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

Abstract

We give an Ulam type stability result for the following functional equation: 𝑓(𝛼𝑥−𝛼𝑥^′+𝑥₀)=𝛽𝑓(𝑥)−𝛽𝑓(𝑥^′)+𝑦₀(forall𝑥,𝑥^′∈𝑋) under a suitable condition. We also give a concrete stability result for the case taking up 𝛿‖𝑥‖^𝑝‖𝑥^′‖^𝑞 as a control function.

1. Introduction

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

Let 𝑋 and 𝑌 be normed spaces over 𝕂, which denotes either the real field ℝ or the complex field ℂ. Throughout the paper, we fix scalars 𝑎,𝑏,𝑐,𝑑∈𝕂⧵{0} and vectors 𝑥₀∈𝑋 and 𝑦₀∈𝑌. We say that a mapping 𝑓 of 𝑋 into 𝑌 is (𝑎,𝑏,𝑐,𝑑;𝑥₀,𝑦₀)-additive if 𝑓𝑎𝑥+𝑏𝑥^′+𝑥₀=𝑐𝑓(𝑥)+𝑑𝑓𝑥^′+𝑦₀ (1) for all 𝑥,𝑥^′∈𝑋. When 𝑥₀=𝑦₀=0, we say it to be (𝑎,𝑏,𝑐,𝑑)-additive. Aczél [5] specified what this generalized Cauchy equation is. The Ulam type stability problem for such an 𝑓 has 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 case 𝑎+𝑏=𝑐+𝑑=0. In Section 2, we state the details of (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mappings (Theorem 3). In Section 3, we give our main results about the stability for them (see Theorems 7–10). In the final section, we apply the results to some concrete examples, where we take up 𝛿‖𝑥‖^𝑝‖𝑥^′‖^𝑞 as a control function 𝜀(𝑥,𝑥^′) (see Corollaries 11–14).

2. (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-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. Let 𝑓 and 𝑔 be two mappings of 𝑋 into 𝑌 such that 𝑔(𝑥)=𝑓(𝑥+𝑥₀)−𝑦₀ for all 𝑥∈𝑋. Then the following three statements are equivalent: (i)𝑓  is (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive, (ii)𝑔  is (𝑎,−𝑎,𝑐,−𝑐)-additive, (iii)𝑔  is additive and 𝑔(𝑎𝑥)=𝑐𝑔(𝑥)   for all 𝑥∈𝑋.

Proof. (i)⇔(ii) Since 𝑓𝑎𝑥−𝑎𝑥^′+𝑥₀=𝑔𝑎𝑥−𝑎𝑥^′+𝑦₀=𝑔𝑎𝑥−𝑥₀−𝑎𝑥^′−𝑥₀+𝑦₀,𝑐(𝑥)−𝑐𝑓𝑥^′+𝑦₀=𝑐𝑓(𝑥)−𝑦₀−𝑐𝑓𝑥^′−𝑦₀+𝑦₀=𝑐𝑔𝑥−𝑥₀−𝑐𝑔𝑥^′−𝑥₀+𝑦₀ (2) for all 𝑥,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(forall𝑥,𝑥^′∈𝑋) ⇔𝑔(𝑎𝑥−𝑎𝑥^′)=𝑐𝑔(𝑥)−𝑐𝑔(𝑥^′)(forall𝑥,𝑥^′∈𝑋) ⇔𝑔∶(𝑎,−𝑎,𝑐,−𝑐)-additive.
(ii)⇒(iii) Suppose that 𝑔𝑎𝑥−𝑎𝑥^′=𝑐𝑔(𝑥)−𝑐𝑔𝑥^′ (3) for all 𝑥,𝑥^′∈𝑋. When 𝑥=𝑥^′, we have 𝑔(0)=0. Using this, 𝑔(𝑎𝑥)=𝑐𝑔(𝑥) and also 𝑔(−𝑎𝑥)=−𝑐𝑔(𝑥) for all 𝑥∈𝑋. Therefore, 𝑔𝑥+𝑥^′𝑎𝑥=𝑔𝑎−𝑥−𝑎^′𝑎𝑥=𝑐𝑔𝑎−𝑥−𝑐𝑔^′𝑎𝑥=𝑐𝑔𝑎+()𝑔−𝑥−𝑐^′𝑎=𝑔(𝑥)+𝑔𝑥^′ (4) for all 𝑥,𝑥^′∈𝑋.
(iii)⇒(ii) Because 𝑔(−𝑥)=−𝑔(𝑥) for all 𝑥∈𝑋 (see also the following remark), it is trivial.

