Advances in Pure Mathematics, 2012, 2, 139-148
http://dx.doi.org/10.4236/apm.2012.23021 Published Online May 2012 (http://www.SciRP.org/journal/apm)
Stable Perturbed Algorithms for a New Class of
Generalized Nonlinear Implicit Quasi Variational
Inclusions in Banach Spaces
Salahuddin, Mohammad Kalimuddin Ahmad
Department of Mathematics, Aligarh Muslim University, Aligarh, India
Email: salahuddin12@mailcity.com, ahmad_kalimuddin@yahoo.co.in
Received August 11, 2011; revised October 11, 2011; accepted October 20, 2011
ABSTRACT
In this work, a new class of variational inclusion involving T-accretive operators in Banach spaces is introduced and
studied. New iterative algorithms for stability for their class of variational inclusions and its convergence results are
established.
Keywords: T-Accretive Operators; Variational Inclusions; Iterative Algorithms; Stability Conditions; Convergence;
Strong Accretivity; Banach Spaces
1. Introduction
Variational inequality theory provides us with a simple,
natural, general and unified framework for studying a
wide range of unrelated problems arising in mechanics,
physics, optimization and control theory nonlinear pro-
gramming, economics, transportation, equilibrium and
engineering sciences.
In recent years, variational inequality has been ex-
tended and generalized in different direction. A useful
and important generalization of the variational inequality
is called variational inclusions see [1-7].
Suppose E is a real Banach space with dual space E*,
norm . and dual pairing .,.
:2
, 2E is the family of all
nonempty subsets of E, CB(E) is the family of all non-
empty closed bounded subset of E and the generalized
duality mapping
E
q
JE
is defined by

**
1
*
:,
,
q
q
** ,
,
J
ufEu
fu


f fu
uE
 
where q > 1 is a constant. In particular, 2
J
is the usual
normalized duality mapping. It is known that, in general
 
2
2
q
q
J
uu
0
Ju
for all and q
u
J
is simple
single valued, if E* is strictly convex. The modulus of
smoothness of E is the function
, 0, :0
E
defined by


1
sup 1:
2
Etuvuv

1, .uvt



A Banach space E is called uniformly smooth if
0
lim 0
E
t
t
t
>0
. E is called q-uniformly smooth, if there
such that exists a constant
,1.
q
Ettq

>0
q
c,uvE
Note that Jq is single valued, if E is uniformly smooth.
Xu and Roach [8] and Xu [9] proved the following re-
sults.
Lemma 1.1. Let E be a real uniformly smooth Banach
space. Then E is q-uniformly smooth if and only if there
exists a constant such that for all

,.
qq q
qq
uvu qvJu cv 
:TE E
:2
Definition 1.1. [10] Let be a single-valued
operator and
E
ME be a multivalued operator.
M is said to be T-accretive if M is accretive and
TMEE
>0 hold for all
.
Remark 1.1. 1) From [11] it is easily establish that if
TI
(the identity map on E), then the definition of I-
accretive operator is that of m-accretive operator.
2) Example 2.1 in [11] shows that an m-accretive
operator need not be T-accretive for some T.
Let ,
:TE E:NE EE
:2
be two single valued
mappings. Let
E
MEE
tE

., :2
be a set-valued mapping
such that for each fixed ,
E
MtE
,,:
be a
T-accretive operator. For given
f
gp EE
uE
are map-
pings, consider the following problem of finding
such that
 
0,,,
f
uNuuMpugu  (1)
C
opyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD
140
which is called the generalized nonlinear implicit quasi
variational inclusions.
Special Cases:
1) If E is a Hilbert space, then problem (1) is equiva-
lent to finding such that
uE

 
,,

 
Dom .,
0,
puM gu
f
uNuu
 Mpugu

,
(2)
which is called the generalized nonlinear implicit quasi
variational inclusions, considered by Ding [12] and Fang
et al. [13].
2) If

M
ut ,utE
uE
Mu for all , then problem
(2) is equivalent to finding
such that

 
Dom
0,
pu M
fu Nu

 
,u Mpu
:2
(3)
where
E
ME

,
is a maximal monotone mapping.
The problem (3) was considered by Huang [14].
3) If

