In this paper, we prove the Hyers-Ulam stability of the Cauchy additive functional equation and the quadratic functional equation in matrix normed spaces. MSC: Primary 47L25; 39B82; 46L07; 39B52
LetX,Ybe vector spaces andka fixed positive integer. It is shown that a mappingf(kx+y)+f(kx-y)=2k2f(x)+2f(y)for allx,y∈Xif and only if the mappingf:X→Ysatisfiesf(x+y)+f(x-y)=2f(x)+2f(y)for allx,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.
In [32, 33], the fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated. Using the fixed point method, we prove the Hyers-Ulam stability of the following additive-quadraticcubic-quartic functional equation f (x + 2y) + f (x − 2y) = 4 f (x + y) + 4 f (x − y) − 6 f (x) + f (2y) + f (−2y) − 4 f (y) − 4 f (−y) (1) in fuzzy Banach spaces.
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