Remark 2. We denote by ℚ the field of all rational numbers. It is well known that if 𝑔 is additive, then 𝑔(𝑞𝑥)=𝑞𝑔(𝑥) for every 𝑞∈ℚ and 𝑥∈𝑋, that is, 𝑔 is ℚ-linear. Hence, if 𝑔 is additive and continuous, 𝑔 must be ℝ-linear. On the other hand, when 𝕂=ℂ, we have a lot of continuous additive nonlinear mappings by considering the composition of linear transformations on ℝ² and the ℝ-linear isometry (𝑥,𝑦)↦𝑥+𝑖𝑦.

The constant 𝑦₀ 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) If 𝑎=𝑐, then there exists a discontinuous (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mapping of 𝑋 into 𝑌.
(II) If 𝑎≠𝑐, then the followings hold: (i)If both 𝑎 and 𝑐 are transcendental numbers, then there exists a discontinuous (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mapping of 𝑋 into 𝑌. (ii)If one of 𝑎 and 𝑐 is transcendental and the other is algebraic, then the constant 𝑦₀ is a unique (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mapping of 𝑋 into 𝑌. (iii)If both 𝑎 and 𝑐 are algebraic with a common minimal polynomial, then there exists a discontinuous (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mapping of 𝑋 into 𝑌. (iv)If 𝑎 and 𝑐 are algebraic with distinct minimal polynomials, then the constant 𝑦₀ is a unique (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mapping of 𝑋 into 𝑌. Moreover, when 𝕂=ℝ, there is no nontrivial continuous (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mapping 𝑓 of 𝑋 into 𝑌. On the other hand, when 𝕂=ℂ, if 𝑎 is not real and 𝑎=𝑐 (the complex conjugate of 𝑐), then the mapping 𝑥↦𝑓(𝑥+𝑥₀)−𝑦₀ must be conjugate linear for every continuous (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mapping 𝑓 of 𝑋 into 𝑌.

In order to show Theorem 3, we need some lemmas for 𝕂-valued (𝑎,−𝑎,𝑐,−𝑐)-additive functions defined on 𝕂. For any 𝑥∈𝕂, we denote by ℚ(𝑥) the subfield of 𝕂 generated by 𝑥 over ℚ.

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

Lemma 4. Any (𝑎,−𝑎,𝑐,−𝑐)-additive 𝜑 of 𝕂 into itself satisfies 𝜑(𝑝(𝑎)𝑥)=𝑝(𝑐)𝜑(𝑥) for all 𝑥∈𝕂 and 𝑝(𝑋)∈ℚ[𝑋].

Proof. Note that 𝜑 is additive and 𝜑(𝑎𝑥)=𝑐𝜑(𝑥) for each 𝑥∈𝕂 by Proposition 1. Let 𝑝(𝑋)=𝑎₀+𝑎₁𝑋+⋯+𝑎_𝑛𝑋^𝑛 with 𝑎₀,𝑎₁,…,𝑎_𝑛∈ℚ. Since 𝜑 is ℚ-linear, 𝜑𝑝(𝑎)𝑥=𝑎₀𝜑(𝑥)+𝑎₁𝜑(𝑎𝑥)+⋯+𝑎_𝑛𝜑𝑎^𝑛𝑥=𝑎₀𝜑(𝑥)+𝑎₁𝑐𝜑(𝑥)+⋯+𝑎_𝑛𝑐^𝑛𝜑(𝑥)=𝑝(𝑐)𝜑(𝑥) (5) for all 𝑥∈𝕂.

Lemma 5. Let 𝐸 and 𝐹 be subfields of 𝕂 and 𝜓 an isomorphism of 𝐸 onto 𝐹. Then, 𝜓 has an additive bijective extension 𝜑 to the full space 𝕂 such that 𝜑(𝜀𝑥)=𝜓(𝜀)𝜑(𝑥) (6) for all 𝜀∈𝐸 and 𝑥∈𝕂. Moreover, one has a discontinuous one whenever ℝ⧵𝐸≠∅.

