𝑞𝜑(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)²−|𝑐|^1/𝑞𝑞,(𝑟+𝑠)^2𝑛𝑞𝑡^𝑞=𝑁^′2^𝑞𝜑(𝑥)(𝑟+𝑠)²−|𝑐|^1/𝑞𝑞,(𝑟+𝑠)^2𝑞|𝑐|^𝑛𝑡^𝑞()⟶1as𝑛⟶∞,𝑟+𝑠^2𝑞|𝑐|>1 (58) for all 𝑡>0 and all 𝑥∈𝑋, that is; the mapping 𝑄∶𝑋→𝑌 given by 𝑁−lim_𝑛→∞𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^2𝑛=𝑄(𝑥) (59) is welldefined for all 𝑥∈𝑋. In addition, it follows from conditions (47), (48), and (𝑁₄) that 𝑁𝐷_𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦(𝑟+𝑠)^2𝑛,𝑡≥𝑁^′𝜑(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^2𝑛𝑞𝑡^𝑞2^𝑞=𝑁^′|𝑐|^𝑛𝜑(𝑥),(𝑟+𝑠)^2𝑛𝑞𝑡^𝑞2^𝑞=𝑁^′𝜑(𝑥),(𝑟+𝑠)^2𝑞|𝑐|^𝑛𝑡^𝑞2^𝑞⟶1as𝑛⟶∞,𝑡>0, (60) for all 𝑥∈𝑋. Therefore we obtain by use of (𝑁₄), (59), and (60) 𝑁𝐷_𝑟𝑠𝑄𝑁𝐷𝑥,𝑦,𝑡≥min_𝑟𝑠𝑄−𝐷𝑥,𝑦^𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦(𝑟+𝑠)^2𝑛,𝑡2,𝑁𝐷_𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦(𝑟+𝑠)^2𝑛,𝑡2𝐷=𝑁^𝑟𝑠𝑓(𝑟+𝑠)^𝑛𝑥,(𝑟+𝑠)^𝑛𝑦(𝑟+𝑠)^2𝑛,𝑡2𝑁forsufficientlylarge𝑛≥min^′𝜑(𝑥),(𝑟+𝑠)^2𝑞|𝑐|^𝑛𝑡^𝑞4^𝑞,𝑁^′𝜑𝑦,(𝑟+𝑠)^2𝑞|𝑐|^𝑛𝑡^𝑞4^𝑞⟶1as𝑛⟶∞,𝑡>0, (61) which implies 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 by (𝑁₂), and so the mapping 𝑄 is quadratic satisfying (6).
(iii) The mapping 𝑄 is a unique fixed point of the operator 𝐽 in the set Δ={𝑔∈Ω∣𝑑_Ω(𝑓,𝑔)<∞}. Thus if we assume that there exists another Euler-Lagrange type quadratic mapping 𝑄^′∶𝑋→𝑌 satisfying (49), then 𝑄^′(𝑥)=𝑄^′((𝑟+𝑠)𝑥)(𝑟+𝑠)²=𝐽𝑄^′(𝑥),𝑑_Ω𝑓,𝑄^′≤2(𝑟+𝑠)²−|𝑐|^1/𝑞<∞, (62) and so 𝑄^′ is a fixed point of the operator 𝐽 and 𝑄^′∈Δ={𝑔∈Ω∣𝑑_Ω(𝑓,𝑔)<∞}. By the uniqueness of the fixed point of 𝐽 in Δ, we find that 𝑄=𝑄^′, which proves the uniqueness of 𝑄 satisfying (49). This ends the proof of the theorem.

Theorem 14. Assume that there exists constant 𝑐∈𝐑 with |𝑐|≠1 and 𝑞>0 satisfying |𝑐|^1/𝑞>(𝑟+𝑠)² such that a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfies the inequality 𝑁𝐷_𝑟𝑠𝑓𝑥,𝑦,𝑡₁+𝑡₂≥min𝑁^′𝜑(𝑥),𝑡^𝑞₁,𝑁^′𝜑𝑦,𝑡^𝑞₂ (63) for all 𝑥,𝑦∈𝑋, 𝑡_𝑖>0  (𝑖=1,2), and 𝜑∶𝑋→𝑍 is a mapping satisfying 𝑁^′𝜑𝑥(𝑟+𝑠),𝑡≥𝑁^′1𝑐𝜑(𝑥),𝑡 (64) for all 𝑥∈𝑋. Then there exists a unique Euler-Lagrange quadratic mapping 𝑄∶𝑋→𝑌 satisfying the equation 𝐷_𝑟𝑠𝑄(𝑥,𝑦)=0 and the inequality 𝑁𝑓(𝑥)−𝑄(𝑥),𝑡≥𝑁^′2^𝑞𝜑(𝑥)|𝑐|^1/𝑞−(𝑟+𝑠)^2𝑞,𝑡^𝑞,𝑡>0, (65) for all 𝑥∈𝑋.

Proof. The proof of this theorem is similar to that of Theorem 13.

