Generalized orthogonal stability of some functional equations

We deal with a conditional functional inequality x perpendicular to y double right arrow parallel to f (x + y) - f (x) - f (y) parallel to <= epsilon(parallel to x parallel to (p) + parallel to y parallel to (p)), where perpendicular to is a given orthogonality relation, epsilon is a given nonnegative number, and p is a given real number. Under suitable assumptions, we prove that any solution f of the above inequality has to be uniformly close to an orthogonally additive mapping g, that is, satisfying the condition x perpendicular to y double right arrow g(x + y) = g(x) + g(y). In the sequel, we deal with some other functional inequalities and we also present some applications and generalizations of the first result. Copyright (C) 2006 Justyna Sikorska.