Proof. Let {𝑒_𝑗∶𝑗∈𝐽} and {𝑓_𝑗∶𝑗∈𝐽^′} be an 𝐸-linear base and an 𝐹-linear base of 𝕂, respectively. Because both of them have same cardinality, we take 𝐽=𝐽^′. Moreover, we may assume without loss of generality that 𝑒_𝑗0=𝑓_𝑗0=1 for some 𝑗₀∈𝐽. For any 𝑥∈𝕂, there exist a finite 𝐼⊂𝐽 and {𝑥_𝑖}_𝑖∈𝐼⊆𝐸 such that 𝑥=_𝑖∈𝐼𝑥_𝑖𝑒_𝑖, (7) and this decomposition is unique. Hence, we can define 𝜑∶𝕂→𝕂 by 𝜑(𝑥)=_𝑖∈𝐼𝜓𝑥_𝑖𝑓_𝑖. (8) This 𝜑 is a desired extension.
In order to get a discontinuous extension, we consider the base {𝑒_𝑗∶𝑗∈𝐽} and {𝑓_𝑗∶𝑗∈𝐽} such that 𝑓_𝑗1≠𝑒_𝑗1∈ℝ⧵𝐸 for some 𝑗₁∈𝐽. Because ℚ⊆𝐸, take a rational sequence {𝜀_𝑛} converging to 𝑒_𝑗1. Suppose that 𝜑 is continuous. Then, 𝜑𝜀_𝑛→𝜑𝑒_𝑗1=𝑓_𝑗1,𝜑𝜀_𝑛=𝜓𝜀_𝑛=𝜀_𝑛→𝑒_𝑗1, (9) as 𝑛→∞. This is contradiction. Thus 𝜑 is discontinuous.

Lemma 6. Suppose that 𝑎≠𝑐. (i)If both 𝑎 and 𝑐 are transcendental numbers, then there exists a discontinuous (𝑎,−𝑎,𝑐,−𝑐)-additive function of 𝕂 into itself. (ii)If one of 𝑎 and 𝑐 is transcendental and the other is algebraic, then the constant 0 is a unique (𝑎,−𝑎,𝑐,−𝑐)-additive function of 𝕂 into itself. (iii)If both 𝑎 and 𝑐 are algebraic with a common minimal polynomial, then there exists a discontinuous (𝑎,−𝑎,𝑐,−𝑐)-additive function of 𝕂 into itself. (iv)If 𝑎 and 𝑐 are algebraic with distinct minimal polynomials, then the constant 0 is a unique (𝑎,−𝑎,𝑐,−𝑐)-additive function of 𝕂 into itself. Moreover, when 𝕂=ℝ, every nontrivial (𝑎,−𝑎,𝑐,−𝑐)-additive function of 𝕂 into itself is discontinuous. On the other hand, when 𝕂=ℂ, if 𝑎 is not real and 𝑎=𝑐, then any continuous (𝑎,−𝑎,𝑐,−𝑐)-additive function 𝑓 of ℂ into itself must be of form 𝑓(𝑥)=𝛼𝑥(𝑥∈ℂ) for some 𝛼∈ℂ.

Proof. (i) Suppose that both 𝑎 and 𝑐 are 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. By Lemma 5, because ℝ⧵ℚ(𝑎)≠∅, 𝜑_𝑎𝑐 has a discontinuous additive extension 𝜑 to 𝕂 such that 𝜑(𝜀𝑥)=𝜑_𝑎𝑐(𝜀)𝜑(𝑥) for every 𝜀∈ℚ(𝑎) and 𝑥∈𝕂. Then, 𝜑(𝑎𝑥)=𝜑_𝑎𝑐(𝑎)𝜑(𝑥)=𝑐𝜑(𝑥) for all 𝑥∈𝕂, and, hence, 𝜑 is (𝑎,−𝑎,𝑐,−𝑐)-additive by Proposition 1.
(ii) Let 𝜑 be any (𝑎,−𝑎,𝑐,−𝑐)-additive function. If 𝑎 is transcendental and 𝑐 is algebraic with nonzero polynomial such that 𝑝(𝑐)=0, then from Lemma 4, we have 𝜑(𝑥)=𝜑𝑝(𝑎)𝑝(𝑎)⁻¹𝑥=𝑝(𝑐)𝜑𝑝(𝑎)⁻¹𝑥=0 (10) for all 𝑥∈𝕂. If 𝑐 is transcendental and 𝑎 is algebraic with nonzero polynomial such that 𝑞(𝑎)=0, then from Lemma 4, we have 𝜑(𝑥)=1𝑞(𝑐)𝑞(𝑐)𝜑(𝑥)=1𝑞(𝑐)𝜑𝑞(𝑎)𝑥=1𝑞(𝑐)𝜑(0)=0 (11) for all 𝑥∈𝕂.
(iii) If 𝑎 is algebraic with minimal polynomial 𝑝(𝑋)∈ℚ[𝑋], then ℚ(𝑎) consists of all polynomials 𝑓(𝑎) in 𝑎 of degree up to deg𝑝−1. So, if 𝑐 is also algebraic with the same minimal polynomial 𝑝, then the substitution 𝑎→𝑐 induces an automorphism 𝜑_𝑎𝑐∶ℚ(𝑎)→ℚ(𝑐)=ℚ(𝑎). As same as (i), 𝜑_𝑎𝑐 has a discontinuous (𝑎,−𝑎,𝑐,−𝑐)-additive extension.
(iv) Suppose that 𝑎 and 𝑐 are algebraic with distinct minimal polynomials 𝑝 and 𝑞 over the field ℚ, respectively. Let 𝜑 be any (𝑎,−𝑎,𝑐,−𝑐)-additive function. To show 𝜑=0, we assume, on the contrary, that there is an 𝑥₀∈𝕂 with 𝜑(𝑥₀)≠0. Then, from Lemma 5, we have 𝑝(𝑐)𝜑(𝑥₀)=𝜑(𝑝(𝑎)𝑥₀)=𝜑(0)=0, and hence 𝑝(𝑐)=0. This contradicts the prerequisite for 𝑎 and 𝑐. Hence, 𝜑 must be zero.
When 𝕂=ℝ, since every continuous additive function is ℝ-linear and 𝑎≠𝑐, there is no continuous (𝑎,−𝑎,𝑐,−𝑐)-additive function by Proposition 1. Now, we consider the case 𝕂=ℂ. Let 𝜑 be a nontrivial continuous (𝑎,−𝑎,𝑐,−𝑐)-additive function. Note that 𝜑 is ℝ-linear. If 𝑎 is not real and 𝑎=𝑐, we can easily see that 𝜑(𝑖𝑥)=−𝑖𝜑(𝑥) for all 𝑥∈ℂ. Thus, 𝜑 is conjugate linear, and hence 𝜑(𝑥)=𝛼𝑥(𝑥∈ℂ), where 𝛼=𝜑(1).

