The Ulam Type Stability of a Generalized Additive Mapping and Concrete Examples

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Volume 2013, Article ID 109754,7pages http://dx.doi.org/10.1155/2013/109754

Research Article

The Ulam Type Stability of a Generalized Additive

Mapping and Concrete Examples

Hiroyoshi Oda,

1

Makoto Tsukada,

1

Takeshi Miura,

2

Yuji Kobayashi,

3

and Sin-Ei Takahasi

4

1_{Department of Information Sciences, Toho University, Funabashi, Chiba 274-8501, Japan} 2_{Yamagata University, Yonezawa, Yamagata 273-0866, Japan}

3_{Toho University, Funabashi, Chiba 274-8501, Japan}

4_{Toho University, Yamagata University, Funabashi, Chiba 273-0866, Japan}

Correspondence should be addressed to Makoto Tsukada; [email protected]

Received 27 December 2012; Accepted 21 February 2013

Academic Editor: Irena Lasiecka

Copyright © 2013 Hiroyoshi Oda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give an Ulam type stability result for the following functional equation:𝑓(𝛼𝑥−𝛼𝑥󸀠+𝑥₀) = 𝛽𝑓(𝑥)−𝛽𝑓(𝑥󸀠)+𝑦₀ (for all𝑥, 𝑥󸀠∈ 𝑋) under a suitable condition. We also give a concrete stability result for the case taking up𝛿‖𝑥‖𝑝‖𝑥󸀠‖𝑞as a control function.

1. Introduction

In 1940, Ulam [1] proposed the following stability problem: “When is it true that a function which satisfies some func-tional equationapproximatelymust be close to one satisfying the equationexactly?” Next year, Hyers [2] gave an answer to this problem for additive mappings between Banach spaces. Furthermore, Aoki [3] and Rassias [4] obtained indepen-dently generalized results of Hyers’ theorem which allow the Cauchy difference to be unbounded.

Let 𝑋and 𝑌be normed spaces over K, which denotes either the real fieldRor the complex field C. Throughout the paper, we fix scalars 𝑎, 𝑏, 𝑐, 𝑑 ∈ K \ {0} and vectors

𝑥₀ ∈ 𝑋and𝑦₀ ∈ 𝑌. We say that a mapping𝑓of𝑋into𝑌 is(𝑎, 𝑏, 𝑐, 𝑑; 𝑥₀, 𝑦₀)-additive if

𝑓 (𝑎𝑥 + 𝑏𝑥󸀠+ 𝑥₀) = 𝑐𝑓 (𝑥) + 𝑑𝑓 (𝑥󸀠) + 𝑦₀ (1)

for all𝑥, 𝑥󸀠 ∈ 𝑋. When 𝑥₀ = 𝑦₀ = 0, we say it to be

(𝑎, 𝑏, 𝑐, 𝑑)-additive. Acz´el [5] specified what this generalized Cauchy equation is. The Ulam type stability problem for such an 𝑓has been investigated in [6–8]. However, these results have been obtained in cases where either 𝑎 + 𝑏 ̸= 0 or 𝑐 + 𝑑 ̸= 0 (see Theorems A and B). In this paper, we will investigate the problem for(𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive

mappings, that is, in the case𝑎 + 𝑏 = 𝑐 + 𝑑 = 0. InSection 2, we state the details of(𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive mappings (Theorem 3). InSection 3, we give our main results about the stability for them (see Theorems7–10). In the final section, we apply the results to some concrete examples, where we take up𝛿‖𝑥‖𝑝‖𝑥󸀠‖𝑞as a control function𝜀(𝑥, 𝑥󸀠)(see Corollaries

11–14).

2. (𝑎,−𝑎,𝑐,−𝑐;𝑥

₀

,𝑦

₀

)

-Additive Mappings

The following result asserts that any (𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀) -additive mapping is transformed into some (𝑎, −𝑎, 𝑐, −𝑐) -additive mapping by a certain translation and that any

(𝑎, −𝑎, 𝑐, −𝑐)-additive mapping is an additive mapping in usual sense with some extra condition.

Proposition 1. Let𝑓and𝑔be two mappings of𝑋into𝑌such

that𝑔(𝑥) = 𝑓(𝑥 + 𝑥₀) − 𝑦₀for all𝑥 ∈ 𝑋. Then the following three statements are equivalent:

(i)𝑓is(𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive,

(ii)𝑔is(𝑎, −𝑎, 𝑐, −𝑐)-additive,

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Proof. (i)⇔(ii) Since

𝑓 (𝑎𝑥 − 𝑎𝑥󸀠+ 𝑥0)

= 𝑔 (𝑎𝑥 − 𝑎𝑥󸀠) + 𝑦₀

= 𝑔 (𝑎 (𝑥 − 𝑥₀) − 𝑎 (𝑥󸀠− 𝑥₀)) + 𝑦₀,

𝑐 (𝑥) − 𝑐𝑓 (𝑥󸀠_{) + 𝑦} 0

= 𝑐 (𝑓 (𝑥) − 𝑦0) − 𝑐 (𝑓 (𝑥󸀠) − 𝑦0) + 𝑦0

= 𝑐𝑔 (𝑥 − 𝑥₀) − 𝑐𝑔 (𝑥󸀠− 𝑥₀) + 𝑦₀

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for all𝑥, 𝑥󸀠∈ 𝑋, it follows that

𝑓 : (𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive

⇔ 𝑔(𝑎(𝑥−𝑥₀)−𝑎(𝑥󸀠−𝑥₀))+𝑦₀= 𝑐𝑔(𝑥−𝑥₀)−𝑐𝑔(𝑥󸀠− 𝑥₀) + 𝑦₀ (for all𝑥, 𝑥󸀠∈ 𝑋)

