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Journal of Applied Mathematics and Stochastic Analysis Volume 2007, Article ID 78343, 6 pages doi:10.1155/2007/78343

Research Article

Nonlinear Vector Variational Inequality Problems for ^-Pseudomonotone Maps

A. P. Farajzadeh

Received 14 March 2007; Accepted 5 July 2007

We consider a new class of complementarity problems for ^-pseudomonotone maps and obtain an existence result for their solutions in real Hausdorff topological vector spaces. Our results extend the same previous results in this literature.

Copyright © 2007 A. P. Farajzadeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Variational inequalities were introduced and considered by Stampacchia [1] in early sixties. It has been shown that a wide class of linear and nonlinear problems arising in various branches of mathematical and engineering sciences can be studied in the unified and general framework of variational inequalities. Variational inequalities have been generalized and extended in several directions using new techniques. Giannessi [2] introduced a new class of variational inequalities, which is vector variational inequality. Vector vari-ational inequalities have many applications in vector optimization, approximate vector optimization, and other areas (see, e.g., [3]). Noor [4] introduced a class of variational inequalities involving two operators, which are called general variational inequalities. It has been shown that nonsymmetric and odd-order obstacle, free, moving, and equilibrium problems can be studied via the general variational inequalities. For the applications, formulation, and numerical methods for solving variational inequalities, see [5-8] and the references therein.

Inspired and motivated by the recent research activities going on in this dynamic field, we introduce a new class of complementarity problems for ^-pseudomonotone maps. Moreover, we obtain an existence result for their solutions in real Hausdorff topological vector spaces setting for a moving cone by relaxing continuity and compactness. This is done by using a new version of famous Ky Fan lemma which is due to Ben-El-Mechaiekh

et al. [9]. Our results represent an improvement and refinement of the recent results obtained in [10].

In the rest of this section, we recall some definitions and preliminaries results which are used in the next section.

We will denote by 2A the family of all subsets of A and by ^(A) the family of all nonempty finite subsets of A. Let X be a real Hausdorff topological vector space (t.v.s.). A nonempty subset P of X is called convex cone if (i) P + P = P, (ii) XP c P, for all X > 0. Cone P is said to be pointed whenever P n - P = {0}. Let Y be a t.v.s. and let P c Y be a cone. The cone P induces an ordering on Y (in this case the pair (Y, P) is called an ordered t.v.s.) which is defined as follows:

x < y ^^ y - x e P. (1.1)

This ordering is antisymmetric if P is pointed. Let K be a nonempty convex subset of a t.v.s. X and let K0 be a subset of K. A multivalued map r : K0 ~^2K is said to be a KKM map if

coA c y xeAr(x), VA e (1.2)

where co denotes the convex hull.

Definition 1.1 (see [9]). Consider a subset A of a topological vector space and a topological space Y. A family {Ci, Ki}ieI of pairs of sets is said to be coercing for a map G : A^2Y if and only if

(i) for each i e K, Ci is contained in a compact convex subset of A, and Kj is a compact subset of Y;

(ii) for each i, j, there exists k e I such that Ci u Cj c Ck;

(iii) for each i e I, there exists k e I with HxeCkG(x) c Ki.

Theorem 1.2 (see [9]). Let F : K~^2Y be a KKM map with compactly closed (in K) values. If F admits a coercing family, then f) xeKF (x) = 0.

2. Main results

Throughout this section we let X and Y be two topological vector spaces, K a nonempty convex subset of X, C : K~^2Y with convex cone values, and let q : K X K~^L(X, Y) and T : K~^L(X, Y) be two nonlinear mappings.

We consider two following problems; the first is called nonlinear vector variational inequality (NVVI) problem with respect to q that consists in finding x e K such that

(T(x),n(y,x)) e C(x), Vy e K. (2.1)

The second problem is called dual nonlinear vector variational inequality (DNVVI) problem with respect to n that consists in finding x e K such that

(T(y),n(x,y)) e- C(y), Vy e K. (2.2)

We denote the solution set of (2.1) and (2.2) with NVVIS and DNVVIS, respectively.

Definition 2.1. T is C-pseudomonotone with respect to n if, for all x,y G K, the following implication holds:

(T(x),n(y,x)> G C(x) (T(y),n(x,y)) G - C(y). (2.3)

Remark that the definition of monotonicity of T with respect to n given in [10] implies C-pseudomonotonicity of T with respect to n, for a constant cone C, that is C(x) = C, for all x G K.

Definition 2.2. T is said to be C-upper sign continuous with respect to n if, for all x,y G K, the following holds:

(T(u),n(y,u)> G C(u), Vu G ]x,y [ (T(x),n(x,y)) G C(x). (2.4)

Let us recall that the above definition is a very weak kind of continuity. This notion is introduced by Hadjisavvas [11] in the framework of variational inequalities and later by Bianchi and Pini [12] for real bifunctions.

