Stable Perturbed Algorithms for a New Class of Generalized Nonlinear Implicit Quasi Variational Inclusions in Banach Spaces

(1)

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 nonempty closed bounded subset of E and the generalized

duality mapping E

q

J E  is defined by

 





* *

1 *

: ,

, q

q

* * _,

,

J u f E u

f u 

 

 

f f u

u E

 



where q > 1 is a constant. In particular, J2 is the usual

normalized duality mapping. It is known that, in general

 

2

 

2

q q

J u  u  J u for all u0 and Jq is simple single valued, if E*_{is strictly convex. The modulus of}

smoothness of E is the function E : 0



, 





0,



defined by

 

sup 1





1:

2

E t u v u v

  _     1, .u  v t_

 

A Banach space E is called uniformly smooth if

0

lim E 0

t t t



 

> 0

. E is called q-uniformly smooth, if there

 such that exists a constant

 

q, 1.

E t t q

  

> 0 q

c u v E,

Note that Jq is single valued, if E is uniformly smooth. Xu and Roach [8] and Xu [9] proved the following results.

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 

 

, .

q q q

q q

u v  u q v J u c v

:

T EE

: 2

Definition 1.1. [10] Let be a single-valued

operator and _{M E}_ E_{be a multivalued operator.}

M is said to be T-accretive if M is accretive and



TM E

 

E hold for all > 0.

Remark 1.1. 1) From [11] it is easily establish that if

T I (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 T E: E, N E E:  E

: 2

be two single valued mappings. Let _{M E E}_{ } E

t E

 

., : 2 be a set-valued mapping

such that for each fixed , _M _{t E}_ E

, , :

be a

T-accretive operator. For given f g p EE

u E

are mappings, consider the following problem of finding  such that

 





0f u N u u, M pu gu, , (1)

(2)

which is called the generalized nonlinear implicit quasi variational inclusions.

Special Cases:

1) If E is a Hilbert space, then problem (1) is equivalent to finding u E such that







,



,





 

Dom .,

0 ,

pu M gu

f u N u u



  M pu gu

 

,

(2)

which is called the generalized nonlinear implicit quasi variational inclusions, considered by Ding [12] and Fang

et al. [13].

2) If M u t 

 

u t E, 

u E

M u for all , then problem

(2) is equivalent to finding  such that

 



Dom

0 ,

pu M

f u N u



  u



M pu

 

,

: 2

(3)

where _{M E}_ E

 

,

is a maximal monotone mapping. The problem (3) was considered by Huang [14].

3) If M u t  

 

, E

u E

u t for each t , then problem (2) is equivalent to finding such that









 



 

,



,

Dom .,

, , ,

pu gu

f u N u u v pu pu

  

   gu  v gu

 

R

    t E

(4)

where :E E such that for each  ,

is a proper convex lower semi- continuous function with

 

., :t E R

  

 



 



 .,t



 .

0

f 

u E

 









.,

, ,

u M pu u



:

T EE

: 2

Range p Dom  (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 problem (1) is equivalent to finding such that

 

Dom

0 ,

pu M

N u u



  (6)

which is called the generalized strongly nonlinear implicit 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

M E E 

 

, .

be a T-accretive operator. Fang and Huang [11] defined the resolvent operator

 



 



1

(., )

, .,

M t T

J _ v  TM t  v  v E

:

T EE

: 2

(7)

By Theorem 2.2 in [11], we know that if is a strictly accretive operator and _{M E E}_{ } E

(., ) , :

M t T

is a

T-accretive operator, then the operator J _ EE

:

T E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 t E , _{M E E}: _{ }2E

(., ) , :

M t T

be a T-accretive operator then the operator J  EE is Lipschitz continuous with constant 1 , i.e.,



 

(., ) (., )

, ,

1 _, _.

M t M t

T T

J _ u J _ v u v u v E



    



_{a b}_



q _2q



_aq__bq









 





 







2 max , 2 max ,

2 .

q q

q q q

a b a b a b

a b

  

 

(8)

Lemma 1.3. Let a and b be two nonnegative real

numbers. Then

. (9) Proof.



