AQCQ-Functional Equation in Non-Archimedean Normed Spaces

and Applied Analysis 3 Theorem 1.2 Rassias 18 . Let X be a real normed linear space and Y a real complete normed linear space. Assume that f : X → Y is an approximately additive mapping for which there exist constants θ ≥ 0 and p, q ∈ R such that r p q / 1 and f satisfies the inequality ∥ ∥f ( x y ) − f x − f(y)∥∥ ≤ θ‖x‖p∥∥y∥∥q 1.5 for all x, y ∈ X. Then there exists a unique additive mapping L : X → Y satisfying ∥ ∥f x − L x ∥ ≤ θ |2r − 2| ‖x‖ r 1.6 for all x ∈ X. If, in addition, f : X → Y is a mapping such that the transformation t → f tx is continuous in t ∈ R for each fixed x ∈ X, then L is an R-linear mapping. Very recently, Rassias 26 in inequality 1.5 replaced the bound by a mixed one involving the product and sum of powers of norms, that is, θ{‖x‖p‖y‖p ‖x‖2p ‖y‖2p }. For more details about the results concerning such problems and mixed product-sum stability Rassias Stability the reader is referred to 27–42 . The functional equation f ( x y ) f ( x − y) 2f x 2f(y) 1.7 is related to a symmetric biadditive function 43, 44 . It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation 1.7 is said to be a quadratic function. It is well known that a function f between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function B1 such that f x B1 x, x for all x. The biadditive function B1 is given by B1 ( x, y ) 1 4 ( f ( x y ) − f(x − y)). 1.8 The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof 45 . In 46 , Czerwik proved the Hyers-Ulam-Rassias stability of 1.7 . Later, Jung 47 has generalized the results obtained by Skof and Czerwik. Jun and Kim 48 introduced the following cubic functional equation: f ( 2x y ) f ( 2x − y) 2f(x y) 2f(x − y) 12f x , 1.9 and they established the general solution and the generalized Hyers-Ulam stability for the functional equation 1.9 . They proved that a function f between two real vector spaces X and Y is a solution of 1.9 if and only if there exists a unique function C : X × X × X → Y 4 Abstract and Applied Analysis such that f x C x, x, x for all x ∈ X; moreover, C is symmetric for each fixed variable and is additive for fixed two variables. The function C is given by C ( x, y, z ) 1 24 ( f ( x y z ) f ( x − y − z) − f(x y − z) − f(x − y z)) 1.10 for all x, y, z ∈ X see also 47, 49–55 . Lee et al. 56 considered the following functional equation: f ( 2x y ) f ( 2x − y) 4f(x y) 4f(x − y) 24f x − 6f(y). 1.11 In fact, they proved that a function f between two real vector spaces X and Y is a solution of 1.11 if and only if there exists a unique symmetric biquadratic function B2 : X × X → Y such that f x B2 x, x for all x. The biquadratic function B2 is given by B2 ( x, y ) 1 12 ( f ( x y ) f ( x − y) − 2f x − 2f(y)). 1.12 Obviously, the function f x cx4 satisfies the functional equation 1.11 , which is called the quartic functional equation. Eshaghi Gordji and Khodaei 49 have established the general solution and investigated the Hyers-Ulam-Rassias stability for a mixed type of cubic, quadratic, and additive functional equation briefly, AQC-functional equation with f 0 0, f ( x ky ) f ( x − ky) k2f(x y) k2f(x − y) 2 ( 1 − k2 ) f x 1.13 in quasi-Banach spaces, where k is nonzero integer with k /∈ {0,±1}. Obviously, the function f x ax bx2 cx3 is a solution of the functional equation 1.13 . Interesting new results concerning mixed functional equations have recently been obtained by Najati et al. 57–59 and Jun and Kim 60, 61 as well as for the fuzzy stability of a mixed type of additive and quadratic functional equation by Park 62 . The stability of generalizedmixed type functional equations of the form f ( x ky ) f ( x − ky) k2(f(x y) f(x − y)) ( k2 − 1 ) ( k2 12 ( f̃ ( 2y ) − 4f̃(y) ) − 2f x )