Stability of exponential functional equations with involutions

Let s be a commutative semigroup if not otherwise specified and f:s→. in this paper we consider the stability of exponential functional equations |f(x+σ(y))-g(x)f(y)|≤φ(x) or φ(y),|f(x+σ(y))-f(x)g(y) |≤φ(x) or φ(y) for all x,y s and where σ:s→s is an involution. as main results we investigate bounded and unbounded functions satisfying the above inequalities. as consequences of our results we obtain the ulam-hyers stability of functional equations (chung and chang (in press); chávez and sahoo (2011); houston and sahoo (2008); jung and bae (2003)) and a generalized result of albert and baker (1982). © 2014 jaeyoung chung and soon-yeong chung.