(C) 2000 OPA (Overseas Publishers Association) N.V.
Published by license under
the Gordon and Breach Science
Publishers imprint.
Printed in Singapore.
J. oflnequal. & Appl., 2000, Vol. 5, pp. 381-395
Reprints available directly from the publisher
Photocopying permitted by license only
Perturbed Iterative Algorithms with
Errors for Completely Generalized
Strongly Nonlinear Implicit
Quasivariational Inclusions
S.H. SHIM a, S.M. KANG a, N.J. HUANG b and Y.J. CHO a,.
a Department
of Mathematics, Gyeongsang National University,
Chinju 660-701, South Korea; b Department of Mathematics,
Sichuan University, Chengdu, Sichuan 610064, RR. China
(Received 4 May 1999; Revised 27 July 1999)
In this paper, we introduce a new class of completely generalized strongly nonlinear implicit
quasivariational inclusions and prove its equivalence with a class of fixed point problems
by using some properties of maximal monotone mappings. We also prove the existence of
solutions for the completely generalized strongly nonlinear implicit quasivariational inclusions and the convergence of iterative sequences generated by the perturbed algorithms
with errors.
Keywords: Completely generalized strongly nonlinear implicit quasivariational
inclusion; Mann type iterative sequence with errors; Ishikawa type iterative
sequence with errors
1991 Mathematics Subject
1
Classification: 49J40, 47H06
INTRODUCTION
It is well known that variational inequality theory and complementarity problem theory are very powerful tool of the current mathematical
technology. In recent years, classical variational inequality and complementarity problem have been extended and generalized to study a
* Corresponding author.
381
S.H. SHIM et al.
382
wide class of problems arising in mechanics, physics, optimization and
control, nonlinear programming, economics and transportation equilibrium and engineering sciences, etc. Various quasi-(implicit)variational
inequalities, generalized quasi-(implicit)variational inequalities, quasi(implicit)complementarity problem and generalized quasi-(implicit)complementarity problem are very important generalizations of these
classical problems. For details we refer to [1,3-18,20-29] and the
references therein.
Recently, Huang [15,16] introduced and studied the Mann and
Ishikawa type perturbed iterative sequences with errors for the generalized implicit quasivariational inequalities and inclusions. Inspired
and motivated by recent research works in this field, in this paper, we
introduce a new class of completely generalized strongly nonlinear
implicit quasivariational inclusions and prove its equivalence with a
class of fixed point problems by using some properties of maximal
monotone mappings. We also show the existence of solutions for this
completely generalized strongly nonlinear implicit quasivariational
inclusions and the convergence of iterative sequences generated by the
perturbed algorithms with errors.
2
PRELIMINARIES
Let H be a real Hilbert space endowed with a norm [l" and an inner
product (.,.}. Letf, p, g, rn H H and N: H x H H be single-valued
mappings. Suppose that M" H x H 2z is a set-valued mapping such
that, for each fixed y E H, M(., y) H 2 n is a maximal monotone mapping and Range(g m) fq dom(M(., y)) :/: for each y E H. We consider
the following problem:
Find u H such that
0
-
(g- m)(u)fq dom(M(., u)) 0,
N(f(u),p(u)) + M((g- m)(u), u),
where g- m is defined as
(g- m)(u)
g(u)
(2.1)
re(u)
for each u E H. The problem (2.1) is called the completely generalized
strongly nonlinear implicit quasivariational inclusion.
NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS
383
Now, we give some special cases of the problem (2.1) as follows:
(I) If M(., y) 0qo(., y) for each y H, where p H x H R U { +oe}
such that for each fixed y H, (., y) H R t3 {+oc} is a proper
convex lower semicontinuous function on H and Range(g-m)
for each y H and 0(., y) denotes the subdifferendom(0(., y))
tial of function (., y), then the problem (2.1) is equivalent to finding
u H such that
(g m)(u) dom(0(., u))
(U(f (u), p(u) ), v (g m)(u)) >_ p( (g m)(u), u)
q(v, u)
(2.2)
for all v E H, which is called the generalized strongly nonlinear implicit
quasivariational inclusion.
