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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 905605, 12 pages doi:10.1155/2009/905605 Research Article Some Maximal Elements’ Theorems in FC-Spaces Rong-Hua He 1, 2 and Yong Zhang 1 1 Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610103, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Rong-Hua He, ywlcd@cuit.edu.cn Received 30 March 2009; Accepted 1 September 2009 Recommended by Nikolaos Papageorgiou Let I be a finite or infinite index set, let X be a topological space, and let Y i ,ϕ N i  i∈I be a family of FC-spaces. For each i ∈ I,letA i : X → 2 Y i be a set-valued mapping. Some new existence theorems of maximal elements for a set-valued mapping and a family of set-valued mappings involving a better admissible set-valued mapping are established under noncompact setting of FC-spaces. Our results improve and generalize some recent results. Copyright q 2009 R H. He and Y. Zhang. 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 It is well known that many existence theorems of maximal elements for various classes of set-valued mappings have been established in different spaces. For their applications to mathematical economies, generalized games, and other branches of mathematics, the reader may consult 1–12 and the references therein. In most of the known existence results of maximal elements, the convexity assumptions play a crucial role which strictly restrict the applicable area of these results. In this paper, we will continue to study existence theorems of maximal elements in general topological spaces without convexity structure. We introduce a new class of generalized G B -majorized mappings A i : X → 2 Y i for each i ∈ I which involve a set-valued mapping F ∈BY, X. The notion of generalized G B -majorized mappings unifies and generalizes the corresponding notions of G B -majorized mappings in 4; L S -majorized mappings in 2, 13; H-majorized mappings in 14. Some new existence theorems of maximal elements for generalized G B -majorized mappings are proved under noncompact setting of FC-spaces. Our results improve and generalize the corresponding results in 2, 4, 14–16. 2 Journal of Inequalities and Applications 2. Preliminaries Let X and Y be two nonempty sets. We denote by 2 Y and X the family of all subsets of Y and the family of all nonempty finite subsets of X, respectively. For each A ∈X,we denote by |A| the cardinality of A.LetΔ n denote the standard n-dimensional simplex with the vertices {e 0 , ,e n }.IfJ is a nonempty subset of {0, 1, ,n}, we will denote by Δ J the convex hull of t he vertices {e j : j ∈ J}. Let X and Y be two sets, and let T : X → 2 Y be a set-valued mapping. We will use the following notations in t he sequel: i Tx{y ∈ Y : y ∈ Tx}, ii TA  x∈A Tx, iii T −1 y{x ∈ X : y ∈ Tx}. For topological spaces X and Y ,asubsetA of X is said to be compactly open resp., compactly closed if for each nonempty compact subset K of X, A∩K is open resp., closed in K. The compact closure of A and the compact interior of A see 17 are defined, respectively, by ccl A    B ⊂ X : A ⊂ B, B is compactly closed in X  , cint A    B ⊂ X : B ⊂ A, B is compactly open in X  . 