2019 International Conference on Applied Mathematics, Modeling, Simulation and Optimization (AMMSO 2019) ISBN: 978-1-60595-631-2
Auxiliary Principle and Iterative Algorithm for a Generalized Nonlinear
Mixed Quasi-variational-like Inequality
Ya-li ZHAO
*, Xin LIU and Dong-xue HAN
College of physics and Mathematics, Bohai University, Jinzhou, Liaoning 121013, P.R. China *Corresponding author
Keywords: Generalized nonlinear mixed quasi-variational-like inequality, Auxiliary principle technique, Minimizing sequence, Iterative algorithm, Convergence.
Abstract. In this paper, we employ the auxiliary principle technique to study a generalized nonlinear mixed quasi-variational-like inequality in Hilbert spaces. Fist, we establish the existence and uniqueness of solution of the corresponding auxiliary generalized nonlinear mixed quasi-variational-like inequality by making use of minimizing sequence of a convex function. Then based on the existence result, we construct an iterative algorithm for finding approximate solution to the exact solution of the generalized nonlinear mixed quasi-variational-like inequality. Our results extend, improve and unify some known results in the literature.
Introduction and Preliminaries
Throughout the paper, let R ( , ),H be a real Hilbert space endowed an inner product , and a norm ,respectively. Let 2H
be the family of all nonempty subsets of H. LetK H: 2Hbe set-valued mapping such that for eachxH,K(x)is a closed convex subset of
.
H Letb:HH Rsatisfy the following conditions:
(i) bis linear in the first argument; (ii)bis convex in the second argument;
(iii)bis bounded, that is, there exists a constantl 0satisfyingb(u,v)l u v,u,vK; (iv) b(u,v)b(u,w)b(u,vw),u,v,wK.
It is easy to see that from (iii) and (iv) that b(u,v)b(u,w)l u vw,u,v,wK,which implies thatbis continuous in the second argument.
Let A,B,C,D,E:H H,N:HHHH,M,:HH H be nonlinear single-valued mappings. We consider the following problem: FindxHsuch that xK(x) and
), ( ,
0 ) , ( ) , ( ) , ( ), , ( ) , .
(AxBx Cx M Dx Ex y x b x y b x x y K x
N
(1)
which is called a generalized nonlinear mixed quasi-variational-like inequality (for short, denoted by GNMQVLI. In many important applications,K x( )has following form[1]: K x( )m x( )K,
, x H
where mis a valued mapping, andKis a closed convex subset ofH.
Case 1. If N1:H H H,BxCxEx0, ( , )b x y ( ),y N x y z( , , )M x y( , )x K x, ( )Kfor all(x,y,z)H,whereKis a nonempty closed convex subset ofH, then the GNMQVLI (1) reduces to the mixed variational-like inequality problem studied by Ansari and Yao[2] and Zeng[3]: findxK, such thatAxDx,(y,x)(y)(x)0,yK.
Case 2. IfBxCxDxExb(x,y)M(x,y)0.N(x,y,z)xfor all (x,y,z)H, Kis the same as in case 1. Then the GNMQVLIP (1) reduces to the variational-like inequality problem studied by Yang and Chen [4]: findxK,such that Ax,(y,x)0,yK.
Case 3. IfBxCxDxExM(x,y)b(x,y)0,.N(x,y,z)x,(x,y) yxfor allx,yH, Kis
the same as in case 2. Then the GNMVLIP (1) is equivalent to find xK such that ,
, 0
,y x y K
Ax
Noting that in the GNMQVLI (1), is a bifunction andbis a nonlinear mapping, so the projection method cannot be applied to it. This fact motivated many authors to develop the auxiliary principle technique to study the existence of solutions of generalized mixed type variational methods for solving various variational inequalities, complementarity problems and optimization problems. The auxiliary principle techniques were first introduced in [6], which has become a useful, important and powerful tool for solving various variational-like inequalities. Ansari and Yao[2], Chang and Xiang[7], Ding and Luo[8], Huang and Dong[9], Huang and Fang[10], Liu, Chen, Kang and Ume[11], Zeng[3], Zeng, Lin and Yao[12], Liu, Zhang, Ume and Kang[13] used the auxiliary principle technique to construct of some iterative methods for finding the exact solutions of the variational and variational-like inequalities and discuss the existence and uniqueness of solutions for the variational and variational-like inequalities and the convergence of iterative sequences generated by the iterative methods.
