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two very simple conditions (
4.109
)and(
4.110
). By using just these conditions, we
know that (
4.108
) can only have
one
solution. This is really quite a remarkable
result, but much more remarkable is the fact that it extends to huge systems of equa-
tions. We will encounter systems with billions of equations and unknowns, and we
can use exactly the same argument to show that only one solution is possible.
4.5.5
Existence
We have seen that if h is a contractive mapping, then the equation
v
D
h.
v
/
can have only one solution. Now we will show an even stronger and more remark-
able result: By assuming that (
4.109
)and(
4.110
) hold, we will show that (
4.108
)
indeed has a solution. After doing this, we have answered all the questions (a), (b),
and (c) on page 121. In order to do so we will need two mathematical results that
we will just state here and refer you to read e.g. [6] in order to learn the proof. The
first result is trivial and just states that
n
X
X
1
˛
nC1
1
˛
1
1
˛
˛
k
D
˛
k
D
and
(4.118)
kD1
kD1
for
j
˛
j
<1.
The second result is much more theoretical and is of fundamental importance
in mathematics. This result concerns sequences of real numbers, say
f
v
k
g
.Sucha
collection of numbers is called a
Cauchy sequence
if, for any ">0, there exists an
integer M such that for all m, n
M we have
j
v
m
v
n
j
<":
(4.119)
So for a Cauchy sequence, we can choose a tiny number ",say10
10
, and be assured
that we can find an integer M such that
j
v
m
v
n
j
<"
(4.120)
if m, n
M . It turns out that this property picks up exactly the sequences that are
convergent: A sequence
f
v
k
g
converges if and only if it is a Cauchy sequence.
Our strategy is now to use the simple result (
4.118
) to show that the sequence
generated by
v
kC1
D
h.
v
k
/
(4.121)