⇔ 𝑔(𝑎𝑥 − 𝑎𝑥󸀠_{) = 𝑐𝑔(𝑥) − 𝑐𝑔(𝑥}󸀠_{) (}_{for all}_{𝑥, 𝑥}󸀠_{∈ 𝑋)}

⇔ 𝑔 : (𝑎, −𝑎, 𝑐, −𝑐)-additive. (ii)⇒(iii) Suppose that

𝑔 (𝑎𝑥 − 𝑎𝑥󸀠) = 𝑐𝑔 (𝑥) − 𝑐𝑔 (𝑥󸀠) (3)

for all𝑥, 𝑥󸀠 ∈ 𝑋. When𝑥 = 𝑥󸀠, we have𝑔(0) = 0. Using this,𝑔(𝑎𝑥) = 𝑐𝑔(𝑥)and also𝑔(−𝑎𝑥) = −𝑐𝑔(𝑥)for all𝑥 ∈ 𝑋. Therefore,

𝑔 (𝑥 + 𝑥󸀠) = 𝑔 (𝑎𝑥_𝑎− 𝑎 (−𝑥_𝑎󸀠))

= 𝑐𝑔 (𝑥

𝑎) − 𝑐𝑔 (− 𝑥󸀠

𝑎)

= 𝑐𝑔 (𝑥_𝑎) + (−𝑐) 𝑔 (−𝑥_𝑎󸀠) = 𝑔 (𝑥) + 𝑔 (𝑥󸀠)

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for all𝑥, 𝑥󸀠∈ 𝑋.

(iii)⇒(ii) Because𝑔(−𝑥) = −𝑔(𝑥)for all𝑥 ∈ 𝑋(see also the following remark), it is trivial.

Remark 2. We denote byQthe field of all rational numbers. It is well known that if𝑔is additive, then𝑔(𝑞𝑥) = 𝑞𝑔(𝑥)for every𝑞 ∈ Qand𝑥 ∈ 𝑋, that is,𝑔isQ-linear. Hence, if𝑔is additive and continuous,𝑔must beR-linear. On the other hand, whenK = C, we have a lot of continuous additive nonlinear mappings by considering the composition of linear transformations onR2 and theR-linear isometry(𝑥, 𝑦) 󳨃→

𝑥 + 𝑖 𝑦.

The constant𝑦₀ is a trivial(𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive mapping of 𝑋 into 𝑌. The following theorem says that unless it is a unique (𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive mapping, discontinuous one always exists.

Theorem 3. (I) If𝑎 = 𝑐, then there exists a discontinuous

(𝑎, −𝑎, 𝑐, −𝑐; 𝑥0, 𝑦0)-additive mapping of𝑋into𝑌.

(II)If𝑎 ̸= 𝑐, then the followings hold:

(i)If both𝑎and𝑐are transcendental numbers, then there exists a discontinuous (𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive mapping of𝑋into𝑌.

(ii)If one of 𝑎and𝑐 is transcendental and the other is algebraic, then the constant 𝑦₀ is a unique (𝑎, −𝑎, 𝑐,

−𝑐; 𝑥₀, 𝑦₀)-additive mapping of𝑋into𝑌.

(iii)If both𝑎and𝑐are algebraic with a common minimal polynomial, then there exists a discontinuous (𝑎, −𝑎,

𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive mapping of𝑋into𝑌.

(iv)If𝑎 and𝑐 are algebraic with distinct minimal poly-nomials, then the constant 𝑦₀ is a unique (𝑎, −𝑎, 𝑐,

−𝑐; 𝑥0, 𝑦0)-additive mapping of𝑋into𝑌.

Moreover, when K = R, there is no nontrivial continuous

(𝑎, −𝑎, 𝑐, −𝑐; 𝑥0, 𝑦0)-additive mapping𝑓of𝑋into𝑌. On the

other hand, when K = C, if 𝑎is not real and 𝑎 = 𝑐(the complex conjugate of𝑐), then the mapping𝑥 󳨃→ 𝑓(𝑥 + 𝑥₀) −

𝑦₀ must be conjugate linear for every continuous (𝑎, −𝑎, 𝑐,

−𝑐; 𝑥₀, 𝑦₀)-additive mapping𝑓of𝑋into𝑌.

In order to showTheorem 3, we need some lemmas for

K-valued(𝑎, −𝑎, 𝑐, −𝑐)-additive functions defined onK. For any𝑥 ∈K, we denote byQ(𝑥)the subfield ofKgenerated by

𝑥overQ.

By the following proofs of Lemma 6and Theorem 3, if there is a discontinuous (𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive map-ping, then there are sufficiently many such mappings in the sense that there exists such a mapping which separates any

Q(𝑎)-linear independent points of𝑋.

Lemma 4. Any(𝑎, −𝑎, 𝑐, −𝑐)-additive𝜑ofKinto itself satisfies

𝜑(𝑝(𝑎)𝑥) = 𝑝(𝑐)𝜑(𝑥)for all𝑥 ∈Kand𝑝(𝑋) ∈Q[𝑋].

Proof. Note that𝜑is additive and𝜑(𝑎𝑥) = 𝑐𝜑(𝑥)for each

𝑥 ∈KbyProposition 1. Let𝑝(𝑋) = 𝑎₀+𝑎₁𝑋+⋅ ⋅ ⋅+𝑎_𝑛𝑋𝑛with

𝑎₀, 𝑎₁, . . . , 𝑎_𝑛∈Q. Since𝜑isQ-linear,

𝜑 (𝑝 (𝑎) 𝑥) = 𝑎0𝜑 (𝑥) + 𝑎1𝜑 (𝑎𝑥) + ⋅ ⋅ ⋅ + 𝑎𝑛𝜑 (𝑎𝑛𝑥)

= 𝑎₀𝜑 (𝑥) + 𝑎1𝑐𝜑 (𝑥) + ⋅ ⋅ ⋅ + 𝑎𝑛𝑐𝑛𝜑 (𝑥)

= 𝑝 (𝑐) 𝜑 (𝑥)

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for all𝑥 ∈K.