Motivated and inspired by the results in [1-13], we introduce and study a new generalized nonlinear mixed quasi-variational-like inequality in Hilbert spaces. By applying the minimizing sequence of a convex function, we show the existence and uniqueness theorem of solution for auxiliary generalized nonlinear mixed quasi-variational-like equality. For finding the approximate solution to the GNMVLIP, we suggest an iterative algorithm by the auxiliary problem. Under certain conditions, we get the existence and arzquenes results of solution for the GNMQVLI and prove the convergence of iterative sequence generated by the iterative algorithm. Our results improve and generalize many known results. Let recall the following concepts and lemmas.
Definition 1. LetA,B,C:H H,N:HHHand:HHHbe mappings.
(1) A is said to be Lipschitz continuous if there exist a constant l0 such that .
,
, u v H
v u l Av
Au
(2) N is said to be relaxed Lipschitz with respect to A,BandC , if there exists a constant 0, such that N(Au,Bu,Cu)N(Av,Bv,Cv),(u,v) uv 2,u,vH.
(3)M is said to be generalized pseudo contractive with respect toAandB, if there exists a constant 0such thatM(Au,Bu)M(Av,Bv),(u,v) uv 2,u,vH.
(4) N is said to be relaxed Lipschitz with respect to A,B and C , if there exist a constant 0such that N(Au,Bu,Cu)N(Av,Bv,Cv),uv uv 2,u,vH. 0
(5)Nis said to be Lipschitz continuous in the third argument if there exist a constant 0such that N(x,y,u)N(x,y,v) uv,x,y,u,vH.
(6) is said to be Lipschitz continuous if there exists a constant 0 such that .
, , )
,
(u v uv u vH
(7)M is said to be Lipschitz continuous with respect to A and B if there exists a constant 0 such that M(Au,Bu)M(Av,Bv) uv ,u,vH.
Assumption 2. LetA,B,C,D,E:HH,N:HHHH,M,:HHH be mappings such that
(i)(x,y)(x,z)(z,y),x,y,zH;(ii)(uv,w)(wu,v),u,v,wH;
(iii)the functionalyN(Ax,By,Cx)M(Dx,Ex),(y,x)are continuous and linear for all yk. Remark 3. It follows from Assumption 2 (i) that(x,x)0and(x,y)(y,x)for allx,yH.
Auxiliary Problem and Iterative Algorithm
Now, we consider the following auxiliary generalized nonlinear mixed variational-like inequality (for short, denoted by AGNMQVLIP) with respect to the GNMQVLIP(1): for each
) (x K
x findwK(x)and some constant 0such that
, , ( , , ) ( , ), ( , )
w y w x y w N Ax Bx Cx M Dx Ex y w
(2) Theorem 4. LetHbe a real Hilbert space andK:H2Hbe a set-valued mapping such that for all xH,K(x) is a nonempty closed convex subset of H. Let N:HHHH,
, :
, H H H
M A,B,C,D,E:H Hbe nonlinear single-valued mappings,b:HHRbe a bifunction such that for any(x,y)H,the functionyb(x,y)is a proper convex and lower semicontinuous functional. If Assumption 6 holds, then the AGNMQVLI (2) has a unique solution. Proof. For any givenxH, we define the functionalJ:K(x)Ras , ( ),
2 1 )
(y y y j y
J where
. , ) , ( ) , ( ), , ( ) , , ( )
(y N Ax Bx Cx M Dx Ex y x b x y x y
j First, we claim thatJhas a unique
minimum point y, in K(x). In fact, by Assumption6(ii) the functional
N(Ax,Bx,Cx) M(Dx,Ex), (y,x)
y is continuous and linear. Since y b(x,y)is convex
lower semicontinuity onK(x).It is easy to see that j(y)is proper convex lower semicontinuity onK(x)and soJ(y)is a strictly convex and lower semicontinuity functional onK(x). Thus,j(y)is bounded from below by a hyper planeu,y , whereuH andR . Hence we have
, 2 1 2 1 , 2 1 ) ( , 2 1 )
(y y y j y y2u y yu2 u2
J which implies that
) (y
J (as y ) (3) Now let
yn be a minimizing sequence ofJonK(x), that is, limJ1(y ) inf J(y).K y n
n We state
that
yn is bounded. If it is not true, then there exists its subsequence
ynK suchthat y K,K 1,2,
K
n . By (3) we haveJ(ynK), which contradicts the fact that
yn is aminimizing sequence ofJonK(x). By the Weierstrass theorem(see [21]) there existsy1K(x)such
that ( ) min ( )
) (
1 J y
y J
x K y
.Again from the strict convexity ofJ , we obtain thatJ has a unique minimum pointy1inK(x). Next, Let us showy1K(x)is a unique solution of the AGNMQVLI(2).
