System of Split General Variational Inequality Problems in Semi inner Product Spaces

2019 International Conference on Applied Mathematics, Modeling, Simulation and Optimization (AMMSO 2019) ISBN: 978-1-60595-631-2

System of Split General Variational Inequality Problems in Semi-inner

Product Spaces

Ya-li ZHAO

and Xin LIU

College of Mathematics and Physics, Bohai University, Jinzhou, Liaoning 121013, China *Corresponding author

Keywords: System of split general variational inequality problems, Sunny retraction mapping, Semi-inner product spaces, Generalizedadjoint operator.

Abstract. In this paper, we introduce a new system of split general variational inequality problems which is a natural extension of system of split variational inequality problems and split variational inequality problems in semi-inner product spaces. We employ the retraction technique to propose an iterative algorithm for computing the approximate solution of the system of split general variational inequality problems. Moreover, the convergence analysis of the iterative algorithm is also discussed. Several special case which can be obtained from the main results are also presented.

Preliminaries

We recall the following concepts and results, which are needed to define the problem and to prove the main result.

Definition1[1] LetV be a vector space over the field R of real numbers. A functional R

V

V 

 ,]:

[ is called a semi-inner product if it satisfies the following conditions:

. , ], , ][ , [ ] , [ ) 4 ( ; 0 ,

0 ] , )[ 3 (

]; , [ ] , )[ 2 ( ; , , ], , [ ] , [ ] , [ ) 1 (

2

V v u v v u u v u u

for u

u

v u v u V

w v u w v w u w v u

  

 

 

 



  

The pair (V,[,]) is called a semi-inner product space. As it is observed in [1] that

[ , ], ,

u  u u  u V is a norm onV . Hence every semi-inner product space is a normed linear space. On the other hand, in a normed linear space, one can generate semi-inner product in infinitely many different ways. Further, it is noted that a Hilbert spaceHcan be made into a semi-inner product space, which a semi-inner product is an inner product if and only if the norm it induced verifies the parallelogram law. LetY be a semi-inner product space and letT:VYbe an arbitrary operator.

Definition 2[2] The generalized adjoint operatorTof an operatorTis defined as follows: The domainD(T)ofTconsists of thoseyY for which there existszVsuch that [Tx,y]_Y [x,z]_V for eachxVandzTy.

Remark 3 As it is observed in [3] that ifV andY are Hilbert spaces then the generalized adjoint operator is the usual adjoint operator. In general,Tis not linear even forTis a bounded linear operator.

In 1967, Giles [5] proved that if the underlying semi-inner produce spaceV is a uniformly Convex smooth Banach space then it is possible to define a semi-inner product uniquely which has the following properties:

(i) [u,v]0for someu,vVif and only ifvis orthogonal to .u

(ii)The semi-inner product is continuous, i.e., for each u,vV, we have .

0 ]

, [ ] ,

[v uv  u v as 

(iii)[ ,u v][ , ],u v    R, u v V,  ,i.e., the semi-inner product is with the homogeneity property.

there is a unique vectorvVsuch thatf(u)[u,v],uV.

Now, we give the properties of the generalized adjoint operator.

Proposition 4[2] LetX andYbe 2-uniformly convex smooth Banach space and let T:X Ybe a bounded linear operator. Then

(i)D(T)Y; _(ii)Tis bounded, and to holds that Ty  T y,yY.

Definition 5 [7] LetDbe a subset ofCandQ_Cbe a mapping ofCintoD. ThenQ_Cis said to be sunny ifQ_C(Qut(uQ_Cu))Q_Cu,wheneverQ_Cut(uQ_Cu)CforuCandt0.

Definition 6 [6] A subsetDofCis called a sunny nonexpansive retract ofCif there exists a sunny nonexpansive retraction fromCintoD.

Proposition 7 [6] LetEbe a smooth Banach space and letCbe a nonempty subset ofE. Let

C E

Q_C:  be a retraction. Then the following are equivalent:

(i)Q_Cis sunny and nonexpansive;

(ii) QCuQCv 2 uv,J(QCuQCv),u,vE; (iii)uQCu,J(vQCu)0,uE,vC.

Lemma 8 [4] Let 1be a real number andEbe a smooth Banach space. Then the following statements are equivalent:

(i)Eis 2-uniformly smooth; (ii)There is a consistc0such that for everyu,vE,the following inequality holds: uv 2  u 22v,J(u)cv 2.

