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,2 and 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
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 that Xis a real Hilbert space endowed with a norm · and an inner product·,·, respectively, 2X stands 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: find u∈Xsuch that
f ∈NAu, Bu−MCu, DuWg−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∈NAu, BuWg−mu, u, 2.2 which is called the completely generalized strongly nonlinear implicit quasivariational inclusionin20.
BIff 0,EI,Nx, y Mx, y xfor anyx, y ∈X, then problem2.1is equivalent to findingu∈Xsuch that
0∈Au−Cu Wg−mu, u, 2.3 which is called thegeneralized nonlinear implicit quasivariational inclusionin10. 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 Wg−mu, 2.4
DIf f 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 element f, 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 beLipschitz continuousandstrongly monotoneif there exist positive constants
sandtsatisfying, respectively,
gx−gy≤sx−y,
gx−gy, x−y≥tx−y2, ∀x, y∈X; 2.6 a2W is said to bemaximal monotoneifWis monotone andIρWX Xfor anyρ >0; a3W is said to beH-monotoneifW is monotone andHρWX Xfor anyρ >0; a4b is called strongly monotonewith respect to H and the first argument of N if there
exists a positive constantssatisfying
Nbx, u−Nby, u, Hx−Hy≥sx−y2, ∀x, y, u∈X; 2.7 a5bis calledrelaxed Lipschitzwith respect toHand the first argument ofNif there exists
a positive constantssatisfying
Nbx, u−Nby, u, Hx−Hy≤ −sx−y2, ∀x, y, u∈X; 2.8 a6b is calledrelaxed monotonewith respect toH and the second argument ofN if there
exists a positive constantssatisfying
Nu, bx−Nu, 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
Similarly, we can define the Lipschitz continuity of N with respect to the second arg-ument. On the other hand, if Nx, y x for any x, y ∈ X, then Definition 2.1 reduces 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 ifW is
H-monotone andHis strictly monotone, thenW is maximal monotone.
Definition 2.2see7. LetH : X→Xbe a strictly monotone operator and letW :X →2X be
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 in X. Suppose that
Fg {x∈ X : x gx}/∅and{xn}n≥0converges to some u∈Fg. Let{zn}n≥0 ⊂ Xand
nzn1−fg, znfor alln≥0. Iflimn→∞n0 implies that limn→∞zn u, then the iteration procedure defined byxn1fg, xnis said to beg-stableorstablewith respect tog.
Lemma 2.4see35. Let{an}n≥0,{bn}n≥0, and{cn}n≥0be nonnegative sequences satisfying
an1≤
1−tnantnbncn, ∀n≥0, 2.13
where{tn}n≥0⊂0,1,n∞0tn∞,limn→∞bn 0, and ∞
n0cn<∞. Thenlimn→∞an0.
