On the stability of a mixed type functional equation in generalized functions

• Published in: Advances in difference equations Vol. 2012; no. 1; pp. 1 - 11
• Main Author: Lee, Young-Su
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 16.02.2012; Springer Nature B.V; BioMed Central Ltd
• Subjects: Analysis; Classification; Difference and Functional Equations; Difference equations; Fourier analysis; Functional Analysis; Functions (mathematics); Heat equations; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Stability; quadratic functional equation; additive functional equation; Gauss transform; heat kernel; stability
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/1687-1847-2012-16

We reformulate the following mixed type quadratic and additive functional equation with n -independent variables 2 f ∑ i = 1 n x i + ∑ 1 ≤ i , j ≤ n i ≠ j f ( x i - x j ) = ( n + 1 ) ∑ i = 1 n f ( x i ) + ( n - 1 ) ∑ i = 1 n f ( - x i ) as the equation for the spaces of generalized functions. Using the fundamental solution of the heat equation, we solve the general solution and prove the Hyers-Ulam stability of this equation in the spaces of tempered distributions and Fourier hyperfunctions. Mathematics Subject Classification 2000: 39B82; 39B52.