Remark 9 (a) It follows from [1,5,7] that every normed linear space is a semi-inner product space. In fact, by Hahn Banach theorem, for eachvEthere exists at least are functional fuE* such thatu,fu u 2.Given any such mappingf fromEinto

*

E , we can verify that[v,u]v,fu defines a semi-inner product. Hence, we write the inequality given in Lemma8 as

. , , ]

, [

2 2

2 2

E v u v c u v x

v

u      The constantCis chosen with best possible minimum value. We callCas the constant of smoothness ofE.

(b)The inequalities given in proposition 7(ii) and (iii) can be written as

. , ,

] ,

)[ ( ; , ], ,

[ )

(ii QCuQCv 2  uv QCuQCv u vE iii uQCu vQCu  uE vC

In the following, otherwise special statement for eachi{1,2}, suppose thatQ_C:E_i C_iis a sunny and nonexpansive retraction, whereEis a smooth Banach space andC_ibe a nonempty subset ofEi. LetE1andE2be 2-uniformly convex, smooth Banach spaces and for eachi{1,2}, letCi Ei be a nonempty closed and convex set and let _{: 2} 1*

1 1

E

E

J  and _{: 2} *2

2 2

E

E

J  be the normalized duality mappings. LetF_i,f_i,G_i,g_i:C_i E_ibe nonlinear mappings, and letA:E₁E₂be a bounded linear operator. We introduced the following system of split general variational inequality problems (in short,SS_PGVIP): Find(u,v)C₁C₁such that

, ,

0 ) (

), ( ;

, 0 ) (

),

( _{1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1}

1 1 1

1v u f v J w u w C Gu v g u J w v w C

F             

 

and such that(u₂,v₂)withu₂Au₁C₂,v₂ Av₁C₂solves

, ,

0 ) (

), ( ;

, 0 ) (

),

( _{2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2}

2 2 2

2v u f v J w u w C G u v g u J w v w C

F             

 

for any, 0.AboveSS_PGVIPis equivalent to find(u₁,v₁)C₁C₁such that

1 1 1 1 1 1 1 1 1

[F v  u f v( ),w u ]  0, w C; (1)

and such that(u₂,v₂)withu₂Au₁C₂,v₂ Av₁C₂solves

2 2 2 2 2 2 2 2 2

[F v  u f v( ),w u ]  0, w C ;

(3)

2 2 2 2 2 2 2 2 2

[G u  v g u( ),w v ]  0, w C; (4) for any, 0.If for eachi{1,2},f_i,g_i I_i,whereI_iis identity mapping onC_i,thenSS_PGVIP

(1)-(4) reduced to the following system of SS_PGVIP _{introduced and studied by Kazmi and Furkan}

[9]: Find(u₁,v₁)C₁C₂such that

1 1 1 1 1 1 1 1

[F v  u v w, u ]  0, w C; (5)

1 1 1 1 1 1 1 1

[G u  v u w, v]  0, w C; (6) and such that(u₂,v₂)withu₂Au₁C₂,v₂ Av₁C₂solves

2 2 2 2 2 2 2 2

[F v  u v w, u ]  0, w C; (7)

2 2 2 2 2 2 2 2

[G u  v u w, v ]  0, w C ; (8) for any, 0.It is worth mentioning that Kazmi and Furkan in [8] firstly introduced and studied

VIP

SS_P in the setting Banach spaces. They used the retraction technique to propose and analyze an iterative algorithm for computing the approximate solution of SS_PVIP(5-8) in 2-uniformly convex smooth Banach spaces. Inspired and motivated by the recent research work [8,10-12] in this paper, we introduced and study a new system of split general variational inequality problems(SS_PVIP

(1)-(4)) employing the retraction technique, we propose an iterative algorithm with errors for computing the approximate solution ofSS_PVIP(1)-(4) in 2-uniformly convex smooth Banach spaces. Further, convergence analysis of the iterative algorithm is discussed. Several special cases which can be obtained from the main result are also presented. The problems and the results obtained here are new and different from the existing problems and results in the literature.

Iterative Algorithms

By making use of proposition 7, we noted thatSS_PVIP(1)-(4) can be formulated as follows: Find

1 1 1 1, )

(u v C C with(u₂,v₂)(Au₁,Au₂)C₂C₂such that

(9)

1

1 C( 1 1 1),

v Q v G u (10)

2

2 C ( 2 2 2),

u Q v F v (11)

2

2 C ( 2 2 2),

v Q u G u (12) for any, 0.Based on above arguments, we propose the following iterative algorithm for approximating a solution toSS_PVIP(1) - (4). Let{n}(0,1)be a sequence such that .