2.1 It is easy to see that cclX \ AX \ cint A, int A ⊂ cint A ⊂ A, A ⊂ ccl A ⊂ clA, A is compactly open resp., compactly closed in X if and only if A  cint A resp., A  ccl A. For each nonempty compact subset K of X, ccl A  K  cl K A  K and cint A  K  int K A  K, where cl K A  Kresp., int K A  K denotes the closure resp., interior of A  K in K.A set-valued mapping T : X → 2 Y is transfer compactly open valued on X see 17 if for each x ∈ X and y ∈ Tx, there exists x  ∈ X such that y ∈ cint Tx  .LetA i i  1, ,m be transfer compactly open valued, then  m i1 cint A i  cint  m i1 A i . It is clear that each transfer open valued correspondence is transfer compactly open valued. The inverse is not true in general. The definition of FC-space and the class BY, X of better admissible mapping were introduced by Ding in 8. Note that the class BY, X of better admissible mapping includes many important classes of mappings, for example, the class BY, X in 18, U k c Y, X in 19 and so on as proper subclasses. Now we introduce the following definition. Definition 2.1. An FC-space Y, ϕ N  is said to be an CFC-space if for each N ∈Y , there exists a compact FC-subspace L N of Y containing N. Y, ϕ N  be a G-convex space, let the notion of CG-convex space was introduced by Ding in 4. Lemma 2.2 8. Let I be any index set. For each i ∈ I,letY i ,ϕ N i  be an FC-space, Y   i∈I Y i and ϕ N   i∈I ϕ N i .ThenY, ϕ N  is also an FC-space. Let X be a topological space, and let I be any index set. For each i ∈ I,letY i ,ϕ N i  i∈I be an FC-space, and let Y   i∈I Y i such that Y, ϕ N  is an FC-space defined as in Lemma 2.2. Journal of Inequalities and Applications 3 Let F ∈BY, X and for each i ∈ I,letA i : X → 2 Y i be a set-valued mapping. For each i ∈ I, 1 A i : X → 2 Y i is said to be a generalized G B -mapping if a for each N  {y 0 , ,y n }∈Y and {y i 0 , ,y i k }⊂N, Fϕ N Δ k     k j0 cint A −1 i π i y i j   ∅, where π i is the projection of Y onto Y i and Δ k  co{e i j : j  0, ,k}; b A −1 i y i {x ∈ X : y i ∈ A i x} is transfer compactly open in Y i for each y i ∈ Y i ; 2 A x,i : X → 2 Y i is said to be a generalized G B -majorant of A i at x ∈ X if A x,i is a generalized G B -mapping and there exists an open neighborhood Nx of x in X such that A i z ⊂ A x,i z for all z ∈ Nx; 3 A i is said to be a generalized G B -majorized if for each x ∈ X with A i x /  ∅, there exists a generalized G B -majorant A x,i of A i at x, and for any N ∈{x ∈ X : A i x /  ∅}, the mapping  x∈N A −1 x,i is transfer compactly open in Y i ; 4 A i is said to be a generalized G B -majorized if for each x ∈ X, there exists a generalized G B -majorant A x,i of A i at x. Then {A i } i∈I is said to be a family of generalized G B -mappings resp., G B -majorant mappings if for each i ∈ I, A i : X → 2 Y i is a generalized G B -mapping resp., G B -majorant mapping. If for each i ∈ I,letY i ,ϕ N i  be a G-convex space, a family of G B -mappings resp., G B - majorant mappings were introduced by Ding in 4. Clearly, each family of generalized G B - mappings must be a family of generalized G B -majorant mappings. If F  S is a single-valued mapping and A i x is an FC-subspace of Y i for each x ∈ X, then condition y i / ∈ A i Sy for each y ∈ Y implies that condition a in 1 holds. Indeed, if a is false, then there exist N  {y 0 , ,y n }∈Y , {y i 0 , ,y i k }⊆N,andy ∈ ϕ N Δ k  such that FySy ∈  k j0 A −1 i π i y i j  and hence π i y i j  ∈ A i Sy for each j  0, ,k. It follows from y ∈ ϕ N Δ k  that π i y ∈ ϕ N i Δ k  where N i  π i N. It follows from A i Sy being an FC-subspace of Y i that π i y ∈ ϕ N i Δ k  ⊂ A i Sy which contradicts condition y i / ∈ A i Sy for each y ∈ Y . Hence each L S -mapping resp., L S -majorant mapping introduced by Deguire et al. see 2, page 934 must be a generalized G B -mapping resp., G B -majorant mapping. The inverse is not true in general. 