Volume 2008, Article ID 634921,13pages doi:10.1155/2008/634921
Research Article
The Solvability of a Class of General Nonlinear
Implicit Variational Inequalities Based on Perturbed Three-Step Iterative Processes with Errors
Zeqing Liu,1Shin Min Kang,2and Jeong Sheok Ume3
1Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, Liaoning 116029, China
2Department of Mathematics and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, South Korea
3Department of Applied Mathematics, Changwon National University, Changwon 641-733, South Korea
Correspondence should be addressed to Shin Min Kang,[email protected] Received 23 October 2007; Accepted 25 January 2008
Recommended by Mohammed Khamsi
We introduce and study a new class of general nonlinear implicit variational inequalities, which includes several classes of variational inequalities and variational inclusions as special cases.
By applying the resolvent operator technique and fixed point theorem, we suggest a new perturbed three-step iterative algorithm with errors for solving the class of variational inequalities.
Several existence and uniqueness results of solutions for the general nonlinear implicit variational inequalities, and convergence and stability results of the sequence generated by the algorithm are obtained. The results presented in this paper extend, improve, and unify a host of results in recent literatures.
Copyrightq2008 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In recent years, various extensions and generalizations of the variational inequalities have been considered and studied. For details, we refer to1–33, and the references therein. It is well known that one of the most interesting and important problems in the variational inequality theory is the development of an efficient iterative algorithm to compute approximate solutions of various variational inequalities and inclusions. In 1994, Hassouni and Moudafi 8 introduced a perturbed algorithm for solving a class of variatioanl inclusions. In 2003, Fang and Huang7 introduced the definitions ofH-monotone operator and its resolvent operator, established the Lipschitz continuity of the resolvent operator, constructed an iterative
algorithm, and obtained the existence of solutions for a class of variational inclusions and convergence of the iterative algorithm. In 2004, Liu and Kang19established several existence and uniqueness theorems and convergence and stability results of perturbed three-step iterative algorithm with errors for a class of completely generalized nonlinear quasivariational inequalities.
Inspired and motivated by the recent research works in 1–28, in this paper, we introduce and study a new class of general nonlinear implicit variational inequalities, which includes the variational inequalities and variational inclusions in 1–28 as special cases.
By applying the resolvent operator technique and fixed point theorem, we suggest a new perturbed three-step iterative process with errors for solving the general nonlinear implicit variational inequalities. Several existence and uniqueness results of solutions for the general nonlinear implicit variational inequalities involvingH-monotone, strongly monotone, relaxed monotone, relaxed Lipschitz and generalized pseudocontractive operators, and convergence and stability results of the perturbed three-step iterative process with errors are given. The results presented in this paper extend, improve, and unify a host of results in recent literatures.
2. Preliminaries
Throughout this paper, we assume thatX is a real Hilbert space endowed with a norm · and an inner product·,·, respectively, 2Xstands for the family of all the nonempty subsets ofX, andIdenotes the identity operator onX. Assume thatH, g, m, A, B, C, D, E:X→Xand N, M:X×X→Xare operators, andW:X×X→2Xis a multivalued operator. Givenf ∈X, we consider the following problem: findu∈Xsuch that
f∈N
Au, Bu
−M
Cu, Du W
g−mu, Eu
, 2.1
which is called the general nonlinear implicit variational inequality, whereg−mx gx−mx for allx∈X.
Some special cases of problem2.1are as follows.
AIff M0,EI, then problem2.1reduces to the following problem: findu∈H such that
0∈N
Au, Bu W
g−mu, u
, 2.2
which is called the completely generalized strongly nonlinear implicit quasivariational inclusion in20.
BIff0,EI,Nx, y Mx, y xfor anyx, y∈X, then problem2.1is equivalent to findingu∈Xsuch that
0∈Au−Cu W
g−mu, u
, 2.3
which is called the generalized nonlinear implicit quasivariational inclusion in10.
CIff 0,Nx, y Mx, y x, andWx, y Wxfor anyx, y, z∈X, then problem 2.1collapses to seekingu∈Xsuch that
0∈Au−Cu W
g−mu
, 2.4
which is called the generalized equation by Uko23.
DIff M 0,g−m I,Nx, y x, andWx, y Wxfor anyx, y ∈ X, then problem2.1is equivalent to findingu∈Xsuch that
0∈Au Wu, 2.5
which was introduced and studied by Fang and Huang7.