Definition 1.2. Let Mn



0,1, 2,

n

and M be a maximal mono- tone mappings for  . The sequence



_Mn



G n

is said to be graph converges to M (write M M)

if for every

 

u v, Graph M

 

, there exists a sequences



_{u v},



__{Graph M}n



n

u u v_nv n 

n

n n such that and

as .

Lemma 1.4. [3] Let M and M be the maximal mono-

tone mappings for _n__{0,1, 2,}_{. Then}_Mn_G _M

 

,

n

M M

if and only if

J u J u u E

(10)





1

M

J_ I M 

  .

 and

for every  > 0, where

 

n ,

 

bn and



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





1 1 , for 0

n n n n n n

a   t a b t c n n (11)

 

0,1 n t 

0 n

n t





,

where



  lim_n_bn0

0 n

n c

, and











. Then a_n0 as n .



0



inf a n n:

  n  . Then 0.

Proof. Let  

> 0.

Sup- pose that  Then an > 0 for all n n 0. It

follows from (11), that

1

1 1

2 2

n n n n n n

n n n n n

a a t t b c

a b t t c



 

    

 

 _  _  

 

.

n n n0 n ,

1 0 n n

(12)

for all ₀ Since b as there exists

such that

1

, for all .

(3)





Combining (11) and (12), we have _, _{0, if} _;

q

Tu Tv J u v u v

1

a_ a 1

2

n  n tncn

1 n n

for all , which implies that

1

2 n n

n n

t a





 

1 1

. n n n

c

 

 

 



0 This is a contradiction. Therefore,  



an_j 0

j

and so there exists a subsequence

 

such that

as . It follows from (11) that



n a

j

n

a 

 

1

j  j b tn nj j cnj

.

j 

j k1

E

: 2

n n

a _ a

and so as A simple induction leads

to as for all and this means

that as n . This completes the proof.

1 0

j n

a _  0 

0  j k n

a

a  

 

:

T E n

Lemma 1.6.Let be a strictly accretive op-

erator and for a fixed tE , _{M E E}_{ } E



 

,

be a T-accretive operator in the first variable. If u is a solu- tion of the problem (1) if and only if

 

(., )

 



,

M gu

T

g u J  T pu f u N u u 

> 0

where  is a constant and









1

., .

M gu 

u E



 



 

, , ,

0 ,

, .

pu gu u T pu

u u

N u u

 



 

 

 _

, :

T g EE (., )

,

M gu

T

J _  T

Proof. is a solution of (1)

 



 



 





 









 



(., ) ,

0 ,

0

,

.,

M gu

T

f u N u u M pu gu

f u N u u M

T pu f u N u

M pu gu

T pu f u N

T M gu pu

pu J _ T pu f 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

Tu Tv J u v  0, u v E, 

or, equivalently





2

, 0

Tu Tv J u v   , u v E,  ;

2) Strictly accretive if T is accretive and

   

> 0

r





3) Strongly accretive if there exists a constant such that

, q q, ,

Tu Tv J u v  r u v u v E





or, equivalently

2 2

, , , ;

Tu Tv J u v  r u v u v E

4) Lipschitz continuous if there exists a constant s > 0 such that

, , ;

Tu Tv s u v u v E

> 0

5) Strongly accretive with respect to g if there exists a constant  such that





, q q, ,

Tu Tv J gu gv   gu gv u v E





or, equivalently

2 2

, , , .

Tu Tv J gu gv   gu gv u v E :

N E E E :

Definition 2.2. Let and g EE

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, ,

N u N v J u v  u v u v E

 

  



or, equivalently

2 2

,. ,. , , , ;

N u N v J u v  u v u v E

> 0

2) N is said to relaxed accretive with respect to g if there exists a constant  such that

 

,.

  

,. ,



,

, ;

q q

N u N v J gu gv gu gv

u v E



    

 

> 0

3) N is Lipschitz continuous in first argument if there exists a constant  such that

 

,.

 

,. , , .

