Volume 2009, Article ID 124953,16pages doi:10.1155/2009/124953
Research Article
Generalized Strongly Nonlinear Implicit
Quasivariational Inequalities
Salahuddin and M. K. Ahmad
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
Correspondence should be addressed to Salahuddin,[email protected] Received 11 February 2009; Accepted 17 June 2009
Recommended by Ram U. Verma
We prove an existence theorem for solution of generalized strongly nonlinear implicit quasivaria-tional inequality problems and convergence of iterative sequences with errors, involving Lipschitz continuous, generalized pseudocontractive and generalized g-pseudocontractive mappings in Hilbert spaces.
Copyrightq2009 Salahuddin and M. K. Ahmad. 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
Variational inequality was initially studied by Stampacchia1in 1964. Since then, it has been extensively studied because of its crucial role in the study of mechanics, physics, economics, transportation and engineering sciences, and optimization and control. Thanks to its wide applications, the classical variational inequality has been well studied and generalized in various directions. For details, readers are referred to2–5and the references therein.
2. Preliminaries
LetHbe a real Hilbert space with norm · and inner product·,·. For a nonempty closed
convex subsetK⊂H, letPKbe the projection ofHontoK. LetK:H → 2Hbe a set valued
mapping with nonempty closed convex values,F, g, G, A:H → HandN:H×H×H → H
be the mappings. We consider the following problem. Findx∈H, such thatgx∈Kxand
gx−NAx, Gx, Fx, y−gx≥0, ∀y∈Kx. 2.1
The problem2.1is called thegeneralized strongly nonlinear implicit quasi-variational inequality problem.
Special Cases
iIfKx mx K, for allx∈H, whereKis a nonempty closed convex subset ofHand
m: H → His a mapping, then the problem2.1is equivalent to findingx∈ Hsuch that
gx−mx∈Kand
gx−NAx, Gx, Fx, y−gx≥0, ∀y∈Kmx, 2.2
the problem2.2is called generalized nonlinear quasi-variational inequality problem.
iiIf we assume A, G, Fas identity mappings, then2.1reduces to the problem of findingx∈Hsuch thatgx∈Kxand
gx−Nx, x, x, y−gx≥0, ∀y∈Kx, 2.3
which is known as general implicit nonlinear quasi-variational inequality problem.
iiiIf we assumeNx, x, x Nx, x, then2.3reduces to the following problem of findingx∈Hsuch thatgx∈Kxand
gx−Nx, x, y−gx≥0, ∀y∈Kx, 2.4
which is known as generalized implicit nonlinear quasi-variational inequality problem, a variant form as can be seen in20, equation2.6.
ivIf we assumegx−Nx, x x−Nx, x, then2.4reduces to the following problem of findingx∈Hsuch thatgx∈Kxand
x−Nx, x, y−gx≥0, ∀y∈Kx. 2.5
The problem 2.5 is called the generalized strongly nonlinear implicit quasi-variational inequality problem, considered and studied by Cho et al.9.
vIfg ≡ I,I an identity mapping, then2.5is equivalent to findingx∈Kxsuch that
Problem2.6is called generalized strongly nonlinear quasi-variational inequality problem, see special cases of Cho et al.9.
viIfKx K,Ka nonempty closed convex subset ofHandNx, x Txfor all
x ∈ H, where T : H → H a nonlinear mapping, then the problem 2.6is equivalent to findingx∈Hsuch that
x−Tx, y−x ≥0, ∀y∈K, 2.7
which is a nonlinear variational inequality, considered by Verma17.
viiIfx−Tx Tx, for allx ∈ H, then 2.7reduces to the following problem for findingx∈Hsuch that
Tx, y−x ≥0, ∀y∈K, 2.8
which is a classical variational inequality considered by1,4,5.
Now, we recall the following iterative process due to Ishikawa13, Mann14, Noor
15and Liu21.
1LetKbe a nonempty convex subset ofHandT :K → Xa mapping. The sequence {xn}, defined by
x0∈K, 2.9
xn1 1−αnxnαnTyn
yn
1−βn
xnβnTzn
zn
1−γn
xnγnTxn,
2.10
n ≥ 0, is called the three-step iterative process, where {αn},{βn}, and {γn} are three real
sequences in0,1satisfying some conditions.