3. Maximal Elements In order to obtain our main results, we need the following lemmas. Lemma 3.1 3. Let X and Y be topological spaces, let K be a nonempty compact subset of X, and let G : X → 2 Y be a set-valued mapping such that Gx /  ∅ for each x ∈ K. Then the following conditions are equivalent: 1 G have the compactly local intersection property; 2 for each y ∈ Y, there exists an open subset O y of X (which may be empty) such that O y  K ⊂ G −1 y and K   y∈Y O y  K; 3 there exists a set-valued mapping F : X → 2 Y such that for each y ∈ Y, F −1 y is open or empty in X, F −1 y  K ⊂ G −1 y, ∀y ∈ Y, and K   y∈Y F −1 y  K; 4 Journal of Inequalities and Applications 4 for each x ∈ K,thereexistsy ∈ Y such that x ∈ cint G −1 y  K and K   y∈Y cint G −1 y  K  y∈Y G −1 y  K; 5 G −1 : Y → 2 X is transfer compactly open valued on Y . Lemma 3.2 8. Let X be a topological space, and let Y, ϕ N  be an FC-space, F ∈BY, X and A : X → 2 Y such that i for each N  {y 0 , ,y n }∈Y  and for each {y i 0 , ,y i k }⊆N, F  ϕ N  Δ k    ⎛ ⎝ k  j0 cint A −1  y i j  ⎞ ⎠  ∅, 3.1 ii A −1 : Y → 2 X is transfer compactly open valued; iii there exists a nonempty set Y 0 ⊂ Y and for each N  {y 0 , ,y n }∈Y,thereexistsa compact FC-subspace L N of Y containing Y 0 ∪ N such that K   y∈Y 0 cint A −1 y c is empty or compact in X,wherecint A −1 y c denotes the complement of cint A −1 y. Then there exists a point x ∈ X such that Ax∅. Theorem 3.3. Let X be a topological space, let K be a nonempty compact subset of X, and let Y, ϕ N  be an FC-space, F ∈BY, X and A : X → 2 Y be a generalized G B -mapping such that i for each N  {y 0 , ,y n }∈Y , there exists a compact FC-subspace L N of Y containing N such that for each x ∈ X \ K, L N  cint Ax /  ∅. Then there exists a point x ∈ K such that A x∅. Proof. Suppose that Ax /  ∅ for each x ∈ X. Since A is a generalized G B -mapping, A −1 is transfer compactly open valued. By Lemma 3.1, we have K   y∈Y  cint A −1  y   K  . 3.2 Since K is compact, there exists a finite set N  {y o , ,y n }∈Y  such that K  n  i0  cint A −1  y i   K  . 3.3 By condition i and F ∈BY, X, there exists a compact FC-subspace L N of Y containing N and FL N  is compact in X, and hence we have F  L N    y∈L N  cint A −1  y   F  L N   . 3.4 By using similar argument as in the proof of Lemma 3.2, we can show that there exists x ∈ X such that A x∅. Condition i implies that x must be in K. This completes the proof. Journal of Inequalities and Applications 5 Remark 3.4. Theorem 3.3 generalizes in 4, Theorem 2.2 in the following several aspects: a from G-convex space to FC-space without linear structure; b from G B -mappings to generalized G B -mappings. Theorem 3.5. Let X be a topological space, and let Y, ϕ N  be an FC-space. Let F ∈BY, X and A : X → 2 Y be a generalized G B -majorized mapping such that i there exists a paracompact subset E of X such that {x ∈ X : Ax /  ∅} ⊂ E; ii there exists a nonempty set Y 0 ⊂ Y and for each N  {y 0 , ,y n }∈Y,there exists a compact FC-subspace L N of Y containing Y 0 ∪ N such that the set K   y∈Y 0 cint A −1 y c is empty or compact. Then there exists a point x ∈ X such that A x∅. Proof. Suppose that Ax /  ∅ for each x ∈ X. Since A is a generalized G B -majorized, for each x ∈ X, there exists an open