Ternary Derivations, Stability and Physical Aspects
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Ternary Derivations, Stability and Physical Aspects Mohammad Sal Moslehian
Received: 18 October 2006 / Accepted: 9 November 2007 / Published online: 27 November 2007 © Springer Science+Business Media B.V. 2007
Abstract Ternary algebras and modules are vector spaces on which products of three factors are defined. In this paper, we present several physical applications of ternary structures. Some recent results on the stability of ternary derivations are reviewed. Using a fixed point method, we also establish the generalized Hyers–Ulam–Rassias stability of ternary derivations from a normed ternary algebra into a Banach tri-module associated to the generalized Jensen functional equations and prove a superstability result. Keywords Generalized Hyers–Ulam–Rassias stability · Superstability · Ternary algebra · Tri-module · Ternary derivation · Generalized Jensen functional equation Mathematics Subject Classification (2000) Primary 39B8 · Secondary 39B52 · 17A40 · 17A36 · 46H25 · 81T60
1 Introduction A classical question in the theory of functional equations is the following: “When is it true that a function, which approximately satisfies a functional equation E must be close to an exact solution of E ?” If the problem accepts a solution, we say that the equation E is stable. The first stability problem concerning group homomorphisms was raised from a famous talk given by S.M. Ulam [40] in 1940.
The author was in part supported by a grant from IPM (No. 85390031). M.S. Moslehian () Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran e-mail: [email protected] M.S. Moslehian Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran e-mail: [email protected]
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We are given a group G and a metric group G with metric ρ(·, ·). Given > 0, does there exist a number δ > 0 such that if f : G → G satisfies ρ(f (xy), f (x)f (y)) < δ for all x, y ∈ G, then a homomorphism h : G → G exists with ρ(f (x), h(x)) < for all x ∈ G? In the next year 1941, Ulam’s problem was affirmatively solved by D.H. Hyers [16] for Banach spaces: Suppose that E1 is a normed space, E2 is a Banach space and a mapping f : E1 → E2 satisfies the inequality f (x + y) − f (x) − f (y) ≤
(x, y ∈ E1 ),
where > 0 is a constant. Then the limit T (x) := lim 2−n f (2n x) exists for each x ∈ E1 and T is the unique additive mapping satisfying f (x) − T (x) ≤
(x ∈ E1 ).
(1.1)
If f is continuous at a single point of E1 , then T is continuous everywhere in E1 . Moreover (1.1) is sharp. In 1950, T. Aoki [3] generalized Hyers’ theorem for approximately additive mappings. In 1978, Th.M. Rassias [34] extended Hyers’ theorem by obtaining a unique linear mapping under certain continuity assumption when the Cauchy difference is allowed to be unbounded (see [28]): Suppose that E is a real normed space, F is a real Banach space and f : E → F is a mapping such that for each fixed x ∈ E the mapping t → f (tx) is continuous on R. Let there exist ε ≥ 0 and p ∈ [0, 1) such that f (
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