Beitrge zur Numerischen Mathematik 6 (1977), 7-10 Summary: Beiträge zur Numerischen Mathematik 6 (1977), 7-10 An exclusion theorem for the solutions of operator equations GÖTZ ALEFELD Let F be a nonlinear mapping from a Banach space X into a Banach space Y. We prove a theorem which yields a ball containing no solution of the equation F(x) = O. This ball is obtained after having performed one step of Newton's method. 1. Introd uction Let F be a nonlinear mapping of the Banach space X into the Banach space Y. There are many resuIts which assure the existence of a solution of the (nonlinear) equation F(x) = O.In many cases there are also error estimates for such a solution. Famous resuIts in this direction foIlow from the contraction mapping principle. Another weIl known result is the Kantorovic theorem on Newton's method (RALL [3]). Thereby one step of Newton's method is performed and then under certain conditions we get a ball in which a solution of the equation F(x) = 0 exists and to which Newton's method will converge. Another interesting result in this direction was proved by W. BURMEISTERin [2]. Correspondingly in this note we give a ball which does not contain any solution of the equation F(x) = O. This ball is also obtained after having performed one step of Newton's method. Of course every ball K of the Banach space X, not containing a solution of the equation F(x) = 0, Collections: Mathematics