When is a function not a function?

When is a function
not a function?
John Lindsay Orr
1
When is a function not a function?
When it’s a number
When it’s a vector
When it’s a point
2
Early on we learn. . .
“Functions are things that map numbers to
other numbers”
3
But we also know. . .
There are transformations of functions into other functions;
and, in general,
or, put more carefully,
where 4
Another Example: Differentiation
For example,
or,
!" # !
$&%(') !* +-, $.) !*
5
Another Example: Shifting
/ 01
23
where
2456 7(8
/945 :
;<6
Illustrated by a picture like;
=
6
Moral
So, there are “functions” that map ordinary functions
into other ordinary functions.
These “super-functions” are usually called operators.
7
What’s the domain of an operator?
Typically operators act on vector spaces of functions.
For example:
> ? @BACEDGF is the set of all continuous functions on the
interval @BACEDGF .
> HJI@BACEDGF is the set of all functions on @BACGDGF where
K
I S exists and is finite.
L MON9PQGR&M Q
8
Norms on Vector Spaces
If functions are like vectors then we can also think of
them as being like points in space. And measure their
distance apart.
The norm measures how far apart two functions are.
So,
TVU W XYT Z
distance between
U
and
X
9
Example
If
[ and \ are in ] B^ _`EaGb then we define
c
c (e f g h.i
[ d \
jlkm-konYp [9qrsd \qrs p
tvuxwzy
v
{}|x~z
10
Another Example
If
and are in JBEG then we define

(

vx z¡
¢}£x¤z¥
11
Banach Spaces
Both of these examples share an important technical
property:
Let ¦¨§ be a list of vectors in a vector space (such as
either of our last examples).
Suppose that the sequence of real numbers
¬ ¬
¦¨§ satisfies:

§¯® °
© § converges
Then the series of vectors,
a vector).
© § ª(«

§¯® °
¦¨§ also converges (to
12
Here’s Our Main Application
Consider the first order differential equation,
&± ²
±&³ ´
µ ¶
³o·²¸
with initial condition
² ³ ¹¯¸
¶
´
²-¹»º
We’d like to know if there is a solution to this initial
·E¾G¿
value problem on the interval ¼B½
. And if there is a
solution, is it the only one?
13
Assumptions
We’ll assume
Æ Ç(È
À ÁÂoÃÄÅ is continuous on the rectangle:
É
Ç Ê Ë
Ë Ì Ë
Ë Î<Ï
ÁÂoÃÄÅ
Â
ÃGÍ
Ä
Lipschitz Condition
Ð so that:
Ñ
Ñ Ë
Ñ
Ñ
À ÁÂoÃÄ»ÒÅÓ À ÁÂoÃÄ&Ô<Å
Ð Ä»Ò Ó Ä&Ô
Ê Ë
Ë Ì
Ë
Ë Î
for all
Â
and Í
Ä Ò ÃÄ Ô
.
We’ll also assume that there’s a constant
14
Example
&Õ Ö
Õ&× Ø
Consider
with
Ö
×Ù Ú Ö<Ù
×
Û when Ø Ü
×
Restricting
attention to ×o
Ý áÛ¯ÖÜ â ã Þ × Ù Þ Ú ¯Û Ö Ü Ù and Ý Û¯Ü Þ
Ö
Þ Û¯Ü , we see that ß à
Ø
satisfies the
Ø
Lipschitz Condition:
ä
ß à
×oáÖ»åâ
×oáÖ â ä
Ù
Ø
Ý ß à
Ø
Þ
Ö<Ù ä
ä Ö<Ùå
Ù
Ý
ä Ö å Ú Ö Ù »ä ä Ö å Ö Ù ä
Öä »å Ö ä Ý
Ù
æÜ
Ý
15
Goal
We want to find a function
ç èêéëì í
ç so that;
î éëoïEç9éëì¨ì
and
ç9éëð¯ì í
ñ-ð
16
Picard’s Theorem
We can find a subinterval òBó-ôöõE÷Gôùø of òBóõE÷Gø (also containing úû ) on which there is a function ü such that:
ü êô ý ú þ ÿ
ý úoõ ü ý ú þ¨þ for all ó ô ú ÷ ô
and such that
ü ýú û þ ÿ
û
17
18
Example Revisited
This says there is a unique solution
that
such
!
and
#" $&%
at least on an interval
' ( ( )
where
* $+" , ' , "
and
" , ) , $+"
19
Goal
We want to find a function
-
so that;
-/.1023 4 5 02768-02393
and
-02;:+3 4 <=:
20
Idea
Write
>?@ACB >?@;D+A E
F >/I1?JAKLJME
HF G
F N ?J#O#>?JA9APKLJ
HF G
and so
>?@A E Q D R
F N ?J#O8>?JA9AKLJ
F G
21
The Operator
Focus on the transformation:
Y [ \ ]#^ S \]_9_`L]
HY Z
\S _
S
In other words, for any function , write a
function b where
\c_ dfe V=W X
Y [ \]#^8g\]_9_`L]
b
YHZ
S T;U
V=W X
for the
22
So this means
We need to find a function
i j
h
so that:
hk l h
23
The Contraction Mapping Theorem
Let m be a closed subset of a Banach space and let n
be a function that maps m back into itself.
Suppose also that there’s a number o , which is smaller
than 1, and such that
p
n qsrPtvu n qw=t
p x
p
p
o r u w
(Anything with this property is called a contraction.)
Then there is a single point
y
in
m
at which
n qy t z y
24
Why?
Pick a point (any point) and call it {| . Then look at
{7}
{&
{&
~f
f~
f~
s{|+
s{ } s { | 9
s{ s {7}9 s{|!99
and in general
{/ ~f s{ ! } 9 s{|!99
25
Suppose the sequence converges
Let
f f

