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Generalized Variational-Like Inequalities for Pseudo-Monotone Type III Operators

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Generalized Variational-Like Inequalities for Pseudo-Monotone Type III Operators Cent. Eur. J. Math. • 6(4) • 2008 • 526-536 DOI: 10.2478/s11533-008-0049-1 Central European Journal of Mathematics Generalized variational-like inequalities for pseudo-monotone type III operators Research Article Mohammad S.R. Chowdhury1∗, Kok-Keong Tan2 1 Department of Mathematics, Lahore University of Management Sciences (LUMS), Phase II, Opposite Sector U, DHA, Lahore Cantt., Lahore - 54792, Pakistan 2 Department of Mathematics & Stattistics, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5 Received 3 April 2008; accepted 31 July 2008 Abstract: Our aim in this paper is mainly to prove some existence results for solutions of generalized variational-like inequal- ities with (η; h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces. Keywords: generalized variational-like inequalities • 0-diagonally concave relation • pseudo-monotone type III operators © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction Browder [4] and Hartman and Stampacchia [20] first introduced variational inequalities and Minty [26, 27] first introduced the theory of monotone nonlinear operators. Since then, there have been many generalizations on the theory of monotone nonlinear operators as well as in variational inequality problems. In 1996, Chowdhury and Tan first obtained generalized variational inequalities for pseudo-monotone type I operators [8]. Moreover, Chowdhury and Tan’s (1997) results on generalized variational inequalities for quasi-monotone operators were obtained in [9]. The recent results of Chowdhury and Tarafdar on generalized variational inequalities for pseudo-monotone type III operators were established in [10]. There are wide applications of variational inequalities in nonlinear elliptic boundary value problems, obstacle problems, complementarity problems, mathematical programming, mathematical economics and in many other areas. There are too many citations of papers concerning variational inequalities. Some references are [20], [23] and [31] where citations of many further references can be found. The topic on monotone variational inequalities is discussed in [35, 36]. In [22], Karamardian showed that the complementarity problems can be reduced to the variational inequality problems. In [25], Mancino and Stampacchia showed the relationship between mathematical programming and the variational inequalities. ∗ E-mail: [email protected] 526 Mohammad S.R. Chowdhury, Kok-Keong Tan In [28] and [34], Moreau and Rockafellar respectively showed the relationship between variational inequalities and convex functions. The relationship between variational inequalities and optimization is given by Peng in [33] and by Yamashita, Taji and Fukushima in [37]. Moreover, the relationship between the variational inequalities and the equilibrium point of Walrasian economy in Riesz spaces was shown by Aliprantis and Brown in [1]. In the year 2000, Ding and Tarafdar [16] obtained generalized variational-like inequalities (see definition below) for (η; h)-pseudo-monotone set-valued mappings, i.e., for (η; h)-pseudo-monotone type I operators. These operators are simply generalizations of Chowdhury and Tan’s pseudo-monotone type I operators (see [8] and also [6]). Note that in [8] pseudo-monotone type I operators were named as set-valued pseudo-monotone (and demi-monotone) operators given below. Definition 1.1. E∗ ∗ Let X be a nonempty subset of a topological vector space E, and let T : X → 2 (where E denotes the topological E∗ ∗ dual space of E and 2 denotes the family of all nonempty subsets of E ). If h : X → R, then T is said to be an h-pseudo-monotone (respectively, h-demi-monotone) operator y ∈ X {y } X if for each and every net α α∈Γ in converging to y (respectively, weakly to y) with Rehu; yα − yi h yα − h y ≤ ; lim sup[u∈ infT yα + ( ) ( )] 0 α∈Γ ( ) we have Rehu; yα − xi h yα − h x lim sup[u∈ infT yα + ( ) ( )] α∈Γ ( ) ≥ inf Rehw; y − xi + h(y) − h(x) for all x ∈ X; w∈T (y) T is said to be pseudo-monotone (respectively, demi-monotone) if T is h-pseudo-monotone (respectively, h-demi- monotone) with h ≡ 0. Later in [6], the above set-valued pseudo-monotone operators were renamed by M.S.R. Chowdhury as pseudo-monotone type I operators. There are many classes of problems where η(x; y) =6 x − y. As an example we can mention the classes of problems of system of generalized mixed implicit quasi-variational inclusions of Ding-Lee-Yu in [12]. In 2005, we introduced generalized variational-like inequalities for (η; h)-pseudo-monotone type II operators [11]. These operators are also generalizations