Distributional Methods for a Class of Functional Equations and Their Stabilities

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

“…In this section we consider a distributional version of the Hyers-Ulam stability of the Pexider type functional equation in half-planes U k,d := {(x, y) ∈ R 2 : x + ky > d} for some fixed k ̸ = 0, 1, d ∈ R. Like in [9][10][11] we prove the Hyers-Ulam stability of the inequality…”

Functional equations in Schwartz distributions and their stabilities

“…3) f (x + y + xy) = g(x) + h(y) + g(x) h(y), x, y ∈ I n , (1. 4) f (x + y + xy) = f (x) + f (y) + f (x) f (y), x, y ∈ I n .…”

“…As the first approach, using regularizing sequence of test functions and converting given distributional versions of functional equations to classical ones [4,5] we obtain the solutions. In particular, this approach is useful to consider the Hyers-Ulam stability problems of functional equations in Schwartz distributions.…”

ON AN L∞-VERSION OF A PEXIDERIZED QUADRATIC FUNCTIONAL INEQUALITY

“…Recently, some of such stability problems have been studied in the sense of Schwartz distributions [2,3,4]. However, the author guesses that the Schwartz theory of distributions would not be interested for the readers.…”

“…However, the author guesses that the Schwartz theory of distributions would not be interested for the readers. For the reason, in the present article, making use of the same methods as in [2,3,4] with possible change of terminologies we consider an L ∞ -version of the stability of generalized quadratic functional equation which would be more interested for the readers(see P. W. Cholewa [1] and F. Skof [7] for classical Hyers-Ulam stability of quadratic functional equations). Throughout this article, we denote by L 1 loc (R n ) the space of all locally integrable functions f : R n → C. Let f ∈ L 1 loc (R n ).…”