The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them.
"Recent developments of the conditional stability of the homomorphism equation." Banach J. Math. Anal. 9 (3) 278 - 326, 2015. https://doi.org/10.15352/bjma/09-3-20