Hyers-Ulam stability with respect to gauges

We suggest a somewhat new approach to the issue of Hyers-Ulam stability. Namely, let A, B be (real or complex) linear spaces, L:A→B be a linear operator, N:=kerL, and ρA and ρB be semigauges on A and B, respectively. We say that L is HU-stable with constant K≥0 if for each x∈A such that ρB(Lx)≤1 there exists z∈N with ρA(z−x)≤K. With that definition we obtain quite general outcomes concerning approximate solutions to some differential equations.