For anyyK(x)andt(0,1), since jis convex andJis lower semicontinuous onK(x).we have
1 1 1 1 1 1 1
2
1 1 1 1 1 1 1 1
2
1 1 1 1 1 1
1
( ) , ( ) ( (1 ) ) ( ) ( ) 2 1 , , , ( ) ( ( ) ( )) 2 2 ( ) , , ( ( ) ( )), 2
J y y y J y J ty t y J y t y y
t
y y t y y y y y y y j y t j y J y
t
j y y y y y t y y y t j y J y
which implies that , , ( ) ( ) 0.
2yy1 yy1y1 yy1 j y j y1
t
Letting t0 in the cotter inequality, we get
. 0 , ) , ( ) , ( ), , ( ) , , ( , ) , ( ) , ( ), , ( ) , , ( , ) ( ) ( , 1 1 1 1 1 1 1 y x y x b x y Ex Dx M Cx Bx Ax N y x y x b x y Ex Dx M Cx Bx Ax N y y y y j y j y y y
1 1 1 1
1
, , ( , , ) ( , ), ( , )
( , ) ( , ), ( ).
y y y x y y N Ax Bx Cx M Dx Ex y y b x y b x y y K x
So,y1is a solution of the AGNMQVLI (2). Last, we show thaty1is the unique solution of the AGNMQVLI (2). Lety1,y2be solutions of the AGNMQVLI (5), then for allxK(x), we have
1 1 1 1
1
, , ( , , ) ( , ), ( , )
[ ( , ) ( , )], ( ).
y y y x y y N Ax Bx Cx M Dx Ex y y b x y b x y y K x
(4)
and
2 2 2 2
2
, , ( , , ) ( , ), ( , )
[ ( , ) ( , )], ( ).
y y y x y y N Ax Bx Cx M Dx Ex y y b x y b x y y K x
(5)
Takingy y2in (4) andy y1in (5), we get that
1 2 1 2 1 2 1
1 2
, , ( , , ) ( , ), ( , )
[ ( , ) ( , )], ( ).
y y y x y y N Ax Bx Cx M Dx Ex y y b x y b x y y K x
(6)
And
2 1 2 1 2 1 2
2 1
, , ( , , ) ( , ), ( , )
[ ( , ) ( , )], ( ).
y y y x y y N Ax Bx Cx M Dx Ex y y b x y b x y y K x
(7)
Adding (6) and (7), and it follows from Remark3 that y1y2,y2y10,that is, y1y2 0, implying thaty1 y2. Hencey1is the unique solution of the AGNMQVLI (2). This complete the proof.
Corollary 5. LetH,K,N,M,A,B,C,D,E,J,i be as in Theorem4. Letb:KKRbe a bifunction such that for any givenx,yH, the functionalyb(x,y)is proper convex and lower semicontinuous. If Assumption6 holds, then the solution of the AGNMQVLI (2) is also the unique minimum point ofJ.