2In particular, ifγn0 for alln≥0, then{xn}, defined by
x0∈K,
xn1 1−αnxnαnTyn
yn
1−βn
xnβnTxn,
2.11
n≥0, is called the Ishikawa iterative process, where{αn}and{βn}are two real sequences in
0,1satisfying some conditions.
3In particular, ifβn0 for alln≥0, then{xn}defined by
x0∈K
xn1 1−αnxnαnTxn,
2.12
Recently Liu 21introduced the concept of three-step iterative process with errors which is the generalization of Ishikawa13and Mann14iterative process, for nonlinear strongly accretive mappings as follows.
4For a nonempty subsetKof a Banach spacesXand a mappingT : K → X, the sequence{xn}, defined by
x0∈K,
xn1 1−αnxnαnTynun,
yn
1−βn
xnβnTznvn,
zn
1−γn
xnγnTxnwn,
2.13
n≥0, is called the three-step iterative process with errors. Here{un},{vn},and{wn}are three
summable sequences inXi.e.,∞n0un<∞,
∞
n0vn<∞and
∞
n0wn<∞, and
{αn},{βn},and{γn}are three sequences in0,1satisfying certain restrictions.
5In particular, ifγn0 forn≥0 andwn 0. The sequence{xn}defined by
x0∈K,
xn1 1−αnxnαnTynun,
yn
1−βn
xnβnTznvn,
2.14
n0,1,2, . . ., is called the Ishikawa iterative process with errors. Here{un}and{vn}are two
summable sequences inXi.e.,∞n0un<∞and∞n0vn<∞;{αn}and{βn}are two
sequences in0,1satisfying certain restrictions.
6In particular, ifβn0 andvn0 for alln≥0. The sequence{xn},defined by
x0∈K, 2.15
xn1 1−αnxnαnTynun 2.16
forn0,1,2, . . ., is called the Mann iterative process with errors, where{un}is a summable
sequence inXand{αn}a sequence in0,1satisfying certain restrictions.
However, in a recent paper 19Xu pointed out that the definitions of Liu21are against the randomness of the errors and revised the definitions of Liu21as follows.
7 Let K be a nonempty convex subset of a Banach space X and T : K → X a mapping. For any givenx0∈K, the sequence{xn},defined by
x0∈K, 2.17
xn1αnxnβnTynγnun,
ynαx nβTz nγnvn,
znαxnβnTxnγnwn
forn0,1,2, . . ., is called the three-step iterative process with errors, where{un},{vn},and
{wn}are three bounded sequences inKand{αn},{βn},{γn},{αn},{βn},{γn},{αn},{βn},and
{γn}are nine sequences in0,1satisfying the conditions
αnβnγn1, αnβnγn1, αnβnγn1 forn≥0. 2.19
8Ifβnγn0 forn0,1,2. . .the sequence{xn},defined by
x0∈K,
xn1αnxnβnTynγnun,
ynαx nβTx nγnvn
2.20
forn0,1,2, . . ., is called the Ishikawa iterative process with errors, where{un}and{vn}are
two bounded sequences inK,{αn},{βn},{γn},{αn},{βn},and{γn}are six sequences in0,1
satisfying the conditions
αnβnγn1, αnβnγn1 forn≥0. 2.21
9Ifβnγn0 forn0,1,2. . ., the sequence{xn}defined by
x0∈K,
xn1αnxnβnTxnγnun,
2.22
forn0,1,2, . . . ,is called the Mann iterative process with errors. For our main results, we need the following lemmas.
Lemma 2.1see3. IfK ⊂ H is a closed convex subset andx ∈ H a given point, thenz ∈ K
satisfies the inequality
x−z, y−x≥0, ∀y∈K, 2.23
if and only if
xPKz, 2.24
wherePKis the projection ofHontoK.
Lemma 2.2see10. The mappingPKdefined by2.24is nonexpansive, that is,
Lemma 2.3see10. IfKu mu Kand K ⊂ H is a closed convex subset, then for any
u, v∈H, one has
PKuv mu PKv−mu. 2.26
Lemma 2.4see21. Letan,bnandcnbe three nonnegative real sequences satisfying
an1 1−tnanbncn, forn≥0,
tn∈0,1,
∞
n0
tn∞, bnOtn,
∞
n0
cn<∞.
