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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,
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