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\title{FIXED POINT THEOREMS \\ and \\
ERROR BOUNDS FOR SOLUTIONS OF EQUATIONS}

\author{G. Rex \\
Institute of Mathematics, University of Leipzig \\
Augustusplatz 11, D-04109 Leipzig, Germany}
\date{}
\maketitle

We consider a system of equations $h(x)=0$, and $h$ be Gateaux-differentiable 
on an open convex set.
A computation of a solution of $h(x)=0$ by numerical algorithms being executed 
on digital computers in finite accuracy usually give a good approximation $z$ 
of the exact solution $x$, but no verified error bounds.

\vskip 2truemm

On the basis of the fixed point theorem of Brouwer, we give new theorems and 
algorithms for the computation of error bounds for the computed $z$ and verify 
that within the error bounds the exact solution $x$ exists and possibly is 
unique; special if $h$ is a linear system of equations. On the other hand we 
give connections to well-known error bound estimations.
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