Lemma 2.5 see 7. Let H : X → X be a strongly monotone operator with constant r and let
W : X → 2X be anH-monotone operator. Then the resolvent operator RH
W,ρ : X → Xis 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 Hg−mx−ρNAx, BxρMCx, Dxρf, ∀x∈X, 3.1
where H, 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
b3the mappingG:X→Xdefined by
Gx 1−λxλx−g−mx RHW·,Ex,ρYx , ∀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 on Lemma 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→ 2X satisfy that for anyx∈X, W·, ExisH-monotone andWn·, ExisHn-monotone for eachn≥0. Givenf, u0∈X, the iterative sequence{un}n≥0is defined by
wn1−cnuncnun−g−munRHn
Wn·,Eun,ρ
Yun rn,
vn
1−bn
unbn
wn−g
wn
mwn
RHn
Wn·,Ewn,ρ
Ywn qn,
un1
1−anunanvn−g−mvnRHn
Wn·,Evn,ρ
Yvn 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→∞qnnlim→∞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 from Algorithm 3.3converge strongly to the unique solutionu ∈ X of 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
monotone with constantδwith respect toHg−mand the second argument ofM. Let
P
1−2pt2qηe, J i2a2−T2,
T jb
h2tq2−2γ k2c2
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−anxnanyn−g−mynRHn
Wn·,Eyn,ρ
Yyn pn,
yn1−bnxnbnzn−g−mznRHn
Wn·,Ezn,ρ
Yzn qn,
zn1−cnxncnxn−g−mxnRHn
Wn·,Exn,ρ
Yxn rn, ∀n≥0,
3.7
whereY is defined by3.1. If there exist positive constantsρ, η, andηnsatisfying
RH
W·,x,ρz−RHW·,y,ρz≤ηx−y, ∀x, y, z∈X, 3.8 RHn
Wn·,x,ρz−R Hn
Wn·,y,ρz≤ηnx−y, ∀x, y, z∈X, n≥0, 3.9
lim
n→∞R
Hn Wn·,Ex,ρ
Yx−RHW·,Ex,ρYx0, ∀x∈X, 3.10 lim
n→∞ηnη, nlim→∞sn s, 3.11
Ps−1ρT <1, 3.12
and one of the following conditions:
ρ−KJ−1< J−1K2−LJ, J >0,|K|>LJ; 3.13
ρ−KJ−1>−J−1K2−LJ, J <0, 3.14
then for any given f ∈ X, the general nonlinear implicit variational inequality 2.1 has a unique solutionu∈Xand the sequence{un}n≥0defined byAlgorithm 3.3converges strongly tou. Moreover, if there exists a constantβ >0satisfying
an≥β, ∀n≥0, 3.15
thenlimn→∞xnuif and only iflimn→∞n 0.
Proof. First of all, we claim that the mappingGdefined by3.3has a unique fixed pointu∈X, whereλis a constant in0,1. Letx, y be two arbitrary elements inX. Note thatg is Lipschtiz continuous and strongly monotone with constantstandp, respectively. It follows that
x−y−
gx−gy ≤
SinceAis strongly monotone with constantαwith respect toHg−mand the first argument of N, Cis relaxed Lipschitz with constantγ with respect toHg−mand the first argument of M, andD is relaxed monotone with constantδ with respect to Hg −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
≤Hg−mx−Hg−my−ρNAx, Bx−NAy, Bx
ρNAy, Bx−NAy, By
ρHg−mx−Hg−myMCx, Dx−MCy, Dx
ρHg−mx−Hg−my−MCy, DxMCy, Dy ≤Hg−mx−Hg−my2
−2ρNAx, Bx−NAy, Bx, Hg−mx−Hg−my
ρ2NAx, Bx−NAy, Bx2 1/2ρjbx−y
ρHg−mx−Hg−my2
2MCx, Dx−MCy, Dx, Hg−mx−Hg−my
MCx, Dx−MCy, Dx2 1/2
ρHg−mx−Hg−my2
−2MCy, Dx−MCy, Dy, Hg−mx−Hg−my
MCy, Dx−MCy, Dy2 1/2 ≤h2tq2−2αρρ2i2a2ρTx−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−myλRWH·,Ex,ρYx−RHW·,Ey,ρYy ≤1−λ1−
1−2pt2−qx−yλRH W·,Ex,ρ
Yx−RHW·,Ey,ρYx
λRHW·,Ex,ρYx−RHW·,Ey,ρYy ≤1−λ1−
1−2pt2−q−ηex−yλs−1Yx−Yy
≤1−λ1−θx−y,
3.18
where
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
Thus 3.18implies that G is a contraction mapping, and henceG has a unique fixed point
u ∈ X. ByLemma 3.1, we conclude that the general nonlinear implicit variational inequality 2.1possesses a unique solutionu∈Xand
u1−cnucnu−g−mu RHW·,Eu,ρYu
1−bnubnu−g−mu RHW·,Eu,ρYu
1−anuanu−g−mu RHW·,Eu,ρYu , ∀n≥0.