The following proposition improves Theorem 3.1 in [10].

Proposition 2.3. If n is antisymmetric, that is, n(x,y) = —n(y,x), the set {y G K : (T(x), n(y,x)} = 0} = {x}, and T is C-pseudomonotone with respect to n, then the solution set of (NVVI) is empty or singleton.

Proof. Let x1, x2 be two solutions of (NVVI). Hence

(Tfa),nfe,x^) G C(x^, (T(x^,n(xux^} G C(x2). (2.5)

From C-pseudomonotone with respect to n ofT, n is antisymmetric, and from (2.1), we get

(T (x2), n(x1, x2)> G C(x2) n — C(x2) = {0}. (2.6)

x1 G {y G K : (T(x2),n(y,x2)> = 0} = {x2}. (2.7)

This completes the proof. □

The following theorem generalizes Theorem 3.2 in [10].

Theorem 2.4. Let T : K~^L(X, Y) and n : K X X~^L(X, Y) be two mappings satisfying the following conditions:

(i) T is C-pseudomonotone with respect to n;

(ii) n is convex in the first variable with n(x, x) = 0, for all x G K;

(iii) T is C-upper sign continuous with respect to n. Then, NVVIS = DNVVIS.

Proof. By the definition of C-pseudomonotone with respect to n, we have

NVVIS ç DNVVIS. (2.8)

Conversely, let x0 G DNVVIS and x G K. By letting xt = x0 + t(x - x0), for t G ]0,1[, from (2.2), we have

{Txt,n(x0,xt)) G- C(xt). (2.9)

If (T(xs),n(x,xs)} G C(xs), for some s G ]0,1[, then it is obvious from (2.9) and (ii) that

0 = (T(xs),(1 - s)n(x0,xs)+sn(x,xs) - n(xs,xs)) G C(xs), (2.10)

which is a contradiction, since C(xs) is a pointed convex cone and 0 G C(xs). Hence we have

{T(xs),n(x,xs)) G C(xs), Vs G ]0,1[. (2.11)

Now, (iii) entails the result. □

Theorem 2.5. Assume that

(i) for each x G K, n(x,x) = 0, and any compact subset W of K, the set {y G W : (Ty, n(x,y)} G C(y)} is closed in W;

(ii) for each finite subset A of K and any y G coA\A, there exists x G A such that (Ty,n(x,y)} G C(y);

(iii) there exist compact subset B and compact convex subset D of K such that for all x G K\B, 3y G D; (Tx,q(y,x)} G C(x).

Then the NVVIS is nonempty and compact.

Proof. We define r : K~^2K as follows:

r(y) = {x G K : {Tx,n(y,x)) G C(x)}. (2.12)

By (i), r has compactly closed values. We claim that r is a KKM mapping. Indeed, if it is false, then there exist elements y1,y2,...,yn of K and z G co({y1,y2,...,yn}) such that z G U n=1r(yi). Thus by the definition of r, we have (Tz,q(y,,z)} G C(z), for i = 1,2,...,n, which is a contradiction (by (ii)). It is clear that {(D,B)} is a coercing family for r. Now, by Theorem 1.2, NVVIS = HxGKr(x) = 0. Using (iii), we obtain

NVVIS =H xGKr(x) ç B, (2.13)

and hence

NVVIS = HxGKr(x) = HxGK (r(x) n B), (2.14)

which is closed in B (by (i)), and so a compact subset of B. □

Theorem 2.6. Assume that

(i) for each x G K, n(x,x) = 0, and any compact subset W of K, the set {y G W : (Ty, n( y, x)} G - C(y)} is closed in W ;

(ii) for each finite subset A of K and any y G coA\A, there exists x G A such that (Ty,n(y,x)} G - C(x);

(iii) there exist compact subset B and compact convex subset D of K such that for all x G K\B, 3y G D; (Ty,n(x,y)} G — C(y).

Then the DNVVIS is nonempty and compact.

Proof. We define r : K~^2K as follows:

r(y) = {x G K : (Ty,n(x,y)> G — C(y)}. (2.15)

By (i), r has compactly closed values. By (ii), r is a KKM mapping. It is obvious that

{(D,B)} is a coercing family for r. Now, by Theorem 1.2, DNVVIS = HxGKT(x) = 0.

Moreover, using (iii),

DNVVIS = f| xGKr(x) ç B, (2.16)

and hence

DNVVIS = HxGKr(x) = HxGK(r(x) n B), (2.17)

which is closed in B (by (i)), and so a compact subset of B. □

Acknowledgment

The author was in part supported by the Razi University in Kermanshah, Iran.

References

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A. P. Farajzadeh: Department of Mathematics, Razi University, Kermanshah 67149, Iran Email address: ali-ff@sci.razi.ac.ir

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