Lemma 5. Let𝐸and𝐹be subfields ofKand𝜓an isomorphism

of𝐸onto𝐹. Then,𝜓has an additive bijective extension𝜑to the full spaceKsuch that

𝜑 (𝜀𝑥) = 𝜓 (𝜀) 𝜑 (𝑥) (6)

for all𝜀 ∈ 𝐸and𝑥 ∈K. Moreover, one has a discontinuous one wheneverR\ 𝐸 ̸= 0.

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For any𝑥 ∈K, there exist a finite𝐼 ⊂ 𝐽and{𝑥_𝑖}_𝑖∈𝐼⊆ 𝐸such that

𝑥 = ∑

𝑖∈𝐼

𝑥_𝑖𝑒_𝑖, ₍₇₎

and this decomposition is unique. Hence, we can define𝜑 :

K → Kby

𝜑 (𝑥) = ∑

𝑖∈𝐼

𝜓 (𝑥_𝑖) 𝑓_𝑖. ₍₈₎

This𝜑is a desired extension.

In order to get a discontinuous extension, we consider the base{𝑒_𝑗 : 𝑗 ∈ 𝐽}and{𝑓_𝑗 : 𝑗 ∈ 𝐽}such that𝑓_𝑗1 ̸= 𝑒_𝑗1 ∈ R\ 𝐸 for some𝑗₁∈ 𝐽. BecauseQ⊆ 𝐸, take a rational sequence{𝜀_𝑛} converging to𝑒_𝑗1. Suppose that𝜑is continuous. Then,

𝜑 (𝜀_𝑛) 󳨀→ 𝜑 (𝑒_𝑗1) = 𝑓_𝑗1,

𝜑 (𝜀_𝑛) = 𝜓 (𝜀_𝑛) = 𝜀_𝑛󳨀→ 𝑒_𝑗1, (9)

as𝑛 → ∞. This is contradiction. Thus𝜑is discontinuous.

Lemma 6. Suppose that𝑎 ̸= 𝑐.

(i)If both𝑎and𝑐are transcendental numbers, then there exists a discontinuous(𝑎, −𝑎, 𝑐, −𝑐)-additive function ofKinto itself.

(ii)If one of𝑎 and𝑐 is transcendental and the other is algebraic, then the constant0is a unique(𝑎, −𝑎, 𝑐, −𝑐) -additive function ofKinto itself.

(iii)If both 𝑎 and𝑐 are algebraic with a common min-imal polynomial, then there exists a discontinuous

(𝑎, −𝑎, 𝑐, −𝑐)-additive function ofKinto itself.

(iv)If𝑎and𝑐are algebraic with distinct minimal polyno-mials, then the constant0 is a unique (𝑎, −𝑎, 𝑐, −𝑐) -additive function ofKinto itself.

Moreover, whenK=R, every nontrivial(𝑎, −𝑎, 𝑐, −𝑐)-additive function ofKinto itself is discontinuous. On the other hand, whenK = C, if𝑎is not real and𝑎 = 𝑐, then any continuous

(𝑎, −𝑎, 𝑐, −𝑐)-additive function𝑓ofCinto itself must be of form

𝑓(𝑥) = 𝛼𝑥 (𝑥 ∈C)for some𝛼 ∈C.

Proof. (i) Suppose that both𝑎and𝑐are transcendental. Then,

Q(𝑎)(resp.,Q(𝑐)) is isomorphic to the rational function field in indeterminate 𝑎(resp., 𝑐). So, the substitution 𝑎 → 𝑐 induces an isomorphism 𝜑_𝑎𝑐 : Q(𝑎) → Q(𝑐) of fields. ByLemma 5, becauseR\Q(𝑎) ̸= 0,𝜑_𝑎𝑐has a discontinuous additive extension𝜑toKsuch that𝜑(𝜀𝑥) = 𝜑_𝑎𝑐(𝜀)𝜑(𝑥)for every𝜀 ∈ Q(𝑎)and𝑥 ∈ K. Then,𝜑(𝑎𝑥) = 𝜑_𝑎𝑐(𝑎)𝜑(𝑥) =

𝑐𝜑(𝑥)for all𝑥 ∈K, and, hence,𝜑is(𝑎, −𝑎, 𝑐, −𝑐)-additive by

Proposition 1.

(ii) Let 𝜑be any(𝑎, −𝑎, 𝑐, −𝑐)-additive function. If 𝑎is transcendental and𝑐is algebraic with nonzero polynomial such that𝑝(𝑐) = 0, then fromLemma 4, we have

𝜑 (𝑥) = 𝜑 (𝑝 (𝑎) 𝑝(𝑎)−1𝑥) = 𝑝 (𝑐) 𝜑 (𝑝(𝑎)−1𝑥) = 0 (10)

for all𝑥 ∈ K. If𝑐is transcendental and𝑎is algebraic with nonzero polynomial such that𝑞(𝑎) = 0, then fromLemma 4, we have

𝜑 (𝑥) = _{𝑞 (𝑐)}1 𝑞 (𝑐) 𝜑 (𝑥) = _{𝑞 (𝑐)}1 𝜑 (𝑞 (𝑎) 𝑥) = _{𝑞 (𝑐)}1 𝜑 (0) = 0

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for all𝑥 ∈K.

(iii) If 𝑎is algebraic with minimal polynomial 𝑝(𝑋) ∈

Q[𝑋], then Q(𝑎) consists of all polynomials 𝑓(𝑎) in 𝑎of degree up to deg𝑝 − 1. So, if𝑐is also algebraic with the same minimal polynomial𝑝, then the substitution𝑎 → 𝑐induces an automorphism𝜑_𝑎𝑐 : Q(𝑎) → Q(𝑐) = Q(𝑎). As same as (i),𝜑_𝑎𝑐has a discontinuous(𝑎, −𝑎, 𝑐, −𝑐)-additive extension.

