Algorithm of Iterative Process for Some Mappings and
Iterative Solution of Some Diffusion Equation
Liu Wenjun
Department of Mathematics
Jiujiang University
Jiujiang , China
Meng Jinghua
Department of Mathematics
Jiujiang University
Jiujiang , China
Abstract—In Hilbert spaces , through improving some corresponding conditions in some literature and extending some
recent relevent results, a strong convergence theorem of some implicit iteration process for pesudocon-traction mappings
and explicit iteration process for nonexpansive mappings were established. And by using the result, some iterative solution
for some equation of response diffusion were obtained.
Keywords—pesudocon-traction mappings; nonexpansive mappings; implit iteration process; explicit iteration process;
diffusion equation.
1. Introduction
Let
E
be Banach spaceˈand k be a nonempty closed
convex subset of
E
. Suppose that
T
is a mapping from
K
to
K ,and
)(TF
is a set of fixed point of T with
() FT I z
.
Assume that
E
EJ 2: o is regular dual mapping on
E
, and
^`
.,,,)( *ExxffxfxEfxJ  ! 
As
HE
is
Hilbert
spaceˈthe internal product of
H
is donate by the symbol
!xx ,
, and the norm of
H
is
designated by symbol x
.
Definition 1 Mapping
KKT o: is said to be pseudo
contraction if for arbitrary
,xy K
, there exits
)()( yxJyxj 
such that
2
)(, yxyxjTyTx !d .
T is said to be strong pseudo contraction if there is
(0,1)k
such that
2
)(, yxkyxjTyTx !d for arbitrary
,xy K
.
Definition 2 Mapping
KKT o:
is said to be nonexpanxive
if for abitrary
,xy K
, there is
Tx Tyxyd
.
As we all know, that
T
is pseudo contraction is equivalent
to that for every
0s! and every
Kyx ,
,there is
])()[( yTIxTIsyxyx d
(1)
When
HE
is Hilbert space,
E
EJ 2: o is single value ,
and for abitrary
,xy K
,there Is
!! yxyxyxjyx ,)(,
.
Obviously, nonexpansive mapping is pseudo contraction.
2. Lemmas and Methods
Lemma 1
[1,2]
Let
E
be a real Banach space , and
K
be
nonempty closed convex subset of
E
. Assume that
KKT o:
is continuous strong pseudo constraction mapping . Then T is
unique fixed point in
K
.
Lemma2
[3]
Let
E
be a real reflexive Banach space
satisfying
opial
condition, and
K
be a nonempty closed
convex subset of
E
. Supposethat
KKTo:
is continuous
strong pseudo constraction mapping . Then for
abitrary ^` Ex
n

,
n
x weakly converge to
y
, and
0
nn
xTxo
. So there is
0)( xTI
.
Lemma 3[4] Let
1, 0pr!!
be two certain real number,
then Banach space is
()0ITx
if and only if there is a
strictly
increasing continuous function
:[0, )[0, )gf of
,
(0) 0g
,
such that
(1 )(1 )()()
pp p
p
xyx yWgxy OOO OO  d  ,
for all
,r xy B
, where
[0,1] O 
, and
r
B
is a closed spheroid
which center is origin and radius is r , and
()(1 )(1 )
pp
p
W OOOOO 
.
Lemma 4
[5]
Let nonnegative real sequence
^`
n
a
satisfy the
inequaltiy :
1
(1 )
nnnn
aa JG

d 
,
0nt
,where
[0,1),
n
J 
1
,
n
n
J
f
f
¦
lim 0
n
nn
G
J
of
or
1
,
n
n
G
f
f
¦
then
lim 0
n
na of
.
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
62 Cop
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In Hilbert space,
[6]
Moudaf has get strong convergence
theorem of implicit iteration process of nonexpansive mapping,
and
[7]
Xu
has improved and extended some relative results in
Reference [7].
In this paper, by applying a new implicit iteration
sequence
nnnnnnn
Txxxfx JED  )( ,and explicit iterative
sequence
nnnnnnn
Tyyyfy JED 

