- Research Article
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Generalized Strongly Nonlinear Implicit Quasivariational Inequalities
Journal of Inequalities and Applications volume 2009, Article number: 124953 (2009)
Abstract
We prove an existence theorem for solution of generalized strongly nonlinear implicit quasivariational inequality problems and convergence of iterative sequences with errors, involving Lipschitz continuous, generalized pseudocontractive and generalized -pseudocontractive mappings in Hilbert spaces.
1. Introduction
Variational inequality was initially studied by Stampacchia [1] in 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 to [2–5] and the references therein.
It is known that one of the most important and difficult problems in variational inequality theory is the development of an efficient and implementable approximation schemes for solving various classes of variational inequalities and variational inclusions. Recently, Huang [6–8] and Cho et al. [9] constructed some new perturbed iterative algorithms for approximation of solutions of some generalized nonlinear implicit quasi-variational inclusions (inequalities), which include many iterative algorithms for variational and quasi-variational inclusions (inequalities) as special cases. Inspired and motivated by recent research works [1, 9–19], we prove an existence theorem for solution of generalized strongly nonlinear implicit quasi-variational inequality problems and convergence of iterative sequences with errors, involving Lipschitzian, generalized pseudocontractivity and generalized -pseudocontractive mappings in Hilbert spaces.
2. Preliminaries
Let be a real Hilbert space with norm
and inner product
. For a nonempty closed convex subset
, let
be the projection of
onto
. Let
be a set valued mapping with nonempty closed convex values,
and
be the mappings. We consider the following problem.
Find , such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ1_HTML.gif)
The problem (2.1) is called the generalized strongly nonlinear implicit quasi-variational inequality problem.
Special Cases
-
(i)
If
, for all
, where
is a nonempty closed convex subset of
and
is a mapping, then the problem (2.1) is equivalent to finding
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ2_HTML.gif)
the problem (2.2) is called generalized nonlinear quasi-variational inequality problem.
-
(ii)
If we assume
as identity mappings, then (2.1) reduces to the problem of finding
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ3_HTML.gif)
which is known as general implicit nonlinear quasi-variational inequality problem.
-
(iii)
If we assume
, then (2.3) reduces to the following problem of finding
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ4_HTML.gif)
which is known as generalized implicit nonlinear quasi-variational inequality problem, a variant form as can be seen in [20, equation ( 2.6)].
-
(iv)
If we assume
, then (2.4) reduces to the following problem of finding
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ5_HTML.gif)
The problem (2.5) is called the generalized strongly nonlinear implicit quasi-variational inequality problem, considered and studied by Cho et al. [9].
-
(v)
If
,
an identity mapping, then (2.5) is equivalent to finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ6_HTML.gif)
Problem (2.6) is called generalized strongly nonlinear quasi-variational inequality problem, see special cases of Cho et al. [9].
-
(vi)
If
,
a nonempty closed convex subset of
and
for all
, where
a nonlinear mapping, then the problem (2.6) is equivalent to finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ7_HTML.gif)
which is a nonlinear variational inequality, considered by Verma [17].
-
(vii)
If
, for all
, then (2.7) reduces to the following problem for finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ8_HTML.gif)
which is a classical variational inequality considered by [1, 4, 5].
Now, we recall the following iterative process due to Ishikawa [13], Mann [14], Noor [15] and Liu [21].
-
(1)
Let
be a nonempty convex subset of
and
a mapping. The sequence
, defined by
(29)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_IEq48_HTML.gif)
, is called the three-step iterative process, where ,
, and
are three real sequences in [ 0,1] satisfying some conditions.
-
(2)
In particular, if
for all
, then
, defined by
(211)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_IEq55_HTML.gif)
, is called the Ishikawa iterative process, where and
are two real sequences in [ 0,1] satisfying some conditions.
-
(3)
In particular, if
for all
, then
defined by
(212)
for , is called the Mann iterative process.
Recently Liu [21] introduced the concept of three-step iterative process with errors which is the generalization of Ishikawa [13] and Mann [14] iterative process, for nonlinear strongly accretive mappings as follows.
-
(4)
For a nonempty subset
of a Banach spaces
and a mapping
, the sequence
, defined by
(213)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_IEq66_HTML.gif)
, is called the three-step iterative process with errors. Here ,
and
are three summable sequences in
(i.e.,
,
and
), and
,
and
are three sequences in [ 0,1] satisfying certain restrictions.