2.27
Then
lim
n→ ∞an0. 2.28
By Lemma 2.1, we know that the generalized strongly nonlinear implicit quasi-variational inequality 2.1has a unique solution if and only if the mappingQ : H → H
by
Qx x−gx PKx gx−tgx−NAx, Gx, Fx 2.29
has a unique fixed point, wheret >0 is a constant.
3. Main Results
In this section, we establish an existence theorem for solution of generalized strongly nonlinear implicit quasi-variational inequality problems and convergence of the iterative sequences generated by2.18. First, we give some definitions.
Definition 3.1. A mappingT : H → His said to be generalized pseudo-contractive if there exists a constantr >0 such that
Tx−Ty2≤r2x−y2Tx−Ty−rx−y2, ∀x, y∈H. 3.1
It is easy to check that3.1is equivalent to
Tx−Ty, x−y≤rx−y2. 3.2
Forr 1 in3.1, we get the usual concept of pseudo-contractive ofT, introduced by Browder and Petryshyn10, that is,
Definition 3.2. LetA:H → HandN:H×H×H → Hbe the mappings. The mappingN
is said to be as follows.
iGeneralized pseudo-contractive with respect toAin the first argument ofN, if there exists a constantp >0 such that
NAx, z, z−NAy, z, z, x−y≤px−y2 ∀x, y, z∈H. 3.4
iiLipschitz continuous with respect to the first argument of N if there exists a constants >0 such that
Nx, z, z−Ny, z, z≤sx−y ∀x, y, z∈H. 3.5
In a similar way, we can define Lipschitz continuity of N with respect to the second and third arguments.
iiiAis also said to be Lipschitz continuous if there exists a constantη >0 such that Ax−Ay≤ηx−y ∀x, y∈H. 3.6
Definition 3.3. Letg, G:H → Hbe the mappings. A mappingN:H×H×H → His said to be the generalizedg-pseudo-contractive with respect to the second argument ofN, if there exists a constantq >0 such that
Nz, Gx, z−Nz, Gy, z, gx−gy≤qgx−gy2 ∀x, y, z∈H. 3.7
Definition 3.4. LetK :H → 2Hbe a set-valued mapping such that for eachx∈H,Kxis a
nonempty closed convex subset ofH. The projectionPKxis said to be Lipschitz continuous
if there exists a constantξ >0 such that
PKxz−PKyz ≤ξx−y, ∀x, y, z∈H. 3.8
Remark 3.5. In many important applications,Kuhas the following form:
Kx mx K, 3.9
wherem :H → His a single-valued mapping andK a nonempty closed convex subset of
H. Ifmis Lipschitz continuous with constantχ >0, then fromLemma 2.3,PKxis Lipschitz
continuous with Lipschitz constantξ2χ.
Now, we give the main result of this paper.
Theorem 3.6. LetHbe a real Hilbert space andK:H → 2Ha set-valued mapping with nonempty
continuous with positive constantsλandμ,respectively. A trimappingN : H×H×H → H is generalized pseudo-contractive with respect toAin the first argument ofNwith constantp >0and generalizedg-pseudo-contractive with respect toGin the second argument ofNwith constantq >0, Lipschitz continuous with respect to the first, second, and third arguments with positive constants
s, δ, ζ,respectively. Suppose thatPKxis Lipschitz continuous with constantξ >0. Let{un},{vn},
and{wn}be the three bounded sequences inHand{αn},{βn},{γn},{αn},{βn},{γn},{αn},{βn},
and{γn}are sequences in0,1satisfying the following conditions:
1αnβnγnαnβnγnαnβnγn1, n≥0,
2limn→ ∞γnlimn→ ∞γnlimn→ ∞γn0,
3n∞0βn∞,
∞
n0γn<∞.
If the following conditions hold:
t−hΩ−p−h s2η2−h2
<
4Ω−p−h2−s2η2−h2Ω2−Ω
s2η2−h2 ,
hΩ> phsη−hsηhΩ2−Ω, hΩ> ph, h < sη
3.10
whereΩ 2λξ, hϕζd,andθ Ω 12ptt2s2η2th <1.
