Fixed Points, Inner Product Spaces, and Functional Equations
© Choonkil Park. 2010
Received: 1 February 2010
Accepted: 5 July 2010
Published: 20 July 2010
Rassias introduced the following equality , , for a fixed integer . Let be real vector spaces. It is shown that, if a mapping satisfies the following functional equation for all with , which is defined by the above equality, then the mapping is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.
1. Introduction and Preliminaries
The stability problem of functional equations originated from a question of Ulam  concerning the stability of group homomorphisms. Hyers  gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias  has provided a lot of influence on the development of what we call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by G vruţa  by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.
is called a quadratic functional equation. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof  for mappings , where is a normed space and is a Banach space. Cholewa  noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik  proved the generalized Hyers-Ulam stability of the quadratic functional equation. The generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by C dariu and Radu .
for all with (see ).
which is called a functional equation of quadratic type. In fact, in satisfies the functional equation of quadratic type. In particular, every solution of the functional equation of quadratic type is said to be a quadratic-type mapping. One can easily show that if satisfies the quadratic functional equation then satisfies the functional equation of quadratic type. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [11–24]).
We recall a fundamental result in fixed point theory.
In 1996, Isac and Rassias  were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [17, 28–31]).
2. Fixed Points and Functional Equations Associated with Inner Product Spaces
We investigate the functional equation (1.4).
This implies that inequality (2.7) holds.
This implies that inequality (2.28) holds.
Combining Theorems 2.2 and 2.5, we obtain the following result.
Combining Remarks 2.4 and 2.7, we obtain the following result.
This paper was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).
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