,
M
ut
 E
uE
ut for each t, then prob-
lem (2) is equivalent to finding such that


 
, ,
Dom .,
,, ,
pu gu
f
uNuuvpu pu

gu vgu


R tE
(4)
where such that for each
:EE
,
is a proper convex lower semi-
continuous function with

., :tER

 

 
., .t

0f
uE


 
.,
,,
u
M puu
:TE E
:2
Range Domp (5)
The problem (4) was considered by Ding [15] for g to
be an identity mapping.
4) If and g is the identity mapping, then prob-
lem (1) is equivalent to finding such that

Dom
0,
pu M
Nuu
 (6)
which is called the generalized strongly nonlinear im-
plicit quasi variational inclusions, considered by Shim et
al. [16].
Remark 1.2. For a suitable choice of f, g, p, N, M and
the space E, a number of classes of variational inequali-
ties, complementarity problems and the variational in-
clusions can be obtained as special cases of the general-
ized nonlinear implicit quasivariational inclusions (1).
Let be a strictly monotone operator and
E
MEE

, .
be a T-accretive operator. Fang and
Huang [11] defined the resolvent operator
 

1
(., )
,.,
Mt
T
J
vTMt
 v vE
:TE E
:2
(7)
By Theorem 2.2 in [11], we know that if
is a strictly accretive operator and
E
MEE
(., )
,:
Mt
T
is a
T-accretive operator, then the operator
J
EE
:TE E
>0
is a single valued. From the proof of Theorem 2.3 in [11],
it is easy to obtain the following result.
Lemma 1.2. [10] Let be a strictly accre-
tive operator with constant
and for each fixed
tE
, :2
E
MEE
(., )
,:
Mt
T
be a T-accretive operator then
the operator
J
EE
is Lipschitz continuous with
constant 1, i.e.,
 
(., )(., )
,,
1,.
Mt Mt
TT
J
uJvuv uvE




2
qqq q
aba b 
 




2max,2 max,
2.
qq
qq
qq q
a babab
ab
 

(8)
Lemma 1.3. Let a and b be two nonnegative real
numbers. Then
. (9)
Proof.
Definition 1.2. Let
n
M
0,1, 2,n
and M be a maximal mono-
tone mappings for . The sequence
n
M
G
n
is said to be graph converges to M (write
M
M)
if for every
, there exists a sequences
,uvGraph M
,n
uvGraph Mn
uun
vv
n
n
nn such that and
as .
Lemma 1.4. [3] Let
M
and M be the maximal mono-
tone mappings for G
n
. Then 0,1, 2,n
M
M
 
,
n
MM
if
and only if
J
uJu

uE
(10)

1
M
JIM
 .
and for every >0
, where
n,
n and b
Lemma 1.5. Let a
n be three se-
quences of nonnegative numbers satisfying the following
condition. There exists a positive integers such that
c
0
n
10
1, for
nnnnnn
atabtcnn
  (11)
0,1
n
t
0
n
n
t
, where

lim 0
n
nb

0
n
n
c
, and

0
n
an. Then as .
0
inf :ann
 0.
n. Then Proof. Let
>0.
Sup-
pose that >0
n
a
Then
0.nn for all It
follows from (11), that
1
11
22
nnnnnn
nnnnn
aattbc
abttc

 

  


.
nn0
n,n
10
nn
(12)
for all 0 Since b as there exists
such that
1
1,forall.
2n
bnn

Copyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD 141
Combining (11) and (12), we have ,0,if;
q
TuTv Juvuv
1
aa
1
2
nn nn
tc
 
1
nn
for all , which implies that
1
1
2nn
nn
ta

11
.
n
nn
c



0
This is a contradiction. Therefore,
0
j
n
a
j
and so there
exists a subsequence

such that
as . It follows from (11) that
n
a
j
n
a

1
j
jjjj
nnn
btc 
.j
j1k
E
:2
nn
aa
and so as A simple induction leads
to as for all and this means
that as .
n This completes the proof.
10
j
n
a
0
0
jk
n
a
a

:TE
n
Lemma 1.6. Let be a strictly accretive op-
erator and for a fixed t, E
E
MEE
 
,
be a
T-accretive operator in the first variable. If u is a solu-
tion of the problem (1) if and only if
 
