Generalized stability of classical polynomial functional equation of order n
© Eungrasamee et al.; licensee Springer 2012
Received: 3 April 2012
Accepted: 23 July 2012
Published: 6 August 2012
We study a general n th order polynomial functional equation on linear spaces and prove its generalized stability.
The problem of the stability of functional equations started in 1940 by S. M. Ulam  when he proposed the question ‘Letbe a group and letbe a metric group with the metric d. Given, does there exist asuch that ifsatisfies the inequalityfor all, then there exists a homomorphismwithfor all?’ In the following year, the answer of this question for the case of a mapping f between Banach spaces and satisfying for all and for some was attested by D. H. Hyers . It was shown that for each , the additive mapping defined by has a property that for all . Furthermore, the mapping A is also unique. Since then, this kind of stability was known as the Hyers-Ulam stability and became a fundamental stability theory concept of functional equations. In 1950, T. Aoki  published a paper on the stability of the additive mappings in Banach spaces, while in 1978, Th. M. Rassias  extended the problem to , for some and some . Subsequently, in 1994, P. Gavruta  generalized the problem to with certain conditions imposed on the function ϕ. This type of stability is referred to as the generalized stability.
In recent years, a number of researchers [6, 7] have investigated stability problem of various types of functional equations which are mostly based on the Cauchy additive functional equation of the form , the classical quadratic functional equation , and the functional equations of higher degree [9, 10]. The stability problem of functional equations can be determined on various domains of functions. There were related efforts on functions being defined on groupoid. A typical work was carried out by A. Gilányi  in 1999, who proved the Hyers-Ulam stability of monomial functional equation on a power-associative, power-symmetric groupoid. Such efforts investigate a viable further investigation on generalizing the stability problem.
where is a function related to the function ϕ.
Basic theorems and lemmas
In this section, we provide some basic theorems and lemmas concerning the difference operators as well as multi-additive functions. For further details and proofs, please refer to the book by S. Czerwik . Throughout the section, we shall let X and Y be two linear spaces and let be an arbitrary function.
We may also write the iterated operators shortly as .
Some properties of the difference operator are shown in the following lemmas.
We then recall the definition of an n-additive function and its diagonalization along with their useful properties.
In particular, when, .
for all and denotes any permutation of .
Later on, we can state the relation between an n-additive function and the difference operator of its diagonalization as shown below.
We thereafter define a polynomial function of order n and then provide the consequent result often used.
for all , will be called a polynomial function of order n.
Generalized polynomial functional equations
In this section, we will show that the general solution of the proposed functional equation (1) is the diagonalization of a symmetric n-additive function.
Theorem 8 Let X and Y be two linear spaces. Let n be a positive integer. A functionsatisfies the functional equation (1) if and only iffor allwhereis the diagonalization of a symmetric n-additive function.
for all . Thus, is additive in the first argument. Taking into account the symmetry of , we conclude that is n-additive.
If we let be the diagonalization of , that is, for all , then the above equation simply states that for all .
which yields the function equation (1). Thus, the proof is complete. □
for all rational numbers r and for all .
In this section, we aim to prove the stability of the functional equation (1). Let us start with some lemmas that will be used in the proof of the main theorem. It should be noted that we will adopt the usual extension of the binomial coefficients, that is for all integers .
Lemma 8 Letbe a function. If a functionsatisfies the inequality
We will now move on to the proof of the generalized stability of the functional equation (1).
In addition, for all.
From the inequality (10), if we let , then we obtain the inequality (8).
Now taking the limit of (11) as and using the conditions (6), we will see that indeed satisfies the functional equation (1).
Taking the limit as , and using the conditions (6), we will have which asserts the uniqueness of . This completes the proof. □
We will also give a stability theorem with the conditions slightly different from those in Theorem 9.
In addition, for all.
The rest of the proof can be carried out in the same fashion as that of Theorem 9. □
The following corollary states the Hyers-Ulam stability of the functional equation (1).
Finally, we will give a result related to the problem extended by T. Aoki and Th. M. Rassias in the following corollary.
Once we evaluate the bound according to Theorem 9 and Theorem 10, we will get the result as desired in the corollary. □
We are very grateful to the referees for their valuable suggestions that improved this article.
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