<#2005#>Proof of the fixed point theorem:<#2005#>
Here the distance #math480##tex2html_wrap_inline3450# is specifically
defined as the <#2008#>p-norm<#2008#> (section #vectorNorms#2009>) of
the vector #math481##tex2html_wrap_inline3452#:
#math482# #tex2html_wrap_indisplay7050#
(114)
where #tex2html_wrap_inline3454#, e.g., #math483##tex2html_wrap_inline3456#. For convenience, we can
drop #tex2html_wrap_inline3458# so that #math484##tex2html_wrap_inline3460#.
- We first prove constructively the existence of a fixed point.
As #math485##tex2html_wrap_inline3462# is a contraction, we have
#math486# #tex2html_wrap_indisplay7057#
(115)
As #tex2html_wrap_inline3464#, we have #math487##tex2html_wrap_inline3466#.
This is a <#2050#>Cauchy sequence<#2050#> (Section #innerProductSpace#2051>)
that converges to some point
#math488##tex2html_wrap_inline3468# also in the space.
We further have
#math489# #tex2html_wrap_indisplay7062#
(116)
i.e., the limit of the Cauchy sequence
#math490##tex2html_wrap_inline3470# is a fixed point.
- We next prove the uniqueness of the fixed point. Let #tex2html_wrap_inline3472#
and #tex2html_wrap_inline3474# be two fixed points of #math491##tex2html_wrap_inline3476#, then we have
#math492# #tex2html_wrap_indisplay7068#
(117)
For any #tex2html_wrap_inline3478#, the above holds only if
#math493##tex2html_wrap_inline3480#,
i.e., #math494##tex2html_wrap_inline3482# is the unique fixed point.
QED
<#2095#>Proof of the Contraction Mapping theorem:<#2095#>
Consider the Taylor expansion (Section #TalorSeries#2096>)
of the function #math495##tex2html_wrap_inline3484# in the neighborhood of #tex2html_wrap_inline3486#:
#math496# #tex2html_wrap_indisplay7075#
(118)
where #math497##tex2html_wrap_inline3488# is the remainder composed of second
and higher order terms of #math498##tex2html_wrap_inline3490#.
Subtracting #math499##tex2html_wrap_inline3492# and taking any p-norm on both sides,
we get
#math500# #tex2html_wrap_indisplay7080#
(119)
When #math501##tex2html_wrap_inline3494#, the second and higher order terms
of #tex2html_wrap_inline3496# disappear and #math502##tex2html_wrap_inline3498#, we
have
#math503# #tex2html_wrap_indisplay7085#
(120)
The inequality is due to the <#2162#>Cauchy-Schwarz inequality<#2162#>
(Section #innerProductSpace#2163>) if #math504##tex2html_wrap_inline3500#,
the function #math505##tex2html_wrap_inline3502# is a contraction at #tex2html_wrap_inline3504#.
QED