(., )
,
Mgu
T
g
uJTpu f

u Nuu

>0
where
is a constant and


1
., .Mgu
uE
 
 

,
,
,
0,
,.
pu gu
u Tpu
uu
Nu u



,:Tg E E
(., )
,
Mgu
T
JT

Proof. is a solution of (1)
 

 

 



 
(., )
,
0,
0,
0
,
.,
Mgu
T
fuNu uMpugu
fuNu uM
TpufuNu
Mpugu
Tpufu N
TMgupu
puJTpuf u
 


 
 
 
 

 
2. Existence and Uniqueness Theorems
In this section, we show the existence and uniqueness of
solutions for the problem (1) in terms of Lemma 1.6.
Definition 2.1. Let E be a real uniformly smooth Ba-
nach space and be two single valued op-
erators; T is said to be
1) Accretive if

,
q
TuTv Juv0,,u vE
or, equivalently

2
,0TuTv Juv,,;u vE
2) Strictly accretive if T is accretive and
 
>0r

3) Strongly accretive if there exists a constant
such that
,,,
q
q
TuTvJu vru vuvE
 

or, equivalently
2
2
,, ,;TuTvJ u vruvuvE
4) Lipschitz continuous if there exists a constant s > 0
such that
,,;TuTvsuvuvE 
>0
5) Strongly accretive with respect to g if there exists a
constant
such that

,,,
q
q
TuTvJgu gvgu gvuvE
 

or, equivalently
2
2
,,,.TuTvJ gu gvgu gvuvE
 
:NE E E :
Definition 2.2. Let and
g
EE
be the maps, then
1)
.,.N
>0
is said to be strongly accretive with respect
to first argument if there exists a constant
such
that
 
,.,. ,,,
q
q
NuNvJu vu vuvE
 
 
or, equivalently
2
2
,.,. ,,,;NuNvJu vuvuvE
 
>0
2) N is said to relaxed accretive with respect to g if
there exists a constant
such that
 
,.,. ,,
,;
q
q
NuNvJgu gvgu gv
uv E


>0
3) N is Lipschitz continuous in first argument if there
exists a constant
such that
,.,., ,.NuNvu vuvE
 
:E
Theorem 2.1. Let E be a q-uniformly smooth Banach
space and TE be a strongly accretive and
Lipschitz continuous with positive constants
and
respectively. Let be the strongly accretive
and Lipschitz continuous with positive constants
:pE E
and
respectively. Let be Lipschitz con-
tinuous with positive constants
,:fgE E
and
respectively.
Let :NE EE
1
p
>0
be relaxed accretive with respect to
in the first and second arguments with constants
and >0
respectively, where 1 is
defined by
:pE E
 
puT puTpu
1 for all uE
.
Assume that N is Lipschitz continuous with respect to
first and second argument with constants >0
and
Copyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD
Copy APM
142
>0
right © 2012 SciRes.
respectively. Let :2
E
MEE be a T-accre-
tive operator with second argument. If there exist con-
stants >0
and >0
such that for all ,, ,uvw E

(., )(.,
,,
Mgu M
TT
)gv
J
wJ

w gugv
 (13)
and
11,QP


(14)
where
1
11
q
q

 


1
2
1.
q
qqqqqq qq
q
Pq C
 



*
uE
. Then the problem (1) has a unique solution
Proof. By Lemma 1.6, it is enough to show that the
mapping EE*
uE has a unique fixed point
F
:
where F is defined as follows.

(., )
,,
Mgu
T
F
uupuJTpufuNuu

 
q
C

Qq 
and
uE
(15)
. From (13) and (15), we have for all
 

 

 




 

 

 

., .,
, ,
., .,
,,
., .,
,,
, ,
,,
,,
Mgu Mgv
TT
Mgu Mgu
TT
Mgu Mgv
TT
FuFvupu JTpufuNuuvpvJTpvfvNvv
uvpupvJT pufuNuuJT pvfvNvv
JT vNvvJTpvfvNvv
u vpupv



 
 
 


 




 
1
pvf (16)

 

 
1
,,
1,(,) .
T puT pvNuuNvvfufvgugv
uvpupvT puT pvNuuNvvfufvuv
 



 
 