1

 



 n

n



Algorithm 10 Given( , ) 1 1,

0 1 0

1 v C C

u   compute the iterative sequence{( 1, 1)} n n

v

u defined by the iteerative schemes:

1

1 C( 1 1 1),

1

1 ( ( )1 1 1 1) 1,

n n n n

C

p Q f v F v  (13)

1

1 ( (1 1) 1 1) 2,

n n n n

C

q Q g u G u  (14)

2

2 ( 2( )2 2 2) 1,

n n n n

C

p Q f v F v  (15)

2

2 ( 2( 2) 2 2) 2,

n n n n

C

q Q g u G u  (16)

1

1 (1 ) 1 ( 1 ( 2 1)),

n n n n n n n

u    u  p A p Ap (17)

1

1 (1 ) 1 ( 1 ( 2 1)),

n n n n n n n

v    v  q A q Aq (18) For alln0,1,2,and,,0,whereAis the generalized adjoint operator ofA,andu₂n Au₁n,

n n

Av

v₂  ₁ for alln,and for eachi {1,2},{ }n ₀ E₁ n

i 

 



 and{ }n ₀ E₂ n

i 

 

 are four sequence to take into account of a possible inexact computation satisfying the following conditions:

. 0 lim

lim  

  



n i n n i

n  

Algorithm 11 Given(u₁0,v₁0)C₁C₂,compute the iterative sequence{(u₁n,v₁n)}defined by the iterative schemes:

1

1 ( 1 1 1),

n n n

C

q Q u G u

2

2 ( 2 2 2),

n n n

C

p Q v F v

2

2 ( 2 2 2),

n n n

C

q Q u G u

1

1 (1 ) 1 ( 1 ( 2 1)),

n n n n n n n

u    u  p A p Ap 1

1 (1 ) 1 ( 1 ( 2 1)),

n n n n n n n

v    v  q A q Aq for alln0,1,2,and,,0,whereAis the generalized adjoint operator ofA,and

n n

Au u2  1,

n n

Av

v2  1 for all .n

Remark 12 Algorithm 11 was proposed by Kamiz and Furkan in [9] to compute the approximate

solution of SS_PVIP(9) - (12), which is special case of Algorithm 10.

Main Results

In order to obtain our main results, we used the following concepts and lemma.

Definition 13 LetF,g:E₁E₁be two mappings.Fis said to be

(1)strongly monotone with respect tog, if there exists a constant0such that

. ,

, ]

,

[ 1 2 1

2 2 1 2

1 2

1 Fu gu gu u u u u E

Fu      

(2)Lipschitz continuous, if there exists a constant 0such that

. ,

, _{1 2 1}

2 1 2

1 Fu u u u u E

Fu     

Lemma 14[9] Let{a_n}_n0,{b_n}_n_0and{c_n}_n0be nonnegative sequences satisfying

, 0 , )

1 (

1     

 a b c n

a_n _{n n} _{n n n}

where{ } [0,1], , ,lim 0.

0 0

0    _ 



 

 







n n

n n

n n

n  c b

 Thenlim 0.

  n

n a

Now, we prove that the sequence of approximate solution of SS_PGVIP(1)-(4) generated Algorithm 10 converges strongly to the solution of SSPGVIP(1)-(4).

1

1 ( 1 1 1),

n n n

C

Theorem 15 For eachi{1,2},letC_ibe a nonempty closed and convex subset of 2-uniformly convex smooth Banach spaceE_iwith constant of smoothnessC_i.LetF₁:C₁E₁be₁strongly monotone with respect to fi:C1 E1and1Lipschitz continuous, letG1:C1E1be2strongly

monotone with respect tog₁:C₁E₁and₂Lipschitz continuous, letF₂:C₂ E₂be₁strongly monotone with respect tog₂and₂Lipschitz continuous, and let f₁,g₁:C₁E₁be₁,₂Lipschitz continuous, respectively, and f₂,g₂:C₂E₂be₁,₂Lipschitz continuous, respectively. Let

2 1

:E E

A  be bounded linear operator. Suppose that(u₁,v₁)C₁C₁be a solution to SS_PGVIP

(1)-(4), then the sequence{( 1, 1)} n n

v

u generated by Algorithm 10 converges strongly to(u₁,v₁) provided that the constant0satisfies the following conditions:

2 2 2 2 2 2 2 2

1 1 1 1

2 _{1 2} 2

1 2

1 1

2 2 2

1

2 2 2

2 2

( ) ( )

max{ } min{ };

1

( ), , , ;

1

2 , 0, 0.

i i i i i i i i

i

i i

i

i i i i i i i

i i i i

C a C a

C C

m

C a a a m A A

m

C



       



 



    

      

   

 



     

 



    



    

(19)

Proof. Given that(u1,v1)is a solution of SSPGVIP(1)-(4), that is,u1,v1satisfy the relations

