Multivalued Variational Inequalities with D J-Pseudomonotone Mappings in Reflexive Banach Spaces

and Applied Analysis 3 the multivalued projection for K; (2) ∇g ⋆ (] 󸀠 ) ∈ K : ⟨∇g ⋆ (] 󸀠 ) − x, ∇g ⋆ (ω 󸀠 )


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 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 [10][11][12]).The results coincide with the corresponding results (see [2,[13][14][15]) 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 N. The field of real (resp., positive real) numbers is denoted by R (resp., R + ).Notation " → " stands for strong convergence and "⇀" for weak convergence.
Throughout this paper, the function  :  → R + ∪ {+∞} is proper, convex, and lower semicontinuous which is also Legendre on int dom().
In light of the above-mentioned discussion, we note that if  −  = ∇, then   is the identity mapping .
The multivalued variational inequality defined by the  mapping (or multivalued   -variational inequality)  −  :  → 2  ⋆ and the set  ⊂  is to find  ∈  such that where  ∈ 2  ⋆ .The multivalued nonlinear complementarity problem defined by the   -mapping (or multivalued nonlinear  complementarity problem)  −  :  → 2  ⋆ and the set  is to find  ∈  such that where  ∈ 2  ⋆ , ] ∈ ( − )(), 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.
Then we have a subsequence denoted by {  } ∈N such that   ⇀  ∈ .Since  −  is bounded, we have ‖]  ‖  ⋆ ≤  for all  ∈ N. 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 lim sup For any  > 0, choose  so large and ũ ∈   such that Therefore, we have Since   ⊂   , we have lim sup Since  is arbitrary, this shows the desired inequality.By the   -pseudomonotonicity of  − , it follows that lim inf for all  ∈ dom  ∩ dom  and  ∈ ( − )().
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.
We are now in a position to state and prove the following theorem.
Proof.Let  ∈ 2  ⋆ satisfy ‖‖  ⋆ <  and The   -coercivity of  −  implies that there exists  > 0 such that 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 ∇ ⋆ () = 0.
The conclusion follows from Theorem 8.