For appropriate and suitable choices of the operators H, g, m, A, B, C, D, E, N, M, W and the elementf, one can obtain various classes of variational inequalities and variational inclusions in1–33as special cases of problem2.1.
We now recall and introduce the following definitions and results.
Definition 2.1. LetN :X×X → X,g, b, c, H :X → Xbe operators and letW :X →2X be a multivalued operator.
a1gis said to be Lipschitz continuous and strongly monotone if there exist positive constants sandtsatisfying, respectively,
gx−gy≤sx−y,
gx−gy, x−y
≥tx−y2, ∀x, y∈X; 2.6 a2Wis said to be maximal monotone ifWis monotone andIρWX Xfor anyρ >0;
a3Wis said to beH-monotone ifWis monotone andHρWX Xfor anyρ >0;
a4b is called strongly monotone with respect toH and the first argument ofN if there exists a positive constantssatisfying
N bx, u
−N
by, u
, Hx−Hy
≥sx−y2, ∀x, y, u∈X; 2.7 a5bis called relaxed Lipschitz with respect toHand the first argument ofNif there exists
a positive constantssatisfying N
bx, u
−N by, u
, Hx−Hy
≤ −sx−y2, ∀x, y, u∈X; 2.8 a6bis called relaxed monotone with respect toH and the second argument ofNif there
exists a positive constantssatisfying N
u, bx
−N
u, by
, Hx−Hy
≥ −sx−y2, ∀x, y∈X; 2.9 a7b is called generalized pseudocontractive with respect to g if there exists a positive
constantssatisfying
bx−by, gx−gy
≤sx−y2, ∀x, y∈X; 2.10 a8N is called Lipschitz continuous with respect to the first argument if there exists a
positive constantssatisfying
Nx, u−Ny, u≤sx−y, ∀x, y∈X. 2.11
Similarly, we can define the Lipschitz continuity ofN with respect to the second arg- ument. On the other hand, if Nx, y x for any x, y ∈ X, then Definition 2.1reduces to the usual concepts of strong monotonicity, relaxed monotonicity, and Lipschitz continuity. It is known that a maximal monotone operator need not beH-monotone for someH, and ifWis H-monotone andHis strictly monotone, thenWis maximal monotone.
Definition 2.2see7. LetH :X →Xbe a strictly monotone operator and letW :X→2Xbe anH-monotone operator. For any givenρ >0, the resolvent operatorRHW,ρ:X →Xis defined by
RHW,ρx HρW−1x, ∀x∈X. 2.12 Definition 2.3see34. Letg:X→Xbe an operator andx0∈X. Assume thatxn1fg, xn define an iteration procedure which yields a sequence of points{xn}n≥0 inX. Suppose that Fg {x∈X :xgx}/∅and{xn}n≥0converges to someu∈Fg. Let{zn}n≥0 ⊂Xand nzn1−fg, znfor alln≥0. Iflimn→∞n0 implies that limn→∞znu, then the iteration procedure defined byxn1fg, xnis said to beg-stable or stable with respect tog.
Lemma 2.4see35. Let{an}n≥0,{bn}n≥0, and{cn}n≥0be nonnegative sequences satisfying an1≤
1−tn
antnbncn, ∀n≥0, 2.13 where{tn}n≥0⊂0,1,∞
n0tn∞, limn→∞bn0, and∞
n0cn<∞. Then limn→∞an0.
Lemma 2.5see 7. LetH : X → X be a strongly monotone operator with constant r and let W : X → 2X be anH-monotone operator. Then the resolvent operator RHW,ρ : X → X is Lipschitz continuous with constantr−1.
3. Existence, convergence, and stability
Now, we use the resolvent operator technique to establish the equivalence between the general nonlinear implicit variational inequality2.1and the fixed point problem.
Lemma 3.1. Letλandρbe two positive constants, letH :X→Xbe a strictly monotone operator, let W:X×X→2Xbe a multivalued operator such that for any fixedx∈X, W·, ExisH-monotone, and
Yx H
g−mx
−ρN
Ax, Bx ρM
Cx, Dx
ρf, ∀x∈X, 3.1 whereH, g, m, A, B, C, D, E : X → X andN, M : X×X → X are operators. Then the following statements are equivalent:
b1the general nonlinear implicit variational inequality2.1possesses a solutiou∈X;
b2there existsu∈Xsatisfying
gu mu RHW·,Eu,ρ Yu
; 3.2
b3the mappingG:X→Xdefined by Gx 1−λxλ
x−g−mx RHW·,Ex,ρ
Yx , ∀x∈X 3.3
has a fixed pointu∈X.