-
(5)
In particular, if
for
and
. The sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_IEq81_HTML.gif)
, is called the Ishikawa iterative process with errors. Here and
are two summable sequences in
(i.e.,
and
;
and
are two sequences in [ 0,1] satisfying certain restrictions.
-
(6)
In particular, if
and
for all
. The sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ16_HTML.gif)
for , is called the Mann iterative process with errors, where
is a summable sequence in
and
a sequence in [ 0,1] satisfying certain restrictions.
However, in a recent paper [19] Xu pointed out that the definitions of Liu [21] are against the randomness of the errors and revised the definitions of Liu [21] as follows.
-
(7)
Let
be a nonempty convex subset of a Banach space
and
a mapping. For any given
, the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ18_HTML.gif)
for , is called the three-step iterative process with errors, where
,
and
are three bounded sequences in
and
,
,
,
,
,
,
,
and
are nine sequences in [ 0,1] satisfying the conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ19_HTML.gif)
 (8) If for
the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ20_HTML.gif)
for , is called the Ishikawa iterative process with errors, where
and
are two bounded sequences in
,
,
,
,
,
and
are six sequences in [ 0,1] satisfying the conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ21_HTML.gif)
 (9) If for
, the sequence
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ22_HTML.gif)
for , is called the Mann iterative process with errors.
For our main results, we need the following lemmas.
Lemma 2.1 (see [3]).
If is a closed convex subset and
a given point, then
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ23_HTML.gif)
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ24_HTML.gif)
where is the projection of
onto
.
Lemma 2.2 (see [10]).
The mapping defined by (2.24) is nonexpansive, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ25_HTML.gif)
Lemma 2.3 (see [10]).
If and
is a closed convex subset, then for any
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ26_HTML.gif)
Lemma 2.4 (see [21]).
Let ,
and
be three nonnegative real sequences satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ27_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ28_HTML.gif)
By Lemma 2.1, we know that the generalized strongly nonlinear implicit quasi-variational inequality (2.1) has a unique solution if and only if the mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ29_HTML.gif)
has a unique fixed point, where 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 by (2.18). First, we give some definitions.
Definition 3.1.
A mapping is said to be generalized pseudo-contractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ30_HTML.gif)
It is easy to check that (3.1) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ31_HTML.gif)
For in (3.1), we get the usual concept of pseudo-contractive of
, introduced by Browder and Petryshyn [10], that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ32_HTML.gif)
Definition 3.2.
Let and
be the mappings. The mapping
is said to be as follows.
(i)Generalized pseudo-contractive with respect to in the first argument of
, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ33_HTML.gif)
(ii)Lipschitz continuous with respect to the first argument of if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ34_HTML.gif)
In a similar way, we can define Lipschitz continuity of N with respect to the second and third arguments.
(iii) is also said to be Lipschitz continuous if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ35_HTML.gif)
Definition 3.3.
Let be the mappings. A mapping
is said to be the generalized
-pseudo-contractive with respect to the second argument of
, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ36_HTML.gif)
Definition 3.4.
Let be a set-valued mapping such that for each
,
is a nonempty closed convex subset of
. The projection
is said to be Lipschitz continuous if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ37_HTML.gif)
Remark 3.5.
In many important applications, has the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ38_HTML.gif)
where is a single-valued mapping and
a nonempty closed convex subset of
. If
is Lipschitz continuous with constant
, then from Lemma 2.3,
is Lipschitz continuous with Lipschitz constant
.
Now, we give the main result of this paper.
Theorem 3.6.
Let be a real Hilbert space and
a set-valued mapping with nonempty closed convex values. Let
be the Lipschitz continuous mappings with positive constants
and
respectively. Let
be the mapping such that
and
are Lipschitz continuous with positive constants
and
respectively. A trimapping
is generalized pseudo-contractive with respect to
in the first argument of
with constant
and generalized
-pseudo-contractive with respect to
in the second argument of
with constant
, Lipschitz continuous with respect to the first, second, and third arguments with positive constants
respectively. Suppose that
is Lipschitz continuous with constant
. Let
,
and
be the three bounded sequences in
and
,
,
,
,
,
,
,
and
are sequences in
satisfying the following conditions:
(1)
(2)
(3)
If the following conditions hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ39_HTML.gif)
where and
.
Then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and
as
, where
is the three-step iteration process with errors defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ40_HTML.gif)
for .
Proof.
We first prove that the generalized strongly nonlinear implicit quasi-variational inequality (2.1) has a unique solution. By Lemma 2.1, it is sufficient to prove the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ41_HTML.gif)
has a unique fixed point in .