(iv) Suppose that 𝑎 and 𝑐 are algebraic with distinct minimal polynomials𝑝and𝑞over the fieldQ, respectively. Let𝜑 be any(𝑎, −𝑎, 𝑐, −𝑐)-additive function. To show𝜑 =

0, we assume, on the contrary, that there is an 𝑥₀ ∈ K with𝜑(𝑥₀) ̸= 0. Then, fromLemma 5, we have𝑝(𝑐)𝜑(𝑥₀) =

𝜑(𝑝(𝑎)𝑥0) = 𝜑(0) = 0, and hence𝑝(𝑐) = 0. This contradicts

the prerequisite for𝑎and𝑐. Hence,𝜑must be zero.

WhenK = R, since every continuous additive function is R-linear and 𝑎 ̸= 𝑐, there is no continuous (𝑎, −𝑎, 𝑐, −𝑐) -additive function byProposition 1. Now, we consider the case

K=C. Let𝜑be a nontrivial continuous(𝑎, −𝑎, 𝑐, −𝑐)-additive function. Note that𝜑isR-linear. If𝑎is not real and𝑎 = 𝑐, we can easily see that𝜑(𝑖𝑥) = −𝑖𝜑(𝑥)for all 𝑥 ∈ C. Thus,

𝜑is conjugate linear, and hence𝜑(𝑥) = 𝛼𝑥 (𝑥 ∈ C), where

𝛼 = 𝜑(1).

Proof ofTheorem 3. (I) We assume without loss of generality that𝑥₀ = 𝑦₀ = 0 with the help ofProposition 1. Given an

(𝑎, −𝑎, 𝑐, −𝑐)-additive function𝜑, take a𝑦₁∈ 𝑌with‖𝑦₁‖ = 1 and a nonzero functionalℎin𝑋∗, the dual space of𝑋. Put

𝑓 (𝑥) = 𝜑 (ℎ (𝑥)) 𝑦1 (𝑥 ∈ 𝑋) . (12)

Then, we can easily see that 𝑓 is an (𝑎, −𝑎, 𝑐, −𝑐)-additive mapping of𝑋into𝑌. Also, if𝜑is discontinuous, then so is

𝑓. In fact, if𝜑is discontinuous, we can find a sequence{𝑎_𝑛} inKsuch that lim_{𝑛 → ∞}𝑎_𝑛 = 0and|𝜑(𝑎_𝑛)| ≥ 1(𝑛 = 1, 2, . . .). Choose an𝑥₁in𝑋withℎ(𝑥₁) = 1and put𝑥_𝑛 = 𝑎_𝑛𝑥₁ for each𝑛 ∈ N. Then,‖𝑥_𝑛‖ = |𝑎_𝑛| ‖𝑥₁‖ → 0as𝑛 → ∞and

‖𝑓(𝑥_𝑛)‖ = |𝜑(𝑎_𝑛)| ≥ 1 (𝑛 = 1, 2, . . .), so𝑓is discontinuous, as required. ThereforeTheorem 3(I), (II)-(i), and (II)-(iii) follow easily from Lemmas4and6.

Given an(𝑎, −𝑎, 𝑐, −𝑐)-additive mapping𝑓of𝑋into𝑌, take𝑥 ∈ 𝑋andℎ ∈ 𝑌∗arbitrarily, and put𝜑(𝑡) = ℎ(𝑓(𝑡𝑥)) for each𝑡 ∈K. Then,𝜑 :K → Kis(𝑎, −𝑎, 𝑐, −𝑐)-additive. If

𝜑 = 0for eachℎ ∈ 𝑌∗and𝑥 ∈ 𝑋, then𝑓 = 0by the Hahn-Banach theorem. Therefore,Theorem 3(II)-(ii) and (II)-(iv) follow easily fromLemma 6. The final assertion in (II) also follows fromLemma 6and its proof.

3. A Stability of Generalized

Additive Mappings

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function𝜀(say acontrol function) on𝑋 × 𝑋and also a certain nonnegative function𝛿on𝑋which depends on𝜀. We say that a system of all(𝑎, 𝑏, 𝑐, 𝑑; 𝑥₀, 𝑦₀)-additive mappings isstrictly

(𝜀, 𝛿)-stablewhenever the following statement is true: “If a mapping𝑓of 𝑋into𝑌satisfies

󵄩󵄩󵄩󵄩

󵄩𝑓 (𝑎𝑥 + 𝑏𝑥󸀠+ 𝑥0) − 𝑐𝑓 (𝑥) − 𝑑𝑓 (𝑥󸀠) − 𝑦0󵄩󵄩󵄩󵄩_{󵄩 ≤ 𝜀 (𝑥, 𝑥}󸀠)

(†)

for all𝑥, 𝑥󸀠 ∈ 𝑋, then there exists a unique(𝑎, 𝑏, 𝑐, 𝑑; 𝑥₀, 𝑦₀) -additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝛿 (𝑥) (‡)

for all𝑥 ∈ 𝑋.”

Throughout the remainder of this paper, we assume that𝑌 is a Banach space. This is because all of our results depend on the following theorems whose proofs need the Banach fixed point theorem.

Theorem A (see [8, Theorem 3.1]). Let𝑎 + 𝑏 ̸= 0and𝐾 ≥ 0

with𝐾|𝑐+𝑑| < 1. One takes a control function𝜀which satisfies

𝜀 (𝑥, 𝑥󸀠) ≤ 𝐾𝜀 ((𝑎 + 𝑏) 𝑥 + 𝑥0, (𝑎 + 𝑏) 𝑥󸀠+ 𝑥0) (13)

for all𝑥, 𝑥󸀠∈ 𝑋and puts

𝛿 (𝑥) = 𝐾𝜀 (𝑥, 𝑥)

1 − 𝐾 |𝑐 + 𝑑| (14)

for each𝑥 ∈ 𝑋.

