Fixed Points and Stability of a Generalized Quadratic Functional Equation
© A. Najati and C. Park. 2009
Received: 26 November 2008
Accepted: 11 February 2009
Published: 24 February 2009
Theorem 1.1 (Th. M. Rassias ).
In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa , who replaced the bounds and by a general control function .
In addition, J. M. Rassias  generalized the Euler-Lagrange quadratic mapping (1.12) and investigated its stability problem. The Euler-Lagrange quadratic mapping (1.12) has provided a lot of influence in the development of general Euler-Lagrange quadratic equations (mappings) which is now known as Euler-Lagrange-Rassias quadratic functional equations (mappings).
Jun and Lee  proved the generalized Hyers-Ulam stability of a pexiderized quadratic equation. 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 [8, 20–47]). We also refer the readers to the books [48–51].
We recall the following theorem by Margolis and Diaz.
Theorem 1.3 (see ).
Throughout this paper, we assume that are nonzero rational numbers with and that is a unital Banach algebra with unit , norm , and . Assume that is a normed left -module and is a (unit linked) Banach left -module. A quadratic mapping is called -quadratic if for all and all .
and using the fixed point method (see [24, 25, 38, 53–55]), we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach -modules associated with the functional equation (1.14). In 1996, Isac and Th. M. Rassias  were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.
2. Fixed Points and Stability of the Generalized Quadratic Functional Equation (1.14)
It is easy to show that is a generalized complete metric space .
For the case , we have the following counterexample which is a modification of the example of Czerwik .
which contradicts (2.88).
The authors would like to thank the referees for bringing some useful references to their attention. The second author was supported by Korea Research Foundation Grant KRF-2008-313-C00041.
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