Báo cáo toán học: Existence Theorems for Some Generalized Quasivariational Inclusion Problems

Trong bài báo này chúng tôi cung cấp cho đủ điều kiện cho sự tồn tại của các giải pháp của vấn đề (P1) (Vấn đề resp. (P2)) của việc tìm kiếm một điểm (z0, x0) ∈ B (z0, x0) × A (x0) như vậy mà F (z0, x0, x) ⊂ C (z0, x0, x0) (resp. F (z0, x0, x0) ⊂ C (z0, x0, x)) với mọi x ∈ A (x0), A, B,C, F được thiết lập giá trị bản đồ giữa các địa phương lồi Hausdorffkhông gian. Một số định lý tồn tại được bao gồm như trường hợp...
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Báo cáo toán học: " Existence Theorems for Some Generalized Quasivariational Inclusion Problems" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:1 (2005) 111–122 RI 0$7+(0$7,&6 ‹ 9$67 Existence Theorems for Some Generalized Quasivariational Inclusion Problems Le Anh Tuan1 and Pham Huu Sach2 1 Ninh Thuan College of Pedagogy, Ninh Thuan, Vietnam 2 Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307, Hanoi, Vietnam Received July 3, 2004 Revised March 9, 2005Abstract. In this paper we give sufficient conditions for the existence of solutions ofProblem (P1 ) (resp. Problem (P2 )) of finding a point (z0 , x0 ) ∈ B (z0 , x0 ) × A(x0 )such that F (z0 , x0 , x) ⊂ C (z0 , x0 , x0 ) (resp. F (z0 , x0 , x0 ) ⊂ C (z0 , x0 , x)) for allx ∈ A(x0 ), where A, B, C, F are set-valued maps between locally convex Hausdorffspaces. Some known existence theorems are included as special cases of the main resultsof the paper.1. IntroductionLet X, Y and Z be locally convex Hausdorff topological vector spaces. LetK ⊂ X and E ⊂ Z be nonempty subsets. Let A : K −→ 2K , B : E × K −→ 2E ,C : E × K × K −→ 2Y and F : E × K × K −→ 2Y be set-valued maps withnonempty values. In this paper, we consider the existence of solutions of thefollowing generalized quasivariational inclusion problems: Problem (P1 ): Find (z0 , x0 ) ∈ E × K such that x0 ∈ A(x0 ), z0 ∈ B (z0 , x0 )and, for all x ∈ A(x0 ), F (z0 , x0 , x) ⊂ C (z0 , x0 , x0 ). Problem (P2 ): Find (z0 , x0 ) ∈ E × K such that x0 ∈ A(x0 ), z0 ∈ B (z0 , x0 )and, for all x ∈ A(x0 ), F (z0 , x0 , x0 ) ⊂ C (z0 , x0 , x).112 Le Anh Tuan and Pham Huu SachObserve that in the above models the set C (z, ξ, x) is not necessarily a convexcone. This is useful for deriving many known results in quasivariational inequal-ities and quasivariational inclusions. We now mention some papers containingresults which can be obtained from the existence theorems of the present pa-per. The generalized quasivariational inequality problem considered in [2, 7]corresponds to Problem (P1 ) where F is single-valued and C (z, ξ, x) ≡ R+ (thenonnegative half-line). The paper [6] deals with Problem (P1 ) where F is single-valued and C (z, ξ, x) equals the sum of F (z, ξ, x) and the complement of thenonempty interior of a closed convex cone. In [11] Problem (P1 ) and (P2 ) areconsidered under the assumption that C (z, ξ, x) is the sum of F (z, ξ, x) and aclosed convex cone. Our main results formulated in Sec. 3 of this paper willinclude as special cases Theorem 3.1 and Corollary 3.1 of [2], Theorem 3 of [7],Theorem 2.1 of [6] and Theorems 3.1 and 3.2 of [11]. It is worth noticing thatTheorems 3.1 and 3.2 of [11] are obtained under the assumptions stronger thanthe corresponding assumptions used in the present paper. This remark can beseen in Sec. 4. Our approach is based on a fixed point theorem of [10] whichtogether with some necessary notions can be found in Sec. 2.2. PreliminariesLet X be a topological space. Each subset of X can be seen as a topologicalspace whose topology is induced by the given topology of X. For x ∈ X, letus denote by U (x), U1 (x), U2 (x), ... open neighborhoods of x. The empty set isdenoted by ∅. A nonempty subset Q ⊂ X is a convex cone if it is convex and ifλQ ⊂ Q for all λ ≥ 0. For a set-valued map F : X −→ 2Y between two topological spaces X andY we denote by im F and gr F the image and graph of F : im F = F (x), x ∈X gr F = {(x, y ) ∈ X × Y : y ∈ F (x)}.The map F is upper semicontinuous (usc) if for any x ∈ X and any open setN ⊃ F (x) there exists U (x) such that N ⊃ F (x ) for all x ∈ U (x). The mapF is lower semicontinuous (lsc) if for any x ∈ X and any open set N withF (x) ∩ N = ∅ there exists U (x) such that F (x ) ∩ N = ∅ for all x ∈ U (x).The map F is continuous if it is both usc and lsc. The map F is closed if itsgraph is a closed set of X × Y. The map F is compact if im F is contained in acompact set of Y. The map F is acyclic if it is usc and if, for any x ∈ X, F (x)is nonempty, compact and acyclic. Here a topological space is called acyclic if ˇall of its reduced Cech homology groups over rationals vanish. Observe thatcontractible spaces are acyclic; and hence, convex sets and star-shaped sets areacyclic. ...