N u N v  u v u v E

: E

Theorem 2.1.Let E be a q-uniformly smooth Banach

space and T E be a strongly accretive and Lipschitz continuous with positive constants  and 

respectively. Let be the strongly accretive and Lipschitz continuous with positive constants p E: E



 and

 respectively. Let be Lipschitz con- tinuous with positive constants

, :

f g EE

 and  respectively. Let N E E:  E

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

:

p EE

  

 

 

p u₁  T p u T pu for all u E . Assume that N is Lipschitz continuous with respect to first and second argument with constants  > 0 and

(4)

> 0

 respectively. Let _{M E E}: _{ }2E_{be a T-accre-}

tive operator with second argument. If there exist con- stants > 0 and > 0 such that for all u v w, , ,E

 

(., ) (.,

, ,

M gu M

T T

)

gv

J  w J  w  gu gv (13) and

1 1,

Q P



 





(14)

where

1

1 q q__ _1













1

2 1.

q

q q q q q q q q

q

P   q     C   



    

 

* u E. Then the problem (1) has a unique solution

Proof. By Lemma 1.6, it is enough to show that the mapping _F_:EE_{has a unique fixed point}_u*__E

where F is defined as follows.

 

(., )

 

, ,

M gu

T

F u  u pu J _ T pu f u N u u

q

C

 

Q q 

and

 

 

u E

(15)  . From (13) and (15), we have

for all

 

_{ }



 

_{ }







 

_{ }

 

_{ }

 

_{ }

_

_

_{ }

 

_{ }





., .,

, ,

., .,

, ,

., .,

, ,

M gu M gv

T T

M gu M gu

T T

M gu M gv

T T

F u F v u pu J T pu f u N u u v pv J T pv f v N v v

u v pu pv J T pu f u N u u J T pv f v N v v

J T v N v v J T pv f v N v v

u v pu pv

 

   

    _   _   _   _

     _   _ _   _

   

   

   

    1

pv f

  (16)

 





 



 



 

1

, ,

1 _, _{( , )} _.

T pu T pv N u u N v v f u f v gu gv

u v pu pv T pu T pv N u u N v v f u f v u v

    

 



 

 

      

           

Now since p is strongly accretive and Lipschitz continuous, we have

















,

1

1 .

q q q

q q

q q q q

q

q q

q

q q

q

u v pu pv u v q pu pv J u v C pu pv

u v q u v C u v

q C u v

u v pu pv q C u v

 

         

     

   

       

(17)

By the strong accretivity of p with constant  , we have









1

,

1

q q

pu pv u v    pu pv J u v   pu pv J u v   u v

u v pu pv



   

that is _{N u u}

_{ }

_, _{N v u}

_{ }

_, _ _{u v} _. (20)

 

,

 

,

N v u N v v pu pv  u v . (18)

.

u v



   (21)

Since f is Lipschitz continuous, we get Similarly, by the strong accretivity of T with constant

 we have

 

f u f v  u v . (22)

   

T pu T pv  pu pv

By

. (19)

Since N E E:  E is a relaxed accretive with re- nd second argumen

spect 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

(5)

 



 



 

, , , q

q

N J T pu T pv

T T pv





  

and similarly

q q

q

q q

u u N v u

pu

pu pv

u v



 



  

(23)





 



 

,



, ,

. q

q

q q

N v u N v v J T pu T pv

u v

 

 

  

,p

 



 



(24)

From (23) and (24), Lipschitz continuity of T , Lemma 1.1 and Lemma 1.3, we have

 



 



 





 









, ,

, , , , ,

2 q

q _q q

q q

q q q q q q q q q q q q

q

q q q q q q q q

T pu T pv N u u N v v

T pu T pv q N u u N v v J T pu T pv C N u u N v v

T pu T pv u v q u v C u v

q C u v



 

      

         

  

      

       

 

_     _ 

(25)

Now from (16), (17), (22) and (25), we have

 

2 q .

q







1 1









1

1 2

1

,

q q

q q q q q q q q q

q q

F u F v q C q C u v

Q P u v u v

             

 

 

 

 _          _ 

 

 

_  _   

 

(26)

where



1 q



1q

q

Q qC   and













1

2 q

q q q q q q q q

q

P   q      C    

 

n n 1



, n



and  Q 1P x x



  .

