A Fixed Point Approach to the Stability of the Functional Equation
© S.-M. Jung and S. Min. 2009
Received: 20 July 2009
Accepted: 30 September 2009
Published: 11 October 2009
By applying the fixed point method, we will prove the Hyers-Ulam-Rassias stability of the functional equation under some additional assumptions on the function and spaces involved.
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 group homomorphisms: "Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?''
and derived Hyers' theorem for the stability of the additive mapping as a special case. Thus in , a proof of the generalized Hyers-Ulam stability for the linear mapping between Banach spaces was obtained. A particular case of Rassias' theorem regarding the Hyers-Ulam stability of the additive mapping was proved by Aoki (see ).
The stability concept that was introduced by Rassias' theorem provided a large influence to a number of mathematicians to develop the notion of what is known today with the term Hyers-Ulam-Rassias stability of the linear mapping. Since then, the stability of several functional equations has been extensively investigated by several mathematicians. The terminology Hyers-Ulam-Rassias stability originates from these historical backgrounds. The terminology can also be applied to the case of other functional equations. For more detailed definitions of such terminologies, we can refer to [5–10].
were investigated in [11, Section 2.2]. The stability problem for a general equation of the form was investigated by Cholewa  (see also ). Indeed, Cholewa proved the superstability of that equation under some additional assumptions on the functions and spaces involved.
In this paper, we will apply the fixed point method to prove the Hyers-Ulam-Rassias stability of the functional equation (1.2) for a class of functions of a vector space into a Banach space. To the best of authors' knowledge, no one has yet applied the fixed point method for studying the stability problems of (1.2). So, one of the aims of this paper is to apply the fixed point theory to this case.
Throughout this paper, let denote either or . Let and be a vector space over and a Banach space over , respectively.
Let be a set. A function is called a generalized metric on if and only if satisfies
() if and only if ;
() for all ;
() for all .
Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. We now introduce one of fundamental results of fixed point theory. For the proof, refer to . For an extensive theory of fixed point theorems and other nonlinear methods the reader is referred to the book of Hyers et al. .
Let be a generalized complete metric space. Assume that is a strictly contractive operator with the Lipschitz constant . If there exists a nonnegative integer such that for some , then the followings are true:
(a)the sequence converges to a fixed point of ;
Recently, Cădariu and Radu  applied the fixed point method to the investigation of the Cauchy additive functional equation [17, 18]. Using such a clever idea, they could present a short, simple proof for the Hyers-Ulam-Rassias stability of Cauchy and Jensen functional equations.
We remark that Isac and Rassias  were the first mathematicians who apply the Hyers-Ulam-Rassias stability approach for the proof of new fixed point theorems.
3. Main Results
In this section, by using an idea of Cădariu and Radu (see [16, 17]), we will prove the Hyers-Ulam-Rassias stability of the functional equation under the assumption that is a bounded linear transformation.
for all .
for every .
for all , that is, in view of (3.6), for any , where is the Lipschitz constant with . Thus, is strictly contractive.
for every , that is, .
which implies the validity of (3.5). According to Theorem 2.1(b), is the unique fixed point of with .
which proves the validity of (3.13) for all .
which ends our proof.
Obviously, for nonnegative constants and , satisfies the conditon (3.3).
for all .
4. An Example
Then it is easy to show that satisfies the condition (3.1).
which implies the boundedness of the linear transformation .
for any .
The authors would like to express their cordial thanks to the referees for their useful comments 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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