(II) If N(u, v) u v for all u, v E H, then the problem (2.1) is equivalent to finding u H such that
(g m)(u) f) dom(M(., u)) q),
0 f(u) p(u) + M((g m)(u), u),
(2.3)
which is called the generalized nonlinear implicit quasivariational inclusion, which was considered by Huang [16].
(III) If N(u, v) u v for all u, v H, m 0 and M(x, y) M(x) for
all y H, where M: H 2z-/is a maximal monotone mapping, then the
problem (2.1) is equivalent to finding u E H such that
:
g(u) fq dom(M(u)) O,
0 e f(u) -p(u) + M(g(u)),
(2.4)
which was studied by Adly [1].
(IV) If N(u, v) u v for all u, v E H and M(., y) 0(., y) for each
yEH, where :HxHRt_J{+oc} such that, for each fixed yEH,
o(., y):H R t_l {+oc} is a proper convex lower semicontinuous function on H and Range(g m) N dom(0o(., y)) 13 for each y E H and
0(., y) denotes the subdifferential of function o(., y), then the problem
(2.1) is equivalent to finding u E H such that
-
(g m)(u) f3 dom(0qo(., u)) I,
(f(u) -p(u), v (g- m)(u)) > qo((g- m)(u), u)
p(v, u)
(2.5)
S.H. SHIM et al.
384
for all v E H, which is called the generalized quasivariational inclusion,
which was considered by Ding [10].
(V) If N(u, v) u v for all u, v E H and M(., y) 0 for all y H,
where 0 denotes the subdifferential of a proper convex lower semicontinuous function :H RU {+c}, then the problem (2.1) is equivalent to finding u H such that
,
(g- m)(u) f3 dom(0) :/:
(f(u) -p(u), v (g- m)(u)) >_ p((g- m)(u))
(v)
(2.6)
for all v E H, which was studied by Kazmi [18].
(VI) If K:H 2H is a set-valued mapping such that, for each x E H,
K(x) is a closed convex subset of H and, for each fixed y E H, M(., y)
Oqo(., y), qo(., y) Ii(y)(.) is the indicator function of K( y) defined by
IK(y)
if x K( y),
+, ifxK(y),
j’ 0,
then the problem (2.1) is equivalent to finding u H such that
g(u) m(u) K(u),
(N(f(u),p(u)), v (g- m)(u)) >_ O,
(2.7)
for all v K(u).
It is clear that the completely generalized strongly nonlinear implicit
quasivariational inclusion problem (2.1) includes many kinds of
variational inequalities, quasivariational inequalities, complementarity
problems and quasi-(implicit)complementarity problems of [1,6,1316,25-27] as special cases.
PERTURBED ITERATIVE ALGORITHMS
3
LEMMA 3.1 u H is a solution of the problem (2.1)/f and only if there
H such that
exists u
u
F(u)
def
u
(g m)(u) + jy(.,u)((g_ m)(u)
aN(f(u),p(u))),
(3.1)
where a >0 is a constant and
proximal mapping on H.
J("u)(I+ aM(.,u)) -1
is the so-called
NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS
385
Proof Let u E H be a solution of the problem (3.1). From the definition of the proximal mapping j(.,u), it follows that
(g- m)(u)
aN(f(u),p(u))
(g- m)(u) + aM((g- m)(u), u)
and so
N(p(u),f (u)) e M((g m)(u), u).
-
Thus u H is a solution of the problem (2.1).
Conversely, if u H is a solution of (2.1), we have u H such that
(g m)(u) fq dom(M(., u))
and
0
N(f(u),p(u)) + M((g- m)(u), u).
Hence we have
(g- m)(u)
0
aN(f(u),p(u)) e (g- m)(u) + aM((g m)(u), u).
From the definition of the proximal mapping jff(.,u), we know that
u E H is a solution of the problem (3.1). This completes the proof.