Now since p is strongly accretive and Lipschitz continuous, we have
 


,
1
1.
qqq
qq
qq q
q
q
q
q
q
q
q
q
uvpu pvuvqpu pvJuvCpu pv
uvq uvCuv
qC uv
uvpu pvqCuv



 
  
 
 
(17)
By the strong accretivity of p with constant
, we have
 
1,
1
q q
qq
pupvu vpupvJu vpupvJu vu v
 
u vpupv
 
that is
,, .NuuNvuuv
 (20)
,,Nvu Nvv
pupvu v
 . (18) .u v
 (21)
Since f is Lipschitz continuous, we get
Similarly, by the strong accretivity of T with constant
we have
f
.ufvuv
 (22)
 
T puT pvpupv

By
. (19)
Since :NE EE
is a relaxed accretive with re-
nd second argumenspect to p in the first a
>0
-Lipschitz continuity of N with respect to first
ent and
t with constant
argum
-Lipschitz continuity of N with respect
nd arguments, we have
and >0
re
have
spectively, from (18) and (19), we
to seco
SALAHUDDIN, M. K. AHMAD 143
 
 
,,,
q
q
NJT puT pv
T Tpv
 
and similarly
q
q
q
qq
uuNvu
pu
pu pv
uv


 
 
(23)


,,,
.
q
q
qq
NvuNvvJTpuT pv
uv
 

 
,p
 

(24)
From (23) and (24), Lipschitz continuity of T,
Lemma 1.1 and Lemma 1.3, we have
  

 

 


,,
,,, ,,
2
q
q q
q
qq
qqq q
qqqqq qqq
q
q
qqq qqqqq
TpuTpvNuuNvv
Tpu Tpvq NuuNvvJTpu TpvCNuuNvv
TpuTpvuvqu vCu v
qCuv

 

 
 
 

 

(25)
Now from (16), (17), (22) and (25), we have
 
2.
q
q




1
11
12
1,
q
q
qqq qqqqqq
qq
F
uFvqCqCuv
Q Puvuv



 
 
 
 




(26)
where

1
1q
q
q
QqC
 
  and



1
2
q
qqqqqq qq
q
Pq C

 

n
1,
nn
and 1
QP
x
x
 .
From (14), we know that 0< <1
. Therefore, there
uE such that

exists a unique
F
uu

. This
3.
truurbed iterative
solving the problem (1) and
tive se-
ith er-
completes the proof.
Perturbed Algorithm
In this section, we consct some new pert
algorithms with errors for
s and Stability
prove the convergence and stability of the itera
quences generated by the perturbed algorithms w
rors.
Definition 3.1. Let T be a self mapping of E and

1,
nn
x
fTx
define an iterative procedure which
yields a sequence of point
n
x
in E. Suppose that
xE
:Txx and

n
x
converges to a fixed
point
x
of T. Let

n
y
E and let
1.
nn n
yfy

,T
1) If lim 0
n
n
 thatn
iterative procedure
implies lim n
yx
, then the
defined by fTx
is
said to be T-stable or stable with respect to T.
2) If n
0
n

implies that lim n
nyx
 , then the
e iterativprocedure
n
x
is lmost T-stable.
stability results of iterative algorithms have been
several a19]. As was sh
he stabi
of the theoretical and numerical interest.
Remark 3.1. An iterative procedure

n
said to be a
Some
established by
Ha
uthors [17-own by
rder and Hicks [20], the study on tlity is both
x
which is T-
stable is almost T-stable and an iterative procedure
n
x
which is almost T-stable need not be T-stable [21].
Algorithm 3.1. Let ,,,:fpgTE E and
:NE EE be the five single valued mappings. Let
n of M be the set-valued mapping from EE
M
into
the power set of E such that for each tE,
.,
n
M
t
and
t are T-accretive mappings and
M
.,
., .,
G
n
M
tMt. For any given 0
uE, the per-
turbed iterative sequence
n
u with errors is defined as
follows:
Copyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD
144

.,
1
1
nn
Mgv
nnnnn
uuvpv


  

,
,
nT nn
nnnnnnT
JTpvfv
vuwpw T
T
 
 
  
,
,
,
nnnnn
n nnnnn
n nnnn
Nvvel
pwfwNwwr
pufuNuus
 
 
 