3.22
Next, we prove that limn→∞unu. Set
θnPns−n1
h2tq2−2ραρ2i2a2ρT,
Pn
1−2pt2qeηn,
gnRHn Wn·,Eu,ρ
Yu−RHW·,Eu,ρYu, ∀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
RHn
Wn·,Eun,ρ
Yun−RHW·,Eu,ρYu rn ≤1−cn1−
1−2pt2−qun−u
cnRHn
Wn·,Eun,ρ
Yun−RHn
Wn·,Eun,ρ
YuRHn
Wn·,Eun,ρ
Yun−RHn Wn·,Eu,ρ
Yu
RHn Wn·,Eu,ρ
≤1−cn1−
1−2pt2−qun−u
cns−n1Y
un−YuηnEun−Eugn rn ≤1−cn1−
1−2pt2−qun−u
cns−n1Hg−mun
−Hg−mu−ρNAun, Bun−NAu, Bun
ρNAu, Bun−NAu, Bu
ρHg−mun−Hg−muMCun, Dun−MCu, Dun
ρHg−mun−Hg−mu−MCu, DunMCu, Du
eηnun−ugnrn
≤1−cnun−ucnθnun−ucngnrn ≤un−ucngnrn.
3.25
Similarly, we conclude that
vn−u≤
1−bnun−ubnθnwn−ubngnqn
≤un−ubn2gnrnqn, 3.26 un1−u≤
1−anun−uanθnvn−uangnpn
≤1−1−θnan un−uan3gnqnbnrnpn ≤
1− 1
21−θan
un−uan3gnqnbnrnpn
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
xnanyn−g−mynRHn
Wn·,Eyn,ρ
Yyn pn−u ≤1−1−θnanxn−uan3gnqnbnrnpn ≤
1− 1
21−θβ
xn−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
xnanyn−g−mynRHn
Wn·,Eyn,ρ
Yyn pn−u ≤xn1−u
1− 1
21−θβ
xn−u3gnqnbnrnpn−→0
Conversely, suppose that limn→∞0. It follows from3.7,3.22, and3.28that
xn1−u ≤1−an
xnan
yn−g−m
yn
RHn
Wn·,Eyn,ρ
Yyn pn−un ≤
1−1
21−θβ
xn−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, if 3.15holds, thenlimn→∞xnuif and only iflimn→∞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−11−2pt2qηes−1tq1−2sh2,
T jb12δl2d21−2γk2c2,
L1−s21−P2>0.
3.33
Let A : X → X be Lipschitz continuous with constanta and strongly monotone with constant α
cand relaxed Lipschitz with constant γ with respect toI and 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 3.13and 3.14, then for any given f ∈ 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, if 3.15holds, thenlimn→∞xnuif and only iflimn→∞0.
Proof. Notice that
H
g−mx−Hg−my−ρNAx, Bx−NAy, Bx −ρNAy, Bx−NAy, By ρMCx, Dx−MCy, Dx
ρMCy, Dx−MCy, Dy
≤Hg−mx−Hg−my−g−mx g−my
g−mx−g−my−xy
x−y−ρNAx, Bx−NAy, Bx ρjb
ρx−yMCx, Dx−MCy, Dx
ρx−y−MCy, DxMCy, Dy ≤tq1−2sh2
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 Theorems 3.5, 3.6, and 3.7, we obtain immediately the result below:
Theorem 3.8. Let H, W,{Hn}n≥0,{Wn}n≥0, g, A, B, C, D, E, J, K, T, L,{xn}n≥0, and{n}n≥0 be as in
Theorem 3.7, andmbe as inTheorem 3.6, and
P 1s−11−2p− q2t2ηes−1tq1−2sh2. 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, thenlimn→∞xnuif and only iflimn→∞0.
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 in 19, Theorems 1 and 2 in 23, 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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