From (14), we know that 0 < < 1 . Therefore, there

u E such that

 

exists a unique  F u u. This

3.

tru urbed iterative

solving the problem (1) and tive se- ith er- completes the proof.

Perturbed Algorithm

In this section, we cons ct 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 ,

n n

x  f T x define an iterative procedure which yields a sequence of point

 

xn in E. Suppose that

x E

 :Txx



  and

 

xn converges to a fixed

point x of T. Let

 

yn E and let





1 .

n yn f yn

  _  T, 1) If lim_n_n 0 that _n_

iterative procedure

implies limyn x



 , then the

defined by   f T x is

said to be T-stable or stable with respect to T. 2) If n



0

n

 



implies that lim_n yn x 

  , then the

e

iterativ procedure

 

x_n is lmost T-stable. stability results of iterative algorithms have been

several a 19]. 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 t lity is both

x which is T- stable is almost T-stable and an iterative procedure

 

xn which is almost T-stable need not be T-stable [21].

Algorithm 3.1.Let f p g T E, , , : E and :

N E E E be the five single valued mappings. Let

 

_Mn _{of M be the set-valued mapping from} _{E E}_ _into

the power set of E such that for each t E , _Mn

 

.,_t

and M

 

.,t are T-accretive mappings and

 

., G

 

.,

n

M t M t . For any given u₀E, the per- turbed iterative sequence

 

un with errors is defined as

follows :

(6)





.,

1 1

n n

M gv

n n n n n

u    u  v pv 









,

n T n n

n n n n n n T

J T pv f v

v u w pw T

T











 









 









 





,

n n n n n

n n n n n n

n n n n n

N v v e l

pw f w N w w r

pu f u N u u s

  

  

  

  _ 

  



  

)









n

 

 



(27



₁ 



_JMn.,gwn

   





., 

,

1 Mn gun

n n n n n n T

w  u  u pu J _

 



   _  

for n0,1, 2,, where

 

n ,

 

n and

 

n are three sequences in [0,1],

 

en ,

 

rn ,



sn



and

 

ln are four sequences in E satisfying the following condi-

: tions

0 0

,

n n

n n n

n n



 

 

 

 



lim lim lim 0

. n

n n

s e r

l



  

  (28)

From Algorithm 3.1, we obtain the following algorithm for the problem (3).

Algorithm 3.2.Let f p T E, , : E and :

N E E E be four single valued mappings. Let

 

_Mn _{and M be T-accretive mappings from E into the}

power of E such that _Mn_G _M _{. For any given}

0

u E, define the perturbed iterative sequences

 

u_n with errors as follows:











 















1 1 , ,

, n

n M

n n n n n n T n n n n n n n

n n n

n

u u v pv J T pv f v N v v e l

w

pu u s



    

 



    _     _ 

 

 

 

0,1,

 _n











,

1

1 n

M

n n n T

M

n n n n n n T

v u w p J T

w u u pu J T



 

  _ 



   _   _



 



,





n n n n n n

pw f w N w w _ r (29)



f u

 

n N u



n n n n

2,, then

 

n  ,

 

_n ,

   

_n , e_n ,



r



,

 

n

n s and

 

l_n are same as Algorithm 3.1

Rem n

nd M rithms

.

ark 3.2.For a suitable choice of T f g N M, , , , , Algorithm 3.1 reduces to several known Algo-

[22-24] as special cases.