Proof. It is clear thatb1holds if and only ifYu∈HρW·, Eug−mu,which is equivalent to3.2by the definition of the resolvent operator. On the other hand,3.3means thatGhas a fixed pointu∈Xif and only if3.2holds. This completes the proof.
Remark 3.2. Lemma 3.1extends and improves Lemma 3.1 in1,7,10,12,19–22,32, Theorem 3.2 in6, Lemma 3.2 in25, Theorem 2.1 in8,24,26, and Lemma 2.227.
Based onLemma 3.1, we suggest the following perturbed three-step iterative process with errors for the general nonlinear implicit variational inequality2.1.
Algorithm 3.3. LetA, B, C, D, E, g, m, H, Hn :X →X, N, M:X×X →Xbe operators,W, Wn : X×X→2Xsatisfy that for anyx∈X, W·, ExisH-monotone andWn·, ExisHn-monotone for eachn≥0. Givenf, u0∈X, the iterative sequence{un}n≥0is defined by
wn 1−cn
uncn
un−g−m un
RHWn
n·,Eun,ρ
Y
un rn, vn
1−bn
unbn
wn−g wn
m wn
RHWn
n·,Ewn,ρ
Y
wn qn, un1
1−an
unan
vn−g−m vn
RHWn
n·,Evn,ρ
Y
vn pn, n≥0,
3.4
whereY is defined by3.1,{pn}n≥0,{qn}n≥0, and{rn}n≥0are sequences inX introduced to take into account possible in inexact computation, and the sequences{an}n≥0,{bn}n≥0, and{cn}n≥0are sequences in0.1satisfying
∞ n0
an∞, ∞
n0
pn<∞, lim
n→∞qnlim
n→∞bnrn0. 3.5
Remark 3.4. Algorithm 3.1 in1,7,12,19,21,25,32, Algorithm 2.1 in8,27, and Algorithm 5.1 in9,11, the Ishikawa-type perturbed iterative algorithm in10, the Ishikawa-type perturbed iterative algorithm with errors in 20, Algorithms 3.1 and 3.2 in 22 are special cases of Algorithm 3.3in this paper.
Next, we study those conditions under which the approximate solutions un obtained fromAlgorithm 3.3converge strongly to the unique solutionu ∈ Xof the general nonlinear implicit variational inequality2.1, and the convergence, under suitable conditions, is stable.
Theorem 3.5. Let H : X → X be strongly monotone and Lipschitz continuous with constants s andh, respectively. LetHn :X → X be strongly monotone with constantsnfor eachn≥ 0 and let g:X→Xbe Lipschtiz continuous and strongly monotone with constantstandp, respectively. Assume thatm, A, B, C, D, E,:X→Xare Lipschitz continuous with constantsq, a, b, c, d,ande, respectively.
LetW, Wn :X×X →2X satisfy that for eachx∈X, W·, ExisH-monotone andWn·, Exis Hn-monotone for eachn≥0.LetN:X×X→Xbe Lipschitz continuous with constantsiandjwith respect to the first and second arguments, respectively. LetM:X×X →X be Lipschitz continuous with constantskandlwith respect to the first and second arguments, respectively. Suppose thatAis strongly monotone with constantαwith respect toHg−mand the first argument ofN,Cis relaxed Lipschitz with constantγ with respect to Hg−mand the first argument ofM, andD is relaxed
monotone with constantδwith respect toHg−mand the second argument ofM. Let
P
1−2pt2qηe, Ji2a2−T2, Tjb
h2tq2−2γk2c2
h2tq22δl2d2, Kα−s1−PT, Lh2tq2−s21−P2>0.
