The Ulam Type Stability of a Generalized Additive Mapping and Concrete Examples

We give an Ulam type stability result for the following functional equation: under a suitable condition. We also give a concrete stability result for the case taking up as a control function. 1. Introduction In 1940, Ulam [1] proposed the following stability problem: “When is it true that a function which satisfies some functional equation approximately must be close to one satisfying the equation exactly?” Next year, Hyers [2] gave an answer to this problem for additive mappings between Banach spaces. Furthermore, Aoki [3] and Rassias [4] obtained independently generalized results of Hyers’ theorem which allow the Cauchy difference to be unbounded. Let and be normed spaces over , which denotes either the real field or the complex field . Throughout the paper, we fix scalars and vectors and . We say that a mapping of into is -additive if for all . When , we say it to be -additive. Aczél [5] specified what this generalized Cauchy equation is. The Ulam type stability problem for such an has been investigated in [6–8]. However, these results have been obtained in cases where either or (see Theorems A and B). In this paper, we will investigate the problem for -additive mappings, that is, in the case . In Section 2, we state the details of -additive mappings (Theorem 3). In Section 3, we give our main results about the stability for them (see Theorems 7–10). In the final section, we apply the results to some concrete examples, where we take up as a control function (see Corollaries 11–14). 2. -Additive Mappings The following result asserts that any -additive mapping is transformed into some -additive mapping by a certain translation and that any -additive mapping is an additive mapping in usual sense with some extra condition. Proposition 1. Let and be two mappings of into such that for all . Then the following three statements are equivalent: (i) ？？is -additive, (ii) ？？is -additive, (iii) ？？is additive and ？？for all . Proof. (i) (ii) Since for all , it follows that ？？ -additive ？ ？ ？ -additive. (ii) (iii) Suppose that for all . When , we have . Using this, and also for all . Therefore, for all . (iii) (ii) Because for all (see also the following remark), it is trivial. Remark 2. We denote by the field of all rational numbers. It is well known that if is additive, then for every and , that is, is -linear. Hence, if is additive and continuous, must be -linear. On the other hand, when , we have a lot of continuous additive nonlinear mappings by considering the composition of linear transformations on and the -linear isometry . The constant is a trivial