### Stability of the cosine-sine functional equation with involution

#### Abstract

Let $S$ and $G$ be a commutative semigroup and a commutative group respectively, $\Bbb C$ and $\Bbb R^+$ the sets of complex numbers and nonnegative real numbers respectively, $\sigma : S \to S$ or $\sigma : G \to G$ an involution and $\psi : G \to \Bbb R^+$ be fixed.  In this paper, we first investigate general solutions of the equation $$g(x+ \sigma y)=g(x)g(y)+f(x)f(y)$$ for all $x,y \in S$, where $f, g : S \to \Bbb C$  are unknown functions to be determined. Secondly, we consider the Hyers-Ulam stability of the equation, i.e., we study the functional inequality $$|g(x+\sigma y)-g(x)g(y)-f(x)f(y)|\le \psi(y)$$ for all $x,y \in G$, where $f, g : G \to \Bbb C$.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 531-546.

Dates
Accepted: 11 September 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431727

Digital Object Identifier
doi:10.22034/aot.1706-1190

Mathematical Reviews number (MathSciNet)
MR3730046

Zentralblatt MATH identifier
1374.39034

#### Citation

Chang, Jeongwook; Choi, Chang-Kwon; Kim, Jongjin; Sahoo, Prasanna K. Stability of the cosine-sine functional equation with involution. Adv. Oper. Theory 2 (2017), no. 4, 531--546. doi:10.22034/aot.1706-1190. https://projecteuclid.org/euclid.aot/1512431727

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