Hàm GAP cho bài toán bất đẳng thức tựa biến phân véc tơ tham số hỗn hợp mạnh

126 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 
 
Abstract— The parametric mixed strong vector 
quasivariational inequality problem contains many 
problems such as, variational inequality problems, 
fixed point problems, coincidence point problems, 
complementary problems etc. There are many 
authors who have been studied the gap functions for 
vector variational inequality problem. This problem 
plays an important role in many fields of applied 
mathematics, especially theory of optimization. In this 
paper, we study a parametric gap function without 
the help of the nonlinear scalarization function for a 
parametric mixed strong vector quasivariational 
inequality problem (in short, (SQVIP)) in Hausdorff 
topological vector spaces. (SQVIP) Find 
),( xKx and ),( xTz such that 
),,(,),,(>,<  xKyRxyfxyz n  
where we denote the nonnegative of 
nR by 
}.,1,2,=0,|),,,(={= 21 nitRttttR i
nT
n
n  
Moreover, we also discuss the lower 
semicontinuity, upper semicontinuity and the 
continuity for the parametric gap function for this 
problem. To the best of our knowledge, until now 
there have not been any paper devoted to the lower 
semicontinuity, continuity of the gap function without 
the help of the nonlinear scalarization function for a 
parametric mixed strong vector quasivariational 
inequality problem in Hausdorff topological vector 
spaces. Hence the results presented in this paper 
(Theorem 1.3 and Theorem 1.4) are new and different 
in comparison with some main results in the 
literature. 
Manuscript Received on July 13th, 2016. Manuscript Revised 
December 06th, 2016. 
This research is funded by Ho Chi Minh City University of 
Technology - VNU-HCM under grant number T-KHUD-2016-
107. 
Le Xuan Dai is with Department of Applied Mathematics, 
Ho Chi Minh City University of Technology - VNU-HCM, 
Vietnam National University - Ho Chi Minh City, Vietnam, 
Email: [email protected] 
Nguyen Van Hung is with Department of Mathematics, Dong 
Thap University, Cao Lanh City, Vietnam, Email: 
[email protected] 
Phan Thanh Kieu is with Department of Mathematics, Dong 
Thap University, Cao Lanh City, Vietnam, Email: 
[email protected] 
Index Terms—Vector quasivariational inequality 
problem; parametric gap function; lower 
semicontinuity; upper semicontinuity, continuity. 
1 INTRODUCTION 
et X and  be Hausdorff topological vector 
spaces. Let ),( nRXL be the space of all linear 
continuous operators from X to .nR ,2: XXK  
),(2:
nRXLXT  are set-valued mappings and let 
nRXf  : be continuous single-valued 
mappings. For   consider the following 
parametric mixed strong vector quasivariational 
inequality problem (in short, (SQVIP)). 
 (SQVIP) Find ),( xKx and ),( xTz such that 
),,(,),,(>,<  xKyRxyfxyz n  
where we denote the nonnegative of nR by 
}.,1,2,=0,|),,,(={= 21 nitRttttR i
nT
n
n  
Here the symbol T denotes the transpose. We 
also denote 
}.,1,2,=0,>|),,,(={= 21 nitRttttintR i
nT
n
n  
 For each   we let )},(|{:=)(  xKxXxE 
and X2:  be set-valued mapping such that 
)( is the solution set of (SQVIP). Throughout 
the paper, we always assume that   )( for each 
 in the neighborhood .0   
 The parametric mixed strong vector 
quasivariational inequality problem contains many 
problems such as, variational nequality problems, 
fixed point problems, coincidence point problems, 
complementarity problems etc,. There are many 
authors have been studied the gap functions for 
vector variational inequality problem, see ([2]-[6], 
[8]-[10]) and the references therein. 
 The structure of our paper is as follows. In 
Section 1 of this article, we introduce the model 
vector quasivariational inequality problem and 
recall definitions for later uses. In Section 2, we 
establish the lower semicontinuity, the upper 
semicontinuity and the continuity for the gap 
The GAP function of a parametric mixed strong 
vector quasivariational inequality problem 
Le Xuan Dai, Nguyen Van Hung, Phan Thanh Kieu 
L
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 
127
function of parametric mixed strong vector 
quasivariational inequality problem. 
A. Preliminaries 
 Next, we recall some basic definitions and their 
some properties. 
Definition 1.1 (See [1], [7]) Let X and Z be 
Hausdorff topological vector spaces and ZXF 2: 
be a multifunction. 
i) F is said to be lower semicontinuous (lsc) at 
0x if  UxF )( 0 for some open set ZU  
implies the existence of a neighborhood N of 
0x such that, for all .)(,   UxFNx An 
equivalent formulation is that: F is lsc at 0x 
if ,0xx  ),( 00 xFz  ),( xFz  .0zz 
F is said to be lower semicontinuous in X if 
it is lower semicontinuous at each .0 Xx 
ii) F is said to be upper semicontinuous (usc) at 
0x if for each open set ),( 0xFU  there is a 
neighborhood N of 0x such that ).(NFU  F 
is said to be upper semicontinuous in X if it 
is upper semicontinuous at each .0 Xx 
iii) F is said to be continuous at 0x if it is both 
lsc and usc at .0x F is said to be continuous 
at 0x if it is continuous at each .0 Xx 
iv) F is said to be closed at Xx 0 if and only if 
,0xxn  0yyn  such that ),( nn xFy we 
have ).( 00 xFy F is said to be closed in X if 
it is closed at each .0 Xx 
Lemma 1.1 (See [1], [7]) If F has compact 
values, then F is usc at 0x if and only if, for each 
net Xx }{ which converges to 0x and for each 
net ),(}{ xFy  there are )(xFy and a subnet 
}{ y of }{ y such that .yy  
B. Main Results 
 In this section, we introduce the parametric gap 
functions for parametric mixed strong vector 
quasivariational inequality problem, then we study 
some properties of this gap function. 
Definition 1.2 A function RXh  : is said to 
be a parametric gap function of (SQVIP) if it 
satisfies the following properties [i)] 
i) 0),( xh for all ).(Ex 
i) 0=),( 00 xh if and only if ).( 00  x 
 Now we suppose that ),( xK and ),( xT are 
compact sets for any .),(  Xx  We define 
function RXh  : as follows 
i
xKyxTz
xyfyxzxh )),,(>,(<maxmin=),(
),(),(