Then, the strict(𝜀, 𝛿)-stability holds for the system of(𝑎, 𝑏,

𝑐, 𝑑; 𝑥₀, 𝑦₀)-additive mappings.

Theorem B (see [8, Theorem 3.2]). Let𝑐 + 𝑑 ̸= 0and𝐾 ≥ 0

with𝐾 < |𝑐 + 𝑑|. One takes a control function𝜀which satisfies

𝜀 ((𝑎 + 𝑏) 𝑥 + 𝑥0, (𝑎 + 𝑏) 𝑥󸀠+ 𝑥0) ≤ 𝐾𝜀 (𝑥, 𝑥󸀠) (15)

𝛿 (𝑥) = _{|𝑐 + 𝑑| − 𝐾}𝜀 (𝑥, 𝑥) (16)

Then, the strict (𝜀, 𝛿)-stability holds for the system of

(𝑎, 𝑏, 𝑐, 𝑑; 𝑥₀, 𝑦₀)-additive mappings.

Both of these theorems do not say about (𝑎, −𝑎, 𝑐,

−𝑐; 𝑥₀, 𝑦₀)-additive mappings at all; however, we will get the following stability theorems for them. Theorem 7 is of the case𝑎 ̸= − 1,Theorem 8is of the case𝑏 ̸= − 1, and Theorems

9and10are for(−1, 1, −1, 1; 𝑥₀, 𝑦₀)-additive mappings. These cover all of the systems of(𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive map-pings.

Theorem 7. Let𝑎 + 1 ̸= 0and𝐾 ≥ 0with𝐾|1 + 𝑐| < |𝑐|. One

takes a control function𝜀which satisfies

𝜀 (𝑥, 𝑥󸀠) ≤ 𝐾𝜀 ((𝑎−1+ 1) 𝑥 − 𝑎−1𝑥₀, (𝑎−1+ 1) 𝑥󸀠− 𝑎−1𝑥₀)

(17)

for all𝑥, 𝑥󸀠∈ 𝑋, and puts

𝛿 (𝑥) = 𝐾𝜀 ((𝑎

−1_{+ 1) 𝑥 − 𝑎}−1_𝑥

0, 𝑥)

|𝑐| − 𝐾 |1 + 𝑐| (18)

Then, the strict(𝜀, 𝛿)-stability holds for the system of(𝑎, −𝑎,

𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive mappings.

Proof. Put𝑢 = 𝑎𝑥 − 𝑎𝑥󸀠+ 𝑥₀and𝑢󸀠= 𝑥󸀠for each𝑥, 𝑥󸀠∈ 𝑋. Then,(†)changes into

󵄩󵄩󵄩󵄩

󵄩𝑓 (𝑎−1𝑢 + 𝑢󸀠− 𝑎−1𝑥0) − 𝑐−1𝑓 (𝑢) − 𝑓 (𝑢󸀠) + 𝑐−1𝑦0󵄩󵄩󵄩󵄩_󵄩

≤ 𝜀₁(𝑢, 𝑢󸀠) , (19)

where

𝜀₁(𝑢, 𝑢󸀠) = 1

|𝑐|𝜀 (𝑎−1𝑢 + 𝑢󸀠− 𝑎−1𝑥0, 𝑢󸀠) (20)

for all𝑢, 𝑢󸀠 ∈ 𝑋. By denoting𝑎₁ = 𝑎−1,𝑏₁ = 1,𝑐₁ = 𝑐−1,

𝑑₁ = 1,𝑢₀= −𝑎−1𝑥₀andV₀ = −𝑐−1𝑦₀in (19),(†)changes up to the following estimate of𝑓by the control function𝜀₁:

󵄩󵄩󵄩󵄩

󵄩𝑓 (𝑎1𝑢 + 𝑏1𝑢󸀠+ 𝑢0) − 𝑐1𝑓 (𝑢) − 𝑑1𝑓 (𝑢󸀠) −V0󵄩󵄩󵄩󵄩_{󵄩 ≤ 𝜀}1(𝑢, 𝑢󸀠)

(21)

for all𝑢, 𝑢󸀠∈ 𝑋.

Under these transformations, 𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁, and 𝜀₁ are equipped with

𝑎1+ 𝑏1 ̸= 0, 𝐾 󵄨󵄨󵄨󵄨𝑐1+ 𝑑1󵄨󵄨󵄨󵄨 < 1,

𝜀₁(𝑢, 𝑢󸀠) ≤ 𝐾𝜀₁((𝑎₁+ 𝑏₁) 𝑢 + 𝑢₀, (𝑎₁+ 𝑏₁) 𝑢󸀠+ 𝑢₀) (22)

for all𝑢, 𝑢󸀠 ∈ 𝑋. The latter follows from the inequality in which𝜀must satisfy because by using (20), we get

|𝑐| 𝜀1(𝑢, 𝑢󸀠) = 𝜀 (𝑎−1𝑢 + 𝑢󸀠+ 𝑢0, 𝑢󸀠) = 𝜀 (𝑥, 𝑥󸀠) ,

𝜀 ((𝑎−1+ 1) 𝑥 − 𝑎−1𝑥₀, (𝑎−1+ 1) 𝑥󸀠− 𝑎−1𝑥₀)

= 𝜀 ((𝑎−1+ 1) 𝑥 − 𝑎−1𝑥₀, (𝑎₁+ 𝑏₁) 𝑢󸀠+ 𝑢₀)

= 𝜀 ((𝑎−1+ 1) (𝑎−1𝑢 + 𝑢󸀠+ 𝑢₀)

−𝑎−1𝑥₀, (𝑎₁+ 𝑏₁) 𝑢󸀠+ 𝑢₀)

