Hyers-Ulam-Rassias Stability of a Generalized Quadratic-Additive Functional Equation

and Applied Analysis 3 The functional equation 1.7 was first solved by Kannappan. In fact he proved that a mapping f on a real vector space is a solution of 1.7 if and only if there exists a symmetric biadditive mapping B and an additive mapping A such that f x B x, x A x , for any x see 9 . The stability problem for 1.7 is also studied in 26 . Moreover 1.7 was pexiderized and solved by Kannappan 9 . In 27 , solutions and the generalized Hyers-Ulam-Rassias stability of the functional equation 1.8 have been studied for k 3. The generalized Hyers-Ulam-Rassias stability problem for the functional equation 1.8 was first considered by Bae and Park 28 . Also solutions and the generalized UlamGăvruţa-Rassias stability of this functional equation were studied by Nakmahachalasint 29 . Indeed for its solutions the following theorem is proved. Theorem 1.1 see 29, Theorem 2.1 . Let n > 2 be a positive integer, and let X and Y be vector spaces. A mapping f : X → Y satisfies the functional equation 1.8 if and only if the even part of f , defined by fe x 1/2 f x f −x for all x ∈ X, satisfies the classical quadratic functional equation and the odd part of f , defined by fo x 1/2 f x − f −x for all x ∈ X, satisfies the Cauchy functional equation f x y f x f y . In Section 2 of this paper, we will prove that the functional equation 1.8 is equivalent to the functional equation 1.7 . In Section 3, first we prove the generalized Hyers-UlamRassias stability of the functional equation 1.8 in non-Archimedean normed spaces, and then as a consequence of this result, we prove the generalized Hyers-Ulam-Rassias stability of 1.8 in non-Archimedean normed spaces. Finally using the methods of Theorem 3.1. in 29 , directly the generalized Hyers-Ulam-Rassias stability of 1.8 will be proved in non-Archimedean normed spaces. The stability problem in non-Archimedean case has been studied by many authors, for example, see 30–34 . First we need some preliminaries in non-Archimedean normed space. Let K be a field. A non-Archimedean absolute value on K is a function | · | : K → R such that for any a, b ∈ Kwe have i |a| ≥ 0 and equality holds if and only if a 0, ii |ab| |a||b|, iii |a b| ≤ max{|a|, |b|}. Condition iii is called the strong triangle inequality. By ii , we have |1| | − 1| 1. Thus, by induction, it follows from iii that |n| ≤ 1, for each integer n. We always assume in addition that | · | is nontrivial, that is, iv there is an a0 ∈ K such that |a0|/ 0, 1. Let X be a linear space over a scalar field K with a non-Archimedean nontrivial valuation | · |. A function ‖ · ‖ : X → R is a non-Archimedean norm valuation if it is a norm over K with the strong triangle inequality ultrametric ; namely, ∥ ∥x y ∥ ∥ ≤ max‖x‖,∥y∥, x, y ∈ X. 1.9 Then X, ‖ · ‖ is called a non-Archimedean normed space. 4 Abstract and Applied Analysis By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. Thanks to the inequality ‖xn − xm‖ ≤ max {∥ ∥xj 1 − xj ∥ ∥ : m ≤ j ≤ n − 1 n > m 1.10 a sequence {xn} is Cauchy if and only if {xn 1 − xn} converges to zero in a non-Archimedean space. 2. Solution of the Functional Equation 1.8 Throughout this section, X and Y will be some vector spaces. The following theorem proves that the functional equation 1.8 is equivalent to the functional equation 1.7 , and so every solution of the functional equation 1.8 is a summation of a quadratic and an additive mappings. Theorem 2.1. Let X and Y be common domain and range of the f ’s in the functional equations 1.7 and 1.8 . Then the functional equation 1.8 is equivalent to 1.7 . Proof. We can easily see that 1.8 implies 1.7 . Now, suppose a mapping f : X → Y satisfies 1.7 , for all x, y ∈ X. Using mathematical induction, we are going to show that, for any k ≥ 3,