\(\mu\)-trigonometric functional equations and Hyers-Ulam stability problem in hypergroups trigonometric formulas on hypergroups. (English) Zbl 1248.39044
Rassias, Themistocles M. (ed.) et al., Functional equations in mathematical analysis. Dedicated to the memory of Stanisław Marcin Ulam on the occasion of the 100th anniversary of his birth. Berlin: Springer (ISBN 978-1-4614-0054-7/hbk; 978-1-4614-0055-4/ebook). Springer Optimization and Its Applications 52, 337-358 (2011).
Summary: Let \((X,\ast)\) be a hypergroup and \(\mu \) be a complex bounded measure on \(X\). We determine the continuous and bounded solutions of each of the following three functional equations
\[
\begin{aligned} \langle \delta_x*\mu*\delta_y,f\rangle &= f(x)g(y) \pm g(x)f(y),\;x,y \in X, \\ \langle \delta_x*\mu*\delta_y,g\rangle &= g(x)g(y) + f(x)f(y),\;x,y \in X.\end{aligned}
\]
In addition, when \(\mu = \delta_e\), Hyers-Ulam stability problems for these functional equations on hypergroups are considered. The results obtained in this paper are natural extensions of previous works done in groups especially by Stetkær, Elqorachi, Redouani, and Székelyhidi.
For the entire collection see [Zbl 1225.39001].
For the entire collection see [Zbl 1225.39001].
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