Proof of Theorem 3. (I) We assume without loss of generality that 𝑥₀=𝑦₀=0 with the help of Proposition 1. Given an (𝑎,−𝑎,𝑐,−𝑐)-additive function 𝜑, take a 𝑦₁∈𝑌 with ‖𝑦₁‖=1 and a nonzero functional ℎ in 𝑋^∗, the dual space of 𝑋. 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 of 𝑋 into 𝑌. Also, if 𝜑 is discontinuous, then so is 𝑓. In fact, if 𝜑 is discontinuous, we can find a sequence {𝑎_𝑛} in 𝕂 such that lim_𝑛→∞𝑎_𝑛=0 and |𝜑(𝑎_𝑛)|≥1 (𝑛=1,2,…). Choose an 𝑥₁ in 𝑋 with ℎ(𝑥₁)=1 and put 𝑥_𝑛=𝑎_𝑛𝑥₁ for each 𝑛∈ℕ. Then, ‖𝑥_𝑛‖=|𝑎_𝑛|‖𝑥₁‖→0 as 𝑛→∞ and ‖𝑓(𝑥_𝑛)‖=|𝜑(𝑎_𝑛)|≥1(𝑛=1,2,…), so 𝑓 is discontinuous, as required. Therefore Theorem 3(I), (II)-(i), and (II)-(iii) follow easily from Lemmas 4 and 6.
Given an (𝑎,−𝑎,𝑐,−𝑐)-additive mapping 𝑓 of 𝑋 into 𝑌, take 𝑥∈𝑋 and ℎ∈𝑌^∗ arbitrarily, and put 𝜑(𝑡)=ℎ(𝑓(𝑡𝑥)) for each 𝑡∈𝕂. Then, 𝜑∶𝕂→𝕂 is (𝑎,−𝑎,𝑐,−𝑐)-additive. If 𝜑=0 for each ℎ∈𝑌^∗ and 𝑥∈𝑋, then 𝑓=0 by the Hahn-Banach theorem. Therefore, Theorem 3(II)-(ii) and (II)-(iv) follow easily from Lemma 6. The final assertion in (II) also follows from Lemma 6 and its proof.