Remark 15. In a real space with a fuzzy norm 𝑁(𝑥,𝑡)=𝑁^′(𝑥,𝑡)=𝑡/(𝑡+‖𝑥‖), the stability result obtained by the direct method is somewhat different from the stability result obtained by the fixed point method as follows. Let 𝑋 be a normed space and 𝑌 a Banach space. Let a mapping 𝑓∶𝑋→𝑌 with 𝑓(0)=0 satisfy the inequality 𝐷_𝑟𝑠𝑓𝑥,𝑦≤𝜃₁‖𝑥‖^𝑝1+𝜃₂𝑦^𝑝2 (66) for all 𝑥,𝑦∈𝑋 and 𝑋⧵{0} if 𝑝₁,𝑝₂<0. Assume that there exist real numbers 𝜃₁,𝜃₂≥0 and 𝑝₁,𝑝₂ such that either 𝑝₁,𝑝₂<2, |𝑟+𝑠|>1  (𝑝₁,𝑝₂>2, |𝑟+𝑠|<1, resp.) or 𝑝₁,𝑝₂>2, |𝑟+𝑠|>1  (𝑝₁,𝑝₂<2, |𝑟+𝑠|<1, resp.). Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 which satisfies the inequality: 𝑓(𝑥)(𝑥≤𝜃−𝑄)₁‖𝑥‖^𝑝1(𝑟+𝑠)²−|𝑟+𝑠|^𝑝1+𝜃₂‖𝑥‖^𝑝2(𝑟+𝑠)²−|𝑟+𝑠|^𝑝2,if𝑝₁,𝑝₂||𝑝<2,𝑟+𝑠>1,₁,𝑝₂||,𝜃>2,𝑟+𝑠<1,resp.₁‖𝑥‖^𝑝1|𝑟+𝑠|^𝑝1−(𝑟+𝑠)²+𝜃₂‖𝑥‖^𝑝2|𝑟+𝑠|^𝑝2−(𝑟+𝑠)²,if𝑝₁,𝑝₂||𝑝>2,𝑟+𝑠>1₁,𝑝₂<2,|𝑟+𝑠|<1,resp. (67) for all 𝑥∈𝑋 and 𝑋⧵{0} if 𝑝₁,𝑝₂<0, which is verified by using the direct method together with the following inequality 𝑓(𝑥)−𝑓(𝑟+𝑠)^𝑛𝑥(𝑟+𝑠)^𝑛≤1(𝑟+𝑠)^2𝑛−1_𝑖=0𝜃₁|𝑟+𝑠|^𝑝1𝑖‖𝑥‖^𝑝1|𝑟+𝑠|^2𝑖+𝜃₂|𝑟+𝑠|^𝑝2𝑖‖𝑥‖^𝑝2|𝑟+𝑠|^2𝑖,𝑓(𝑥)−(𝑟+𝑠)^2𝑛𝑓𝑥(𝑟+𝑠)^𝑛≤1(𝑟+𝑠)^2𝑛_𝑖=1𝜃₁|𝑟+𝑠|^2𝑖‖𝑥‖^𝑝1|𝑟+𝑠|^𝑝1𝑖+𝜃₂|𝑟+𝑠|^2𝑖‖𝑥‖^𝑝2|𝑟+𝑠|^𝑝2𝑖, (68) for all 𝑥∈𝑋.
On the other hand, assume that there exist real numbers 𝜃₁,𝜃₂≥0 and 𝑝₁,𝑝₂ such that either max{𝑝₁,𝑝₂}<2, |𝑟+𝑠|>1 (min{𝑝₁,𝑝₂}>2, |𝑟+𝑠|<1, resp.) or min{𝑝₁,𝑝₂}>2, |𝑟+𝑠|>1 (max{𝑝₁,𝑝₂}<2, |𝑟+𝑠|<1, resp.). Then there exists a unique quadratic function 𝑄∶𝑋→𝑌 which satisfies the inequality 𝑓(𝑥)(𝑥≤𝜃−𝑄)₁‖𝑥‖^𝑝1+𝜃₂‖𝑥‖^𝑝2(𝑟+𝑠)²−|𝑟+𝑠|^{max{𝑝1,𝑝2}},ifmax𝑝₁,𝑝₂||𝜃<2,𝑟+𝑠>1,₁‖𝑥‖^𝑝1+𝜃₂‖𝑥‖^𝑝2(𝑟+𝑠)²−|𝑟+𝑠|^{min{𝑝1,𝑝2}},ifmin𝑝₁,𝑝₂||𝜃>2,𝑟+𝑠<1,₁‖𝑥‖^𝑝1+𝜃₂‖𝑥‖^𝑝2|𝑟+𝑠|^{min{𝑝1,𝑝2}}−(𝑟+𝑠)²,ifmin𝑝₁,𝑝₂||𝜃>2,𝑟+𝑠>1,₁‖𝑥‖^𝑝1+𝜃₂‖𝑥‖^𝑝2|𝑟+𝑠|^{max{𝑝1,𝑝2}}−(𝑟+𝑠)²,ifmax𝑝₁,𝑝₂<2,|𝑟+𝑠|<1 (69) for all 𝑥∈𝑋 and 𝑋⧵{0} if 𝑝₁,𝑝₂<0, which is established by using the fixed point method together with ||𝑐=𝑟+𝑠^{max{𝑝1,𝑝2}},ifmax𝑝₁,𝑝₂||||<2,𝑟+𝑠>1,𝑟+𝑠^{min{𝑝1,𝑝2}},ifmin𝑝₁,𝑝₂||||>2,𝑟+𝑠<1,𝑟+𝑠^{min{𝑝1,𝑝2}},ifmin𝑝₁,𝑝₂||||>2,𝑟+𝑠>1,𝑟+𝑠^{max{𝑝1,𝑝2}},ifmax𝑝₁,𝑝₂<2,|𝑟+𝑠|<1. (70) Therefore, we observe that the corresponding subsequential four stability results by the direct method are sharper than the corresponding subsequential four stability results obtained by the fixed point method.

Acknowledgment

This work was supported by research fund of Chungnam National University.