On two functional equations originating from number theory

Abstract

Reducing the functional equations introduced in Proc. Indian Acad. Sci. (Math. Sci.) 113(2) (2003) 91-98 and in Appl. Math. Lett. 21 (2008) 974-977 to equations in complex variables and quaternions, we find general solutions of the equations.We also obtain the stability of the equations.

Stability of Exponential Functional Equations with Involutions

“…Using the results in Section 3 we can also obtain bounded functions : R 2 → R satisfying the inequalities (10), (11) and bounded functions : R 4 → R satisfying (12) and (13) as in [13].…”

“…In this section we consider the stability of (8) and (9) which were dealt with in [13]. Let H = { + + + | , , , ∈ R} be the quaternion group.…”

“…We denote ‖ ‖ = √ * = √ 2 + 2 + 2 + 2 . We first consider unbounded functions : R 2 → R satisfying (10) and (11) and unbounded functions : 12and (13).…”

“…We also refer the reader to [5] for another proof of finding general solutions of (8) and refer to [13] for the stability of inequalities (10)∼(13).…”

Ulam’s Type Stability of Involutional-Exponential Functional Equations

“…As a result, all unbounded functions , satisfying the inequalities (1) and (2) are exactly described only when is a constant function while only one of unbounded functions , satisfying each of (1) and (2) is exactly described when is an arbitrary unbounded function.…”