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Auxiliary Principle for Generalized Strongly Nonlinear Mixed VariationalLike Inequalities
Journal of Inequalities and Applications volume 2009, Article number: 758786 (2009)
Abstract
We introduce and study a class of generalized strongly nonlinear mixed variationallike inequalities, which includes several classes of variational inequalities and variationallike 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 variationallike inequality. The existence of solutions and convergence of sequence generated by the algorithm for the generalized strongly nonlinear mixed variationallike 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 variationallike 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 variationallike inequalities and established the existence results for these variationallike inequalities.
Inspired and motivated by the results in [1–13], in this paper, we introduce and study a class of generalized strongly nonlinear mixed variationallike inequalities. Making use of the auxiliary principle technique, we construct an iterative algorithm for solving the generalized strongly nonlinear mixed variationallike inequality. Several existence results of solutions for the generalized strongly nonlinear mixed variationallike 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 variationallike inequality problem: find such that
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 variationallike inequality (2.1) collapses to seeking such that
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
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 variationallike 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 variationallike inequality, which is the main motivation of this paper.
Definition 2.2.
Let and be mappings.
is said to be relaxed Lipschitz with constant if there exists a constant such that
is said to be cocoercive with constant with respect to in the first argument if there exists a constant such that
is said to be cocoercive with constant with respect to in the first argument if there exists a constant such that
is said to be relaxedcocoercive with respect to in the first argument if there exist constants such that
is said to be Lipschitz continuous with constant if there exists a constant such that
is said to be relaxed Lipschitz with constant with respect to in the second argument if there exists a constant such that
is said to be relaxed Lipschitz with constant with respect to in the second argument if there exists a constant such that
is said to be generalized pseudocontractive with constant with respect to in the second argument if there exists a constant such that
is said to be strongly monotone with constant if there exists a constant such that
is said to be relaxed Lipschitz with constant if there exists a constant such that
is said to be cocoercive with constant if there exists a constant such that
is said to be Lipschitz continuous with constant if there exists a constant such that
is said to be Lipschitz continuous in the first argument if there exists a constant such that
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 ,
(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 variationallike inequality (2.1). To be more precise, we consider the following auxiliary problem associated with the generalized strongly nonlinear mixed variationallike inequality (2.1): given , find such that
where is a constant, is a mapping. The problem is called a auxiliary problem for the generalized strongly nonlinear mixed variationallike 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
We claim that the mappings and satisfy all the conditions of Lemma 2.7 in the weak topology. Note that
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
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
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,
Let and . Replacing by in (3.6) we gain that
Letting in (3.7), we get that
which means that is a solution of (3.1).
Suppose that are any two solutions of the auxiliary problem (3.1). It follows that
Taking in (3.9) and in (3.10) and adding these two inequalities, we get that
Since is relaxed Lipschitz, we find that
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 variationallike 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 variationallike 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
If there exists a constant satisfying
and one of the following conditions:
then the generalized strongly nonlinear mixed variationallike inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to
Proof.
It follows from (3.13) that
Adding (4.7), we obtain that
Since is relaxed Lipschitz and Lipschitz continuous with constants and , and is strongly monotone and Lipschitz continuous with constants and , respectively, we get that
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
Let
It follows from (4.8)–(4.10) that
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
By Assumption 2.4, we deduce that
Since as and is bounded, it follows that
which implies that
In light of (C3) and (m2), we get that
which means that as . Similarly, we can infer that as . Therefore,
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
If there exists a constant satisfying
and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variationallike 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
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
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
If there exists a constant satisfying (4.2) and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variationallike 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
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
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 ggeneralized pseudocontractive with constant with respect to in the second argument, and is cocoercive with constant . Let
If there exists a constant satisfying
and one of (4.3), (4.4), and (4.6), then the generalized strongly nonlinear mixed variationallike 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
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
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
If there exists a constant satisfying
and one of (4.3)–(4.6), then the generalized strongly nonlinear mixed variationallike inequality (2.1) possesses a solution and the sequence defined by Algorithm 3.2 converges to
Proof.
Notice that
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)(20090073655).
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Liu, Z., Chen, L., Ume, J.S. et al. Auxiliary Principle for Generalized Strongly Nonlinear Mixed VariationalLike 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
Keywords
 Hilbert Space
 Variational Inequality
 Iterative Algorithm
 Inequality Problem
 Real Hilbert Space