(9)-(13). SinceF₁:C₁E₁is₁strongly monotone with respect to f₁and₁Lipschitz continuous and f₁:C₁E₁is₁Lipschitz continuous, from Algorithm 10 (13) and (9), we have

1 1

1 1 1 1 1 1 1 1 1 1 1

1

2 ₂ 2

2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1

( ( ) ) ( ( ) )

( ( ) ( ) 2 [ , ( ) ( )] )

,

n n n n

C C

n n n n n

n n

P u Q f v F v Q f v F v

f v f v F v F v f v f v c Fy Fy

v v

  

  

 

     

       

  

(20)

where₁  ₁22₁C₁2₁2.Next, sinceG:C₁E₁is₂strongly monotone with respect tog₁

and₂Lipschitz continuous andg₁:C₁E₁is₂Lipschitz continuous, from Algorithm 10 (18) and (14), we get

1 1 2 1 1 2 ,

n n n

g v  u u   (21) where₂  ₂22₂C₁2₂2.Again, sinceF₂:C₂ E₂is₁strongly monotone with respect to

2 2

2:C E

f  and₁Lipschitz continuous and f₂:C₂ E₂is Lipschitz, from Algorithm 10 (15) and (11), we have

2 2 3 2 2 1 ,

n n n

P u  v v   (22) where₃ ₁22₁C₂2₁2.Since G₂:C₂E₂ is₂ strongly monotone with respect to

2 2

2:C E

g  and₂Lipschitz continuous and g₂:C₂ E₂ is₂ Lipschitz continuous, from Algorithm 10 (16) and (12), we have

2 2 4 2 2 2 ,

n n n

g v  u u   (23)

where 2 22.

2 2 2 2

2

4    

   C Now, using the factAis bounded, we have

1

1 1 1 1 1 1 2 1

1 1 1 3 1 1 1 1 1

(1 ) ( )

(1 ) ( ( )) (1 ).

n n n n n n

n

n n n n n n

n

u u u u p u A p Ap

u u A A v v A A A

  

          

 

  

       

Similarly, we obtain

1

1 1 1 1 2 4 2 1 1

2 2

(1 ) ( ( ))

((1 ) )

n n n n

n

n n

n

v v v v A A u u

A A A

     

    

 

 

       

   (25)

Now, we define the norm

*

 onE₁E₂by ( , ) , ( , ) _{1 2}.

* u v u v E E

v

u      We can easily know

that( , )

* 2

1E 

E is a Banach space. Combining (24) and (25), we have

), }( , max{ ) )( 1 ( ) , ( ) , ( 1 1 1 1 2 1 1 1 1 1 * 1 1 1 1 1 1 v v u u k k v v u u v u v u n n n n n n n n              

wheremax{ ,k k_{1 2}},k₁ ₁ m( ₃ ₁),k₂₂m( ₄ ₂),

3 4 3 4

max{ ,k k},k 1 m k, m , m A A.

A 



     

By condition (19) on, we know that(0,1),letting ( , ) ( , ) , (( , ) , ).

* 2 2 * 1 1 * 1 1 1 1 n n n n n n n

n u v u v b

a        

Since



   1 n n

 and(0,1),thenn(1)(0,1)and (1 ) ,lim lim ( 1 2 1 2 ) 0,

1               



n n n n

n  b    

 , 1  



c it follows from Lemma 14 and the above inequality that, the whole conditions of Lemma 14 holds. So ( , ) ( , ) 0( ),

* 1 1 1 1 1

1   

  n v u v

un n that is,{( , 1)}

1 1 1   n n v

u converges strongly to(u1,v1)as ,

 

n implying that u₁n u₁

andv₁n v₁

,as n.Further, it follows from (20) and (21),

respectively, thatp₁n u₁

andq₁n v₁

asn.Hence, it follows from (22) and (23), respectively,

thatp₂n u₂ Au₁

andq₂n v₂ Av₁

asn.This completes the proof.

Corollary 16 For eachi{1,2},letC_i,E_i,F_i,G_i,Abe same as in Theorem 15. Suppose that

2 1 1 1, )

(u v C C be a solution to SS_PGVIP (5-8), then the sequence{( ₁n, ₁n)}

v

u generated by Algorithm 11 converges strongly to(u₁,v₁)provided that the constant0satisfying the following conditions: . 0 , 0 , 2 1 ; , 1 1 , ) 1 ( }; ) 1 ( { min ) ( { max 2 2 2 2 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 2 1 2 2 1                                               i i i i i i i i i i i i i i i i i i i i c A A m m i m a a c c a c c a c Acknowledgement

This research was financially supported by the National Science Foundation (11371070).

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