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Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit variational-like inclusions

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Abstract

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.
J.
oflnequal.
&
Appl.,
2000,
Vol.
5,
pp.
381-395
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(Overseas
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and
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imprint.
Printed
in
Singapore.
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
inclu-
sions
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
Classification:
49J40,
47H06
1
INTRODUCTION
It
is
well
known
that
variational
inequality
theory
and
complementar-
ity
problem
theory
are
very
powerful
tool
of
the
current
mathematical
technology.
In
recent
years,
classical
variational
inequality
and
com-
plementarity
problem
have
been
extended
and
generalized
to
study
a
*
Corresponding
author.
381
382
S.H.
SHIM
et
al.
wide
class
of
problems
arising
in
mechanics,
physics,
optimization
and
control,
nonlinear
programming,
economics
and
transportation
equilib-
rium
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
gener-
alized
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
2
z
is
a
set-valued
mapping
such
that,
for
each
fixed
y
E
H,
M(.,
y)
H
2
n
is
a
maximal
monotone
map-
ping
and
Range(g
m)
fq
dom(M(.,
y))
:/:
for
each
y
E
H.
We
consider
the
following
problem:
Find
u
H
such
that
(g-
m)(u)fq
dom(M(.,
u))
-
0,
(2.1)
0
N(f(u),p(u))
+
M((g-
m)(u),
u),
where
g-
m
is
defined
as
(g-
m)(u)
g(u)
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)
dom(0(.,
y))
for
each
y
H
and
0(.,
y)
denotes
the
subdifferen-
tial
of
function
(.,
y),
then
the
problem
(2.1)
is
equivalent
to
finding
u
H
such
that
(g
m)(u)
dom(0(.,
u))
(2.2)
(U(f
(u),
p(u)
),
v
(g
m)(u))
>_
p(
(g
m)(u),
u)
q(v,
u)
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
equiva-
lent
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
inclu-
sion,
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
2
z-/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
func-
tion
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)
384
S.H.
SHIM
et
al.
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
semi-
continuous
function
:H
RU
{+c},
then
the
problem
(2.1)
is
equi-
valent
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
2
H
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
j’
0,
if
x
K(
y),
IK(y)
+,
ifxK(y),
then
the
problem
(2.1)
is
equivalent
to
finding
u
H
such
that
g(u)
m(u)
K(u),
(2.7)
(N(f(u),p(u)),
v
(g-
m)(u))
>_
O,
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
[1,6,13-
16,25-27]
as
special
cases.
3
PERTURBED
ITERATIVE
ALGORITHMS
LEMMA
3.1
u
H
is
a
solution
of
th