= 𝜀 (𝑎−1((𝑎₁+ 𝑏₁) 𝑢 + 𝑢₀)

+ ((𝑎₁+ 𝑏₁) 𝑢󸀠+ 𝑢₀)

−𝑎−1𝑥0, (𝑎1+ 𝑏1) 𝑢󸀠+ 𝑢0)

= |𝑐| 𝜀1((𝑎1+ 𝑏1) 𝑢 + 𝑢0, (𝑎1+ 𝑏1) 𝑢󸀠+ 𝑢0)

(23)

for all𝑢, 𝑢󸀠 ∈ 𝑋. Since (21) and (22) hold, it follows from Theorem A that there exists a unique (𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁; 𝑢₀,V₀) -additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝐾𝜀1(𝑥, 𝑥)

(5)

for all𝑥 ∈ 𝑋. However, we can easily see the following two assertions:

(i)𝑓_∞is(𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁; 𝑢₀,V₀)-additive if and only if𝑓_∞is

(𝑎, −𝑎, 𝑐, −𝑐; 𝑥0, 𝑦0)-additive,

(ii) (24) is equivalent to(‡).

This completes the proof.

Theorem 8. Let𝑐 + 1 ̸= 0and𝐾 ≥ 0with𝐾|𝑐| < |𝑐 + 1|. One

takes a control function𝜀which satisfies

𝜀 ((𝑎−1+ 1) 𝑥 − 𝑎−1𝑥₀, (𝑎−1+ 1) 𝑥󸀠− 𝑎−1𝑥₀)

≤ 𝐾𝜀 (𝑥, 𝑥󸀠) (25)

𝛿 (𝑥) = 𝜀 ((𝑎

−1_{+ 1) 𝑥 − 𝑎}−1_𝑥

0, 𝑥)

|𝑐 + 1| − 𝐾 |𝑐| (26)

Then the strict(𝜀, 𝛿)-stability holds for the system of(𝑎, −𝑎,

𝑐, −𝑐; 𝑥₀, 𝑦₀)-additive mappings.

Proof. We consider the same transformations and the same estimate (21) of𝑓by𝜀₁in the proof ofTheorem 7. Under these transformations, we have

𝑐₁+ 𝑑₁ ̸= 0, _{𝐾 < 󵄨󵄨󵄨󵄨𝑐}₁+ 𝑑₁󵄨󵄨󵄨󵄨. (27)

Moreover, for every𝑢, 𝑢󸀠∈ 𝑋we have

𝜀₁((𝑎₁+ 𝑏₁) 𝑢 + 𝑢₀, (𝑎₁+ 𝑏₁) 𝑢󸀠+ 𝑢₀) ≤ 𝐾𝜀₁(𝑢, 𝑢󸀠) , (28)

because

|𝑐| 𝜀1(𝑢, 𝑢󸀠) = 𝜀 (𝑎−1𝑢 + 𝑢󸀠+ 𝑢0, 𝑢󸀠) = 𝜀 (𝑥, 𝑥󸀠) , (29)

|𝑐| 𝜀1((𝑎1+ 𝑏1) 𝑢 + 𝑢0, (𝑎1+ 𝑏1) 𝑢󸀠+ 𝑢0)

= 𝜀 ((𝑎−1+ 1) 𝑥 − 𝑎−1𝑥₀, (𝑎−1+ 1) 𝑥󸀠− 𝑎−1𝑥₀) .

(30)

Since (21), (27) and (28) hold, it follows from Theorem B that there exists a unique(𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁; 𝑢₀,V₀)-additive mapping

𝑓∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 󵄨󵄨󵄨󵄨𝑐𝜀1(𝑥, 𝑥)

1+ 𝑑1󵄨󵄨󵄨󵄨 − 𝐾 (31)

for all𝑥 ∈ 𝑋. This means(‡)and𝑓_∞is(𝑎, −𝑎, 𝑐, −𝑐; 𝑥₀, 𝑦₀) -additive.

Theorem 9. Let0 ≤ 𝐾 < 1/2. One takes a control function𝜀

which satisfies

𝜀 (𝑥, 𝑥󸀠) ≤ 𝐾𝜀 (2𝑥 − 𝑥₀, 2𝑥󸀠− 𝑥₀) (32)

𝛿 (𝑥) = 𝐾𝜀 (𝑥, 2𝑥 − 𝑥_{1 − 2𝐾} 0) (33)

Then the strict (𝜀, 𝛿)-stability holds for the system of

(−1, 1, −1, 1; 𝑥₀, 𝑦₀)-additive mappings.

Proof. Put𝑢 = 𝑥, 𝑢󸀠= −𝑥 + 𝑥󸀠+ 𝑥₀for each𝑥, 𝑥󸀠∈ 𝑋. Then,

(†)changes into the following estimate of 𝑓by the control function𝜀₁:

󵄩󵄩󵄩󵄩

󵄩𝑓 (𝑢 + 𝑢󸀠− 𝑥0) − 𝑓 (𝑢) − 𝑓 (𝑢󸀠) + 𝑦0󵄩󵄩󵄩󵄩_{󵄩 ≤ 𝜀}1(𝑢, 𝑢󸀠) , (34)

where

𝜀₁(𝑢, 𝑢󸀠) = 𝜀 (𝑢, 𝑢 + 𝑢󸀠− 𝑥₀) . (35)

for all𝑢, 𝑢󸀠∈ 𝑋. Put𝑎₁= 𝑏₁= 𝑐₁= 𝑑₁= 1.