3.6
Let{xn}n≥0be any sequence inXand define{n}n≥0⊂0,∞by nxn1−
1−an
xnan
yn−g−m yn
RHWn
n·,Eyn,ρ
Y
yn pn, yn
1−bn
xnbn
zn−g−m zn
RHWn
n·,Ezn,ρ
Y
zn qn, zn
1−cn xncn
xn−g−m xn
RHWn
n·,Exn,ρ
Y
xn rn, ∀n≥0,
3.7
whereY is defined by3.1. If there exist positive constantsρ, η, andηnsatisfying
RHW·,x,ρz−RHW·,y,ρz≤ηx−y, ∀x, y, z∈X, 3.8 RHWn
n·,x,ρz−RHWn
n·,y,ρz≤ηnx−y, ∀x, y, z∈X, n≥0, 3.9
n→∞limRHWn
n·,Ex,ρ
Yx
−RHW·,Ex,ρ
Yx0, ∀x∈X, 3.10
n→∞limηnη, lim
n→∞sns, 3.11
Ps−1ρT <1, 3.12
and one of the following conditions:
ρ−KJ−1< J−1
K2−LJ, J >0,|K|>
LJ; 3.13
ρ−KJ−1>−J−1
K2−LJ, J <0, 3.14 then for any givenf ∈ X, the general nonlinear implicit variational inequality 2.1has a unique solutionu∈Xand the sequence{un}n≥0defined byAlgorithm 3.3converges strongly tou. Moreover, if there exists a constantβ >0 satisfying
an≥β, ∀n≥0, 3.15
then limn→∞xnuif and only if limn→∞n0.
Proof. First of all, we claim that the mappingGdefined by3.3has a unique fixed pointu∈X, whereλis a constant in0,1. Letx, ybe two arbitrary elements inX. Note thatgis Lipschtiz continuous and strongly monotone with constantstandp, respectively. It follows that
x−y−
gx−gy ≤
1−2pt2x−y. 3.16
SinceAis strongly monotone with constantαwith respect toHg−mand the first argument ofN, C is relaxed Lipschitz with constantγ with respect toHg−mand the first argument ofM, and Dis relaxed monotone with constantδ with respect toHg−mand the second argument ofM, it follows from the Lipschitz continuity ofA, B, C, D, andH, and the Lipschitz continuity ofNandMwith respect to the first and second arguments, respectively, that
yx−yy
≤H
g−mx
−H
g−my
−ρ N
Ax, Bx
−N
Ay, Bx ρN
Ay, Bx
−N
Ay, By ρH
g−mx
−H
g−my M
Cx, Dx
−M
Cy, Dx ρH
g−mx
−H
g−my
−M
Cy, Dx M
Cy, Dy
≤H
g−mx
−H
g−my2
−2ρ N
Ax, Bx
−N
Ay, Bx , H
g−mx
−H
g−my ρ2N
Ax, Bx
−N
Ay, Bx2 1/2ρjbx−y ρH
g−mx
−H
g−my2 2
M
Cx, Dx
−M
Cy, Dx , H
g−mx
−H
g−my M
Cx, Dx
−M
Cy, Dx2 1/2 ρH
g−mx
−H
g−my2
−2 M
Cy, Dx
−M
Cy, Dy , H
g−mx
−H
g−my M
Cy, Dx
−M
Cy, Dy2 1/2
≤
h2tq2−2αρρ2i2a2ρT
x−y.
3.17
In view ofLemma 2.5,3.3,3.6,3.8,3.16, and3.17,we deduce that Gx−Gy
≤1−λx−yλx−y−g−mxg−myλRHW·,Ex,ρ Yx
−RHW·,Ey,ρ Yy
≤ 1−λ
1−
1−2pt2−qx−yλRHW·,Ex,ρ Yx
−RHW·,Ey,ρ Yx λRHW·,Ex,ρ
Yx
−RHW·,Ey,ρ Yy
≤ 1−λ
1−
1−2pt2−q−ηex−yλs−1Yx−Yy
≤
1−λ1−θ
x−y,
3.18 where
θPs−1
h2tq2−2ραρ2i2a2ρT
>0. 3.19
In light of3.6,3.12, and3.19, we derive that θ <1⇐⇒
h2tq2−2ραρ2i2a2< s1−P−ρT ⇐⇒Jρ2−2Kρ <−L. 3.20 It follows from one of3.13and3.14that
θ <1. 3.21
Thus3.18implies thatGis a contraction mapping, and hence Ghas a unique fixed point u ∈X. ByLemma 3.1, we conclude that the general nonlinear implicit variational inequality 2.1possesses a unique solutionu∈Xand
u 1−cn
ucn
u−g−mu RHW·,Eu,ρ Yu
1−bn ubn
u−g−mu RHW·,Eu,ρ Yu
1−an uan
u−g−mu RHW·,Eu,ρ
Yu , ∀n≥0.
