- Research Article
- Open Access
Stability of a 2-Dimensional Functional Equation in a Class of Vector Variable Functions
© Won-Gil Park and Jae-Hyeong Bae. 2010
- Received: 16 March 2010
- Accepted: 24 June 2010
- Published: 12 July 2010
- Continuous Function
- Real Number
- Normed Space
- Quadratic Mapping
- Stability Problem
In 1940, Ulam proposed the stability problem (see ):
In 1941, this problem was solved by Hyers  in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability. The authors investigated various functional equations and their Hyers-Ulam stability [3–8]. This Hyers-Ulam stability is a classical type of stability, but there is another kind of stability introduced by Risteski  for functional equations spanned over an -dimensional complex vector space too.
The quadratic form given by is a solution of (1.2). In 2008, the authors of  acquired the general solution and proved the stability of the -dimensional quadratic functional equation (1.2) for the case that and are real vector spaces as follows.
The results of [8, Theor ms 3 and 4] also hold for complex vector spaces and . In this paper, we investigate the stability of (1.2) with two module actions in Banach modules over a unital -algebra.
A unital -algebra is said to have real rank (see ) if the invertible self-adjoint elements are dense in .
For any element , , where and are self-adjoint elements; furthermore, , where and are positive elements (see [12, Lemma ]).
for all and all . For each fixed , let the sequence converge uniformly on . If is continuous in for each fixed , then there exists a unique -dimensional vector variable mapping satisfying (1.2) and (3.10) such that for all and all .
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