Existence Results for System of Variational Inequality Problems with Semimonotone Operators
© Somyot Plubtieng and Kamonrat Sombut. 2010
Received: 14 July 2010
Accepted: 29 October 2010
Published: 31 October 2010
We introduce the system of variational inequality problems for semimonotone operators in reflexive Banach space. Using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of variational inequality problems for semimonotone with finite-dimensional continuous operators in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results for variational inequality problems studied in recent years.
The variational inequality problem (VIP) has been recognized as suitable mathematical models for dealing with many problems arising in different fields, such as optimization theory, game theory, economic equilibrium, mechanics. In the last four decades, since the time of the celebrated Hartman Stampacchia theorem (see [1, 2]), solution existence of variational inequality and other related problems has become a basic research topic which continues to attract attention of researchers in applied mathematics (see, e.g., [3–14] and the references therein). Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the current interest in functional analysis. It is natural to consider a unified approach to these different problems; see, for example, [10, 11].
There are many kinds of mappings in the literature of recent years; see, for example, [12, 13, 15–18]. In 1999, Chen  introduced the concept of semimonotonicity for a single valued mapping, which occurred in the study of nonlinear partial differential equations of divergence type. Recently, Fang and Huang  introduced two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, they obtained the existence of solutions for variational-like inequalities with relaxed monotone mappings in reflexive Banach spaces. Moreover, they present the solvability of variational-like inequalities with relaxed semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed point theorem.
On the other hand, some interesting and important problems related to variational inequalities and complementarity problems were considered in recent papers. In 2004, Cho et al. , introduced, and studied a system of nonlinear variational inequalities. They proved the existence and uniqueness of solution for this problem and constructed an iterative algorithm for approximating the solution of system of nonlinear variational inequalities. In 2000, , systems of variational inequalities were introduced and an existence theorem was obtained by Ky Fan lemma. In 2002, Kassay et al.  introduced and studied Minty and Stampacchia variational inequality systems by the Kakutani-Fan-Glicksberg fixed point theorem. Very recently, Fang and Huang  introduced and studied systems of strong implicit vector variational inequalities by the same fixed point theorem. Zhao and Xia  introduced and established some existence results for systems of vector variational-like inequalities in Banach spaces by also using the Kakutani-Fan-Glicksberg fixed point theorem.
In this paper, we introduce the system of generalized variational inequality in real reflexive Banach space. By using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain some existence results for system of generalized variational inequality for semimonotone and finite dimensional continuous in real reflexive Banach spaces. The results presented in this paper extend and improve the corresponding results of Chen  and many others.
exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for . We recall that is uniformly convex if and only if is uniformly smooth. It is well known that is smooth if and only if is strictly convex.
Definition 2.1 (KKM mapping).
Lemma 2.2 (Fan-KKM Theorem).
(i) is said to be upper semicontinuous at if, for any open set containing , there exists an open set containing such that, for all , ; is said to be upper semicontinuous on if it is upper semicontinuous at all .
(ii) is said to be lower semicontinuous at if, for any open set with , there exists an open set containing such that, for all , ; is said to be lower semicontinuous on if it is lower semicontinuous at all .
Lemma 2.5 (see ).
Lemma 2.6 (see Kakutani-Fan-Glicksberg ).
Lemma 2.7 (see ).
Definition 2.8 (see ).
This completes the proof.
3. The Existence of the System of Generalized Variational Inequality
In this section, we prove two existence theorems for system of variational inequality problems for semimonotone with finite dimensional continuous operators in real reflexive Banach spaces. First, we prove an existence theorem for system of variational inequality problems for continuous mappings as follows.
Step 5 (Show that the problem (1.2) has a solution).
Step 6 (Show that the set of solutions of problem (1.2) is closed).
Let be a nonempty bounded closed convex subset of a real reflexive Banach space with dual space . Suppose that is a lower semicontinuous convex function with . Let and be two mappings satisfying the following conditions:
By (3.10), we know that is nonempty bounded. We observe that , where is the weak* closure of in . Since is reflexive, it follows that is weak* compact. For any , we note that . So has the finite intersection property. Therefore, it follows that . Let , then we have .
This completes the proof.
Corollary 3.4 (see ).
Let be a real reflexive Banach space with dual space , and let be a nonempty unbounded closed convex subset with . Suppose that is a lower semicontinuous convex function with . Let , be two mappings satisfying the following conditions:
This completes the proof.
Similarly as in the proof of Corollary 3.4, we have the following result.
Corollary 3.7 (see ).
The first author is thankful to the Thailand Research Fund for financial support under Grant BRG5280016. Moreover, the second author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant CHE-Ph.D-THA- SUP/86/2550, Thailand.
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