- Research Article
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Auxiliary Principle for Generalized Strongly Nonlinear Mixed Variational-Like Inequalities
Journal of Inequalities and Applications volume 2009, Article number: 758786 (2009)
Abstract
We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities, which includes several classes of variational inequalities and variational-like inequalities as special cases. By applying the auxiliary principle technique and KKM theory, we suggest an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. The existence of solutions and convergence of sequence generated by the algorithm for the generalized strongly nonlinear mixed variational-like inequalities are obtained. The results presented in this paper extend and unify some known results.
1. Introduction
It is well known that the auxiliary principle technique plays an efficient and important role in variational inequality theory. In 1988, Cohen [1] used the auxiliary principle technique to prove the existence of a unique solution for a variational inequality in reflexive Banach spaces, and suggested an innovative and novel iterative algorithm for computing the solution of the variational inequality. Afterwards, Ding [2], Huang and Deng [3], and Yao [4] obtained the existence of solutions for several kinds of variational-like inequalities. Fang and Huang [5] and Liu et al. [6] discussed some classes of variational inequalities involving various monotone mappings. Recently, Liu et al. [7, 8] extended the auxiliary principle technique to two new classes of variational-like inequalities and established the existence results for these variational-like inequalities.
Inspired and motivated by the results in [1–13], in this paper, we introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities. Making use of the auxiliary principle technique, we construct an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. Several existence results of solutions for the generalized strongly nonlinear mixed variational-like inequality involving strongly monotone, relaxed Lipschitz, cocoercive, relaxed cocoercive and generalized pseudocontractive mappings, and the convergence results of iterative sequence generated by the algorithm are given. The results presented in this paper extend and unify some known results in [9, 12, 13].
2. Preliminaries
In this paper, let , let
be a real Hilbert space endowed with an inner product
and norm
, respectively, let
be a nonempty closed convex subset of
. Let
and let
be mappings. Now we consider the following generalized strongly nonlinear mixed variational-like inequality problem: find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ1_HTML.gif)
where is a coercive continuous bilinear form, that is, there exist positive constants
and
such that
(C1)
(C2)Clearly,
Let satisfy the following conditions:
(C3) for each ,
is linear in the first argument;
(C4) is bounded, that is, there exists a constant
such that
(C5);
(C6) for each ,
is convex in the second argument.
Remark 2.1.
It is easy to verify that
(m1)
(m2),
where (m2) implies that for each ,
is continuous in the second argument on
.
Special Cases
(m3) If and
for all
where
, then the generalized strongly nonlinear mixed variational-like inequality (2.1) collapses to seeking
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ2_HTML.gif)
which was introduced and studied by Ansari and Yao [9], Ding [11] and Zeng [13], respectively.
(m4) If for all
where
, then the problem (2.2) reduces to the following problem: find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ3_HTML.gif)
which was introduced and studied by Yao [12].
In brief, for suitable choices of the mappings and
, one can obtain a number of known and new variational inequalities and variational-like inequalities as special cases of (2.1). Furthermore, there are a wide classes of problems arising in optimization, economics, structural analysis and fluid dynamics, which can be studied in the general framework of the generalized strongly nonlinear mixed variational-like inequality, which is the main motivation of this paper.
Definition 2.2.
Let and
be mappings.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq51_HTML.gif)
is said to be relaxed Lipschitz with constant if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq54_HTML.gif)
is said to be cocoercive with constant with respect to
in the first argument if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ5_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq58_HTML.gif)
is said to be -cocoercive with constant
with respect to
in the first argument if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq63_HTML.gif)
is said to be relaxed-cocoercive with respect to
in the first argument if there exist constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq68_HTML.gif)
is said to be Lipschitz continuous with constant if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq71_HTML.gif)
is said to be relaxed Lipschitz with constant with respect to
in the second argument if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq75_HTML.gif)
is said to be -relaxed Lipschitz with constant
with respect to
in the second argument if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq80_HTML.gif)
is said to be -generalized pseudocontractive with constant
with respect to
in the second argument if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq85_HTML.gif)
is said to be strongly monotone with constant if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq88_HTML.gif)
is said to be relaxed Lipschitz with constant if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq91_HTML.gif)
is said to be cocoercive with constant if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq94_HTML.gif)
is said to be Lipschitz continuous with constant if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_IEq97_HTML.gif)
is said to be Lipschitz continuous in the first argument if there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ16_HTML.gif)
Similarly, we can define the Lipschitz continuity of in the second argument.
Definition 2.3.
Let be a nonempty convex subset of
and let
be a functional.
