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We consider the general solution of quartic functional equations and prove the Hyers-Ulam- Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions. [ABSTRACT FROM AUTHOR]

The article offers an overview of the study which examines the stability of a quadratic functional equation in the spaces of generalized functions. Making use of the pullbacks, it reformulated the quadratic equation f (x + y + z) + f (x) + f (y) + f (z) = f(x + y) + f ( y + z ) + f ( z + x ) in the spaces of generalized functions. It also used the fundamental solution of the heat equation, it obtained the general solution and proved the Hyers-Ulam stability of this equation in spaces of generalized functions such as tempered distributions and Fourier hyperfunctions.

The article focuses on the stability of cubic functional equation in the spaces of generalized functions. It is proved that the Hyers-Ulam-Rassias stability theorem cubic functional equation of the cubic functional equation f ( ax + y ) + f ( ax - y ) = af ( x + y ) + af ( x - y ) + 2a(a2 - 1 ) f ( x ) for fixed integer a with a ≠ 0, ± 1 in the spaces of Schwartz tempered distributions and Fourier hyperfunctions. It is mentioned that in 1940, questions concerning the stability of group homomorphisms was solved by Hyers, assuming that G1 and G2 are Banach spaces. It is said that in 1978, Rassias generalized Hyers' result to the unbounded Cauchy difference.