3. A Stability of Generalized Additive Mappings

In this section, we consider a couple of cases which are left out in [8] about the Ulam type stability. We take a nonnegative function 𝜀 (say a control function) on 𝑋×𝑋 and also a certain nonnegative function 𝛿 on 𝑋 which depends on 𝜀. We say that a system of all (𝑎,𝑏,𝑐,𝑑;𝑥₀,𝑦₀)-additive mappings is strictly (𝜀,𝛿)-stable whenever the following statement is true:

“If a mapping 𝑓 of 𝑋 into 𝑌 satisfies𝑓𝑎𝑥+𝑏𝑥^′+𝑥₀−𝑐𝑓(𝑥)−𝑑𝑓𝑥^′−𝑦₀≤𝜀𝑥,𝑥^′ (†)for all 𝑥,𝑥^′∈𝑋, then there exists a unique (𝑎,𝑏,𝑐,𝑑;𝑥₀,𝑦₀)-additive mapping 𝑓_∞ such that𝑓(𝑥)−𝑓_∞(𝑥)≤𝛿(𝑥) (‡)for all 𝑥∈𝑋.”

Throughout the remainder of this paper, we assume that 𝑌 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]). Let 𝑎+𝑏≠0 and 𝐾≥0 with 𝐾|𝑐+𝑑|<1. One takes a control function 𝜀 which satisfies 𝜀𝑥,𝑥^′≤𝐾𝜀(𝑎+𝑏)𝑥+𝑥₀,(𝑎+𝑏)𝑥^′+𝑥₀ (13) for all 𝑥,𝑥^′∈𝑋 and puts 𝛿(𝑥)=𝐾𝜀(𝑥,𝑥)1−𝐾|𝑐+𝑑| (14) for each 𝑥∈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]). Let 𝑐+𝑑≠0 and 𝐾≥0 with 𝐾<|𝑐+𝑑|. One takes a control function 𝜀 which satisfies 𝜀(𝑎+𝑏)𝑥+𝑥₀,(𝑎+𝑏)𝑥^′+𝑥₀≤𝐾𝜀𝑥,𝑥^′ (15) for all 𝑥,𝑥^′∈𝑋 and puts 𝛿(𝑥)=𝜀(𝑥,𝑥)|𝑐+𝑑|−𝐾 (16) for each 𝑥∈𝑋.
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 case 𝑎≠−1, Theorem 8 is of the case 𝑏≠−1, and Theorems 9 and 10 are for (−1,1,−1,1;𝑥₀,𝑦₀)-additive mappings. These cover all of the systems of (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mappings.

Theorem 7. Let 𝑎+1≠0 and 𝐾≥0 with 𝐾|1+𝑐|<|𝑐|. One takes a control function 𝜀 which satisfies 𝜀𝑥,𝑥^′≤𝐾𝜀𝑎⁻¹+1𝑥−𝑎⁻¹𝑥₀,𝑎⁻¹+1𝑥^′−𝑎⁻¹𝑥₀ (17) for all 𝑥,𝑥^′∈𝑋, and puts 𝛿(𝑥)=𝐾𝜀𝑎⁻¹+1𝑥−𝑎⁻¹𝑥₀,𝑥|𝑐|−𝐾|1+𝑐| (18) for each 𝑥∈𝑋.
Then, the strict (𝜀,𝛿)-stability holds for the system of (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive mappings.