Based on Lemma 3.1, we now suggest and analyze the following
new general and unified algorithms for the problem (2.1).
(A) Mann Type Perturbed Iterative Algorithm with
Errors (MTPIAE)
H be mappings. Given u0 H,
the iterative sequence {un} are defined by
Let f,p, g, m H
Un+l
H and N: H x H
"Tn)Un + OZn[Un (g- m)(Un)
+ j"(.,u,)((g m)(un) aN(f(un),p(un)))] + nen
(1
On
(3.2)
for n=0, 1,2,..., where M n" H x H 2 / is a set-valued mapping
such that, for each y H, M"(.,y):H 2n is a maximal monotone
mapping for n=0, 1,2,..., a > 0 is a constant, {en} is a bounded
S.H. SHIM
386
et al.
sequence of the element of H introduced to take into account possible
inexact computation and the sequences {an}, {Tn} in [0, 1] satisfying
the following conditions:
(1) an+Tn_< forn=0,1,2,...,
(2)
(3) En=oOZn 00, -_n=O’)/’n < 00.
(B) Ishikawa Type Perturbed Iterative Algorithm with
Errors (ITPIAE)
H be mappings. Given u0 E H,
the iterative sequence {un} are defined by
H and N: H x H
Let f,p, g, m H
Un+l
(1
an
")’n)Un -+- OZn[ln
-
(g m)(Vn)
+ j"(.,v,)((g_ m)(Vn) oN(f(Vn),P(Vn)))] +")’nen,
n
(1 /n n)bln /n[l’ln (g m)(Un)
+ .j"(.,u,)((g m)(Un) aN(f(un),p(un)))] + Snfn
(3.3)
for n 0, 1,2,..., where M n H x H 2 H is a set-valued mapping such
that, for each y H, Mn(.,y):H 2 /is a maximal monotone mapping for n =0, 1,2,..., a > 0 is a constant, {en}, {fn} are two bounded
sequences in H introduced to take into account possible in exact computation and the sequences {an}, {/3n}, {Tn}, {6n} in [0, 1] satisfying the
following conditions:
(1) an+’Tn<_l,n+6n<_l foralln=O, 1,2,...,
(2) an O, n -’+ 0 as n c,
(3)
(C) Mann Type Perturbed Iterative Algorithm (MTPIA)
Let f, p, g, m H H and N: H H H be mappings. Given u0 H,
the iterative sequence {un} are defined by
un+
n)Un + n[Un (g- m)(un)
+ j"(.,u,)((g_ m)(un) aN(f(Un),p(un)))]
(1
(3.4)
NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS
-
387
for n 0, 1, 2,..., where M n H x H 2 n is a set-valued mapping such
that, for each y E H, M"(., y):H-- 2 /is a maximal monotone mapping for n=0, 1,2,..., a >0 is a constant, the sequence {a} in [0, 1]
satisfying the following conditions:
(1) a,0 as nec,
(2) En=00n
(D) Ishikawa Type Perturbed Iterative Algorithm (ITPIA)
Let f,p, g, m H H and N: H x H-- H be mappings. Given u0 E H,
the iterative sequence {u.} are defined by
Un+,
On)Un + n[Vn
(1
(g- m)(Vn)
+ jM"(.,,,n)((g_ m)(Vn) aU(f(v,,),p(vn)))],
Vn
(1 -/n)Un +/n[Un
(g- m)(un)
+ jM"(.,u,)((g_ m)(Un) aN(f(Un),p(un)))]
for n 0, 1,2,..., where M H x H-- 2 is a set-valued mapping such
that, for each y H, M( y) H 2 t-/is a maximal monotone mapping
for n=0, 1,2,..., a >0 is a constant, the sequences {an}, {/3n}, {7},
{} in [0, 1] satisfying the following conditions:
.,
(1) a- 0 as n
(2)
oc,
Remark 3.1
(1) If /3=n=0 for all n=0,1,2,..., then ITPIAE reduces to
MTPIAE.
