Abstract
We prove the Hyers-Ulam-Rassias stability of the Jensen type quadratic and additive functional equation \(9f ( \frac{x+y+z}{3} ) + 4 [ f (\frac{x-y}{2} ) + f (\frac{y-z}{2} ) + f (\frac{z-x}{2} ) ] = 3 [ f(x)+f(y)+f(z) ]\) under the approximately conditions such as even, odd, quadratic, and additive in Banach spaces.
Highlights
The study of stability problems for functional equations has originally been raised by Ulam [ ]: under what condition does there exist a homomorphism near an approximate homomorphism? In, Hyers [ ] had answered affirmatively the question of Ulam for Banach spaces
We prove the Hyers-Ulam-Rassias stability of the Jensen type quadratic and additive functional equation x–y 2
1 Introduction The study of stability problems for functional equations has originally been raised by Ulam [ ]: under what condition does there exist a homomorphism near an approximate homomorphism? In, Hyers [ ] had answered affirmatively the question of Ulam for Banach spaces
Summary
The study of stability problems for functional equations has originally been raised by Ulam [ ]: under what condition does there exist a homomorphism near an approximate homomorphism? In , Hyers [ ] had answered affirmatively the question of Ulam for Banach spaces. The terminology Hyers-Ulam-Rassias stability originates from these historical backgrounds and this terminology is applied to the case of other functional equations (see [ – ]). ) as the mixed type quadratic and additive functional equation. Jun and Kim [ ] solved the general solutions and proved the stability of the following functional equation, which is a generalization of Najati and Moghimi [ ] introduced another type quadratic and additive functional equation f ( x + y) + f ( x – y) = f (x + y) + f (x – y) + f ( x) – f (x) and investigated the stability of this equation in quasi-Banach spaces. Lee et al [ ] introduced the following Jensen type quadratic and additive functional equation: x+y+z x–y y–z z–x. This proves the validity of inequality ( . ) for the case n +
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