Proof. Put 𝑢=𝑎𝑥−𝑎𝑥^′+𝑥₀ and 𝑢^′=𝑥^′ for each 𝑥,𝑥^′∈𝑋. Then, (†) changes into 𝑓𝑎⁻¹𝑢+𝑢^′−𝑎⁻¹𝑥₀−𝑐⁻¹𝑓(𝑢)−𝑓𝑢^′+𝑐⁻¹𝑦₀≤𝜀₁𝑢,𝑢^′, (19) where 𝜀₁𝑢,𝑢^′=1|𝑐|𝜀𝑎⁻¹𝑢+𝑢^′−𝑎⁻¹𝑥₀,𝑢^′ (20) for all 𝑢,𝑢^′∈𝑋. By denoting 𝑎₁=𝑎⁻¹, 𝑏₁=1, 𝑐₁=𝑐⁻¹, 𝑑₁=1, 𝑢₀=−𝑎⁻¹𝑥₀ and 𝑣₀=−𝑐⁻¹𝑦₀ in (19), (†) changes up to the following estimate of 𝑓 by the control function 𝜀₁: 𝑓𝑎₁𝑢+𝑏₁𝑢^′+𝑢₀−𝑐₁𝑓(𝑢)−𝑑₁𝑓𝑢^′−𝑣₀≤𝜀₁𝑢,𝑢^′ (21) for all 𝑢,𝑢^′∈𝑋.
Under these transformations, 𝑎₁,𝑏₁,𝑐₁,𝑑₁, and 𝜀₁ are equipped with 𝑎₁+𝑏₁≠0,𝐾𝑐₁+𝑑₁𝜀<1,₁𝑢,𝑢^′≤𝐾𝜀₁𝑎₁+𝑏₁𝑢+𝑢₀,𝑎₁+𝑏₁𝑢^′+𝑢₀ (22) for all 𝑢,𝑢^′∈𝑋. The latter follows from the inequality in which 𝜀 must satisfy because by using (20), we get |𝑐|𝜀₁𝑢,𝑢^′=𝜀𝑎⁻¹𝑢+𝑢^′+𝑢₀,𝑢^′=𝜀𝑥,𝑥^′,𝜀𝑎⁻¹+1𝑥−𝑎⁻¹𝑥₀,𝑎⁻¹+1𝑥^′−𝑎⁻¹𝑥₀=𝜀𝑎⁻¹+1𝑥−𝑎⁻¹𝑥₀,𝑎₁+𝑏₁𝑢^′+𝑢₀=𝜀𝑎⁻¹+1𝑎⁻¹𝑢+𝑢^′+𝑢₀−𝑎⁻¹𝑥₀,𝑎₁+𝑏₁𝑢^′+𝑢₀=𝜀𝑎⁻¹𝑎₁+𝑏₁𝑢+𝑢₀+𝑎₁+𝑏₁𝑢^′+𝑢₀−𝑎⁻¹𝑥₀,𝑎₁+𝑏₁𝑢^′+𝑢₀=|𝑐|𝜀₁𝑎₁+𝑏₁𝑢+𝑢₀,𝑎₁+𝑏₁𝑢^′+𝑢₀ (23) for all 𝑢,𝑢^′∈𝑋. Since (21) and (22) hold, it follows from Theorem A that there exists a unique (𝑎₁,𝑏₁,𝑐₁,𝑑₁;𝑢₀,𝑣₀)-additive mapping 𝑓_∞ such that 𝑓(𝑥)−𝑓_∞(𝑥)≤𝐾𝜀₁(𝑥,𝑥)1−𝐾𝑐₁+𝑑₁ (24) for all 𝑥∈𝑋. However, we can easily see the following two assertions: (i)𝑓_∞ is (𝑎₁,𝑏₁,𝑐₁,𝑑₁;𝑢₀,𝑣₀)-additive if and only if 𝑓_∞ is (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive, (ii)(24) is equivalent to (‡).
This completes the proof.

Theorem 8. Let 𝑐+1≠0 and 𝐾≥0 with 𝐾|𝑐|<|𝑐+1|. One takes a control function 𝜀 which satisfies 𝜀𝑎⁻¹+1𝑥−𝑎⁻¹𝑥₀,𝑎⁻¹+1𝑥^′−𝑎⁻¹𝑥₀≤𝐾𝜀𝑥,𝑥^′ (25) for all 𝑥,𝑥^′∈𝑋 and puts 𝛿(𝑥)=𝜀𝑎⁻¹+1𝑥−𝑎⁻¹𝑥₀,𝑥|𝑐+1|−𝐾|𝑐| (26) for each 𝑥∈𝑋.
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) of 𝑓 by 𝜀₁ in the proof of Theorem 7. Under these transformations, we have 𝑐₁+𝑑₁≠0,𝐾<𝑐₁+𝑑₁. (27) Moreover, for every 𝑢,𝑢^′∈𝑋 we have 𝜀₁𝑎₁+𝑏₁𝑢+𝑢₀,𝑎₁+𝑏₁𝑢^′+𝑢₀≤𝐾𝜀₁𝑢,𝑢^′, (28) because |𝑐|𝜀₁𝑢,𝑢^′=𝜀𝑎⁻¹𝑢+𝑢^′+𝑢₀,𝑢^′=𝜀𝑥,𝑥^′,|𝑐|𝜀₁𝑎₁+𝑏₁𝑢+𝑢₀,𝑎₁+𝑏₁𝑢^′+𝑢₀=𝜀𝑎⁻¹+1𝑥−𝑎⁻¹𝑥₀,𝑎⁻¹+1𝑥^′−𝑎⁻¹𝑥₀. (29)(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≤𝜀)¹(𝑥,𝑥)𝑐₁+𝑑₁−𝐾 (31) for all 𝑥∈𝑋. This means (‡) and 𝑓_∞ is (𝑎,−𝑎,𝑐,−𝑐;𝑥₀,𝑦₀)-additive.

