Let A be a unital Banach algebra and M be a unital Banach A-bimodule. The main results characterize a continuous linear map ϕ : A → M that satisfies aϕ(a^−1) = ϕ(1) or aϕ(a^−1) + ϕ(a^−1)a = 2ϕ(1) for all a in principal component of invertible elements of A. The proof is based on the consideration of a continuous bilinear map satisfying a related condition