Under these transformations, 𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁, and 𝜀₁ are equipped with

𝑎₁+ 𝑏₁= 2 ̸= 0, _{𝐾 󵄨󵄨󵄨󵄨𝑐}₁+ 𝑑₁󵄨󵄨󵄨󵄨 = 2𝐾 < 1,

𝜀₁(𝑢, 𝑢󸀠) ≤ 𝐾𝜀₁((𝑎₁+ 𝑏₁) 𝑢 − 𝑥₀, (𝑎₁+ 𝑏₁) 𝑢󸀠− 𝑥₀) (36)

for all 𝑢, 𝑢󸀠 ∈ 𝑋. The latter follows from the following inequality:

𝜀₁(𝑢, 𝑢󸀠) = 𝜀 (𝑢, 𝑢 + 𝑢󸀠− 𝑥₀)

≤ 𝐾𝜀 (2𝑢 − 𝑥₀, 2 (𝑢 + 𝑢󸀠− 𝑥₀) − 𝑥₀)

= 𝐾𝜀 (2𝑢 − 𝑥₀, (2𝑢 − 𝑥₀) + (2𝑢󸀠− 𝑥₀) − 𝑥₀)

= 𝐾𝜀₁(2𝑢 − 𝑥₀, 2𝑢󸀠− 𝑥₀)

(37)

for all 𝑢, 𝑢󸀠 ∈ 𝑋. Since (34), (36) hold, it follows from Theorem A that there exists a unique(𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁; −𝑥₀, −𝑦₀) -additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝐾𝜀1(𝑥, 𝑥)

1 − 𝐾 󵄨󵄨󵄨󵄨𝑐1+ 𝑑1󵄨󵄨󵄨󵄨 (38)

for all𝑥 ∈ 𝑋. However we can easily see the following two assertions:

(i)𝑓_∞ is (𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁; −𝑥₀, −𝑦₀)-additive if and only if

𝑓_∞is(−1, 1, −1, 1; 𝑥₀, 𝑦₀)-additive; (ii) (38) is equivalent to(‡).

This completes the proof.

Theorem 10. Let0 ≤ 𝐾 < 2. One takes a control function𝜀

which satisfies

(6)

𝛿 (𝑥) =𝜀 (𝑥, 2𝑥 − 𝑥_{2 − 𝐾} 0) (40)

Then the strict (𝜀, 𝛿)-stability holds for the system of

(−1, 1, −1, 1; 𝑥₀, 𝑦₀)-additive mappings.

Proof. We consider the same transformation and the same estimate (34) of 𝑓by 𝜀₁ in the proof ofTheorem 9. Under these transformations,𝑐₁, 𝑑₁and𝜀₁are equipped with

𝑐1+ 𝑑1= 2 ̸= 0, 𝐾 < 2 = 󵄨󵄨󵄨󵄨𝑐1+ 𝑑1󵄨󵄨󵄨󵄨, (41)

𝜀₁(2𝑢 − 𝑥₀, 2𝑢󸀠− 𝑥₀)

= 𝜀 (2𝑢 − 𝑥₀, (2𝑢 − 𝑥₀) + (2𝑢󸀠− 𝑥₀) − 𝑥₀)

= 𝜀 (2𝑥 − 𝑥₀, 2 (𝑢 + 𝑢󸀠− 𝑥₀) − 𝑥₀)

= 𝜀 (2𝑥 − 𝑥₀, 2𝑥󸀠− 𝑥₀)

≤ 𝐾𝜀 (𝑥, 𝑥󸀠)

= 𝐾𝜀 (𝑢, 𝑢 + 𝑢󸀠− 𝑥₀)

= 𝐾𝜀1(𝑢, 𝑢󸀠)

(42)

for all𝑢, 𝑢󸀠∈ 𝑋. So it follows that

𝜀₁((𝑎₁+ 𝑏₁) 𝑢 − 𝑥₀, (𝑎₁+ 𝑏₁) 𝑢󸀠− 𝑥₀) ≤ 𝐾𝜀₁(𝑢, 𝑢󸀠) (43)

for all𝑢, 𝑢󸀠∈ 𝑋. Since (34), (41) and (43) hold, it follows from Theorem B that there exists a unique(𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁; −𝑥₀, −𝑦₀) -additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 󵄨󵄨󵄨󵄨𝑐𝜀1(𝑥, 𝑥)

1+ 𝑑1󵄨󵄨󵄨󵄨 − 𝐾 (44)

for all 𝑥, 𝑥󸀠 ∈ 𝑋. This means (‡) and 𝑓_∞ is (−1, 1, −1,

1; 𝑥₀, 𝑦₀)-additive.

4. Concrete Examples

Throughout this section, let𝑥₀= 𝑦₀= 0. We fix nonnegative constants𝑝, 𝑞and𝛿and take the control function𝜀defined by𝜀(𝑥, 𝑥󸀠) = 𝛿‖𝑥‖𝑝‖𝑥󸀠‖𝑞for every𝑥, 𝑥󸀠∈ 𝑋.

Corollary 11. When𝑎 + 1 ̸= 0and|𝑎|𝑝+𝑞|𝑐 + 1| < |𝑐||𝑎 + 1|𝑝+𝑞,

one puts

𝛿󸀠(𝑥) = 𝛿|𝑎 + 1|𝑝|𝑎|𝑞‖𝑥‖𝑝+𝑞

|𝑐| |𝑎 + 1|𝑝+𝑞_{− |𝑐 + 1| |𝑎|}𝑝+𝑞 (45)

Then, the system of (𝑎, −𝑎, 𝑐, −𝑐)-additive mappings is strictly(𝜀, 𝛿󸀠)-stable.

Proof. Put𝐾 = |𝑎−1+ 1|−(𝑝+𝑞). Then𝐾|𝑐 + 1| < |𝑐|and

𝜀 (𝑥, 𝑥󸀠) = 𝐾𝜀 ((𝑎−1+ 1) 𝑥, (𝑎−1+ 1) 𝑥󸀠) (46)

for all𝑥, 𝑥󸀠∈ 𝑋. ByTheorem 7, for a mapping𝑓of𝑋into𝑌 satisfying (†) for𝑎 = −𝑏and𝑐 = −𝑑, there exists a unique

(𝑎, −𝑎, 𝑐, −𝑐)-additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝐾𝛿󵄨󵄨󵄨󵄨󵄨𝑎 −1

+ 1󵄨󵄨󵄨󵄨󵄨𝑝 ‖𝑥‖𝑝+𝑞

|𝑐| − 𝐾 |𝑐 + 1| (47)

for all𝑥, 𝑥󸀠∈ 𝑋. Because of𝐾 = |𝑎−1+ 1|−(𝑝+𝑞), we have the corollary.

