Nonzero Solutions of Generalized Variational Inequalities*

The existence of nonzero solutions for a class of generalized variational inequalities is studied by fixed point index approach for multivalued mappings in finite dimensional spaces and reflexive Banach spaces. Some new existence theorems of nonzero solutions for this class of generalized variational inequalities are established .


Introduction
Variational inequality theory with applications are an important part of nonlinear analysis and have been applied intensively to mechanics, differential equation, cybernetics, quantitative economics, optimization theory and nonlinear programming etc. (see [1][2][3][4]).
Variational inequalities, generalized variational inequalities and generalized quasivariational inequalities were studied intensively in the last 30 years with topological method, variational method, semi-ordering method, fixed point method, minimax theorem of Ky Fan and KKM technique ( [1][2][3][4]).In 1998, motivated by the paper [5], Zhu [6] studied a system of variational inequalities involving the linear operators in re flexive Banach spaces by using the coincidence degree theory due to Mawhin [7].Some existence results of positive solutions for this system of variational inequalities in re flexive Banach spaces were proved.
Let X be a real Banach space, * X its dual and (•, •) the pair between * X and X .Suppose that K is a nonempty closed convex subset of X .
Find u K ∈ , 0 u = / , and ( ) where mapping * : A K X → is nonlinear and * : 2 X g K → is a multi-valued mapping.The existence of nonzero solutions for variational inequalities is an important topic of variational inequality theory.[8] discussed the variational inequality (1) when A is coercive or monotone and g is set-contractive or upper semi-continuous.[9] considered the variational inequality (1) when A is single-valued continuous and g is set-contractive.
On the other hand, recently, under some different conditions, [10,11] obtained some existence theorems of nonzero solutions for a class of generalized variational inequalities by fixed point index approach for mu lti-valued mappings in reflexive Banach space.
Based on the importance of studying the existence of nonzero solutions for variational inequalities, and motivated and inspired by recent research works in this field, in this paper, we discuss the existence of nonzero solutions for a class of generalized variational inequalities as follows: ), where * , : . It is easily seen that the recession cone is indeed a cone and we have that rcK = ∅ / .For a proper lower semicontinuous convex functional : +∞ , in the virtue of [12], the limit 1 lim ( ) ∈ and j ∞ is also a lower semicontinuous convex functional with (0) 0 j ∞ = and with the property that ( ) , then the fixed point index, ( , ) K i T U , is well defined(see [13]).Proposition 1 [13] Let K be a nonempty closed convex subset of real Banach space X and U be an open subset of X .Suppose that : 2 K K T U → is an upper semicontinuous mapping with nonempty compact convex values and ( ) , U U be two open subsets of X with → be an upper semicontinuous mapping with nonempty compact convex values and , ) ( (0, ), ) For every * q X ∈ , let ( ) U q be the set of solutions in K of the following variational inequality ( , ), Obviously, ( ) A K q = ∅ if and only if the variational inequality (3) has no solution in K .

Nonzero Solutions in n R
Lemma 1 Let X be a separable reflexive Banach space.Suppose that is a bounded monotone hemicontinuous mapping (i.e., for every bounded subset D of X , ( ) A D is bounded) and : ( , ] j K → −∞ +∞ is a proper convex lower semicon- tinuous functional.Assume that there exists 0 Then for any given ), .
Proof.Without loss of generality, assume that 0 . Because X is a separable reflexive Banach space, for given r , there exists a closed convex sets sequences , 1, 2, , First, we shall verify that for each m , there exists Because m X is a finite dimensional subspace (deno- ted its inner product by [.,.] ), there exists a linear continuous mapping which is equivalent to the equality ( ) by [2,3], where m J P is an approximate mapping of m J .
Obviously, ( ) : According to Brouwer's fixed point theorem (see [2,3]), there exists m m u K ∈ satisfying the equality (9), that is, m u is a solution of the variational inequality (6).
Second, we shall verify that for each r , there exists In fact, … .Since X is a reflexive and r K is weakly closed, there exists a subsequence when µ is sufficiently large.Thus we have 0 0 This together with A being a monotone hemiconti- nuous mapping implies that inf ( , ) If when µ is sufficiently large.It thus follows from (13) that Because . therefore r u is a solution of the variational inequality (10).
New we shall verify that the variational inequality (5) has a solution.Taking and so it then follows from condition (4) that there exists constant C > 0 such that and so r u is an inner point of r B .
Thus for any given X ω ∈ , we have (1 ) Therefore r u is a solution of the variational inequality (5).
Theorem 1 Let K be a nonempty unbounded closed convex set in . We shall verify that ( , ) 1 Firstly, define a mapping by : [0 It is easily seen that ( , ) H t u is an upper semicontinuous mapping with nonempty compact convex values.We claim that there exists large enough R such that ( , ) u H t u ∉ for all (0,1), t ∈ ( ) ), by Proposition 1(4) and ( 2).Secondly, we shall verify that ( , ) 0 small enough r ( 1 r < ).In fact, there exist constants 1 2 , , 0 C C M > from the boundedness of j , locally boundedness of A and condition (b) such that for all Define a mapping by [0,1] 2 ( , , ) . Then H is an upper semi-continuous mapping with nonempty compact convex values.We claim that there exists a small enough r such that ( , ) ), , we obtain a contradiction.Therefore ( , ) ), 21) and (23).That contradicts to (22).Therefore, ( (1, ), ) 0 It follows from Proposition 1(3) that ( , Therefore there exists a fixed point \ R r u K K ∈ which is a nonzero solution of (2).

Nonzero Solutions in Reflexive Banach Spaces
Theorem 2 Let X be a reflexive Banach space and K X ⊂ a nonempty unbounded closed convex set with 0 K ∈ .Suppose that . Then, , F F A g are hemicontinuous and continuous respective- ly.For 1 2 , ).
‖‖ ‖‖‖‖ Therefore all conditions in Theorem 1 are satisfied on space F and so there exists , 0 ), This together with condition (b) implies that there exists a constant M > 0 such that for all finite dimensional subspace F containing 0 u .Since X is reflexive and K is weakly closed, with a similar argu- ment to that in the proof of Theorem 2 in [10] (also see [8]), we shall show that there exists u K ′ ∈ such that for every finite dimensional subspace F containing 0 u , u′ is in the weak closure of the set where 1 F is a finite dimensional subspace in X .
In fact, since F V is bounded, we know that ( ) w F V (the weak closure of the set F V ) is weakly compact.
On the other hand, let That is to say, there exists u K ′ ∈ such that for every finite dimensional subspace F containing 0 u , u′ is in the weak closure of the set The right side of the above inequality tends to 0, which contradicts to the condition (c).Therefore u′ is a nonzero solution of (2).

*
This work was supported by The Zhejiang Provincial Natural Science Foundation(No.Y7080068) and the Foundation of Department of Education of Zhejiang Province(No.20070628) Since j is a lower semicontinuous function and ε is an arbitrary positive number, we have sup by the condition (c).Let F X ⊂ be a finite dimensional subspace containing 0 u .We shall show that all conditions in Theorem 1 are satisfied on space F .
of A .This means that F A is mono- tone.On the other hand, let v K ∈and F ′ a finite dimensional subspace of X which contains 0 u and v .Since u′ belongs to the weak closure of the set