Solutions of Equations Involving Analytic Functions in a Banach Algebra

In this paper, we discuss the existence and uniqueness of solution of an equation f(x)=α ina Banach algebra. We have the following fundamental lemma: Let A be a Banach algebra with identity, U be an open subset of complex plane C,f be an univalentfunction in U, V=f(U),α∈A. If the spectral set of α,σ(α)=V, then there exists an unique x∈A, suchthatσ(x)U, and f(x)=α Using this lemma, we can get generalizations of corresponding results in[1--8]. Moreover, we alsogive a generalization of the fondamental theorem on special solution in [3].