A PROPERTY OF A FUNCTIONAL INCLUSION CONNECTED WITH HYERS-ULAM STABILITY

We prove that a set-valued map F : X -> P-0(Y) satisfying the functional inclusion F(x) lozenge F(y) subset of F(x * y) admits, in appropriate conditions, a unique selection f : X -> Y satisfying the functional equation f (x) lozenge f (y) = f (x * y), where (X,*), (Y, lozenge) are square-symmetric grupoids and lozenge is the extension of lozenge to the collection P-0(Y) of all nonempty parts of Y.