Approximate n-Lie Homomorphisms and Jordan n-Lie Homomorphisms on n-Lie Algebras

and Applied Analysis 3 Park and Rassias 59 proved the stability of homomorphisms in C∗-algebras and Lie C∗-algebras and also of derivations on C∗-algebras and Lie C∗-algebras for the Jensen-type functional equation μf ( x y 2 ) μf ( x − y 2 ) − fμx 0 1.6 for all μ ∈ T1 : {λ ∈ C; |λ| 1}. In this paper, by using the fixed-point methods, we establish the stability of n-Lie homomorphisms and Jordan n-Lie homomorphisms on n-Lie Banach algebras associated to the following generalized Jensen type functional equation: μf (∑n i 1 xi n ) μ n ∑ j 2 f (∑n i 1,i / j xi − n − 1 xj n ) − fμx1 ) 0 1.7 for all μ ∈ T1/no : {eiθ; 0 ≤ θ ≤ 2π/no} ∪ {1} , where n ≥ 2. Throughout this paper, assume that A, A , B, B are two n-Lie Banach algebras. 2. Main Results Before proceeding to the main results, we recall a fundamental result in fixed point theory. Theorem 2.1 see 60 . Let Ω, d be a complete generalized metric space, and let T : Ω → Ω be a strictly contractive function with Lipschitz constant L. Then for each given x ∈ Ω, either d ( Tx, T 1x ) ∞ ∀m ≥ 0, 2.1 or other exists a natural number m0 such that i d Tx, T 1x < ∞ for all m ≥ m0; ii the sequence {Tmx} is convergent to a fixed point y∗ of T ; iii y∗ is the unique fixed point of T in Λ {y ∈ Ω : d T0x, y < ∞}; iv d y, y∗ ≤ 1/ 1 − L d y, Ty for all y ∈ Λ. We start our work with the main theorem of the our paper. Theorem 2.2. Let n0 ∈ N be a fixed positive integer number. Let f : A → B be a function for which there exists a function φ : A → 0,∞ such that ∥ ∥ ∥ ∥ ∥ ∥ μf (∑n i 1 xi n ) μ n ∑ j 2 f (∑n i 1,i / j xi − n − 1 xj n ) − f μx1 ∥ ∥ ∥ ∥ ∥ ∥ B ≤ φ x1, x2, . . . , xn 2.2 4 Abstract and Applied Analysis for all μ ∈ T1/no : {eiθ; 0 ≤ θ ≤ 2π/no} ∪ {1} and all x1, . . . , xn ∈ A, and that ∥ ∥f x1x2 · · ·xn A − [ f x1 f x2 · · · f xn ] B ∥ ∥ B ≤ φ x1, x2, . . . , xn 2.3 for all x1, . . . , xn ∈ A. If there exists an L < 1 such that φ x1, x2, . . . , xn ≤ nLφ ( x1 n , x2 n , . . . , xn n ) 2.4 for all x1, . . . , xn ∈ A, then there exists a unique n-Lie homomorphism H : A → B such that ∥ ∥f x −H x ∥ ≤ L 1 − L x, 0, 0, . . . , 0 2.5 for all x ∈ A. Proof. Let Ω be the set of all functions from A into B and let d ( g, h ) : inf { C ∈ R : ∥g x − h x ∥∥B ≤ Cφ x, 0, . . . , 0 , ∀x ∈ A } . 2.6 It is easy to show that Ω, d is a generalized complete metric space 61 . Now we define the mapping J : Ω → Ω by J h x 1/n h nx for all x ∈ A. Note that for all g, h ∈ Ω, d ( g, h ) < C ⇒ ∥g x − h x ∥ ≤ Cφ x, 0, . . . , 0 , ∀x ∈ A, ⇒ ∥ ∥ ∥ ∥ 1 n g nx − 1 n h nx ∥ ∥ ∥ ∥ ≤ 1 |n| Cφ nx, 0, . . . , 0 , ∀x ∈ A, ⇒ ∥ ∥ ∥ ∥ 1 n g nx − 1 n h nx ∥ ∥ ∥ ∥ ≤ LCφ x, 0, . . . , 0 , ∀x ∈ A, ⇒ dJg, J h ) ≤ LC. 2.7