Proof. Lety1be a solution of the AGNMQVLI (2), then
1 1 1 1 1 1 1 1
2
1 1 1 1
1 1 1
1 1
( , , ) ( ( ), ( ) , )
2 2
1
, ,
2
, ( , , ) ( , ), ( , ) [ ( , ) ( , )]
y y y y y y y y y y y y
y y y y y y y y
x y y N Ax Bx Cx M Dx Ex y y b x y b x y
By Assumption 2 (i), the latter inequality implies that
1 1 1 1
1
, ( , , ) ( , ), ( , ) ( , ) ,
2 1
, ( , , ) ( , ), ( , ) ( , ) ,
2
y y N Ax Bx Cx M Dx Ex y x b x y x y
y y N Ax Bx Cx M Dx Ex y x b x y x y
That is,J(y)J(y1),yK(x).From the strict convexity ofJ,we conclude thatJhas a unique minimum pointy1K(x),which is completed the proof. Based on Theorem 4, we can construct the following iterative algorithm for solving the GNMQVLI (1).
Algorithm 6. For a given x0H,let the sequence{xn}satisfies the following conditions:
) ( n
n K x
1 1 1 1
1 1
, , ( , , ) ( , ), ( , )
[ ( , ) ( , )], ( ).
n n n n n n n n n n
n n n n
x y x x y x N Ax Bx Cx M Dx Ex y x b x x b x y y K x
for any n0,1,2,, where0is a constant.
Existence and Convergence
In this section, we prove the existence of the solution of problem (1) and the convergence of the iterative sequence generated by Algorithm6.
Theorem 7. Let H,K,N,M,A,B,C,D,E, be as in Theorem4. Let the mapping
H H
m: satisfy (2), and letb:HHRbe real-valued functional satisfying the properties in Theorem8 and properties(i)-(iv). Assume that the following conditions hold:
(1)is Lipschitz continuous;
(2)Nis related Lipschitz with respect toA,BandCin the first, second and third argument; (3)Nis1,2and3Lipschitz continuous in the first, second and third argument, respectively; (4)A,B,C,D,Eis1,2,3,4and5Lipschitz continuous, respectively;
(5)M is generalized pseudocontractive with respect toDand Ein the first and second arguments;
(6)M is Lipschitz continuous with respect toDandE;
(7)mis Lipschitz continuous, Assumption 6 are satisfied and there exist constant 0such that
. 1 ] 2
1 ) (
2 1 1
[ 2 1
1 2 2 2
3 3 2 2 1 1
2
l (8)
Then there existsxK(x)is a solution of the GNMQVLI (1), and the sequence{xn}generated by Algorithm 6 strongly converges tox.
Proof. First, it follows from Algorithm 10 that for allyK(xn),
xn,y xn xn1,y xn N(Axn1,Bxn1,Cxn1) M(Dxn1,Exn1),(y,xn) (9) and for allyK(xn1),
1 1 1 1
1
, , ( , , ) ( , ), ( , )
[ ( , ) ( , )].
n n n n n n n n n n
n n n
x y x x y x N Ax Bx Cx M Dx Ex y x b x x b x y
(10)
Addingm(xn),yxnto the two sides of (9) and then takingym(xn)xn1m(xn1)K(xn),we
have
1 1 1 1 1
1 1 1 1 1 1 1
1 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
x m x m x x x m x x m x m x x x m x
A Ax Bx Cx M Dx Ex m x x M x x
b x x b x m x x m x
(11)
Similarly, adding m x( n1),yxn1 to the two sides of (13) and then takingym x( n1) xn m x( n)K x( n1),we obtain
1 1 1 1 1 1 1
1 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
x m x x m x x m x x m x x m x x m x
N Ax Bx Cx M Dx Ex m x x m x x b x x b x m x x m x
(12)
1 1 1 1 1 1 1
1 1
1 1
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 n n n n n
x m x x m x x m x x m x x m x x m x
N Ax Bx Cx M Dx Ex m x x m x x b x x b x m x x m x
x m x x m x x m x x
1
1 1 1 1
1 1 1 1 1
( )
( ) ( ), ( ) ( )
( , , ) ( , , ), ( ( ) ( ), )
n
n n n n n n n n
n n n n n n n n n n
m x x x m x m x x m x x m x
N Ax Bx Cx N Ax Bx Cx m x x m x x
(13)
By properties (iii) of b, (13) implies that
2
1 1 1 1 1 1
1 1 1 1
1 1 1 1 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 n
n n n n n
x x m x m x x x m x m x x x m x m x
x x N Ax Bx Cx N Ax Bx Cx
x x M Dx Ex M Dx Ex m x x m x x
l x x x x m x
m x( n1) .
