Abstract
In this paper we establish the general solutions of the following mixed type quadratic-additive functional equation: in the class of functions between real vector spaces. Moreover, we prove the generalized Hyers-Ulam-Rassias stability of this equation in Banach spaces. MSC:39B82, 39B52.
Highlights
The stability problems of functional equations go back to when Ulam [ ] proposed the following question: Let f be a mapping from a group G to a metric group G with the metric d(·, ·) such that d f, f (x)f (y) ≤ . does there exist a group homomorphism L : G → G and δ > such that d f (x), L(x) ≤ δ for all x ∈ G ?This question was solved affirmatively by Hyers [ ] under the assumption that G is a Banach space
We introduce the following quadratic-additive functional equation: x+y+z x–y y–z z–x
For real vector spaces X and Y, we prove in Section that a mapping f : X → Y satisfies ( . ) if and only if there exist a quadratic mapping Q : X → Y satisfying
Summary
The stability problems of functional equations go back to when Ulam [ ] proposed the following question: Let f be a mapping from a group G to a metric group G with the metric d(·, ·) such that d f (xy), f (x)f (y) ≤. Kannappan [ ] introduced the following mixed type quadratic-additive functional equation:. Najati and Moghimi [ ] introduced another mixed type quadratic-additive functional equation f ( x + y) + f ( x – y) = f (x + y) + f (x – y) + f ( x) – f (x) and investigated the generalized Hyers-Ulam-Rassias stability of this equation in quasiBanach spaces. ) if and only if there exist a quadratic mapping Q : X → Y satisfying. There exist a unique quadratic mapping Q : X → Y satisfying If φ satisfies the condition (B) (and implies (D)), the proof is analogous to that of Case. Using a similar argument to that of Case , we see that the sequence
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