Stability of functional equations on restricted domains in a group and their asymptotic behaviors

In this paper we consider Hyers–Ulam stability problems for the Pexider equation, the Cauchy equation, and the Jensen equation in general restricted domains in a group. The main purpose of this paper is to find restricted domains such that the functional inequality satisfied in those domains extends to the inequality for the whole domain and such that the Hyers–Ulam stability theorem holds for the inequalities as it does when the inequality holds globally. We also consider a distributional version of the Hyers–Ulam stability of the Pexider equation in restricted domains and its asymptotic behaviors.