 
Summary: Existence of Solution~ and Iterations for
Nonlinear Equations
by G. Alefeld, Karlsruhe University
1. Introduction. In this paper we discuss the use of inter
val analysis in order to prove the existence of solutions of
an equation. In Chapter 2 we repeat the generalization of
the bisection process using interval arithmetic tools. The
use of the Brouwer fixedpoint theorem is demonstrated in
Chapter 3. We show in Example 1 that by using interval
arithmetic it is sometimes possible to improve known
existence statements. since the proof of the Brouwer fixed
point theorem is nontrivial it seems worthwhile to investi
gate if one can prove the existence of fixed points by using
interval arithmetic tools alone. Some ideas in this direc
tion are described in Chapter 4. In the final Chapter 5 the
IntervalNewtonMethod is reconsidered again and a new
statement concerning the order of convergence is given. The
terminology used in this paper is the same as in [5].
2. Bisection. It is wellknown that if for areal continuous
function f : ffi ~ ffi there exist reals a and b , a < b ,