 (1) 
 where ixyfyxz )),,(>,(<  is the i th 
component of ),,,(>,< xyfyxz .,1,2,= ni  
 Since ),( xK and ),( xT are compact sets, 
),( xh is well-defined. 
 In the following, we will always assume that 
0=),,( xxf for all ).(Ex 
Theorem 1.2 The function ),( xh defined by (1) 
is a parametric gap function for the (SQVIP). 
 Proof. We define a function 
nn RRXLXh ),(:1 as follows 
,)),,(>,(<maxmax=),(
1),(
1 i
nixKy
xyfyxzzxh 

where ).,(),(  xTzEx 
i) It is easy to see that 0.),(1 zxh Suppose to the 
contrary that there exists )(0 Ex and ),( 00 xTz 
such that 0,<),( 001 zxh then 
i
nixKy
xyfyxzzxh )),,(>,(0 00
1),0(
001 

),,(,)),,(>,(<max 000
1
 xKyxyfyxz i
ni
  
which is impossible when .= 0xy Hence, 
0,)),,(>,(<maxmax=),(
1),(
1 
i
nixKy
xyfyxzzxh 

where ).,(),(  xTzEx Thus, since ),( xTz is 
arbitrary, we have 
0.)),,(>,(<maxmin=),(
),(),(
i
xKyxTz
xyfyxzxh 

ii) By definition, 0=),( 00 xh if and only if there 
exists ),( 000 xTz such that 0,=),( 001 zxh i.e., 
0,=)),,(>,(<maxmax 0000
1)0,0(
i
nixKy
xyfyxz 