)(
1
, we shall consider
the problem involving the fixed point of strong pseudo
constraction and nonexpansive mapping on closed convex set
K . When exact conditions are satisfied, ^`
n
x and ^`
n
y all
strongly converge to the fixed point of
T . When the
conditions for ^`
n
D and
f
in Reference
[6],[7]
are widened,
and as
0
n
E
, we can obtain the iterative sequence in
Reference
[6],[7]
, and then we improve and extend some
ralative results and obtain some equation of diffusion by
applying the above results.
Let
:TK Ko be continuous pseudo constraction
mapping , and
:fKKo
be continuous strong pseudo
constraction mapping with constant D
(0 1) D 
. Suppose
that
1
nnn
DEJ  for ,, (0,1)
nnn
DEJ 
, and we stucture
mapping
:
n
SK Ko
,
()
nn nn
Sxf xxTx DEJ 
.Then
n
S
is continuous strong pseudo constraction mapping . By virtue
of Lemma 1 ,
n
S
hasunique fixed point
n
x
, then we have
()
nnn nnnnnn
xSxfxx Tx DEJ 
(2)
3. Main Results
Theorem 1 Let
E
be a Hilbert space , and
K
be a
nonempty closed convex subset of
E
. Assume that
:fKKo
is continuous strong pseudo constraction mapping
with constant
D
(0 1) D 
, and
f
is bounded on bounded
set , and
:TK Ko
is continuous pseudo constraction
mapping .Then
(a) If
0
1
n
n
n
D
VE o

or
limsup 1
n
n
V
of

ˈand there
is
()pFT
such that
22
() 0
nn
fxpx po
,
then implict iterative sequence˄2˅strongly converges to the
point of
()FT
.
(b) If T is nonexpansive mapping and
f
is constraction
mapping with constant D ,as
10 nn
nn
VV
DV

o
and
n
D f
¦
,
the explicit iterative sequence
1()
nnnnnnn
yfyyTy DEJ
 
strongly converges to the point of
()FT
.
Proof.
˄a˅Because
()pFT
,
2
n
xp
(( ))()(),
nnnnnn n
fxpxpTxpxp DEJ !
222
()
nnnn nnnn
xpfppxp xpxp DD DE J d ,
we have
() ()
11
nn
n
nnn
xpfpp fpp
DD
DDE JD
d 

.
Hence
^`^`^ `
,(),
nnn
xfxTx are bounded.
If
0
1
n
n
n
D
VE o

,then using formula (2),we can
write
()(1 )
nnn nn
xfx Tx VV  ,and then we obtain
()0
nn n nn
xTxfxTx V  o
. (3)
If
limsup 1
n
n
V
of

and there is
()pFT
such that
2
() 0
nn
fxpx po , then by virtue of formula (1)
and Lemma 3, we obtain
2
n
xp
2
1()
2
n
nnn
n
xp xTx
V
V

d 
2
1(( ))
2
n
nnn
xpfx Tx
V 

2
11
(( ))()
22
nn
fxpx p 
22 111
()( ())
224
nn nn
fxpxpg xfxd
,
and then
22
1(())()0
2nn nn
g xfxfxpxpdo
.
So we have
() 0
nn
xfxo
.
Whereas
()() 0
1
n
nn nnnnn
n
xTxfxTxx fx
V
VV   o

(4)
Because
^`
n
x
is bounded , and E is Hilbert space , we
have that
n
x weakly converge to
qK
.By virtue of formula
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(3)or (4) and Lemma 2, we have
()qFT
.
Because
2
( ())(1)(),
nnn nnn
xqfx qTxqxq VV  !
22
() ,(1)
nnnnn n
xq fqqxqxq DV VV d! ,
we obtain
2
1(),
1
nn
xq fqqxq
D
d !

. Since
n
x
weakly converges to
q
,
n
x
strongly converges to
()qFT
.
(b) Because
1nn
xx


11 1
()(1 )()(1 )
n nnn nnnn
fxTx fxTx VVV V
 

111 111
()(1 )
nnnnnnnnnnnn
xxfx xxTx VDV VVV V
 
d 
,
we obtain
1
1
(1 )
nn
nn
n
xx M
VV
VD



d 
, (5)
where
1
() 2
n
M
fx

d
ˈ
12 n
M
Tx d
.
1nn
yy


() ()
nn nnnnnn nnnn
fyy Tyfxx Tx DEJDEJ 
[1(1 )]
nn nnn nnn nnn n
yxyxyxyx DDEJD D d 
11
[1(1 )][1(1 )]
nnnnnn
yx xx DD DD