Theorem 9. Let 0≤𝐾<1/2. One takes a control function 𝜀 which satisfies 𝜀𝑥,𝑥^′≤𝐾𝜀2𝑥−𝑥₀,2𝑥^′−𝑥₀ (32) for all 𝑥,𝑥^′∈𝑋 and puts 𝛿(𝑥)=𝐾𝜀𝑥,2𝑥−𝑥₀1−2𝐾 (33) for each 𝑥∈𝑋.
Then the strict (𝜀,𝛿)-stability holds for the system of (−1,1,−1,1;𝑥₀,𝑦₀)-additive mappings.

Proof. Put 𝑢=𝑥,𝑢^′=−𝑥+𝑥^′+𝑥₀ for each 𝑥,𝑥^′∈𝑋. Then, (†) changes into the following estimate of 𝑓 by the control function 𝜀₁: 𝑓𝑢+𝑢^′−𝑥₀−𝑓(𝑢)−𝑓𝑢^′+𝑦₀≤𝜀₁𝑢,𝑢^′, (34) where 𝜀₁𝑢,𝑢^′=𝜀𝑢,𝑢+𝑢^′−𝑥₀. (35) for all 𝑢,𝑢^′∈𝑋. Put 𝑎₁=𝑏₁=𝑐₁=𝑑₁=1.
Under these transformations, 𝑎₁,𝑏₁,𝑐₁,𝑑₁, and 𝜀₁ are equipped with 𝑎₁+𝑏₁=2≠0,𝐾𝑐₁+𝑑₁𝜀=2𝐾<1,₁𝑢,𝑢^′≤𝐾𝜀₁𝑎₁+𝑏₁𝑢−𝑥₀,𝑎₁+𝑏₁𝑢^′−𝑥₀ (36) for all 𝑢,𝑢^′∈𝑋. The latter follows from the following inequality: 𝜀₁𝑢,𝑢^′=𝜀𝑢,𝑢+𝑢^′−𝑥₀≤𝐾𝜀2𝑢−𝑥₀,2𝑢+𝑢^′−𝑥₀−𝑥₀=𝐾𝜀2𝑢−𝑥₀,2𝑢−𝑥₀+2𝑢^′−𝑥₀−𝑥₀=𝐾𝜀₁2𝑢−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 mapping 𝑓_∞ such that 𝑓(𝑥)−𝑓_∞(𝑥)≤𝐾𝜀₁(𝑥,𝑥)1−𝐾𝑐₁+𝑑₁ (38) for all 𝑥∈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. Let 0≤𝐾<2. One takes a control function 𝜀 which satisfies 𝜀2𝑥−𝑥₀,2𝑥^′−𝑥₀≤𝐾𝜀𝑥,𝑥^′ (39) for all 𝑥,𝑥^′∈𝑋 and puts 𝛿(𝑥)=𝜀𝑥,2𝑥−𝑥₀2−𝐾 (40) for each 𝑥∈𝑋.
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 𝑓 by 𝜀₁ in the proof of Theorem 9. Under these transformations, 𝑐₁,𝑑₁ and 𝜀₁ are equipped with 𝑐₁+𝑑₁=2≠0,𝐾<2=𝑐₁+𝑑₁,𝜀₁2𝑢−𝑥₀,2𝑢^′−𝑥₀=𝜀2𝑢−𝑥₀,2𝑢−𝑥₀+2𝑢^′−𝑥₀−𝑥₀=𝜀2𝑥−𝑥₀,2𝑢+𝑢^′−𝑥₀−𝑥₀=𝜀2𝑥−𝑥₀,2𝑥^′−𝑥₀≤𝐾𝜀𝑥,𝑥^′=𝐾𝜀𝑢,𝑢+𝑢^′−𝑥₀=𝐾𝜀₁𝑢,𝑢^′ (41)(42) for all 𝑢,𝑢^′∈𝑋. So it follows that 𝜀₁𝑎₁+𝑏₁𝑢−𝑥₀,𝑎₁+𝑏₁𝑢^′−𝑥₀≤𝐾𝜀₁𝑢,𝑢^′ (43) for all 𝑢,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 mapping 𝑓_∞ such that 𝑓(𝑥)−𝑓_∞(𝑥≤𝜀)¹(𝑥,𝑥)𝑐₁+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, let 𝑥₀=𝑦₀=0. We fix nonnegative constants 𝑝,𝑞 and 𝛿 and take the control function 𝜀 defined by 𝜀(𝑥,𝑥^′)=𝛿‖𝑥‖^𝑝‖𝑥^′‖^𝑞 for every 𝑥,𝑥^′∈𝑋.

Corollary 11. When 𝑎+1≠0 and |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) for each 𝑥∈𝑋.
Then, the system of (𝑎,−𝑎,𝑐,−𝑐)-additive mappings is strictly (𝜀,𝛿^′)-stable.