Then there exists a uniquex∈Hsatisfying the generalized strongly nonlinear implicit quasi-variational inequality2.1andxn → xasn → ∞, where{xn}is the three-step iteration process
with errors defined as follows:
x0∈H,
xn1αnxnβn
yn−g
yn
PKyn g
yn
−tgyn
−NAyn, Gyn, Fyn
γnun,
ynαnxnβn
zn−gzn PKzn gzn−t
gzn−NAzn, Gzn, Fzn
γnvn,
znαnxnβn
xn−gxn PKxn gxn−t
gxn−NAxn, Gxn, Fxn
γnwn
3.11
forn0,1,2, . . . .
Proof. We first prove that the generalized strongly nonlinear implicit quasi-variational inequality2.1has a unique solution. ByLemma 2.1, it is sufficient to prove the mapping defined by
Qx x−gx PKx gx−tgx−NAx, Gx, Fx 3.12
Letx, y be two arbitrary points inH. FromLemma 2.2and Lipschitz continuity of
PKuandI−g, we have
Qx−Qy
x−gx PKx gx−t
gx−NAx, Gx, Fx
−ygy−PKy g
y−tgy−NAy, Gy, Fy
≤x−gx−y−gy
PKx gx−tgx−NAx, Gx, Fx−PKx gy−tgy−NAy, Gy, Fy
PKx gy−tgy−NAy, Gy, Fy−PKy gy−tgy−NAy, Gy, Fy
≤2x−gx−y−gyx−ytNAx, Gx, Fx−NAy, Gx, Fx
tgx−gy−NAy, Gx, Fx−NAy, Gy, Fx
tNAy, Gy, Fx−NAy, Gy, Fyξx−y
≤2λx−yx−ytNAx, Gx, Fx−NAy, Gx, Fx
tgx−gy−NAy, Gx, Fx−NAy, Gy, Fx
tNAy, Gy, Fx−NAy, Gy, Fyξx−y
≤2λξx−yx−ytNAx, Gx, Fx−NAy, Gx, Fx
tgx−gy−NAy, Gx, Fx−NAy, Gy, Fx
tNAy, Gy, Fx−NAy, Gy, Fy.
3.13
Since N is generalized pseudo-contractive with respect to A in the first argument ofN and Lipschitz continuous with respect to first argument ofN and alsoAis Lipschitz continuous, we have
x−ytNAx, Gx, Fx−NAy, Gx, Fx2
x−y22tx−y, NAx, Gx, Fx−NAy, Gx, Fx
t2NAx, Gx, Fx−NAy, Gx, Fx2
≤ x−y22tpx−y2t2s2Ax−Ay2
≤ x−y22tpx−y2t2s2η2x−y2
≤12tpt2s2η2x−y2.
Again sinceN is generalized g-pseudo-contractive with respect toG in the second argument ofN and Lipschitz continuous with respect to second argument ofN andG is Lipschitz continuous, we have
gx−gy−NAy, Gx, Fx−NAy, Gy, Fx2
gx−gy2−2gx−gy, NAy, Gx, Fx−NAy, Gy, Fx
NAy, Gx, Fx−NAy, Gy, Fx2
≤μ2x−y2−2qgx−gy2δ2Gx−Gy2
≤μ2x−y2−2qμ2x−y2δ2σ2x−y2
≤μ21−2qδ2σ2x−y2,
3.15
NAy, Gy, Fx−NAy, Gy, Fy ≤ζFx−Fy ≤ζdx−y. 3.16
It follows from3.13–3.16that
Qx−Qy ≤θx−y, ∀x, y∈H, 3.17
where
θ Ω
12ptt2s2η2th,
ϕ
μ21−2qδ2σ2,
hϕζd.
3.18
From3.10, we know that 0< θ <1 and soQhas a unique fixed pointx∈H, which is a unique solution of the generalized strongly nonlinear implicit quasi-variational inequality
2.1.
Now we prove that{xn}converges tox. In fact, it follows from3.11andx ∈ Qx
that
xn1−x
αnxnβn yn−g
yn
PKyn
gyn
−tgyn
−NAyn, Gyn, Fyn
γnun−x
≤αnxnβn yn−g
yn
PKyn
gyn
−tgyn
−NAyn, Gyn, Fyn
γnun
−αnxβn
x−gx PKx gx−tgx−NAx, Gx, Fx−γnx
≤αnxn−xβnQ
yn
−Qxγnun−x.