(d1) is said to be convex if for any
and any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ17_HTML.gif)
(d2) is said to be concave if
is convex;
(d3) is said to be lower semicontinuous on
if for any
, the set
is closed in
;
(d4) is said to be upper semicontinuous on
, if
is lower semicontinuous on
.
In order to gain our results, we need the following assumption.
Assumption 2.4.
The mappings satisfy the following conditions:
(d5)
(d6) for given the mapping
is concave and upper semicontinuous on
.
Remark 2.5.
It follows from (d5) and (d6) that
(m5)
(m6) for any given , the mapping
is convex and lower semicontinuous on
.
Proposition 2.6 (see [9]).
Let be a nonempty convex subset of
. If
is lower semicontinuous and convex, then
is weakly lower semicontinuous.
Proposition 2.6 yields that if is upper semicontinuous and concave, then
is weakly upper semicontinuous.
Lemma 2.7 (see [10]).
Let be a nonempty closed convex subset of a Hausdorff linear topological space
, and let
be mappings satisfying the following conditions:
(a) and
-
(b)
for each
is upper semicontinuous on
-
(c)
for each
the set
is a convex set
-
(d)
there exists a nonempty compact set
and
such that
Then there exists such that
3. Auxiliary Problem and Algorithm
In this section, we use the auxiliary principle technique to suggest and analyze an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality (2.1). To be more precise, we consider the following auxiliary problem associated with the generalized strongly nonlinear mixed variational-like inequality (2.1): given , find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ18_HTML.gif)
where is a constant,
is a mapping. The problem is called a auxiliary problem for the generalized strongly nonlinear mixed variational-like inequality (2.1).
Theorem 3.1.
Let be a nonempty closed convex subset of the Hilbert space
. Let
be a coercive continuous bilinear form with (C1) and (C2), and let
be a functional with (C3)–(C6). Let
be Lipschitz continuous and relaxed Lipschitz with constants
and
, respectively. Let
be Lipschitz continuous with constant
,
and let
satisfy Assumption 2.4. Then the auxiliary problem (3.1) has a unique solution in
.
Proof.
For any , define the mappings
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ19_HTML.gif)
We claim that the mappings and
satisfy all the conditions of Lemma 2.7 in the weak topology. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ20_HTML.gif)
and for any
Since
is convex in the second argument and
is a coercive continuous bilinear form, it follows from Remark 2.1 and Assumption 2.4 that for each
,
is weakly upper semicontinuous on
. It is easy to show that the set
is a convex set for each fixed
Let
be fixed and put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ21_HTML.gif)
Clearly, is a weakly compact subset of
. From Assumption 2.4, the continuity of
and
, and the properties of
and
, we gain that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ22_HTML.gif)
Thus the conditions of Lemma 2.7 are satisfied. It follows from Lemma 2.7 that there exists a such that
for any
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ23_HTML.gif)
Let and
. Replacing
by
in (3.6) we gain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ24_HTML.gif)
Letting in (3.7), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ25_HTML.gif)
which means that is a solution of (3.1).
Suppose that are any two solutions of the auxiliary problem (3.1). It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ27_HTML.gif)
Taking in (3.9) and
in (3.10) and adding these two inequalities, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ28_HTML.gif)
Since is relaxed Lipschitz, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ29_HTML.gif)
which implies that . That is, the auxiliary problem (3.1) has a unique solution in
. This completes the proof.
Applying Theorem 3.1, we construct an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality (2.1).
Algorithm 3.2.
-
(i)
At step
, start with the initial value
.
-
(ii)
At step
, solve the auxiliary problem (3.1) with
. Let
denote the solution of the auxiliary problem (3.1). That is,
(3.13)
where is a constant.
-
(iii)
If, for given
,
stop. Otherwise, repeat (ii).
4. Existence of Solutions and Convergence Analysis
The goal of this section is to prove several existence of solutions and convergence of the sequence generated by Algorithm 3.2 for the generalized strongly nonlinear mixed variational-like inequality (2.1).
Theorem 4.1.
Let be a nonempty closed convex subset of the Hilbert space
. Let
be a coercive continuous bilinear form with (C1) and (C2), and let
be a functional with (C3)–(C6). Let
be Lipschitz continuous with constants
in the first and second arguments, respectively. Let
and
be Lipschitz continuous with constants
respectively, let
be cocoercive with constant
with respect to
in the first argument, let
be relaxed Lipschitz with constant
and let
be strongly monotone with constant
. Assume that Assumption 2.4 holds. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ31_HTML.gif)
If there exists a constant satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ32_HTML.gif)
and one of the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ35_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ36_HTML.gif)
then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence
defined by Algorithm 3.2 converges to
Proof.