 
 
 
)


n


(27


.,
1
nn
Mg
w
J
  


.,
,
1
nn
Mgu
nnnnnnT
wu
upuJ


 
for 0,1,2,n, where
n
,
n
and are
n
three sequences in [0,1],

n
e,
n
r,

n
s
and
n
l
are four sequences in E satisfying the following condi-
: tions
00
,
nn
nnn
nn
 




lim lim lim0
.
n
nn
ser
l


 (28)
From Algorithm 3.1, we obtain the following algorithm
for the problem (3).
Algorithm 3.2. Let ,, :
f
pT EE and
:NE EE be four single valued mappings. Let
n
M and M be T-accretive mappings from E into the
power of E such that G
n
M
M. For any given
0
uE
, define the perturbed iterative sequences
n
u
with errors as follows:
 

 
 
1,
1,
,
n
n
M
nnnnnnTnnnnnnn
nn n
n
uuvpvJTpvfvNvvel
w
puus
 


 


 
0, 1,n

,
,
1
1
n
M
nnn T
M
nnnnnnT
vu
wpJT
wuupuJT




 
,
n nnnnn
pwfwNwwr
 
 
(29)

n nnnn
fuNu
 
 
2, , then

n
,
,

e,
n
n
, n
r,

n
n
s
and

n
l are same as Algorithm 3.1
Rem n
nd M
rithms
.
ark 3.2. For a suitable choice of ,,, ,TfgNM
, Algorithm 3.1 reduces to several known Algo-
[22-24] as special cases.
Theorem 3.1. Let ,,,
a
f
pgT and N be the same as in
rem 2.1. Suppose that

n
M and M are set-valued
ings from EE into the power set of E such that
tE,

.,
n
exists constants >0
and >0
s
,,uvz
uch that for each
E
and 0n


(., )Mu
Theo
mapp
for each
M
t and
.,
M
t are T-accretive
ings and
 
., .,
G
n
mapp
M
tMt. Assume that there

(., )
,,
(.,)(., )
,,
n
Mv
TT
n
Mu Mv
TT
J
zJ zuv
zJzuv



J

(30)
and the condition (13) holds. Let

n
y
E and define a sequence

n
be a sequence in
of real numbers as
follows:

 


  


1
nn
nnn
xy
zpz T

  
 

 


 
.,
1,
.,
,
.,
1,
,
,
nn
nn
n
Mgx
n nT
Mgz
nTn nnnnn
Mgy
nn nnnnn
xpxJT Ne
JpzfzNzzr
pyfyNyys
 
 

 


 

 
(31)
,
1
n
nn
nnnT
zy
ypyJT

where

n
nn n

n
nnnn nnn
pxfxxxl
 
 yy

,

n
,

n
,

n
e,

n
r,
n
s
and

n
l are sdefinedorithm 3.1. Then
the following ho
ame seq
ld
u
s:
ences
in Alg
que

n defined by Algorithm 3.1 con-
e unique solution u of the prob-
lem
1) The se
verges strongly
(1).
nce u
to th
2) If nnnn

 with
0n
lim 0
n
n
, then lim n
nyu

.
3) If lim n
nyu
implies that lim 0
n
n
 .
Proof. Let uE
be the unique solution of the prob-
lem (1). It is easy to see that the conclusion (1) follows
from the conclusion (2). Now we prove that (2) is true. It
follows from Lemma 1.6 that
n
 and
Copyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD 145

 
.,
,
Mgu
nn T
uuupuJTpufu
 
 
  
 

1,.Nuu




(32)
From (27), (31) and (32), we have


  



  

 




 

1uu
pu


  


.,
11 ,
.,
1,
,
1,
1,
,
nn
Mgu
nnnn T
Mgx
nnnnT nnnnnnn
nnnTnnn
nn
yu yuupuJTpufuNuu
yyxpxJTpxfxNxxel
fNxx
 
 
 




 





 




.,
1
nn
n n
Mgx
nn
yxpxJT px

  n
x
 


  


 



 
,
.,
.,
,,
.,
,
1
,
nn
nn
Tn
nn
n nnn
Mgu
Mgx
nTnnnn T
nnn nnn
Mgx
nTn n
JT
uxpx pu
JTpxfxNxxJTpuuNu
x upxpu
JTpxfx