of pseudo-monotone type II operators introduced by M.S.R. Chowdhury (see [5] and also [6]). Note that in [5] pseudo-monotone type II operators were named as demi operators. Later in [6], the demi operators were renamed by M.S.R. Chowdhury as pseudo-monotone type II operators. In this paper we shall define (η; h)-pseudo-monotone type III operators. These operators are generalizations of pseudo-monotone type III operators introduced by Chowdhury and Tarafdar in [10]. Note that in [10] pseudo-monotone type III operators were named as hemi-continuous operators. Later in [6], hemi-continuous operators were renamed by M.S.R. Chowdhury as pseudo-monotone type III operators. In this paper we have used mainly the concepts of pseudo-monotone type III operators which are derived by slight modification of the concepts of pseudo-monotone type I operators given in Definition 1.1. Our classical definition of single-valued pseudo-monotone type I operators was introduced by Brézis, Nirenberg and Satmpacchia in [3]. So, our concept of psedo-monotonocity does not contradict or confuse other classical concepts of pseudo-monotonocity available in the literature. Our aim in this paper is to establish some results on existence theorems for a new class of generalized variational-like inequalities with (η; h)-pseudo-monotone type III operators defined on non-compact sets in topological vector spaces. 527 Generalized variational-like inequalities for pseudo-monotone type III operators We shall begin with some preliminary concepts related to this paper: F Suppose that E and F are two vector spaces over the real or the complex field Φ. Let 2 denote the family of all nonempty subsets of F and let F(E) denote the family of all nonempty finite subsets of E and co(A) denotes the convex hull of A for any subset A ⊂ E. Let X be a nonempty subset of E and h ; i : F × E → Φ be a bilinear functional satisfying the following property: (P) the family {h·; xi}x∈E separates the points of F; i.e., for each y ∈ F with y =6 0, there exists x ∈ E such that hy; xi= 6 0. Throughout this paper, unless otherwise stated, E, F and h·; ·i will be assumed to satisfy the property (P). F With E, F and X as above, let T : X → 2 be a set-valued mapping, f : X → F and η : X × X → E be two single-valued mappings, and h : X × X → R be a real-valued function. The classical variational inequality (in short, VI(f; X; F)) is: find a point yˆ ∈ X such that Rehf(yˆ); yˆ − xi ≤ 0; ∀x ∈ X: (1) The generalized variational-like inequality (in short, GV LI(T ; η; h; X; F)) is: find yˆ ∈ X and wˆ ∈ T (yˆ) such that Rehw;ˆ η(y;ˆ x)i + h(y;ˆ x) ≤ 0; ∀x ∈ X: (2) The GV LI(T ; η; h; X; F) given in (2) was defined in [16] by Ding and Tarafdar and includes various variational inequalities studied in [13–15, 18, 19, 21, 29, 30, 32, 33] as special cases. In particular, when η(y;ˆ x) = yˆ − x and there 0 0 0 exists h : X → R such that h(y;ˆ x) = h (yˆ) − h (x), then the GVLI (2) reduces to Chowdhury and Tarafdar’s GVLI in [10]. E F X x ∈ E > W x {w ∈ F |hw; x i| < } σ F;E Let , and as above. For each 0 and each 0, let ( 0; ) = : 0 . Let ( ) be the topology on F generated by the family {W (x; ): x ∈ E and > 0} as a subbase for the neighbourhood system at 0. Then F becomes a locally convex topological vector space which is not necessarily Hausdorff. Since the bilinear functional h ; i : F × E → Φ satisfies the property (P) above, F also becomes Hausdorff. Throughout this paper, the underlying space F in the problem (2) is equipped with the σ(F;E)-topology. F Let E be also a topological vector space and T : X → 2 be a set-valued mapping. T is said to be upper semi-continuous (in short, u.s.c.) at x ∈ X if for each open set U in F containing T (x), there is an open neighbourhood V of x in X such that T (y) ⊂ U for all y ∈ V . T is called u.s.c. on X if T is u.s.c. at each x ∈ X. Definition 1.2. Let X be a nonempty subset of a topological vector space E over Φ and F be a vector space over Φ which is equipped F with the σ(F;E)-topology. Let T : X → 2 , η : X × X → E and h : X × X → R. Then T is said to be a (η; h)-pseudo- monotone type III (respectively, a strong (η; h)-pseudo-monotone type III) operator if for each x; y ∈ X and every net {y } X y y α α∈Γ in converging to (respectively, weakly to ) with Rehu; η yα ; x i h yα ; x ≤ ; lim sup[u∈ infT yα ( ) + ( )] 0 α∈Γ ( ) we have Rehu; η yα ; y i h yα ; y lim sup[u∈ infT yα ( ) + ( )] α∈Γ ( ) ≥ inf Rehw; η(y; x)i + h(y; x): w∈T (y) T is said to be an h-pseudo-monotone (respectively, a strong h-pseudo-monotone) type III operator, if T is an (η; h)- pseudo-monotone type III (respectively, a strong (η; h)-pseudo-monotone type III) operator with η(x; y) = x − y and for 0 0 0 some h : X → R, h(x; y) = h (x) − h (y) for all x; y ∈ X.
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