(14)
So, it follows from (14) and condition (1), we have
1 1 1
1 1 1 1
1 1 1 1
2 ( ) ( )
[ ( ( , , ) ( , , ))
( ( , ) ( , )) ] .
n n n n n n
n n n n n n n n
n n n n n n n n
x x x x m x m x
x x N Ax Bx Cx N Ax Bx Cx
x x M Dx Ex M Dx Ex l x x
(15)
From (12)-(14) that
2
1 1 1 1
2
2 2
1 1 2 2 3 3 1
( ( , , ) ( , , )
(1 2 ( ) ) .
n n n n n n n n
n n
x x N Ax Bx Cx N Ax Bx Cx
x x
(16)
It follows from (15)-(16) that
2
1 1 1
2
1 1 1 1
2 2
2 2 2
1 1 1
( ( , ) ( , )
2 , ( , ), ( , )
( , ) ( , ) (1 2 ) .
n n n n n n
n n n n n n n n
n n n n n n
x x M Dx Ex M Dx Ex
x x x x M Dx Ex M Dx Ex
N Dx Ex M Dx Ex x x
(17)
It follows from condition (7) and (15)-(17), we obtain
2 2 2 2
1 1 1 2 2 3 3
1
1
[1 ( 1 2 ( ) 1 2 )]
1 2
.
n n
n n
x x l
x x (18)
Where [1 ( 1 2 ( ) 1 2 )].
2 1
1 2 2 2
3 3 2 2 1 1
2
l It follows
from condition (11) that 1. So (18) implies that {xn} is a Cauchy sequence in H. Let{xn}xHas n.Next, we claim thatxis a solution of the GNMQVLI (1). In fact, by Theorem 8 we can assume thatxK(x)is the unique solution of the AGNMQVLI (5), that is
). ( )], , ( ) , ( [ ) , ( ), , ( ) , , ( ,
,y x x y x N Ax Bx Cx M Dx Ex y x b x x b x y y K x
x
Together with (9), and a similar argument as in (15), we easily
get
x
x
x
0x
,
n
n
where is the same as above. Since 1,Acknowledgement
This research was financially supported by the National Science Foundation (11371070).
References
[1] Noor. Generalized set-valued mixed nonlinear quasi variational inequalities, Korean J. Comput. Appl. Math. 1(1998), 72-89.
[2] Q. H. Ansari, J-C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optim. Theory Appl. 108(2001), 527-541.
[3] L. C. Zeng, Iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities. Aeta Math. Appl. Sin. Engl. Ser. 20 (2004), 477-486.
[4] X. Q. Yang. G. Y. Chen, A class of nonconver functions and prevariational inequality. J. Math. Anal. Appl, 169 (1992), 359-373.
[5] J. C. Yao. Variational inequalities with generalized monotone operators. Math. Opera. Res. 19(1994), 691-706.
[6] R. Glowinski. J. L. Lions, R. Tremolieres, Numerical analysis of variational inequalities. North-Holland Publishing Co. Amsterdam-New York. (1981). 1
[7] S. S. Chang, S. W. Xiang, On the existence of solutions for a class of quasi-variational inequality, (In China), J. Systems Sci. Math. Sci, 16 (1996), 136-140.
[8] X. P. Ding. C. L. Luo, Existence and algorithm for solving some generalized mixed variational inequalitues. Comput. Math. Appl. 37(1990), 23-30.
[9] N-J. Huang. C-X. Deng. Auriliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. J. Math. Anal. Appl. 256 (2001), 345-359. [10] N-J. Huang. Y-P. Fang. Auxiliary principle technique for generalized set-valued nonlinear quasi-variational-like mequalities, Math. Inequal. Appl. 6(2003), 339-350.
[11] Z. Liu. Z. S. Chen, S. M. Kang, J. S. Ume, Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces. Nonlinear Anal, 69(2008), 3259-3272.
[12] L-C. Zeng. L. J. Lin. J-C. Yao, Auxiliary problem method for mixed variational-like inequalities. Taiwanese. J. Math. 10(2006). 515-529.