 



 



  
.,Mgu pu

,nn
fuNuu el
 

,nn n
u el


1yu


 
yu
 

  

  

  

  

.,
,
.,
.,
,,
., .,
,,
, ,
, ,
,,
1
nn
n
nn
n
Mgx
nnT
Mgu
Mgx
nT T
Mgu Mgu
nTTnn n
nnn
Nx xJTpufuNuu
JTpufuNuuJTpufuNuu
J
Tpufu NuuJTpufu Nuuel
y



 


 
 
  
  











 




















*
,,
1
,,
1
nn n
n
nnnn
nn nnnnn
nnn nnn
n
nnn
n
nnnnnnnn
n
nn nnn
uxupxpu
Tpx TpufxfuNxxNuu
gx guGel
y uxupxpu
Tpx TpuNxxNuu
fx fuxuGe l
y uxupxpu


 



 




 

 
 

 






**
,,
,
nnn
n
nnnnnnnnn
Tpx TpuNxxNuu
xuxuG el
  

 

(33)
where

  
  
., .,
,,
, ,
n
Mgu Mgu
nT T
GJTpufuNuuJTpu fuN

 
 
 


 0.uu


(34)
It follows from (17) and (25), that
Copyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD
146

1,
q
qn
q
nn
upxpu q

 C xu


x
,
nn
f
xfuxu







1
.
q
qn
x u

  (35)
Substituting (35) into (33), we have
,, 2
qq qq qqqq
nnn
Tpx TpuNxxNuuqC


 


,
nn
n n
xu
l
11
nnn
nn nn
yu yu
Ge


(36)

 

where

1
nnn
nnnnn
xu yu
zuG r


 

(38)
and again




1
1and 1q
q
q
qqqqqqq
q
QP QqC
Pq C

1
,
2.
q
q


 

 
 

(37)
From (14), at 0<<1



1
11
nnn
nn nnnn
nnnnnn
nnnn
zu yu
yu Gs
yu Gs
yuG s





 
 
(39)
where
111
n

we know th
. Similarly, we
have
.
From (38) and (39), we get




11
,
nnnnnnnnnnn
nnnnnnnnn
nnnnnnnn
1
n
x
uyuyuGsGr
yuGsG r
yuGsG r

 
 

 

 
(40)
d (40), we get



since


11 1
n

. From (36) an





2
2
1
1 .
1
nnnnnnnnnnnnnnnn
nnnnnnnnnnnnn
yuG sGrG el
GsG rG el
 
11
11
n nnn
nn
y uyu
yu





 

(41)

 
 
11.
nnnnnn
atabtc
Let

From the assumption, we know that
n
b,

n
a,


2
1
.
nn
nnnnnn
nn n
bG
sGr
Ge
, , 1
1
nn nnnnn
ayu clt

 
 



(42)
We can write (42) as follows:
n
c
and
n
t satisfy the conditions of Lemma 1.5. This im-
nd so n
plies that 0
n
a a
y
u
.
he condition (3). Suppose that Next, we prove t
lim n
nyu

. It follows (28), (38) and (39) that n
zu
and n
x
u
. From (31), we have

   

 


  

.,
1,
.,
1 ,
1,
1,
n
nn
Mgx
nnnnnnnTnnnnnnn
Mgx
nnnnnnnnn Tnnnn
yyxpxJTpxfxNxxel
yuelyxpxJTpxfxN xx
 
 

  



  


As in the proof of (36), we have
.
n
u
(43)


 


.,
,
1,
.
nn
Mgx
nnnnnTnnnn
nn nn nn
yxpxJTpxfxN xxu
xuG
 


 



(44)
It follows from (43) and (44) that
1yu


11.
nn nnnnnnnnn
y ue lxuxuG
 

  (45)
Copyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD 147
That lim 0
n
n
is implies th
 . This completes the proof.
Theorem 3.2. Let f, p, N and T be the same as in Theo-
ccretive mappings
from E into the power set of E such that G
n
rem 3.1. Let

n
M and M be T-a
M.
M

Assume that there exists constant >0
such that
hold. Let
(14)
n
n
be a sequence in E and define
y
as
follows:



 

 

