Solutions of equations involving analytic functions in a Banach algebra

In this paper, we discuss the existence and uniqueness of solution of an equationf(x)=a in a Banach algebra. We have the following fundamental lemma:LetA be a Banach algebra with identity,U be an open subset of complex planeC,f be an univalent function inU, V =f(U), a A. If the spectral set ofa, (a) V, then there exists an uniquex A, such that(x)U, andf (x) = a.Using this lemma, we can get generalizations of corresponding results in 1–8]. Moreover, we also give a generalization of the fundamental theorem on special solution in 3].