Summary: Beiträge zur Numerischen Mathematik
6 (1977), 7-10
An exclusion theorem for the solutions of operator equations
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
). 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 . 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,