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gives good approximations, provided that a solution of (
4.111
) exists. But we do
not know that it exists! We have not discussed existence yet. For a scalar equation
of the form (
4.108
), it is usually sufficient to just graph the functions
v
and h.
v
/
and observe that they cross each other at one and only one point. This ensures both
existence and uniqueness. But we will do this in a much harder way. The reason
for this is that later we will study systems of equations of the form (
4.108
), i.e.,
equations where
v
is a vector and h is a vector of functions. In such cases we cannot
analyze (a) and (b) above just by graphing. So by presenting a more theoretical
argument already in the case of a scalar equation, you will find the argument in
the case of systems easier to comprehend. But keep in mind that for any specific
equation of the form
v
D
h.
v
/;
(4.112)
or, more generally on the form,
F.
v
/
D
0;
(4.113)
it is always instructive to graph the involved functions in order to understand what
kind of problem you have at hand.
4.5.4
Uniqueness
Letusfirstseethat(
4.108
) can only have one solution, provided that h is a contrac-
tive mapping satisfying (
4.109
)and(
4.110
). In order to see this, we simply assume
that we have two solutions
v
and
w
, i.e., we assume that
v
D
h.
v
/
(4.114)
and
w
D
h.
w
/:
(4.115)
Using (
4.109
) with
v
D
v
and
w
D
w
,wehave
j
h.
v
/
h.
w
/
j
ı
j
v
w
j
(4.116)
where ı<1. But by (
4.114
)and(
4.115
)wehave
j
v
w
j
ı
j
v
w
j
;
(4.117)
which can only hold when
v
D
w
, and consequently we can have only one solu-
tion. You should note that this is really highly nontrivial. We have (
4.108
)andjust