. This means;
7
&
&

and so
+¡
H¡
!¡
!¡

¡
26
But how do we know they don’t just bounce around?
¢¤£/¥ ¦ £ ¥!§©¨ ¢
ª
¯
¢#« s¬ £ !¥ §©¨H ¦ « ¬s£ ¥!§7®/ ¢
° ¢¤£ ¥!§©¨ ¦ £ !¥ §7® ¢
¢¤£/¥ ¦ £ ¥!§©¨ ¢
¯
¯
¯
¯
° ¤¢ £ ¥!§©¨ ¦ £ ¥!§7® ¢
° ® ¢¤£ ¥!§7® ¦ £ ¥!§7± ¢
²²²
° ¥ ¢¤£ ¨ ¦ £³;¢
27
So what?
´
So
µ+¶ ·
¸¤¹ µ º ¹ µ!»©· ¸
converges
by term-by-term comparison with
´ the series
¸¤¹ · º ¹¼P¸
µ+¶ ·¾½
µ
And so (by the properties of Banach spaces!), the following series converges;
´
µ+¶ ·
¹ µ º ¹ µ!»©·
¿
¿
Ã
¹ µ º ¹ µ!»©·
ÀÃÅÄ ÁfÂ ´

+µ ¶ ·
¹
ÀÃÅÄ ÁfÂ ´ Ã º ¹ ¼
28
Telescoping Sums
Æ/Ç È Æ !Ç É©Ê
Ë Æ Ç!É©Ê È Æ Ç!É7Ì
Ë Æ !Ç É7Ì È Æ !Ç É7Í
Ë ÎÎÎ
Ë Æ Ê È Æ Ï
29
Where Were We?
For any function Ð , we wrote
where
Õ7ÒÖÔ ×fØ Ù=Ú Û
Ñ ÒÓÐ&Ô
for the function
Ü
HÜ ÝÞ Òß#à8áÒßÔ9ÔâLß
So, is this a contraction?
30
Õ
Probably Not. . .
óó ñ
#ã ä å#æçéè ä åÓê&çLã ë ì íî ó
å ö#÷#æåöç9çvè
å ö#÷¤êPåöç9çùøLö
ó
ïéðñ=ð7ò ó ñHôõ
õ
óó
óó
ó
Remember
ú å ö#÷û&üHçCè åö#÷ûLý!çLú+þ ÿ úû&ü è ûLýPú
õ
õ
and so
ã#ä å#æçvè ä åÓê&çLã
and
ì íî ÿ
éï ðñ=ð7ò
ñ ú æåöçCè êPåöçLúHøLö
Hñ ô
where
ñ ú æåöçCè êPåöçLúHøLö
Hñ ô
ì íî ÿ
éï ðñ=ð7ò
fë
ÿ
ë
ì íî ú è
ì íî ú æåöçCè êPåöçLú
ÿ
ãæ è ê¾ã
;ú
31
Why is
$%$'&)(+*,$%$
!
"
#
32
But. . .
Take
-/.
7
.
0
12
0
1
So that if we restrict to
same, except now
9
>
so that
?A@
BACED
3
@
3
2
.;:
0
B
8
4
56
7
.=<
then everything is the
4
56
D?
F
4
56
G
5
?C
4
3
?
F
33

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