(2) If 3’ 6n 0 for all n 0, 1,2,..., then ITPIAE reduces to ITPIA.
(3) If 7, 0 for all n 0, 1,2,..., then MTPIAE reduces to MTPIA.
(4) Our ITPIAE includes several known algorithms of[1,6,10,15,16,18]
as special cases.
4
EXISTENCE AND CONVERGENCE THEOREMS
In this section, we show the existence of a solution of the problem (2.1)
and the convergence of iterative sequence generated by ITPIAE.
S.H. SHIM et al.
388
DEFINITION 4.1 Let N" H x H-- H and g" H
H.
(1) g is said to be strongly monotone/f there exists some 6 > 0 such that
(g(ul) g(u2), ul
for all u E H,
u2) _> llul
u=ll
1,2;
(2) g & said to be Lipschitz continuous if there exkcts some r > 0 such that
IIg(u) g(u.)[[
u2
for all ui E H,
1,2;
(3) g is said to be strongly monotone with respect to the first argument
of N if there exists some # > 0 such that
u2)
(N(g(u,), .)- N(g(u2), "), Ul
for all ui H,
[[Ul
u2ll 2
1,2;
(4) g
is said to be Lipschitz continuous with respect to the first argument of N if there exists some > 0 such that
N(g(u2), ")11
IIN(g(u), ")
llu u211
for all uiH, i= 1,2.
In a similar way, we can define strong monotonicity and Lipschitz
continuity of g with respect to the second argument of N.
DEFINITION 4.2 Let {S n} and S be maximal monotone mappings for
n= 1,2,... The sequence {S n} is said to be graph-convergence to S
(write S" S) if, for every (x,y) Graph(S), there exists a sequence
(xn, y) Graph(S ") such that xn x and y, y in H.
LEMMA 4.1 [2] Let
n O, 1, 2,... Then S n
x H and A > O.
-
S be maximal monotone mappings for
{S} Sand and
every
only J{" (x) J(x)
if
if
for
LEMMA 4.2 [19] Let {an}, {bn} and {cn) be three nonnegative real
sequences such that, for all n O, 1,2,...,
an+ <_ (1
with O < t <
limn an 0.
y’n=otn
c
tn )an -+- bn + Cn
o
Cn < cxz. Then
bn o(tn) and -]n=0
NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS
389
THEOREM 4.1 Let N: H x H--+ H be a mapping, g, m H-- H be
Lipschitz continuous with constants cr and #, respectively, f, p H-- H be
Lipschitz continuous with constants rl and e with respect to the first and
second arguments of N, respectively, f be strongly monotone with constant/3 with respect to the first argument of N and g-m be strongly
monotone with constant Assume that
.
(m(v) m(u),g(u) g(v)) < llu- vii 2
(4.1)
.
for all u, v E H andfor some constant ,k such that o < < or#, where
Ao inf{s: (m(v) m(u),g(u) g(v)) < s[lu vii 2 for all u, v e H}.
Suppose that there exists a constant > 0 such that, for each x, y, z H,
IIJM(’’x) (2) jM(.,y)(Z)]l ll x YII"
(4.2)
If the following conditions hold."
2
+ e(k 1) < V/(/3 + (k- 1)e) (r/: e2)k(2 k)
72 e2
(4.3)
(4.4)
tee< l-k,
k=.+2V/1-26+cr2+#2+2A,
k<l,
(4.5)
then the problem (2.1) has a unique solution u* H. Moreover, suppose
that M n H H 2 is a set-valued mapping such that, for each y H,
M’( y) H---. 2 is a maximal monotone mapping for n O, 1, 2,...,
M n (., y) M(., y) and, for each x, y, z H,
.,
IlJMn(" x) (z)- jMn(.,y) (z)ll <_C[[x-yll.
Then {u,} strongly convergences to it u*, where {Un} is
ITPIAE.