for )( 00 Ex if and only if, for any ),,( 00 xKy 
0,)),,(>,(<max 0000
1
i
ni
xyfyxz  
namely, there is an index ,1 0 ni such that 
0,)),,(>,(<
00000
 ixyfyxz  which is equivalent 
to 
),,(,),,(>,< 000000  xKyRxyfyxz
n  
that is, ).( 00  x 
Remark 1.1 As far as we know, there have not 
been any works on parametric gap functions for 
mixed strong vector quasiequilibrium problems, 
and hence our the parametric gap functions is new 
and cannot compare with the existing ones in the 
literature. 
128 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 
Example 1.1 Let [0,1],=2,=,= nRX 
[0,1],=),( xK 
 422,3
2
1
=),( xxxT  and 
0.=),,( xyf Now we consider the problem (QVIP), 
finding ),( xKx and ),( xTz such that 
 ))((3),(
2
1
=),,(>,< 422 xyxxxyxyfxyz  
.2 R It follows from a direct computation 
{0}=)( for all [0,1].  Now we show that (.,.)h is 
a parametric gap function of (SQVIP). Indeed, 
taking ,(1,1)= nintRe we have 
i
nixKyxTz
xyfyxzxh )),,(>,(<maxmaxmin=),(
1),(),(


 (0,1],
0=0,
=)))(((3max= 532
422
),( xifxx
xif
yxxx
xKy 


Hence, (.,.)h is a parametric gap function of 
(SQVIP). 
 The following Theorem 1.3 gives sufficient 
condition for the parametric gap function (.,.)h is 
continuous in . X 
Theorem 1.3 Consider (SQVIP). If the following 
conditions hold: 
 i) (.,.)K is continuous with compact values in 
; X 
 ii) (.,.)T is upper semicontinuous with compact 
values in . X 
 Then (.,.)h is lower semicontinuous in . X 
 Proof. First, we prove that (.,.)h is lower 
semicontinuous in . X Indeed, we let Ra and 
suppose that   Xx )},{(  satisfying 
   ,),( axh and ),(),( 00  xx as . It 
follows that =),( xh 
.)),,(>,(<maxmaxmin=
1),(),(
axyfyxz i
nixKyxTz
   

We define the function RRXLXh n  ),(:0 by 
).()),,(>,(<maxmax=),,(
1),(
0 

Exyfyxzzxh i
nixKy
Since g and f are continuous, we have 
ixyfyxz )),,(>,(<  is continuous, and since (.,.)K 
is continuous with compact values in . X Thus, 
by Proposition 19 in Section 3 of Chapter 1 [1] we 
can deduce that ),,(0 zxh is continuous. By the 
compactness of ),,(  xT there exists ),(  xTz 
such that =),( xh 
i
nixKyxTz
xyfyxz )),,(>,(<maxmaxmin=
1),(),(
   
 
=),,(= 0 zxh 
.)),,(>,(<maxmax=
1),(
axyfyxz i
nixKy
  
 
Since (.,.)K is lower semicontinuous in , X for 
any ),,( 000 xKy there exists ),( xKy such 
that .0yy For ),,( xKy we have 
.)),,(>,(<max
1
axyfyxz i
ni
  (2) 
 Since (.,.)T is upper semicontinuous with 
compact values in , X there exists ),( 000 xTz 
such that 0zz (taking a subnet }{ z of }{ z if 
necessary) as . Since 
i
ni
xyfyxz )),,(>,(<max
1
 
 is continuous. Taking the 
limit in (2), we have 
.)),,(>,(<max 000000
1
axyfyxz i
ni
 (3) 
 Since ),( 00 xKy is arbitrary, it follows from 
(3) that 
=),,( 0000 zxh 
.)),,(>,(<maxmax= 0000
1)0,0(
axyfyxz i
nixKy


and so, for any ),,( 00 xTz we have =),( 00 xh 
.)),,(>,(<maxmaxmin= 000
1)0,0()0,0(
axyfyxz i
nixKyxTz


This proves that, for ,Ra the level set 
}),(|),{( axhXx   is closed. Hence, (.,.)h is 
lower semicontinuous in . X 
Theorem 1.4 Consider (SQVIP). If the following 
conditions hold: [i)] 
 1. (.,.)K is continuous with compact values in 
; X 
 2. (.,.)T is continuous with compact values in 
. X 
 Then (.,.)h is continuous in . X 
 Proof. Now, we need to prove that (.,.)h is upper 
semicontinuous in . X Indeed, let Ra and 
suppose that   Xx )},{(  satisfying 
,),( axh  for all and ),(),( 00  xx as 
, then =),( xh 
axyfyxz i
nixKyxTz
)),,(>,(<maxmaxmin=
1),(),(
   

and so 
TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 
129
)(,)),,(>,(<maxmax ,
1),( 
 
  
 xTzaxyfyxz i
nixKy
  
 (4) 
 Since (.,.)T is lower semicontinuous with 
compact values in , X for any ),,( 000 xTz there 
exists ),( xTz such that 0zz as . 
 Since ),,( xTz it follows (4) that 
,)),,(>,(<maxmax
1),(
axyfyxz i
nixKy
  