d 
Since
n
D f
¦
,
1
0
(1 )
nn
nn
VV
DV D

o

, formula (5) and
Lemma 4, we obtain
1
0
nn
yx

o
.
Hence we have
11
0
nnnn
yqy xxq

d  o
, which
means that ^`
n
y
strongly converges to
()qFT
.
Note.
Theorem 1 improves and extends some relative results in
Reference [6] and [7].
As follows, we will discuss iterative solution of some
response diffusion equation .
Let
2
()ELI
{
(, )(,)xts tsI
,
(, )xts
and
2(, )xts
are
Lebsgue intergrable on I},where
[,][, ]Iabcd u
, and
,xy E
,we define
,(,)(,)
I
x yxtsyt sdtds!
³³
.Then E is
Hilbert
space, and
22
,(,)
I
xx xxt s dtds !
³³
,
,(),, ,yjxyxxy E! !
.
Consider the problem involving solution of some first
order
diffusion equation:
0
01
,
(,0)(),(0,)( )
xx
uxGxhx
ts
xsxsxtx t
ww
 
°ww
®
°
¯ˈ
(6)
where G
is continuous mapping on
E
, and
00ut
is
constant , and
(, )0hhts t
.
This problem is equivalent to the integral equation as
follows:
0
00 0000
(, )(, )(, )(, )
xt tsts
xtsds uxtsdthtsxtsdtdsxGxdtds 
³³ ³³³³
001
00
()() 0
st
xs dsuxtdt
³³
˄7˅
Let
^
(, )KxExts  is continuous function on I
`
,
then
K
is nonempty closed convex subset of
E
.
Let
:HK Ko .
00 00000
(, )(,)()()
t s tsts
Hxuxtsdtx tsdsh txx ts dtdsxGxdtds 
³ ³³³³³
001
00
()() .
st
xs dsuxtdt
³³
If G satisfies (A)˖
,, ,xy KxGxyGy d
then let
:,TKKTxHx xo 
.
If G
satisfies (B)˖there is
1
0L!
such that
1
xGx yGyLx yd
for abitrary
,xy K
. Then
H
is
Lipschitz
mapping on K ˈand then we have 0L!
such that ,,xyK HxHyLxyd
.
Let
1
2
HH
L
ˈ
111
:, .TK KTxHxxo 
Theorem 2 Let integral equation˄7˅has solutionˈthen
(i) If G satisfies (A), when
0
1
n
n
n
D
VE o

or
limsup 1
n
n
V
of

,and there is
()pFT
such that
22
() 0
nn
fxpxpo
, Implicit iterative sequence
()
nnn nnnn
xfxxTx DEJ  strongly converges to the fixed
point of T which is solution of equation (7).
(ii) If G satisfies˄B˅,when
n
D f
¦
and
10 nn
nn
VV
DV

o
,
explicit iterative sequence
1() nnnnnnn
yfyyTy DEJ
 
strongly converges to the fixed point of
T which is solution
of equation (7).
Proof.
(i) Now,
,,( )()xyKHxHyxy
is nonnegative on
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^`
1(, )0EtsIxy d
and
^`
2
(, )0EtsIxy t
.
Then we have
,0,HxHy xy !t
that is said that
T is
pseudo constraction mapping on
K
. Using Theorem 1, we
obtain the result .
(ii) Now ,
11
,, 2xyK HxHyxyd 
˄8˅
2
1) 1
()Tx Ty
2
11
[()]Hx xHyy 
22
11 11
()2()()()HxHyyx HyHxxy 
22
11 11
()2()Hy Hxy xHy Hxx y 
2
1111
()( 2)()Hy HxHy Hxyxxy 
If
11
2,Hy Hxy xd
then we obtain
22
11
()()Tx Tyxyd
.
If
11
20,Hy Hxy xtt
then we obtain
2 2
11 1111
()( 2)( 2)()TxTyHy Hxy xHy Hxyxxyd 
222
11
()4()( )HyHxxyxy  .
Hence, by virtue of formula (8), we have
22
11
Tx Tyxyd .
That is said that
1
T
is nonexpansive mapping on
K
.
Using Theorem 1, we obtain the result.
Thereforeˈthrough improving some corresponding
conditions in literature
[6],[7]
ˈand extending some recent
relevent results, Theorem 1 was established. Theorem 1 is a
strong convergence theorem of some implicit iteration process
for pesudocon-traction mappings and explicit iteration process
for nonexpansive mappings . By applying Theorem 1, the
iterative solution for some equation of response diffusion was
obtained, Theorem 2 was established.
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