(4.6)
defined by
S.H. SHIM et al.
390
First we prove that there exists u* E H which is a unique solution of the problem (2.1). By Lemma 3.1, it is enough to show that the
mapping F: H H defined by (3.1) has a unique fixed point u* E H.
In fact, for any u, v H, we have
Proof
F(u)
u
(g m)(u) + j(.,u)((g m)(u)
aN(f(u),p(u)))
F(v)
v
(g m)(v) + j(.,v)((g_ m)(v)
aN(f(v),p(v))).
and
From the definition of j(.,u) and (4.2), we have
liE(u)
F(v)
Ilu (g m)(u) + j(.,u)((g m)(u) aN(f(u),p(u)))
[v (g- m)(v) + j(.,v)((g m)(v) aN(f(v),p(v)))][[
_< Ilu- v- ((g- m)(u) (g- m)(v))ll
/ IIJl,ul((g- m)(u) aN(f(u),p(u)))
j.,ul((g_ m)(v) N(f(v),p(v)))]ll
/ IlJ’,ul((g- m)(v) aN(f(v),p(v)))
J"((g- m)(v) cN(f(v),p(v)))]ll
_< 211u- v- ((g- m)(u) (g- m)(v))]l
+ u v a(N(f(u),p(u)) N(f(v),p(u)))[[
/ llN(f(v),p(u)) N(f(v),p(v))ll / llu vii.
(4.7)
By (4.1), the Lipschitz continuity off, g, m and strong monotonicity of
g- m and f, we obtain
Ilu- v- ((g- m)(u) (g- m)(v))ll 2
Ilu- vii 2 2(u v, (g- m)(u) (g- m)(v))
lira(u) m(v)ll 2 / Ilg(u) -g(v)ll =
+ 2(m(v) m(u),g(u) g(v))
< (1 2 + cr2 + #9. + 2A)IIu vii =
/
(4.8)
NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS
391
and
Ilu
a(N(f(u),p(u)) N(f(v),p(u)))ll 2
< (1 2a/3 + a2r/2) Ilu vii 2,
v-
(4.9)
Further, since p is Lipschitz continuous with respect to the second argument of N, we get
IIN(f(v),p(u))
N(f(v),p(v))ll <_
ellu- vii.
(4.10)
From (4.7)-(4.10), it follows that
IlF(u)- F(v)]l 5 hllu-
(4.11)
where
h
2 V/1
26 + er2 + #2 + 2, + { +
v/1 2c/3 + 022 -1-- Ct.
From (4.3)-(4.5), we know that 0 < h < and so F has a unique fixed
point u* E H. By Lemma 3.1 and (4.11), we know that u* is a unique
solution of the problem (2.1).
Next we prove that the iterative sequence {un} defined by ITPIAE
strongly converges to u*. Since u* is a solution of the problem (2.1),
we have
F(u*)
u*
(g m)(u*) + jM(.,u*)((g_ m)(u*)
aN(f(u*),p(u*))).
(4.12)
It follows from (3.3), (4.12) and Lemma 3.1 that
Ilu+,
u*ll
]1(1 ogn ")/n)Un -I- Ogn[ltn (g- m)(ln) + jaMn("vn)((g- m)(vn)
oU(f(Vn),p(vn)))] + %en (1 on ")’n)U*
Ctn[U* (g- m)(u*) + jM"(.,u*)((g m)(u*)
oU(f (u*),p(u*)))] +
S.H. SHIM
392
< (1
On
(u*
et al.
