This paper is concerned with the study of a class of variational inequalities with multivalued -pseudomonotone mappings in reflexive Banach spaces by using the -antiresolvent technique. An application to the multivalued nonlinear -complementarity problem is also presented. The results coincide with the corresponding results announced by many others for the gradient state.

1. Introduction and Preliminaries

Variational inequalities give a convenient mathematical framework for discussing a large variety of interesting problems appearing in pure and applied sciences. It is well known that the theory of pseudomonotone mappings plays an important part in the study of the above-mentioned variational inequalities.

In recent years, pseudomonotone theory has become an attractive field for many mathematicians (see [18]).

In a very recent paper [9], by using the -antiresolvent technique (where is the duality mapping) devised by the first author, the author introduced a new concept of monotonicity, which is called the -pseudomonotone type.

In the present paper, the concept of multivalued -pseudomonotone mappings in reflexive Banach spaces is used to study a wide class of variational inequalities, called the multivalued -pseudomonotone variational inequalities.

Moreover, the results obtained in this paper can be applied to the multivalued nonlinear -complementarity problem. This problem contains, in particular, a mathematical model arising in the study of the postcritical equilibrium state of a thin plate resting, without friction on a flat rigid support (see [1012]). The results coincide with the corresponding results (see [2, 1315]) in the case of gradient mappings.

Unless otherwise stated, stands for a real reflexive Banach space with norm and stands for the uniformly convex dual of with the dual norm . The duality pairing between and is denoted by . The set of all nonnegative integers is denoted by . The field of real (resp., positive real) numbers is denoted by (resp., . Notation “” stands for strong convergence and “” for weak convergence.

A mapping is said to be a duality mapping (see, e.g., [16]) with gauge function (i.e., is continuous strictly increasing real-valued function satisfying and ) if for every . If is a Hilbert space, then , the identity mapping.

Assume that has a weakly sequentially continuous duality mapping (i.e., if is a sequence in which weakly convergent to a point , then the sequence converges to (see, e.g., [17])).

Let be a function. The domain of is . When , is called proper (see, e.g., [18]). The interior of the domain of is denoted by . The function is said to be Gâteaux differentiable at (see, e.g., [18]), if exists for all .

Let be proper, convex, lower semicontinuous, and Gâteaux differentiable at ; then the gradient of at is the function which is defined by for any . It is known (see, e.g., [19]) that the conjugate is also proper, convex, and lower semicontinuous.

The convex function is said to be of Legendre type (see, e.g., [20]) if the following conditions hold: , is Gâteaux differentiable on and ; , is Gâteaux differentiable on and .

It is well known (see, e.g., [21]) that if is a proper, convex, lower semicontinuous, and Legendre type, then and range .

Throughout this paper, the function is proper, convex, and lower semicontinuous which is also Legendre on .

The Bregman distance (see, e.g., [22]) is the function defined by with It should be pointed out that if is a Hilbert space and , then (the identity mapping) and .

For a multivalued mapping , the associated -antiresolvent (where is the duality mapping) of (see [9]) is the mapping , defined by Such a mapping is known as (see [23]) a -antiresolvent mapping of when (in this case, the mapping is denoted by ).

In light of the above-mentioned discussion, we note that if , then is the identity mapping .

Following [9], the mapping is said to be -pseudomonotone, if for every , and every sequence and the conditions imply that As remarked in [9], the -pseudomonotonicity of the mapping coincides with the pseudomonotonicity (or the -pseudomonotonicity in the sense of Bregman distance ) of the mapping , if .

The multivalued variational inequality defined by the -mapping (or multivalued -variational inequality) and the set is to find such that where .

The multivalued nonlinear complementarity problem defined by the -mapping (or multivalued nonlinear -complementarity problem) and the set is to find such that where , , and .

The multivalued -variational inequality and multivalued nonlinear -complementarity problem are very general in the sense that they include, as special cases, multivalued variational inequality and multivalued nonlinear complementarity problem.

The following definition and results will be used in the sequel.

Definition 1 (see, e.g., [15, p. 84]). The mapping is continuous on finite dimensional subspaces if for any finite dimensional subspace , the restriction of to is weakly continuous.

Corollary 2 (see [24]). Let be the injection mapping. Let be its dual mapping. Then, is continuous.

Corollary 3 (see [25]). Let be a nonempty compact convex set of and let be continuous. Then admits a fixed point.

