On the Hyers-Ulam Stability of a General Mixed Additive and Cubic Functional Equation in n-Banach Spaces

and Applied Analysis 3 Definition 1.4. A sequence {xk} in an n-normed space X is said to converge to some x ∈ X in the n-norm if lim k→∞ ∥ ∥xk − x, y2, . . . , yn ∥ ∥ 0, 1.4 for every y2, . . . , yn ∈ X. Definition 1.5. A sequence {xk} in an n-normed space X is said to be a Cauchy sequence with respect to the n-norm if lim k,l→∞ ∥ ∥xk − xl, y2, . . . , yn ∥ ∥ 0, 1.5 for every y2, . . . , yn ∈ X. If every Cauchy sequence in X converges to some x ∈ X, then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be an n-Banach space. Now we state the following results as lemma see 9 for the details . Lemma 1.6. Let X be an n-normed space. Then, 1 For xi ∈ X i 1, . . . , n and γ , a real number, ∥x1, . . . , xi, . . . , xj , . . . , xn ∥ ∥x1, . . . , xi, . . . , xj γxi, . . . , xn ∥ 1.6 for all 1 ≤ i / j ≤ n, 2 |‖x, y2, . . . , yn‖ − ‖y, y2, . . . , yn‖| ≤ ‖x − y, y2, . . . , yn‖ for all x, y, y2, . . . , yn ∈ X, 3 if ‖x, y2, . . . , yn‖ 0 for all y2, . . . , yn ∈ X, then x 0, 4 for a convergent sequence {xj} in X, lim j→∞ ∥xj , y2, . . . , yn ∥ ∥∥∥ lim j→∞ xj , y2, . . . , yn ∥∥∥ 1.7 for all y2, . . . , yn ∈ X. 2. Approximate Mixed Additive-Cubic Mappings In this section, we investigate the generalized Hyers-Ulam stability of the generalized mixed additive-cubic functional equation in n-Banach spaces. Let X be a linear space and Y an nBanach space. For convenience, we use the following abbreviation for a given mapping f : X → Y : Df ( x, y ) : f ( kx y ) f ( kx − y − kfx y − kfx − y − 2f kx 2kf x 2.1 for all x, y ∈ X. 4 Abstract and Applied Analysis Theorem 2.1. Let X be a linear space and Y an n-Banach space. Let f : X → Y be a mapping with f 0 0 for which there is a function φ : X 1 → 0,∞ such that ∞ ∑ j 0 1 2j φ ( 2x, 2y, u2, . . . , un ) < ∞, 2.2 ∥ ∥Df x, y , u2, . . . , un ∥ ∥ Y ≤ φ ( x, y, u2, . . . , un ) 2.3 for all x, y, u2, . . . , un ∈ X. Then, there is a unique additive mapping A : X → Y such that ∥ ∥f 2x − 8f x −A x , u2, . . . , un ∥ ∥ Y ≤ ∞ ∑ j 0 1 2j 1 φ̃ ( 2x, u2, . . . , un ) 2.4 for all x, u2, . . . , un ∈ X, where φ̃ x, u2, . . . , un : 1 |k3 − k| { |k| 1 φ x, 2k 1 x, u2, . . . , un φ x, 2k − 1 x, u2, . . . , un ] φ 3x, x, u2, . . . , un ( 8k2 1 ) φ x, x, u2, . . . , un φ x, 3kx, u2, . . . , un φ x, kx, u2, . . . , un k2φ 2x, 2x, u2, . . . , un φ 2x, 2kx, u2, . . . , un 2φ x, k 1 x, u2, . . . , un 2φ x, k − 1 x, u2, . . . , un 2φ 2x, x, u2, . . . , un 2φ 2x, kx, u2, . . . , un 8φ ( x 2 , kx 2 , u2, . . . , un ) 8|k|φ ( x 2 , 2k − 1 x 2 , u2, . . . , un ) 8|k|φ ( x 2 , 2k 1 x 2 , u2, . . . , un ) 8φ ( x 2 , 3kx 2 , u2, . . . , un ) |k| 1 |k − 1| 0, k 1 x, u2, . . . , un 8k2 1 |k − 1| φ 0, k − 1 x, u2, . . . , un 2 |k − 1| 0, x, u2, . . . , un |k| |k − 1| 0, 3k − 1 x, u2, . . . , un k2 |k − 1| 0, 2 k − 1 x, u2, . . . , un k2 |k| − 1 |k − 1| φ 0, 2kx, u2, . . . , un 8|k| |k − 1| ( 0, 3k − 1 x 2 , u2, . . . , un ) 8|k| |k − 1| ( 0, k 1 x 2 , u2, . . . , un ) 8k2 2|k| − 8 |k − 1| φ 0, kx, u2, . . . , un } .