(g m)(Vn)
%)llu. u*ll / anllVn
(g m)(u*))[] + %lien
u*[I / CnllJ("((g m)(v)
jy(.,u*)((g_ m)(u*) N(f(u*),p(u*)))[[
<.(1 an 7n)[lUn u*ll / OnllVn U* ((g- m)(Vn)
((g-- m)(v)
(g- m)(u*))ll /’rnlle u*ll + nlI/
oN(f(Vn),p(Vn))) JM"("v)((g m)(u*) cN(f(u*),p(u*)))ll
+ nllJ"("v)((g- m)(u*) aN(f(u*),p(u*)))
.Z_"u*((g m)(u*) N(f(u*),p(u*)))ll
/ CnllJ"u*l((g- m)(u*) aN(f(u*),p(u*)))
jl.,u*l((g m)(u*) N(f(u*),p(u*)))ll
_< (1
%)llu u*ll / 2llv u* ((g m)(v)
(g m)(u*))ll / tnllVn u*ll / %lle u*ll
+ cnllVn u* a(N(f(vn),p(Vn)) N(f(u*),p(v)))l
oN(f(Vn),p(Vn)))
/
cllN(f(u*),p(Vn))
N(f(u*),p(u*))II
+ tnllJ("u*)((g- m)(u*) aN(f(u*),p(u*)))
j.,u*l((g_ m)(u*) cN(f(u*),p(u*)))ll.
(4.13)
By (4.1), the Lipschitz continuity off, g, m and strong monotonicity of
g- rn and f, we obtain
IIv
u*
< (1
(g m)(u*))ll
26 + 02 + #2 + 2,)II Vn u* =
((g- m)(Vn)
(4.14)
and
[IVn U* ce(N(f(Vn),p(vn)) N(f(u*),p(Vn)))ll 2
< (1 2c/3 + o2T]2)[lvn u*ll 2.
(4.15)
NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS
393
Further since p is Lipschitz continuous with respect to the second argument of N, we get
]lN(f(u*),p(v))
N(f(u*),p(u*))[] <_ e[[Vn
u*[[.
(4.16)
From (4.13)-(4.16), it follows that
u*ll (1
Ilu.+,
/
on
u*ll + hcnl]Vn u*ll
)llu.
o,,b, / .),nile.
(4.17)
u*ll,
where
b.
IlJM"("*/((g- m)(u*) ozN(f(u*),p(u*)))
j(.,u*)((g_ m)(u*) aU(f(u*).p(u*)))l I.
Similarly, we have
u*ll + hllu u*ll
//nbn / n IIf U* I1"
( -/ 6n)llUn
[Iv. u*ll
(4.18)
It follows from (4.17) and (4.18) that
[lUn+l--U*II
Let M1
(1--OZn--’n / hc.(1 -n-6n
+ hon(nbn / 6nllfn u*ll) / .b. / 7.lie.
u*ll: n > O}
sup{IlL
and M2
sup{lien
u*ll: n > 0}.
Since
0 < h < 1, we have
I[Un+l
u*ll
(1 ce,(1 h))llun u*ll
+ an(h6nM1 + (hfln + 1)bn)/ ynM2.
Using Lemma 4.1, we know that bn
(4.19) and Lemma 4.2 that un --* u* as n
(4.19)
.
0 as n cx. If follows from
This completes the proof.
Remark 4.1
(1) If/3n 6n 0 for all n 0, 1,2,..., then Theorem 4.1 gives the conditions under which the sequence {u} defined by MTPIAE strongly
converges to u*.
S.H. SHIM et al.
394
(2) If’),n 6n 0 for all n 0, 1, 2,..., then Theorem 4.1 gives the conditions under which the sequence {un} defined by ITPIA strongly
,
converges to u*.
(3) If’),, =/3= =0 for all n=0, 1,2,..., then Theorem 4.1 gives the
conditions under which the sequence {u,} defined by MTPIA
strongly converges to u*.
Acknowledgements
The third author was supported in part by ’98 APEC Post-Doctor
Fellowship (KOSEF) while he visited Gyeongsang National University, and the second and fourth authors wish to acknowledge the financial support of the Korea Research Foundation made in the program
year of 1998, Project No. 1998-015-D00020.
References
[1] S. Adly, Perturbed algorithm and sensitivity analysis for a general class of variational inclusions. J. Math. Anal. Appl. 201 (1996), 609-630.
[2] H. Attouch, Variational convergence for functions and operators, in Appl. Math.