Proof. Put 𝐾=|𝑎⁻¹+1|^{−(𝑝+𝑞)}. Then 𝐾|𝑐+1|<|𝑐| and 𝜀𝑥,𝑥^′=𝐾𝜀𝑎⁻¹+1𝑥,𝑎⁻¹+1𝑥^′ (46) for all 𝑥,𝑥^′∈𝑋. By Theorem 7, for a mapping 𝑓 of 𝑋 into 𝑌 satisfying (†) for 𝑎=−𝑏 and 𝑐=−𝑑, there exists a unique (𝑎,−𝑎,𝑐,−𝑐)-additive mapping 𝑓_∞ such that 𝑓(𝑥)−𝑓_∞(𝑥≤𝑎)𝐾𝛿⁻¹+1^𝑝‖𝑥‖^𝑝+𝑞|𝑐|−𝐾|𝑐+1| (47) for all 𝑥,𝑥^′∈𝑋. Because of 𝐾=|𝑎⁻¹+1|^{−(𝑝+𝑞)}, we have the corollary.

Corollary 12. When 𝑐+1≠0 and |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) for each 𝑥∈𝑋.
Then the system of (𝑎,−𝑎,𝑐,−𝑐)-additive mappings is strictly (𝜀,𝛿^′)-stable.

Proof. Put 𝐾=|𝑎⁻¹+1|^𝑝+𝑞. Then 𝐾|𝑐|<|𝑐+1| and 𝜀𝑎⁻¹+1𝑥,𝑎⁻¹+1𝑥^′=𝐾𝜀𝑥,𝑥^′ (49) for all 𝑥,𝑥^′∈𝑋. By Theorem 8, for a mapping 𝑓 of 𝑋 into 𝑌 satisfying (†) for 𝑎=−𝑏 and 𝑐=−𝑑, there exists a unique (𝑎,−𝑎,𝑐,−𝑐)-additive mapping 𝑓_∞ such that 𝑓(𝑥)−𝑓_∞(𝑥≤𝛿𝑎)⁻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. When 𝑝+𝑞>1, one puts 𝛿^′(𝑥)=2^𝑞𝛿‖𝑥‖^𝑝+𝑞2^𝑝+𝑞−2 (51) for each 𝑥∈𝑋.
Then, the system of (−1,1,−1,1)-additive mappings is strictly (𝜀,𝛿^′)-stable.

Proof. Put 𝐾=2^{−(𝑝+𝑞)}. Then, 𝜀𝑥,𝑥^′=𝐾𝜀2𝑥,2𝑥^′ (52) for all 𝑥,𝑥^′∈𝑋. Since 𝑝+𝑞>1, we also have 0≤𝐾<1/2. By Theorem 9, for a mapping 𝑓 of 𝑋 into 𝑌 satisfying (†) for 𝑎=𝑐=−1 and 𝑏=𝑑=1, there exists a unique (−1,1,−1,1)-additive mapping 𝑓_∞ such that 𝑓(𝑥)−𝑓_∞(𝑥≤2)^𝑞𝛿𝐾‖𝑥‖^𝑝+𝑞1−2𝐾 (53) for all 𝑥∈𝑋. Because of 𝐾=2^{−(𝑝+𝑞)}, we have the corollary.

Corollary 14. When 𝑝+𝑞<1, one puts 𝛿^′(𝑥)=2^𝑞𝛿‖𝑥‖^𝑝+𝑞2−2^𝑝+𝑞 (54) for each 𝑥∈𝑋.
Then, the system of (−1,1,−1,1)-additive mappings is strictly (𝜀,𝛿^′)-stable.

Proof. Put 𝐾=2^𝑝+𝑞. Then, 𝜀2𝑥,2𝑥^′=𝐾𝜀𝑥,𝑥^′ (55) for all 𝑥,𝑥^′∈𝑋. Since 𝑝+𝑞<1, we also have 0≤𝐾<2. By Theorem 10, for a mapping 𝑓 of 𝑋 into 𝑌 satisfying (†) for 𝑎=𝑐=−1 and 𝑏=𝑑=1, there exists a unique (−1,1,−1,1)-additive mapping 𝑓_∞ such that 𝑓(𝑥)−𝑓_∞(𝑥≤2)^𝑞𝛿‖𝑥‖^𝑝+𝑞2−𝐾 (56) for all 𝑥∈𝑋. Because of 𝐾=2^𝑝+𝑞, we have the corollary.

Remark 15. In Corollary 14, taking 𝑝=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 Promotion of Science.