Estimates to the stability of functional equations

Summary.

Given a function f mapping a groupoid (X, ${\diamond}$) into a metric groupoid (Y, * ,d) and satisfying the inequality

$$ d(f(x \diamond y),f(x)*f(y))\leq \varepsilon(x,y)\quad (x,y \in X), $$

the problem of stability in the sense of Hyers-Ulam is to construct a solution g of the functional equation

$$ g(x \diamond y) = g(x)*g(y) \quad (x,y \in X) $$

and to obtain estimates for the pointwise distance between g and f. Applying the so-called direct method, the stability problem for more general functional equations is also investigated.