Ser., Pitman, London, 1974.
[3] C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, Application to Free Boundary Problems, Wiley, New York, 1984.
[4] A. Bensoussan, Stochastic Control by Functional Analysis Method, North-Holland,
Amsterdam, 1982.
[5] A. Bensoussan and J.L. Lions, Impulse Control and Quasivariational Inequalities,
Gauthiers-Villers, Bordas, Paris, 1984.
[6] S.S. Chang, Variational Inequality and Complementarity Problem Theory with Applications, Shanghai Scientific and Tech. Literature Publishing House, Shanghai, 1991.
[7] R.W. Cottle, J.P. Pang and R.E. Stone, The Linear Complementarity Problem,
Academic Press, London, 1992.
[8] J. Crank, Free and Moving Boundary Problems, Clarendon, Oxford, 1984.
[9] V.F. Demyanov, G.E. Stavroulakis, L.N. Polyakova and P.D. Panagiotopoulos,
[10]
[11]
[12]
[13]
[14]
Quasidifferentiability and Nonsmooth Modeling in Mechanics, Engineering and
Economics, Kluwer Academic, Holland, 1996.
X.P. Ding, Perturbed proximal point for generalized quasivariational Inclusions,
J. Math. Anal. Appl. 210 (1997), 88-101.
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag,
Berlin, 1976.
F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium
Problems, Plenum, New York, 1995.
P.T. Harker and J.S. Pang, Finite-dimensional variational inequality and nonlinear
complementarity problems: A survey of theory, algorithms and applications, Math.
Programming 48 (1990), 161-220.
A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions,
J. Math. Anal. Appl. 185 (1994), 706-712.
NONLINEAR IMPLICIT QUASIVARIATIONAL INCLUSIONS
395
[15] N.J. Huang, On the generalized implicit quasivariational inequalities, J. Math.
Anal. Appl. 216 (1997), 197-210.
[16] N.J. Huang, Mann and Ishikawa type perturbed iterative algorithms for generalized
nonlinear implicit quasi-variational inclusions, Computers Math. Appl. 35(10) (1998),
1-7.
[17] G. Isac, Complementarity problems, Lecture Notes in Math. 1528, Springer-Verlag,
Berlin, 1992.
[18] K.R. Kazmi, Mann and Ishikawa and perturbed iterative algorithms for generalized
quasi-variational inclusions, J. Math. Anal. Appl. 209 (1997), 572-584.
[19] L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly
accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114-125.
[20] U. Mosco, Implicit variational problems and quasi-variational inequalities, Lecture
Notes in Math. 543, Springer-Verlag, Berlin, 1976.
[21] M.A. Noor, Some recent advances in variational inequalities (I), New Zealand J.
Math. 26 (1997), 53-80.
[22] M.A. Noor, Some recent advances in variational inequalities (II), New Zealand J.
Math. 26 (1997), 229-255.
[23] M.A. Noor, Generalized set-valued variational inclusions and resolvent equations,
J. Math. Anal. Appl. 228 (1998), 206-220.
[24] M.A. Noor, An implicit method for mixed variational inequalities, Appl. Math.
Lett. 11(4)(1998), 109-113.
[25] A.H. Siddiqi and Q.H. Ansari, Strongly nonlinear quasivariational inequalities,
J. Math. Anal. Appl. (1990), 444-340.
[26] A.H. Siddiqi and Q.H. Ansari, General strongly nonlinear variational inequalities,
J. Math. Anal. Appl. 166 (1992), 386-392.
[27] L.U. Uko, Strongly nonlinear generalized equations, J. Math. Anal. Appl. 220
(1998), 65-76.
[28] J.C. Yao, The generalized quasivariational inequality problem with applications,
J. Math. Anal. Appl. 158 (1991), 139-160.
[29] George Xian-Zhi Yuan, KKM Theory and Applications in Nonlinear Analysis,
Marcel Dekker, Inc., 1999.