Stability in Generalized Functions

Abstract

We consider the following additive functional equation with 𝑛-independent variables: ∑𝑓(𝑛𝑖=1𝑥𝑖∑)=𝑛𝑖=1𝑓(𝑥𝑖∑)+𝑛𝑖=1𝑓(𝑥𝑖−𝑥𝑖−1) in the spaces of generalized functions. Making use of the heat kernels, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the mollifiers, we extend these results to the space of distributions.