- Research Article
- Open Access
© Soon-Mo Jung. 2009
- Received: 2 July 2009
- Accepted: 5 November 2009
- Published: 8 November 2009
- Vector Space
- Linear Equation
- General Solution
- Functional Equation
- Regularity Condition
In 1940, Ulam gave a wide-ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems . Among those was the question concerning the stability of homomorphisms.
In the following year, Hyers affirmatively answered in his paper  the question of Ulam for the case where and are Banach spaces.
for any (see [3, Definition ]).
This terminology is also applied to the case of other functional equations. It should be remarked that we can find in the books [4–7] a lot of references concerning the stability of functional equations (see also [8–18]).
In this paper, we will solve the functional equation
In this section, let be either a real vector space if or a complex vector space if . In the following theorem, we investigate the general solution of the functional equation (1.7).
for all and . If we substitute for in (2.3) and divide the resulting equations by , respectively, , and if we substitute for in the resulting equations, then we obtain the equations in (2.3) with in place of , where . Therefore, the equations in (2.3) are true for all and .
which completes the proof.
It should be remarked that the functional equation (1.7) is a particular case of the linear equation with and . Moreover, a substantial part of proof of Theorem 2.1 can be derived from theorems presented in the books [19, 20]. However, the theorems in [19, 20] deal with solutions of the linear equation under some regularity conditions, for example, the continuity, convexity, differentiability, analyticity and so on, while Theorem 2.1 deals with the general solution of (1.7) without regularity conditions.
We can prove the Hyers-Ulam stability of the functional equation (1.7) as we see in the following theorem.
On the other hand, it also follows from (3.1) that
By (3.9) and (3.15), we have
By (3.24) and (3.25), we have
for any , that is, for all . Therefore, we conclude that for any . (The presented proof of uniqueness of is somewhat long and involved. Indeed, the referee has remarked that the uniqueness can be obtained directly from [21, Proposition ].)
The functional equation (1.7) is a particular case of the linear equations of higher orders and the Hyers-Ulam stability of the linear equations has been proved in [21, Theorem ]. Indeed, Brzdęk et al. have proved an interesting theorem, from which the following corollary follows (see also [22, 23]):
Hence, the estimation (3.2) of Theorem 3.1 is better in these cases than the estimation (3.28).
As we know, is the Fibonacci sequence. So if we set and in Theorems 2.1 and 3.1, then we obtain the same results as in [24, Theorems , , and ].
The author would like to express his cordial thanks to the referee for useful remarks which have improved the first version of this paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0071206).
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