From3.17and3.19, it follows that
xn1−x ≤αnxn−xθβnyn−xγnun−x. 3.20
Similarly, we have
yn−x
αnxnβn zn−gzn PKzn
gzn−t
gzn−NAzn, Gzn, Fzn
γnvn−x
αnxn−x βn zn−gzn PKzn
gzn−t
gzn−NAzn, Gzn, Fzn
−β x−gx PKxgx−tgx−NAx, Gx, Fxγnvn−x
≤αnxn−xβnQzn−Qxγnvn−x
≤αnxn−xβnθzn−xγnvn−x.
3.21
Again,
zn−x ≤αnxn−xβnθxn−xγnwn−x. 3.22
Let
Mmax
sup
n
un−x ,sup n
vn−x,sup n
wn−x, n≥0
. 3.23
ThenM <∞and
zn−x ≤αnxn−xβnθxn−xγnM. 3.24
Similarly, we deduce from3.21the following:
yn−x ≤αnxn−xβnθαnxn−xβnβnθ2xn−xβnθγnMγnM, 3.25
xn1−x ≤αnxn−xβnθyn−xMγn ∀n≥0. 3.26
From the above inequalities, we get
xn1−x ≤αnxn−xβnθαnxn−xβnθ2βnαnxn−x
βnθ3βnβnxn−xβnθ2βnγnMβnθMγnMγn
≤αnθβn
αnθβn
αnθβn
xn−xMθβn
γnβnγnθ
Mγn
≤anxn−xbncn,
where
anαnθβn
αnθβn
αnθβn
, bnθMβn
γnβnγnθ
, cMγn. 3.28
Since 0< θ <1, it follows from conditions1and3that
anαnθβn
αnβnθ
αnθβn
≤1−βnθβn≤1−1−θβn, 3.29
θMβn
γnβnγnθ
≤Mγnγn
βn. 3.30
Therefore,
xn1−x ≤
1−1−θβn
xn−xM
γnγn
γnM. 3.31
From 3.29-3.31and Lemma 2.4, we know that{xn} converges to the solutionx.
This completes the proof.
Remark 3.7. We now deduceTheorem 3.6in the direction of Ishikawa iteration.
Theorem 3.8. Let H be a real Hilbert space and K : H → 2H a set-valued mapping with the
nonempty closed convex values. LetA, G, F, g,andNbe the same as inTheorem 3.6. Suppose that
PKxis Lipschitz continuous with constantξ >0. Let{un}and{vn}be the two bounded sequences
inHand{αn},{βn},{γn},{αn},{βn},and{γn}be six sequences in0,1satisfying the following
conditions:
1αnβnγn1, αnβnγn1, n≥0
2limn→ ∞βnlimn→ ∞βnlimn→ ∞γn0,
3n∞0βn∞, n∞0γn<∞.
If the following conditions holds:
θ Ω
12ptt2s2η2th <1, hϕζd, Ω 2λζ,
ϕ
μ21−2qδ2σ2
t−hΩ−p−h s2η2−h2
<
hΩ−p−h2−s2η2−h2Ω2−Ω
s2η2−h2 ,
hΩ> phsη−hsηhΩ2−Ω, hΩ> ph, h < sη.
Then there exists a uniquex∈Hsatisfying the generalized strongly nonlinear implicit quasi-variational inequality2.1andxn → xas n → ∞, where{xn}is the Ishikawa iteration process
with errors defined as follows:
x0∈H,
xn1αnxnβn
yn−g
yn
PKy
n g
yn
−tgyn
−NAyn, Gyn, Fyn
γnun,
ynαnxnβn
xn−gxn PKxn gxn−t
gxn−NAxn, Gxn, Fxn
γnvn
3.33
forn0,1,2, . . . .
Remark 3.9. We can also deduceTheorem 3.6in the direction of2.16.
Theorem 3.10. Let H, K, N, G, A, F, g, and PKx be the same as inTheorem 3.6. Let{un} be a
bounded sequence inHand{αn},{βn},and{γn}be three sequences in0,1satisfying the following
conditions:
1αnβnγn1forn≥0,
2limn→ ∞βn0,
3n∞0βn∞andn∞0γn<∞.