3.22
Next, we prove that limn→∞unu. Set θnPns−1n
h2tq2−2ραρ2i2a2ρT , Pn
1−2pt2qeηn, gnRHWn
n·,Eu,ρ
Yu
−RHW·,Eu,ρ
Yu, ∀n≥0.
3.23
In terms of3.11,3.19, and3.21, we know that limn→∞θnθ <1. Hence there exists some positive integerQsatisfying
θn<1
21θ<1, ∀n≥Q. 3.24
UsingLemma 2.5,Algorithm 3.3,3.22, and3.24, we know that forn > Q, wn−u
≤
1−cnun−ucnun−u−g−m un
g−mu RHWn
n·,Eun,ρ
Yun
−RHW·,Eu,ρ
Yu rn
≤ 1−cn
1−
1−2pt2−qun−u cnRHWn
n·,Eun,ρ
Yun
−RHWnn·,Eun,ρ
YuRHWn
n·,Eun,ρ
Yun
−RHWnn·,Eu,ρ Yu RHWn
n·,Eu,ρ
Yu
−RHW·,Eu,ρ
Yu rn
≤ 1−cn
1−
1−2pt2−qun−u cn
s−1n Y un
−YuηnE un
−Eugn rn
≤ 1−cn
1−
1−2pt2−qun−u cn
s−1n H
g−m un
−H
g−mu
−ρ N
A un, B
un
−N
Au, B un ρN
Au, B un
−N
Au, Bu ρH
g−m un
−H
g−mu M
C un
, D un
−M
Cu, D un ρH
g−m un
−H
g−mu
−M
Cu, D un
M
Cu, Du eηnun−ugn
rn
≤
1−cnun−ucnθnun−ucngnrn
≤un−ucngnrn.
3.25 Similarly, we conclude that
vn−u≤
1−bnun−ubnθnwn−ubngnqn
≤un−ubn
2gnrnqn, 3.26
un1−u≤
1−anun−uanθnvn−uangnpn
≤ 1−
1−θn
an un−uan
3gnqnbnrnpn
≤ 1−1
21−θan
un−uan
3gnqnbnrnpn 3.27 forn > Q. It is easy to see that limn→∞un−u0 byLemma 2.4,3.5,3.10, and3.27.
Assume that3.15holds. As in the proof of3.27, we easily deduce that 1−an
xnan
yn−g−m yn
RHWn
n·,Eyn,ρ
Y
yn pn−u
≤ 1−
1−θn
anxn−uan
3gnqnbnrnpn
≤
1−1
21−θβxn−u3gnqnbnrnpn
3.28
forn > Q.
Suppose that limn→∞xnu. By virtue of3.5,3.7,3.10, and3.28, we see that n≤xn1−u1−an
xnan
yn−g−m yn
RHWn
n·,Eyn,ρ
Y
yn pn−u
≤xn1−u
1−1
21−θβxn−u3gnqnbnrnpn−→0
3.29 asn→ ∞. Therefore, limn→∞n0.
Conversely, suppose that limn→∞0. It follows from3.7,3.22, and3.28that xn1−u
≤1−an
xnan
yn−g−m yn
RHWn
n·,Eyn,ρ
Y
yn pn−un
≤
1−1
21−θβxn−u3gnqnbnrnpnn
3.30
for n > Q. Using 3.5, 3.10, 3.30, and Lemma 2.4, we infer that limn→∞xn u. This completes the proof.
Theorem 3.6. LetH, W,{Hn}n≥0,{Wn}n≥0, g, A, B, C, D, E, J, T, L,{xn}n≥0, and{n}n≥0 be as inTheorem 3.5and
P
1−2p− q2t2ηe. 3.31
Letm:X→Xbe generalized pseudocontractive with constantwith respect toI−gand be Lipschitz continuous with constantq. If there exist positive constantsρ, η, andηnsatisfying3.8–3.12and one of 3.13and3.14, then for any givenf ∈X, the general nonlinear implicit variational inequality 2.1 has a unique solution u ∈ X and the sequence {un}n≥0 defined by Algorithm 3.3 converges strongly tou. Moreover, if3.15holds, then limn→∞xnuif and only if limn→∞n0.