Theorem 3.1. Let , , ,

a

f p g T and N be the same as in rem 2.1. Suppose that

 

_Mn _{and M are set-valued}

ings from E E into the power set of E such that t E , n

 

.,

exists constants > 0 and > 0 s

, ,

u v z

uch that for each E

 and n0

 

(., )

M u

Theo mapp

for each M t and M

 

.,t are T-accretive ings and n

 

., G

 

.,

mapp M t M t . Assume that there

 

(., )

, ,

(., ) (., )

, ,

n

M v

T T

n

M u M v

T T

J z J z u v

z J z u v

 



   

J    (30)

and the condition (13) holds. Let

 

y_n E and define a sequence

 

_n

be a sequence in

 of real numbers as follows:

 

_

_

_

_











 











1

n n n n n

x  y  z pz T





   _ 

 

 

_

_

_

_{ }

_

_

_

 

_

_

_{ }

_

_

_

.,

1 ,

., ,

.,

1 ,

,

, n

n

n n

n

M gx

n n T

M gz

n T n n n n n n

M gy

n n n n n n n

x px J T N e

J pz f z N z z r

py f y N y y s



  

  

 _ _ _

 

 



   _

 _ _ _



(31)



,

1 n

n n n n n T

z  y  _y py J  T

where

 

_n

n n n

  y_   y_n  px_n f x_n x x_n _n _n_n l_n

 ,

 

_n ,

 

_n ,

 

e_n ,

 

r_n ,

 

s_n and

 

ln are s defined orithm 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 _n_n _n _n with

0

n







lim n 0

n  , then nlimyn u 

  .

3) If _nlimyn u

_{implies that}_lim ₀ n

n 



   .

Proof. Let uE be the unique solution of the problem (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

(7)





 

.,

 

,

M gu

n n T

u _{ }₁  u_ u_pu_J _ 



T pu _f u ___{N u u}



_, 





_.

 

  (32)

From (27), (31) and (32), we have





 



 











 

_

_

_

_{ }

_

_

_









 

_

_

_

_{ }

_

_

_



1 



u  u pu

    

 



 

.,

1 1 ,

.,

1 ,

,

1 ,

, n

n

M gu

n n n n T

M gx

n n n n T n n n n n n n

n n n T n n n

n n

y u y u u pu J T pu f u N u u

y y x px J T px f x N x x e l

f N x x



   

    

 



       

 



  

  _   _     __

 

 

 

    _     _ 

 

 

 

 ., 

1 n n

n n

M gx

n yn x px J T px

  

     x_n 

 











 

_

_

_

_{ }

_

_

_

 

   













 

_

_

_{ }

,

., .,

, ,

., ,

1

, n

n

n n

T n

n n n n n n

M gu

M gx

n T n n n n T

n n n n n n

M gx

n T n n

J T

u x px pu

J T px f x N x x J T pu u N u

x u px pu

J T px f x



 



  

    

 



 

  

 



 

 

 

     

 

    _   _

  

  





 





.,

M gu

pu



 

 f u N u u,   en ln

 _  _ _

,u n en ln

  _ _

1 y u

   

    

y  u 

 





 





 

_{ }

_

_

 

_{ }

_

_





., ,

., .,

, ,

., .,

, ,

1

n n

n

M gx

n n T

M gu

M gx

n T T

M gu M gu

n T T n n n

n n n

N x x J T pu f u N u u

J T pu f u N u u J T pu f u N u u

J T pu f u N u u J T pu f u N u u e l

y



 

  

    

     

 



 

   

       

 

    

   

   

 _   _ _   _

   

 _   _ _   _  

   









 



 























 









 









*

, ,

1

, ,

1

n n n

n

n n n n

n n n n n n n

n n n n n n

n

n n n

n

n n n n n n n n

n

n n n n n

u x u px pu

T px T pu f x f u N x x N u u

gx gu G e l

y u x u px pu

T px T pu N x x N u u

f x f u x u G e l

y u x u px pu



 _ _



   

  

 _



 _ _{ } _ _





 

  

   



  



  

   

     

    

       

   

      

       





 











 



* *

, ,

,

n n n

n

n n n n n n n n n

T px T pu N x x N u u

x u x u G e l

 

  _{ } _ _



  

  

         

(33) where

 

.,

_{ }

_

_

 

.,

_{ }

_

_

, , , ,

n

M gu M gu

n T T

G  J _  _T pu f u N u u  _J _  _T pu f u N u u  _ 0. (34) It follows from (17) and (25), that

(8)





1 q ,

q n

q

n u pxn pu q

 

     C  x _u

 

x f x_n f u  x_n_u ,





 

















1

. q

q xn u

  _ _  ₍₃₅₎

Substituting (35) into (33), we have

, , q q q q 2q q q q

n n n

T px _T pu _ N x x _N u u  _   _q    _ _ C











 



,

n n

x u

l



1 1

n n n

n n n n

y u y u

G e

 

 

 

   (36)

 

     

  

where





1

n n n

n n n n n

x u y u

z u G r



  

 



   

    (38)

and again





















1

and 1 q q

q

q q q q q q q

q

Q P Q q C

P q C

 



1

,

2 q q .