It follows from (3.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ37_HTML.gif)
Adding (4.7), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ38_HTML.gif)
Since is relaxed Lipschitz and Lipschitz continuous with constants
and
, and
is strongly monotone and Lipschitz continuous with constants
and
, respectively, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ39_HTML.gif)
Notice that is Lipschitz continuous in the first and second arguments,
and
are both Lipschitz continuous, and
is cocoercive with constant
with respect to
in the first argument. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ40_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ41_HTML.gif)
It follows from (4.8)–(4.10) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ42_HTML.gif)
From (4.2) and one of (4.3)–(4.6), we know that . It follows from (4.12) that
is a Cauchy sequence in
. By the closedness of
there exists
satisfying
. In term of (3.13) and the Lipschitz continuity of
, we gain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ43_HTML.gif)
By Assumption 2.4, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ44_HTML.gif)
Since as
and
is bounded, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ45_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ46_HTML.gif)
In light of (C3) and (m2), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ47_HTML.gif)
which means that as
. Similarly, we can infer that
as
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ48_HTML.gif)
This completes the proof.
Theorem 4.2.
Let and
be as in Theorem 4.1. Assume that
,
are Lipschitz continuous with constants
and
, respectively,
is relaxed Lipschitz with constant
, and
is relaxed Lipschitz with constant
with respect to
in the second argument. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ49_HTML.gif)
If there exists a constant satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ50_HTML.gif)
and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence
defined by Algorithm 3.2 converges to
Proof.
As in the proof of Theorem 4.1, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ51_HTML.gif)
Because is relaxed Lipschitz and Lipschitz continuous,
is relaxed Lipschitz with respect to
in the second argument and Lipschitz continuous, and
is Lipschitz continuous in the second argument, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ52_HTML.gif)
The rest of the argument is the same as in the proof of Theorem 4.1 and is omitted. This completes the proof.
Theorem 4.3.
Let and
be as in Theorem 4.1, and let
be as in Theorem 4.2. Assume that
is
-cocoercive with constant
with respect to
in the first argument and Lipschitz continuous with constant
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ53_HTML.gif)
If there exists a constant satisfying (4.2) and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution
and the sequence
defined by Algorithm 3.2 converges to
Proof.
As in the proof of Theorem 4.1, we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ54_HTML.gif)
Because is Lipschitz continuous,
is Lipschitz continuous in the first argument, and
is
-cocoercive with with respect to
in the first argument and Lipschitz continuous, we gain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ55_HTML.gif)
The rest of the proof is identical with the proof of Theorem 4.1 and is omitted. This completes the proof.
Theorem 4.4.
Let and
be as in Theorem 4.1. Let
and
be as in Theorems 4.2 and 4.3, respectively. Assume that
,
are Lipschitz continuous with constants
,
and
, respectively,
is g-generalized pseudocontractive with constant
with respect to
in the second argument, and
is cocoercive with constant
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ56_HTML.gif)
If there exists a constant satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ57_HTML.gif)
and one of (4.3), (4.4), and (4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence
defined by Algorithm 3.2 converges to
Proof.
By a similar argument used in the proof of Theorem 4.1, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ58_HTML.gif)
Since is
-generalized pseudocontractive with respect to
in the second argument and Lipschitz continuous,
is Lipschitz continuous and
is Lipschitz continuous in the second argument,
is cocoercive and Lipschitz continuous, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ59_HTML.gif)
The rest of the argument follows as in the proof of Theorem 4.1 and is omitted. This completes the proof.
Theorem 4.5.
Let and
be as in Theorem 4.1. Assume that
are Lipschitz continuous with constants
,
, respectively,
is relaxed
-cocoercive with respect to
in the first argument,
is
-relaxed Lipschitz with constant
with respect to
in the second argument. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ60_HTML.gif)
If there exists a constant satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ61_HTML.gif)
and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variational-like inequality (2.1) possesses a solution and the sequence
defined by Algorithm 3.2 converges to
Proof.
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F758786/MediaObjects/13660_2009_Article_2004_Equ62_HTML.gif)
The rest of the proof is similar to the proof of Theorem 4.1 and is omitted. This completes the proof.
Remark 4.6.
Theorems 4.1–4.5 extend, improve, and unify the corresponding results in [9, 12, 13].
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Acknowledgments
The authors thank the referees for useful comments and suggestions. This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2009A419) and the Korea Research Foundation (KRF) grant funded by the Korea government (MEST)(2009-0073655).
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Liu, Z., Chen, L., Ume, J.S. et al. Auxiliary Principle for Generalized Strongly Nonlinear Mixed Variational-Like Inequalities. J Inequal Appl 2009, 758786 (2009). https://doi.org/10.1155/2009/758786
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DOI: https://doi.org/10.1155/2009/758786