1,
,
,
, ,
1,
1,
nn nnnTnnn nnn
n
M
nnnT nnnnnn
n
M
nnnnnnTnnnnn
xpxTxNxxel
xy pzJTpzfzNzzr
zyypyJTpyfyNyys
 
 
 

 


 



 
(46)
(1 )
nn
nnn
yy
z


 
 
n
M
J
n
pxf
n


where
n
,
n
,
n
,
n
e,
n
r,
n
s
and
n
l
same in Algorithm 3.2, then
are
1) The sequence
n
u defineorithm 3.2, con-
verges s
d by Alg
trongly to unique solution u* of the problem (3),
2) If nnnn

 with

and
0
n
n
,
3) n
yu implies that lim 0
n
n
lim
n 0
n
 , then
*
*
n
yulim
n
lim
n
 .
ve or is to establish existence and
zed nonlinear implicit quasi
in Banach spaces. We de-
ped the T-rrator with T-accretive map-
by using thof Fang and Huang [11] and
Pe] and proved that the problem (1) is equivalent
to a fixed point problem. On the basis of fixed point
formulation we suggested perturbed iterative algorithm
with errors and by the theory of Hick and Harder [
proved the convergence and stability of iterativ
quences generated by algorithms.
A further attention is required for the study of varia-
tioe useful mathematical
ols to deal with the problems arising in mathematical
sciences.
REFERENCES
[1] S. Adly, “Perturbed Algorithms and Sensitivity Analysis
Inclusions,” Journal of
Mathematical Analysis and Applications, Vol. 201, No. 2,
1996, pp. 609-630. doi:10.1006/jmaa.1996.0277
4. Conclusions
f this pape
lts of generali
problem
The objecti
uniqueness resu
variational inclusion
velo
ing
esolvent ope
e concepts p
ng [10
20] we
e se-
nal inclusions that might provid
to
for a General Class of Variational
[2] Ya. Alber, “Metric and Generalized Projection Operators
in Banach Spaces: Properties and Applications,” In: A.
ations of Nonlinear
e Type, Marcel Dek-
ker, New York, 1996, pp. 15-50.
[3] H. Attouch, “Variational Convergence for Functions and
icable Mat
“Variational and Quasi-
ication to Free Boundary
, 1984.
“Iterative Algorithms for
l Inclusions Systems with (A,
ch Spaces,” Advances in
Nonlinear Variational Inequalities, Vol. 11, No. 1, 2006,
pp. 15-30.
[6] R. U. Verma, “Generalized System for Relaxed Coercive
Variational Inequalities and Projection Methods,” Journal
of Optimization Theory and Applications, Vol. 121, No. 1,
2004, pp. 203-210.
doi:10.1023/B:JOTA.0000026271.19947.05
Kartsatos, Ed., Theory and Applic
Operators of Monotone and Accretiv
Operators,” Applhematics Series, Pitman, Mas-
sachusetts, 1984.
[4] C. Baiocchi and A. Capelo,
Variational Inequalities Appl
oblems,” Wiley, New York
[5]
Nonlinear Fuzzy Variationa
η)-accretive Mappings in Bana
Pr
H. Y. Lan and R. U. Verma,
[7] R. U. Verma, “Approximation Solvability of a New Class
of Nonlinear Set-Valued Variational Inclusions Involving
(A, η)-Monotone Mappings,” Journal of Mathematical
Analysis and Applications, Vol. 337, No. 2, 2008, pp.
975. doi:10.1016/j.jmaa.2007.01.114
969-
[8] Z. B. Xu and G. F. Roach, “Charactertic Inequalities
Uniformly Convex and Uniformly Smoothanach Spaces,”
Apcatiool.
157,
.116/0019014
is
B
Journal of Mathematical Analysis andplins, V
No. 1, 1991, pp. 189-210.
doi:10022-247X(9 )4-O
[9]
tions,” 991, pp
1127-1
Z. B. Xu, “Inequalities in Banach Spaces with Applica-
Nonlinear Analysis, Vol. 16, No. 12, 1.
138. doi:10.1016/0362-546X(91)90200-K
W. Pengl Inclusions with T-