If the conditions of 3.10hold, then there exists a uniquex ∈ Hsatisfying the generalized strongly nonlinear implicit quasi-variational inequality2.1andxn → xasn → ∞, where{xn}is
the Mann iterative process with errors defined as follows:
x0∈H
xn1αnxnβn
xn−gxn PKxn gxn−t
gxn−NAxn, Gxn, Fxn
γnun
3.34
forn0,1,2, . . . .
Our results can be further improved in the direction of2.25.
Theorem 3.11. LetHbe a real Hilbert space andK:H → 2Ha set-valued mapping with nonempty
closed convex values. LetA, G, F : H → H be the Lipschitz continuous mapping with respect to positive constantsη, σ and d,respectively. Let g : H → H be the mapping such thatI −g
andgbe Lipschitz continuous with respect to positive constantsλandμ,respectively. A trimapping
N:H×H×H → His generalized pseudo-contractive with respect to mapAin first argument ofN
with constantp >0and generalizedg-pseudo-contractive with respect toGin the second argument of
Nwith constantq >0, Lipschitz continuous with respect to first, second, and third arguments with positive constantss, δ, ζ, respectively. Suppose thatm : H → H is a Lipschitz continuous with positive constantχ > 0. Let{un},{vn},and {wn}be three bounded sequences in 0,1satisfying
uniquex∈Hsatisfying2.2andxn → xasn → ∞, where{xn}is the three step iteration process
with errors defined as follows:
x0 ∈H
xn1 αnxn
βn yn−g
yn
myn
PK
gyn
−tgyn
−NAyn, Gyn, Fyn
− myn
γnun
ynαnxn
βn zn−gzn mzn PK
gzn−tgzn tNAzn, Gzn, Fzn− mzn
γnvn
znαnx
βn xn−gxn mxn PK
gxn−tgxn tNAxn, Gxn, Fxn−mxn
γnwn
3.35
forn0,1,2, . . . .
Now, we deduceTheorem 3.6for three step iterative process in terms of2.10.
Theorem 3.12. LetH, K, N, G, A, F, gandPKxbe the same as inTheorem 3.6. Let{αn},{βn},
{αn},{βn},{αn},and{βn}be six sequences in0,1satisfying conditions:
1αnβn1, αnβn1, αnβn1,forn≥0,
2limn→ ∞βnlimn→ ∞βnlimn→ ∞βn0,
3n∞0βn∞.
If the conditions of 3.10 hold, then there existsx ∈ H satisfying2.1and xn → xas
n → ∞, where the three-step iteration process{xn}is defined by
x0∈H,
xn1αnxnβn
yn−g
yn
PKyngyn
−tgyn
−NAyn, Gyn, Fyn
,
ynαnxnβn zn−gzn PKzn
gzn−t
gzn−NAzn, Gzn, Fzn
,
znαnxnβn xn−gxn PKxn
gxn−t
gxn−NAxn, Gxn, Fxn
3.36
forn0,1,2, . . . .
Next, we state the results in terms of iterations2.10and2.25.
Theorem 3.13. LetH, K, N, g, G, A, F,andmbe the same as in theTheorem 3.11. Let{αn},{βn},
{αn},{βn},{αn},and{βn}be six sequences in0,1satisfying conditions (1)–(3) ofTheorem 3.6. If
nonlinear implicit quasi-variational inequality2.2andxn → xasn → ∞, where the three-step
iteration process{xn}is defined by
x0∈H
xn1 αnxnβn yn−g
yn
myn
PK
gyn
−tgyn
tNAyn, Gyn, Fyn
−myn
ynαnxnβn zn−gzn mzn PK
gzn−tgzn tNAzn, Gzn, Fzn
−mzn
znαnxnβn xn−gxn mxn PK
gxn−tgxn tNAxn, Gxn, Fxn
−mxn
3.37
forn0,1,2, . . . .
Remark 3.14. Theorem 3.13can also be deduce for Ishikawa and Mann iterative process.
Acknowledgment
The authors thank the editor Professor R. U. Verma and anonymous referees for their valuable useful suggestions that improved the paper.
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