Proof. Becausemis generalized pseudocontractive with constantwith respect toI−gand Lipschitz continuous with constantq, g is Lipschtiz continuous and strongly monotone with constantstandp, respectively, it follows that
I−gx−I−gy mx−my
mx−my22mx−my,I−gx−I−gyI−gx−I−gy2 1/2
≤
q22x−y2x−y2−2gx−gy, x−ygx−gy2 1/2
≤
1−2p− q2t2x−y, ∀x, y∈X.
3.32 The rest of the proof now follows that as in the proof ofTheorem 3.5. This completes the proof.
Theorem 3.7. Let H, W,{Hn}n≥0, {Wn}n≥0, g, m, B, E, J, K, {xn}n≥0, and {n}n≥0 be as in Theorem 3.5, and
P
1s−1
1−2pt2q
ηes−1tq
1−2sh2, T jb
12δl2d2
1−2γ k2c2, L1−s21−P2>0.
3.33
LetA : X → X be Lipschitz continuous with constant a and strongly monotone with constantα with respect toI and the first argument ofN. LetC:X →Xbe Lipschitz continuous with constant
cand relaxed Lipschitz with constantγ with respect toIand the first argument ofM. Assume that D:X→Xis Lipschitz continuous with constantdand relaxed monotone with constantδwith respect toIand the second argument ofM. If there exist positive constantsρ, η, andηnsatisfying3.8–3.12 and one of 3.13and3.14, then for any givenf ∈ X, the general nonlinear implicit variational inequality 2.1 has a unique solution u ∈ X and the sequence {un}n≥0 defined by Algorithm 3.3 converges strongly tou. Moreover, if3.15holds, then limn→∞xnuif and only if limn→∞0.
Proof. Notice that H
g−mx
−H
g−my
−ρ N
Ax, Bx
−N
Ay, Bx
−ρ N
Ay, Bx
−N
Ay, By ρ M
Cx, Dx
−M
Cy, Dx ρ
M
Cy, Dx
−M
Cy, Dy
≤H
g−mx
−H
g−my
−g−mx g−my g−mx−g−my−xy
x−y−ρ N
Ax, Bx
−N
Ay, Bx ρjb ρx−yM
Cx, Dx
−M
Cy, Dx ρx−y−M
Cy, Dx M
Cy, Dy
≤
tq
1−2sh2
1−2pt2q
1−2ραρ2i2a2ρTx−y
3.34
for anyx, y∈X. The rest of the proof is identical with the proof ofTheorem 3.5. This completes the proof.
Following similar arguments as in the proof of Theorems3.5,3.6, and 3.7, we obtain immediately the result below:
Theorem 3.8. LetH, W,{Hn}n≥0,{Wn}n≥0, g, A, B, C, D, E, J, K, T, L,{xn}n≥0, and{n}n≥0be as in Theorem 3.7, andmbe as inTheorem 3.6, and
P
1s−1
1−2p− q2t2ηes−1tq
1−2sh2. 3.35 If there exist positive constantsρ, η, andηnsatisfying3.8–3.12and one of 3.13and3.14, then for any givenf ∈X, the general nonlinear implicit variational inequality2.1has a unique solution u∈Xand the sequence{un}n≥0defined byAlgorithm 3.3converges strongly tou. Moreover, if 3.15 holds, then limn→∞xnuif and only if limn→∞0.
Remark 3.9. Theorems3.5–3.8establish both the existence and uniqueness of solutions for the general nonlinear implicit variational inclusion2.1and show the convergence and stability of the perturbed three-step iterative process with errors under certain conditions.
Remark 3.10. Theorems3.5–3.8extend, improve, and unify Theorem 3.4 in1,6, Theorem 2.1 in8, Theorem 3.1 in7,12,21,22,25,32, Theorem 2.3 in24, Theorem 2.2 in26, Theorem 5.19,11, Theorem 4.1 in10,20, Theorems 4.1–4.3 in19, Theorems 1 and 2 in23, and Theorems 3.1–3.6 in3,13.
Acknowledgment
This work was supported by the Science Research Foundation of Educational Department of Liaoning Province20060467and the Korea Research Foundation Grant funded by the Korean GovernmentMOEHRD, Basic Research Promotion Fund KRF-2006-312-C00026.
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