2. Main Results

Theorem 4. Let be a closed convex set in and let be a multivalued mapping. Then the following are equivalent: the multivalued projection for ;

Proof. Assume that (1) holds. Let and . For every and , we have This implies Hence, implies (2).
On the other hand, assume that (2) holds. For every , we have This implies (1).

Remark 5. In the particular situation when Theorem 4 coincides (in gradient setting) with Theorem 2.3 in [15] and also with Proposition 2.1 (1) and (2) in [2].

Theorem 6. In addition to conditions on , and , one assumes that is separable, is nonempty closed and convex, are weakly continuous mappings, and either or is continuous. Moreover, assume that the mapping is a bounded -pseudomonotone mapping and that, for each , there exist , and such that Then there exists a solution to the multivalued -variational inequality (7).

Proof. Suppose that is an infinite dense set in and , is the linear span of .
Let . Let be the injection mapping and let be its restriction dual. Observe that is dense in , for .
Now, fix an integer and consider the finite dimensional problem.
Find such that for each , there exist ,  and   such that The equivalent form of problem (15) is to find such that for each , there exist ,   and   such that Using the identification of with and and Theorem 4 (with ), we see that (16) is equivalent to .
Let be any closed ball containing . It is well known (see, e.g., [26, p. 54, 224]) that is compact and convex in ; thus it is weakly closed.
From Corollary 2, is continuous; hence the function is continuous from into .
Hence, by Corollary 3, this equation admits a solution. If the closed convex set is assumed to be bounded, then by the reflexivity of it is weakly compact (by employing the Banach -Alaoglu theorem (see, e.g., [16, p. 3]).
Then we have a subsequence denoted by such that . Since is bounded, we have for all . Since is weakly continuous and since either or is continuous by hypothesis, it follows that is weakly continuous by [27, Lemma 1]. So, we have .
Now, we prove that For any , choose so large and such that Therefore, we have Since , we have Since is arbitrary, this shows the desired inequality.
By the -pseudomonotonicity of , it follows that for all and .
If , we have Hence for every in .
Since is dense in , so we have that is a solution to (7).
Now, to complete the proof, we consider the case when is unbounded.
In this case we consider the set , where .
Since is bounded, there exists at least one : for and .
Since , we have This, together with (14), implies that .
To clarify that is also a solution to original problem on , for any , set = for is sufficiently small, where and . Consequently This completes the proof.

Remark 7. In the particular situation when , Theorem 6 coincides with the Brezis Theorem (see, e.g., [13, 14]) for the case of gradient mapping.
We are now in a position to state and prove the following theorem.

Theorem 8. Let all assumptions of Theorem 6 hold, except for condition (14) let it be replaced by the -coercive condition: for , Suppose further that has the following property (W): for all and .
Then for every there exist , such that for all .

Proof. Let satisfy and The -coercivity of implies that there exists such that for with .
So we conclude The second part of (28) thus follows from Theorem 6.
To prove the first part of (28), observe that we can choose a point in and and assume that .
Therefore, from Theorem 6, we have for all .
On the other hand, setting , where , we get This implies So, This completes the proof of (28).

The following proposition gives a characterization of the sum of two -Pseudomonotone mappings.

Proposition 9. Let , and be as above and let be weakly continuous mappings. If , are -pseudomonotone mappings such that , then is -pseudomonotone.

Proof. Let , with and Now, we prove for , that If (note that otherwise, by symmetry), then there exists a subsequence such that This implies that From the -pseudomonotonicity of , we get for all Letting , we obtain a contradiction.
Hence, This holds for any subsequence, so (37) holds and the proof follows immediately by the superadditivity of the .

3. Application to Multivalued Nonlinear -Complementarity Problem

As applications of Theorem 8 we consider the multivalued nonlinear -complementarity problem (8) with , where are two nonlinear multivalued mappings from to , and .

Theorem 10. Let , and be the same as in Theorem 6, and suppose that has the property (W). Let the mappings and be weakly continuous and let either or both be continuous. Let be two bounded -pseudomonotone mappings.

Let Be such that , where and is the gauge function. Then for every problem (8) with has a solution in .

Proof. By Proposition 9, is -pseudomonotone for every . Set . Then This implies that the mapping is -coercive.

The conclusion follows from Theorem 8.