Corollary 12. When𝑐 + 1 ̸= 0and|𝑎 + 1|𝑝+𝑞|𝑐| < |𝑐 + 1||𝑎|𝑝+𝑞,

one puts

𝛿󸀠(𝑥) = 𝛿|𝑎 + 1|𝑝|𝑎|𝑞‖𝑥‖𝑝+𝑞

|𝑎|𝑝+𝑞|𝑐 + 1| − |𝑎 + 1|𝑝+𝑞|𝑐| (48)

Then the system of (𝑎, −𝑎, 𝑐, −𝑐)-additive mappings is strictly(𝜀, 𝛿󸀠)-stable.

Proof. Put𝐾 = |𝑎−1+ 1|𝑝+𝑞. Then𝐾|𝑐| < |𝑐 + 1|and

𝜀 ((𝑎−1_{+ 1) 𝑥, (𝑎}−1_{+ 1) 𝑥}󸀠_{) = 𝐾𝜀 (𝑥, 𝑥}󸀠₎ ₍₄₉₎

for all𝑥, 𝑥󸀠 ∈ 𝑋. ByTheorem 8, for a mapping𝑓of𝑋into

𝑌satisfying (†) for𝑎 = −𝑏and𝑐 = −𝑑, there exists a unique

(𝑎, −𝑎, 𝑐, −𝑐)-additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝛿󵄨󵄨󵄨󵄨󵄨𝑎 −1

+ 1󵄨󵄨󵄨󵄨󵄨𝑝 ‖𝑥‖𝑝+𝑞

|𝑐 + 1| − 𝐾 |𝑐| (50)

for all 𝑥 ∈ 𝑋. Because of 𝐾 = |𝑎−1+ 1|𝑝+𝑞, we have the corollary.

Corollary 13. When𝑝 + 𝑞 > 1, one puts

𝛿󸀠(𝑥) = 2𝑞₂_𝑝+𝑞𝛿‖𝑥‖_{− 2}𝑝+𝑞 (51)

Then, the system of (−1, 1, −1, 1)-additive mappings is strictly(𝜀, 𝛿󸀠)-stable.

Proof. Put𝐾 = 2−(𝑝+𝑞). Then,

𝜀 (𝑥, 𝑥󸀠) = 𝐾𝜀 (2𝑥, 2𝑥󸀠) (52)

for all𝑥, 𝑥󸀠 ∈ 𝑋. Since𝑝 + 𝑞 > 1, we also have0 ≤ 𝐾 < 1/2. ByTheorem 9, for a mapping𝑓of𝑋into𝑌satisfying (†) for

𝑎 = 𝑐 = −1and𝑏 = 𝑑 = 1, there exists a unique(−1, 1, −1, 1) -additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤2

𝑞_{𝛿𝐾 ‖𝑥‖}𝑝+𝑞

1 − 2𝐾 (53)

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Corollary 14. When𝑝 + 𝑞 < 1, one puts

𝛿󸀠(𝑥) = 2𝑞_{2 − 2}𝛿‖𝑥‖_𝑝+𝑞𝑝+𝑞 (54)

Then, the system of (−1, 1, −1, 1)-additive mappings is strictly(𝜀, 𝛿󸀠)-stable.

Proof. Put𝐾 = 2𝑝+𝑞. Then,

𝜀 (2𝑥, 2𝑥󸀠) = 𝐾𝜀 (𝑥, 𝑥󸀠) (55)

for all𝑥, 𝑥󸀠 ∈ 𝑋. Since𝑝 + 𝑞 < 1, we also have0 ≤ 𝐾 < 2. ByTheorem 10, for a mapping𝑓of𝑋into𝑌satisfying (†) for

𝑎 = 𝑐 = −1and𝑏 = 𝑑 = 1, there exists a unique(−1, 1, −1, 1) -additive mapping𝑓_∞such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓∞(𝑥)󵄩󵄩󵄩󵄩 ≤ 2

𝑞 _{𝛿 ‖𝑥‖}𝑝+𝑞

2 − 𝐾 (56)

for all𝑥 ∈ 𝑋. Because of𝐾 = 2𝑝+𝑞, we have the corollary.

Remark 15. InCorollary 14, taking𝑝 = 𝑞 = 0, we can easily observe that the corollary is just the stability result due to Hyers [2, Theorem 1].

Acknowledgment

The third and fifth authors are partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promo-tion of Science.

References

[1] S. M. Ulam,A Collection of Mathematical Problems, vol. 8 of

Interscience Tracts in Pure and Applied Mathematics,, Inter-science Publishers, New York, NY, USA, 1960.

[2] D. H. Hyers, “On the stability of the linear functional equation,”

Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.

[3] T. Aoki, “On the stability of the linear transformation in Banach spaces,”Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.

[4] T. M. Rassias, “On the stability of the linear mapping in Banach spaces,”Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.

[5] J. Acz´el, Lectures on Functional Equations and Their Appli-cations, vol. 19 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1966.

[6] S.-E. Takahasi, T. Miura, and H. Takagi, “On a Hyers-Ulam-Aoki-Rassias type stability and a fixed point theorem,”Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 423–439, 2010. [7] S. -E. Takahasi, T. Miura, H. Takagi, and K. Tanahashi, “Ulam

type stability problems for alternative homomorphisms”. [8] M. Todoroki, K. Kumahara, T. Miura, and S.-E. Takahasi,

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