Nearly Ternary Quadratic Higher Derivations on Non-Archimedean Ternary Banach Algebras: A Fixed Point Approach

and Applied Analysis 3 for x, y, z ∈ A. A Banach non-Archimedean ternary algebra is a normed non-Archimedean ternary algebra such that the normed non-Archimedean vector space with norm ‖ · ‖ is complete. The ternary algebras have been studied in nineteenth century. Their structures appeared more or less naturally in various domains of mathematical physics and data processing. The discovery of the Nambu mechanics and the progress of quantum mechanics 14 , as well as work of Okubo 15 on Yang-Baxter equation gave a significant development on ternary algebras see also 16–20 . We say that a functional equation ξ is stable if any function g satisfying the equation ξ approximately is near to true solution of ξ . We say that a functional equation ξ is superstable if every approximately solution of ξ is an exact solution of it see 21 . The stability of functional equations was first introduced by Ulam 22 in 1940. In 1941, Hyers 23 gave a first affirmative answer to the question of Ulam for Banach spaces. In 1978, Rassias 24 generalized the theorem ofHyers for linearmappings by considering the stability problem with unbounded Cauchy differences ‖f x y − f x − f y ‖ ≤ ‖x‖ ‖y‖ , > 0, p ∈ 0, 1 . In 1991, Gajda 25 answered the question for the case p > 1, which was raised by Rassias. This new concept is known as generalized Hyers-Ulam stability of functional equations see 6–12, 17, 21, 25–58 . In 1982–1994, Rassias see 59–66 solved the Ulam problem for different mappings and for many Euler-Lagrange type quadratic mappings, by involving a product of different powers of norms. In addition, Rassias considered the mixed product sum of powers of norms control function 67 . In 1949, Bourgin 41 proved the following result, which is sometimes called the superstability of ring homomorphisms. Suppose that A and B are Banach algebras with unit. If f : A → B is a surjective mapping such that ∥f ( x y ) − f x − f ( y )∥ ≤ , ∥f ( xy ) − f x f ( y )∥ ≤ δ 1.3 for some ≥ 0, δ ≥ 0 and for all x, y ∈ A, then f is a ring homomorphism. Badora 68 and Miura et al. 69 proved the Ulam-Hyers stability and the Isaac and Rassias type stability of derivations 30 . The functional equation f ( x y ) f ( x − y ) 2f x 2f ( y ) , 1.4 is related to a symmetric biadditive function 2, 27 . It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation 1.4 is said to be a quadratic mapping. It is well known that a mapping f between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive mapping B1 such that f x B1 x, x for all x. The bi-additive mapping B1 is given by B1 x, y 1/4 f x y − f x − y . The stability problem of quadratic functional equation 1.4 was proved by Skof 37 for functions f : A → B, where A is normed space and B Banach space see also 42, 43 . 4 Abstract and Applied Analysis Definition 1.4. A mapping H : A → B is called a ternary quadratic homomorphism between non-Archimedean ternary algebras A,B if 1 H is a quadratic function, 2 H xyz H x H y H z , for all x, y, z ∈ A. For instance, let A be commutative ternary algebra, then the function f : A → A defined by f a a2 a ∈ A is a quadratic homomorphism. Definition 1.5. A mapping D : A → A is called a non-Archimedean ternary quadratic derivation on ternary non-Archimedean algebra A if 1 D is a quadratic function, 2 D xyz D x y2z2 x2D y z2 x2y2D z , for all x, y, z ∈ A. For example, consider the algebra of 2 × 2 matrices A {[ c1 c2 0 0 ] : c1, c2 ∈ C } , 1.5 then it is easy to see that A is a ternary algebra. Moreover, the function f : A → A defined by f ([ c1 c2 0 0 ]) [ 0 c2 2 0 0 ] , 1.6 is a ternary quadratic derivation. We note that ternary quadratic derivations and ternary ring derivations are different. As another example, LetA be a Banach algebra. Then we take T ⎡ ⎢ ⎣ 0 A A 0 0 A 0 0 0 ⎤ ⎥ ⎦, 1.7 where T is a ternary Banach algebra equipped with the usual matrix-like operations and the following norm: ∥∥∥∥∥∥∥ ⎡ ⎢ ⎣ 0 a b 0 0 c 0 0 0 ⎤ ⎥⎥ ⎦ ∥∥∥∥∥∥∥ ‖a‖ ‖b‖ ‖c‖ a, b, c ∈ A . 1.8 Abstract and Applied Analysis 5and Applied Analysis 5