Let be two arbitrary points in
. From Lemma 2.2 and Lipschitz continuity of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ42_HTML.gif)
Since is generalized pseudo-contractive with respect to
in the first argument of
and Lipschitz continuous with respect to first argument of
and also
is Lipschitz continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ43_HTML.gif)
Again since is generalized
-pseudo-contractive with respect to
in the second argument of
and Lipschitz continuous with respect to second argument of
and
is Lipschitz continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ44_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ45_HTML.gif)
It follows from (3.13)–(3.16) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ46_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ47_HTML.gif)
From (3.10), we know that and so
has a unique fixed point
, which is a unique solution of the generalized strongly nonlinear implicit quasi-variational inequality (2.1).
Now we prove that converges to
. In fact, it follows from (3.11) and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ48_HTML.gif)
From (3.17) and (3.19), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ49_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ50_HTML.gif)
Again,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ51_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ52_HTML.gif)
Then and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ53_HTML.gif)
Similarly, we deduce from (3.21) the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ54_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ55_HTML.gif)
From the above inequalities, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ56_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ57_HTML.gif)
Since , it follows from conditions (1) and (3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ58_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ59_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ60_HTML.gif)
From (3.29)-(3.31) and Lemma 2.4, we know that converges to the solution
. This completes the proof.
Remark 3.7.
We now deduce Theorem 3.6 in the direction of Ishikawa iteration.
Theorem 3.8.
Let be a real Hilbert space and
a set-valued mapping with the nonempty closed convex values. Let
and
be the same as in Theorem 3.6. Suppose that
is Lipschitz continuous with constant
. Let
and
be the two bounded sequences in
and
,
,
,
,
and
be six sequences in
satisfying the following conditions:
(1)
(2)
(3)
If the following conditions holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ61_HTML.gif)
Then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and
as
, where
is the Ishikawa iteration process with errors defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ62_HTML.gif)
for .
Remark 3.9.
We can also deduce Theorem 3.6 in the direction of (2.16).
Theorem 3.10.
Let and
be the same as in Theorem 3.6. Let
be a bounded sequence in
and
,
and
be three sequences in
satisfying the following conditions:
(1) for
,
(2),
(3)and
If the conditions of (3.10) hold, then there exists a unique satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.1) and
as
, where
is the Mann iterative process with errors defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ63_HTML.gif)
for .
Our results can be further improved in the direction of (2.25).
Theorem 3.11.
Let be a real Hilbert space and
a set-valued mapping with nonempty closed convex values. Let
be the Lipschitz continuous mapping with respect to positive constants
and
respectively. Let
be the mapping such that
and
be Lipschitz continuous with respect to positive constants
and
respectively. A trimapping
is generalized pseudo-contractive with respect to map
in first argument of
with constant
and generalized
-pseudo-contractive with respect to
in the second argument of
with constant
, Lipschitz continuous with respect to first, second, and third arguments with positive constants
, respectively. Suppose that
is a Lipschitz continuous with positive constant
. Let
,
and
be three bounded sequences in
satisfying the conditions (1)–(3) of Theorem 3.6. If the conditions of (3.10) hold for
, then there exists a unique
satisfying (2.2) and
as
, where
is the three step iteration process with errors defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ64_HTML.gif)
for .
Now, we deduce Theorem 3.6 for three step iterative process in terms of (2.10).
Theorem 3.12.
Let and
be the same as in Theorem 3.6. Let
,
,
,
,
and
be six sequences in
satisfying conditions:
(1) for
(2),
(3)
If the conditions of (3.10) hold, then there exists satisfying (2.1) and
as
, where the three-step iteration process
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ65_HTML.gif)
for .
Next, we state the results in terms of iterations (2.10) and (2.25).
Theorem 3.13.
Let and
be the same as in the Theorem 3.11. Let
,
,
,
,
and
be six sequences in
satisfying conditions (1)–(3) of Theorem 3.6. If the conditions of (3.10) hold for
, then there exists
satisfying the generalized strongly nonlinear implicit quasi-variational inequality (2.2) and
as
, where the three-step iteration process
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F124953/MediaObjects/13660_2009_Article_1887_Equ66_HTML.gif)
for .
Remark 3.14.
Theorem 3.13 can also be deduce for Ishikawa and Mann iterative process.
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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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Salahuddin, Ahmad, M.K. Generalized Strongly Nonlinear Implicit Quasivariational Inequalities. J Inequal Appl 2009, 124953 (2009). https://doi.org/10.1155/2009/124953
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DOI: https://doi.org/10.1155/2009/124953