Page 110 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

dy
Unit 6: Existence Theorem for the Solution of the Equation = f(x, y)
dx

Notes
y
x
t  ( , )
k sup ( , ) D ...(9)
y
x

y 
where k is a limit superior.
Then the mean value theorem implies that (8) holds for f(x, y). By Assumption 1, f(x, y) is
continuous on the bounded domain D, therefore |f(x, y)| is bounded on D, that is,
y
SUP ( , ) D f ( , ) M ...(10)
x
y
x

Set
= Min (h, k/m) ...(11)
Let us define a sequence of functions {y (x)} for |x x | ,
n 0
successively by
x
y 0 ( ) y 0
x
t
x
y 1 ( ) y 0 f ( , y 0 )dt
0 x
x
y 2 ( ) y 0 f ( , y 1 )dt
x
t
0 x
..........................................
..........................................
...(12)
..........................................
x
x
t
y n ( ) y 0 f , t y n 1 ( ) dt
x 0
Theorem: That {y (x)} converges uniformly on the internal |x x | , and the limit y(x) of the
n 0
sequence is a solution of (5) which satisfies (3).
Picard’s Method of Successive Approximation
The above theorem is proved by Picard’s method of successive approximation as follows. We
here give this proof as shown by K. Yosida.
Proof: According to (10) and (11), we obtain
|y (x) y | M k
1 0
x
t
for |x x | . Therefore f ( , y ( )dt can be defined for |x x | h, and
t
0 1 0
0 x
|y (x) y | M K
2 0
In the same manner, we can define y (x),......y (x) for |x x | and obtain
3 n 0
|y (x) y | M K, for K = 1, 2, ...n
k 0
using assumption (2), we have
x
t
t
–
x
|y k 1 ( ) y k ( )| K | | ( ) y k 1 ( )|dt
x
y
k
0 x
for |x x | . Therefore, if we assume that for k = 1, 2, ......n
0
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