 (5) 
 Since f and g are continuous, so 
i
ni
xyfyxz )),,(>,(<max
1
 
 is continuous. By the 
compactness of (.,.)K there exists ),(  xKy 
such that 
.)),,(>,(<max
1
axyfyxz i
ni
  (6) 
 Since (.,.)K is upper semicontinuous with 
compact values, there exists ),( 000 xKy such that 
0yy (taking a subnet }{ y of }{ y if 
necessary) as . Since 
i
ni
xyfyxz )),,(>,(<max
1
 
 is continuous. Taking 
limit in (6), we have 
.)),,(>,(<max 000000
1
axyfyxz i
ni
 (7) 
 For any ),,( 00 xKy we have 
.)),,(>,(<maxmax 0000
1)0,0(
axyfyxz i
nixKy


 (8) 
 Since ),( 00 xTz is arbitrary, it follows from (8) 
that =),( 00 xh f 
axyfyxz i
nixKyxTz
)),,(>,(<maxmaxmin= 000
1)0,0()0,0(


This proves that, for ,Ra the level set 
}),(|),{( axhXx   is closed. Hence, (.,.)h is 
upper semincontinuous in . X 
2 CONCLUSION 
To the best of our knowledge, until now there 
have not been any paper devoted to the lower 
semicontinuity, continuity of the gap function 
without the help of the nonlinear scalarization 
function for a parametric mixed strong vector 
quasivariational inequality problem in Hausdorff 
topological vector spaces. Hence our results, 
Theorem 1.3 and Theorem 1.4 are new. 
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130 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 
Tóm tắt - Bài toán bất đẳng thức tựa biến phân 
véc tơ tham số hỗn hợp mạnh bao gồm nhiều 
vấn đề như bài toán bất đẳng thức biến phân, 
bài toán điểm bất động, bài toán điểm trùng lặp, 
bài toán bù nhau, v.v. Có nhiều tác giả đang 
nghiên cứu tìm hàm gap cho bài toán bất đẳng 
thức biến phân véc tơ. Bài toán này đóng vai trò 
quan trọng trong nhiều lĩnh vực toán ứng dụng, 
đặc biệt là lý thuyết tối ưu. Trong bài báo này, 
chúng tôi nghiên cứu hàm gap tham số với sự hỗ 
trợ của hàm phi tuyến vô hướng cho bài toán 
bất đẳng thức tựa biến phân véc tơ tham số hỗn 
hợp mạnh (viết tắt (SQVIP)) trong không gian 
tô pô véc tơ Hausdorff. (SQVIP) Tìm 
),( xKx và ),( xTz sao cho 
),,(,),,(>,<  xKyRxyfxyz n  
với 
}.,1,2,=0,|),,,(={= 21 nitRttttR i
nT
n
n  
Ngoài ra, chúng tôi cũng thảo luận tính nửa liên 
tục dưới, nửa liên tục trên và tính liên tục của 
hàm gap tham số cho bài toán này. Theo những 
hiểu biết của mình, chúng tôi cho rằng tới nay 
chưa từng có bài báo nào nghiên cứu tính nửa 
liên tục dưới, tính liên tục của hàm gap mà 
không cần sự trợ giúp của hàm phi tuyến vô 
hướng đối với bài toán bất đẳng thức tựa biến 
phân véc tơ tham số hỗn hợp mạnh trong không 
gian tô pô véc tơ Hausdorff. Do đó những kết 
quả được trình bày trong bài báo này (Định lý 
1.3 và Định lý 1.4) là mới và khác biệt so với một 
số kết quả chính trong tài liệu tham khảo 
Từ khóa - Bài toán bất đẳng thức tựa biến phân 
véctơ; hàm gap tham số; tính nửa liên tục dưới; tính 
nửa liên tục trên, tính liên tục. 
Hàm GAP cho bài toán bất đẳng thức tựa biến 
phân véc tơ tham số hỗn hợp mạnh 
Lê Xuân Đại, Nguyễn Văn Hưng, Phan Thanh Kiều