A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

In this paper, we prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations.In Section 4, we apply our uniqueness theorem to complement stability theorems of the papers [4,5] where the uniqueness has not been proved.Indeed, this uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for various quadratic-additive type functional equations.

Main Result
Let  be a real vector space and  a real normed space.For any mapping  :  → ,   and   denote the even part and the odd part of , respectively.
In the following theorem, we prove that if, for any given mapping , there exists a mapping  (near ) with some properties possessed by additive or quadratic or quadraticadditive mappings, then the mapping  is uniquely determined.

2
Journal of Function Spaces Theorem 1.Let  > 1 be a real constant, let Φ :  \ {0} → [0, ∞) be a function satisfying one of the following conditions, lim lim for all  ∈  \ {0}, and let  :  →  be an arbitrarily given mapping.If there exists a mapping  :  →  such that for all  ∈  \ {0} and for all  ∈ , then  is determined by for all  ∈  \ {0}.In other words,  is the unique mapping satisfying (5) and (6).

Applications
In general, it is not easy to apply Theorem 1 in practical applications.Hence, we introduce two corollaries which are easily applicable to investigating the uniqueness problems in the generalized Hyers-Ulam stability of functional equations.
For the exact definition of the generalized Hyers-Ulam stability, we refer the reader to [6].
Proof.If  satisfies ( 16), then we have lim For the case of (17), it holds that lim that is, Φ satisfies condition (4) for all  ∈  \ {0}.Hence, our assertion is true in view of Theorem 1.
Similarly, we have for all  ∈ \{0}.If we make change of the summation indices in the last equality with  =  + 2 and  =  − 2, then we get for any  ∈  \ {0}.Thus, we obtain lim for each  ∈  \ {0}.Thus, it holds that lim for each  ∈  \ {0}.Theorem 1 implies that our conclusion for this corollary is true.
In the following corollary, we prove that if for any given mapping  there exists an additive or a quadratic or a quadratic-additive mapping  near , then the mapping  is uniquely determined.
The proofs of the following two corollaries immediately follow from Corollaries 2 and 3, respectively, because each of additive, quadratic, and quadratic-additive mappings satisfies both conditions in (6) for any given rational number  > 1.
Since each of additive, quadratic, and quadratic-additive mappings satisfies the conditions in ( 6), using Corollary 6, we can easily prove the following corollary.for all  ∈  \ {0}, then  is uniquely determined.
for all  ∈ .