Accretive Operators in Banach Spaces,” Applied Mathe-
matics Letters, Vol. 19, No. 3, 2004, pp. 273-282.
doi:10.1016/j.aml.2005.04.009
[10] , “Set Valued Variationa
[11] Y. P. Fang and N. J. Huang, “H-accretive Opera
Resolvent Operator Technique for Solving Variational
Inclusions in Banach Spaces,” Applied Mathematics Let-
ters, Vol. 17, No. 6, 2004, pp. 647-653.
doi:10.1016/S0893-9659(04)90099-7
tors and
[12] X. P. Ding, “Generalized Implic
sions with Fuzzy Set Valued Ma
matics with Applications, Vol. 38, No. 1, 1999, pp. 71-79.
[13] Y. P. Fang, N. J. Huang, J. M. Kang and Y. J. Cho, “Gen-
eralized Nonlinear Implicit Quasivariational Inclusions,”
Journal Inequalities and Ap
261-275.
[1
it Quasivariational Inclu-
ppings,” Computer Mathe-
plications, Vol. 3, 2005, pp.
4] N. J. Huang, “Mann and Ishikawa Type Perturbed Algo-
rithms for Generalized Nonlinear Implicit Quasivaria-
Copyright © 2012 SciRes. APM
SALAHUDDIN, M. K. AHMAD
148
tional Inclu Journal of Mathematical Analysis and
Application 210, No. 1, 1997, pp. 88-101.
sions,”
, Vol.
[15] X. P. Ding, “Perturbed Proximal Point Algorithms for
Generalized Quasivariational Inclusions,” Journal of Mathe-
matical Analysis and Applications, Vol. 210, No. 1, 1997,
pp. 88-101. doi:10.1006/jmaa.1997.5370
[16] S. H. Shim, S. M. Kang, N. J. Huang and Y. J. Cho,
“Perturbed Iterative Algorithms with Errors for Com-
pletely Generalized Strongly Nonlinear Implicit Quasivaria-
tional Inclusions,” Journal of Inequalities and Applica-
tions, Vol. 5, No. 4, 2000, pp. 381-395.
[17] S. S. Chang, “On Chidume’s Open Questions and Ap-
proximate Solution of Multivalued Strong Mapping Equa-
ns inanacpac Journal of Math
1994-1
tio Bh Ses,”ematical Analy-
sis and Applications, Vol. 216, No. 1,7, pp. 911.
doi:10.1006/jmaa.1997.5661
[18] N. J. Huang, M. R. Bai, Y. J. Cho and S. M. Kang, “Gen-
eralized Nonlinear Mixed Quasivariational Inequalities,”
Cotics withs, Vol. 4
2-315.
/S0898-1221(00)00154-1
mputer Mathema Application0, No.
, 2000, pp. 205-2
doi:10.1016
[19] M. O., “Stable Itn Procedures for Strong
tors of the Ac
atical
ion ,
.100
Osilikeeratio
Pseudo-Contractions and Nonlinear Opera-
cretive Type,” Journal of Mathem Analysis and Ap-
plicats Vol. 204, No. 3, 1996, pp. 677-692.
doi:10 6/jmaa.1996.0461
[20] A. M. Harder and T. L. Hicks, “Stability Results for Fixed
Point Iteration Procedures,” Mathematics Journal, Vol.
33, 1988, pp. 693-706.
[21] N. J. Huang and Y. P. Fang, “A Stable Perturbed Proxi-
mal Point Algorithm for a New Class
Strongly Nonlinear Quasivariational Like
Press.
[22] W. R. Mann, “Mean Value Methods in Iteration,” Pro-
ceeding of American Mathematical Society, Vol. 4, 1953,
ding of American
of Generalized
Inclusions,” in
pp. 506-510.
[23] Ishikawa, “Fixed Points and Iteration of a Non-Expansive
Mapping in Banach Spaces,” Procee
Mathematical Society, Vol. 59, No. 1, 1976, pp. 65-71.
doi:10.1090/S0002-9939-1976-0412909-X
[24] N. J. Huang, “Mann and Ishikawa Type Perturbed Itera-
tive Algorithms for Generalized Nonlinear Implicit Qua-
sivariational Inclusions,” Computer Mathematics with Ap-
plications, Vol. 35, No. 10, 1998, pp. 1-7.
Copyright © 2012 SciRes. APM