On the Stability of Quadratic Double Centralizers and Quadratic Multipliers: A Fixed Point Approach
© Abasalt Bodaghi et al. 2011
Received: 3 December 2010
Accepted: 18 January 2011
Published: 8 February 2011
We prove the superstability of quadratic double centralizers and of quadratic multipliers on Banach algebras by fixed point methods. These results show that we can remove the conditions of being weakly commutative and weakly without order which are used in the work of M. E. Gordji et al. (2011) for Banach algebras.
Găvruţa then generalized the Rassias's result in .
is called quadratic functional equation. In addition, every solution of functional eqaution (1.6) is said to be a quadratic mapping. A 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. Indeed, Czerwik in  proved the Cauchy-Rassias stability of the quadratic functional equation. Since then, the stability problems of various functional equation have been extensively investigated by a number of authors (e.g, [8–13]).
One should remember that the functional equation is called stable if any approximately solution to the functional equation is near to a true solution of that functional equation, and is supersuperstable if every approximately solution is an exact solution of it (see ). Recently, the first and third authors in  investigated the stability of quadratic double centralizer: the maps which are quadratic and double centralizer. Later, Eshaghi Gordji et al. introduced a new concept of the quadratic double centralizer and the quadratic multipliers in , and established the stability of quadratic double centralizer and quadratic multipliers on Banach algebras. They also established the superstability for those which are weakly commutative and weakly without order. In this paper, we show that the hypothesis on Banach algebras being weakly commutative and weakly without order in  can be eliminated, and prove the superstability of quadratic double centralizers and quadratic multipliers on a Banach algebra by a method of fixed point.
2. Stability of Quadratic Double Centralizers
A linear mapping is said to be left centralizer on if , for all . Similarly, a linear mapping satisfying , for all is called right centralizer on . A double centralizer on is a pair , where is a left centralizer, is a right centralizer and , for all . An operator is said to be a multiplier if , for all .
Throughout this paper, let be a complex Banach algebra. Recall that a mapping is a quadratic left centralizer if is a quadratic homogeneous mapping, that is is quadratic and , for all and , and , for all . A mapping is a quadratic right centralizer if is a quadratic homogeneous mapping and , for all . Also, a quadratic double centralizer of an algebra is a pair where is a quadratic left centralizer, is a quadratic right centralizer and , for all (see  for details).
Before proceeding to the main results, we will state the following theorem which is useful to our purpose.
Theorem 2.1 (The alternative of fixed point ).
Suppose that we are given a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then for each given , either , for all , or else exists a natural number such that
Now, we establish the superstability of double quadratic centralizers on Banach algebras as follows
3. Stability of Quadratic Multipliers
Assume that is a complex Banach algebra. Recall that a mapping is a quadratic multiplier if is a quadratic homogeneous mapping, and , for all (see ). We investigate the stability of quadratic multipliers.
Using Theorem 3.1, we establish the superstability of quadratic multipliers on Banach algebras.
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