 

        

     

     

(37) From (14), at 0 < < 1













1

1 1

n n n

n n n n n n

n n n n

z u y u

y u G s



   



 



   

   

     

   

(39)

where



1 _n



1 



1

we know th  . Similarly, we

have From (38) and (39), we get    .





































1 1

,

n n n n n n n n n n n

n n n n n n n n n

n n n n n n n n

1 n

x u y u y u G s G r

y u G s G r

      

     

   

  



       

     

   

(40)

d (40), we get

 

 

since



1n



1



1. From (36) an















 





















 



2

1

1 .

1

n n n n n n n n n n n n n n n n

n n n n n n n n n n n n n

y u G s G r G e l

G s G r G e l

         

1 1

n n n n

n n

y u y u

y u

 

         





          

 

  _        _  

(41)

 





    

    





1 1 .

n n n n n n

a  t a b t c

Let    

From the assumption, we know that

 

bn ,

 





an ,













2

1

.

n n n n n n n n

n n n

b G s G r

G e

, , 1

1

n n n n n n n

a  y u c  l  t  

      

 _   



   

(42)

We can write (42) as follows:

n c

and

 

tn satisfy the conditions of Lemma 1.5. This im- nd so n

plies that an0 a y u.

he condition (3). Suppose that Next, we prove t

lim n

n y u



  . It follows (28), (38) and (39) that zn u





and x_n u. From (31), we have

 









 













 

_

_

_{ }

_

_

.,

1 ,

.,

1 ,

n

n n

M gx

n n n n n n n T n n n n n n n

M gx

n n n n n n n n n T n n n n

y y x px J T px f x N x x e l

y u e l y x px J T px f x N x x



     

    



 

 

    __     __ 

 

       _     _

 

As in the proof of (36), we have

. n

u



(43)





 

_

_

_{ }

_

_





., ,

1 ,

. n

n

M gx

n n n n n T n n n n

n n n n n n

y x px J T px f x N x x u

x u G



   

  



 

  _     _

 

  

(44)

It follows from (43) and (44) that

1  y u 

  





1 1 .

n yn u n en ln n xn u n xn u nGn

        



(9)

This implies that lim_n_n0. 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 n G

rem 3.1. Let

 

_Mn _{and M be T-a}

M

 .

M 

Assume that there exists constant > 0 such that hold. Let

(14)

 

yn be a sequence in E and define

 

n as

follows:



























 

















 







1 ,

,

, ,

1 ,

n n n n n T n n n n n n

n M

n n n T n n n n n n

n M

n n n n n n T n n n n n

x px T x N x x e l

x y pz J T pz f z N z z r

z y y py J T py f y N y y s



  

  

    

  

  _   _ 

 

  _    _

 

 

       

(46) (1 n) n

n n n

y y

z

 

 

 

 

n

M J

 px_n f

n

 

 

where

 

_n ,

 

_n ,

 

_n ,

 

e_n ,

 

r_n ,

 

s_n and

 

l_n

same in Algorithm 3.2, then

are

1) The sequence

 

u_n define orithm 3.2, converges s

d by Alg

trongly to unique solution u*_{of the problem (3),}

2) If nn n n with 



 



and

0 n

n

,

3) ynu implies that lim_n n 0 lim

n n 0, then

*

n

y u

lim n

lim

n   .

ve o r is to establish existence and zed nonlinear implicit quasi in Banach spaces. We de- ped the T-r rator with T-accretive map-

by using th of 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-

tio e useful mathematical

ols to deal with the problems arising in mathematical sciences.

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