On a certain functional identity involving inverses in division rings and local rings

Let [Formula: see text] be a division ring, [Formula: see text] an automorphism of [Formula: see text] and [Formula: see text] a fixed element. We determine the forms of additive maps [Formula: see text] satisfying the identity [Formula: see text] for every nonzero [Formula: see text]. We show that [Formula: see text] and [Formula: see text] are both generalized [Formula: see text]-derivations of [Formula: see text] except possibly when [Formula: see text] is the identity automorphism and [Formula: see text] is an imperfect field of characteristic [Formula: see text]. We further study the same rational identity in the context of local rings under some mild assumptions.