Fixed Points and Random Stability of a Generalized Apollonius Type Quadratic Functional Equation
© Min June Kim et al. 2011
Received: 23 December 2010
Accepted: 28 February 2011
Published: 14 March 2011
Using the fixed-point method, we prove the generalized Hyers-Ulam stability of a generalized Apollonius type quadratic functional equation in random Banach spaces.
The stability problem of functional equations was 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 Th. M. Rassias  for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias  has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa  by replacing the unbounded Cauchy difference by a general control function in the spirit of the Th. M. Rassias' approach.
On the other hand, in 1982–1998, J. M. Rassias generalized the Hyers' stability result by presenting a weaker condition controlled by a product of different powers of norms.
The functional equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability of 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 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 [23–44]).
in Banach spaces.
Definition 2.1 (see ).
Definition 2.2 (see ).
Theorem 2.4 (see ).
Let be a generalized metric space. An operator : satisfies a Lipschitz condition with Lipschitz constant if there exists a constant such that for all . If the Lipschitz constant is less than 1, then the operator is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Diaz and Margolis.
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. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [61–67]).
In this paper, we prove the generalized Hyers-Ulam stability of the generalized Apollonius type quadratic functional equation (1.3) in random Banach space by using the fixed point method.
3. Generalized Hyers-Ulam Stability of the Quadratic Functional Equation (1.3) in RN-Spaces
where, as usual, . It is easy to show that is a generalized complete metric space (see [68, Lemma 2.1]).
for all and all . Letting in (3.13), we find that for all , which implies . By [45, Lemma 2.1], the mapping, : is quadratic.
The rest of the proof is similar to the proof of Theorem 3.1.
The fifth author 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) and by Hi Seoul Science Fellowship from Seoul Scholarship Foundation. This paper was dedictated to Professor Sehie Park.
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