neighborhood Nx of x in X and a generalized G B -mapping A x : X → 2 Y such that a Az ⊂ A x z for each z ∈ Nx, b for each N  {y 0 , ,y n }∈Y and {y i 0 , ,y i k }⊆N, Fϕ N Δ k     k j0 cint A −1 x y i j   ∅, c A −1 x is transfer compactly open in Y , d for any N ∈{x ∈ X : Ax /  ∅}, the mapping  x∈N A −1 x is transfer compactly open in X. Since Ax /  ∅ for each x ∈ X, it follows from condition i that X  {x ∈ X : Ax /  ∅}  E is paracompact. By Dugundji in 20, Theorem VIII.1.4, the open covering {Nx : x ∈ X} has an open precise locally finite refinement {Ox : x ∈ X}, and for each x ∈ X, Ox ⊂ Nx since X is normal. For each x ∈ X, define a mapping B x : X → 2 Y by B x  z   ⎧ ⎨ ⎩ A x  z  , if z ∈ Ox, Y, if z ∈ X \ Ox. 3.5 Then for each y ∈ Y , we have B −1 x  y    z ∈ Ox : y ∈ A x  z     z ∈ X \ Ox : y ∈ Y    A −1 x  y   Ox    X \ Ox    A −1 x  y    X \ O  x     O  x    X \ O  x    A −1 x  y   X \ O  x  . 3.6 Hence B −1 x y is transfer compactly open in Y by c. Now define a mapping B : X → 2 Y by B  z    x∈X B x  z  , ∀z ∈ X. 3.7 6 Journal of Inequalities and Applications We claim that B is a generalized G B -mapping and Az ⊂ Bz for each z ∈ X. Indeed, for any nonempty compact subset C of X and each y ∈ Y with B −1 y ∩ C /  ∅, we may take any fixed u ∈ B −1 y ∩ C. Since {Ox : x ∈ X} is locally finite, there exists an open neighborhood V u of u in X such that {x ∈ X : V u ∩ Ox /  ∅}  {x 1 , ,x n } is a finite set. If x / ∈{x 1 , ,x n }, then ∅  V u ∩ OxV u ∩ Ox, and hence B x zY for all z ∈ V u which implies that Bz  x∈X B x z  n i1 B x i z for all z ∈ V u . It follows that for each y ∈ Y, B −1  y    z ∈ X : y ∈ B  z   ⊃  z ∈ V u : y ∈ B  z     z ∈ V u : y ∈ n  i1 B x i  z    V u   n  i1 B −1 x i  y   . 3.8 For any nonempty compact subset C of X and each y ∈ Y ,ifv ∈ V u ∩   n i1 B −1 x i y  C ⊂ B −1 y  C. Since V u is open in X, it follows from d that there exists y  ∈ Y such that v ∈ V u  cint  n  i1 B −1 x i  y     C  cint  V u  n  i1 B −1 x i  y     C  cint B −1  y    C. 3.9 This proves that B −1 : Y → 2 X is transfer compactly open valued in Y . On the other hand, for each N  {y 0 , ,y n }∈Y and N 1  {y i 0 , ,y i k }⊆N,ift ∈  k j0 cint B −1 y i j , then N 1 ⊂ cint Bt. Since there exists x 0 ∈ X such that t ∈ Ox 0  and N 1 ⊂ cint Bt ⊂ cint B x 0 tcint A x 0 t, we have t ∈  k j0 cint A −1 x 0 y i j , and hence t / ∈ Fϕ N Δ k  by b. Hence we have F  ϕ N  Δ k    ⎛ ⎝ k  j0 cint B −1  y i j  ⎞ ⎠  ∅ 3.10 for each N  {y 0 , ,y n }∈Y  and N 1  {y i 0 , ,y i k }⊆N. This shows that B is a generalized G B -mapping. For each z ∈ X,ify / ∈ Bz, then there exists an x 0 ∈ X such that y / ∈ B x 0 zA x 0 z and z ∈ Ox 0  ⊂ Nx 0 . It follows from a that y / ∈ Az. Hence we have Az ⊂ Bz for each z ∈ X. By condition ii, there exists a nonempty set Y 0 ⊂ Y and for each N  {y 0 , ,y n }∈Y, there exists a compact FC-subspace L N of Y containing Y 0 ∪ N such that the set K   y∈Y 0 cint A −1 y c is empty or compact. Note that Az ⊂ Bz for each z ∈ X implies cint B −1 y c ⊂ cint A −1 y c for each y ∈ Y . Hence K    y∈Y 0 cint B −1 y c ⊂ K and K  is empty or compact. By Lemma 3.2, there exists a point x ∈ X such that Bx∅,and hence A x∅ which contradicts the assumption that Ax /  ∅ for each x ∈ X. Therefore, there exists x ∈ X such that A x∅. Journal of Inequalities and Applications 7 Theorem 3.6. Let X be a topological space, let K be a nonempty compact subset of X and Y, ϕ N  be an FC-space. Let F ∈BY, X and A : X → 2 Y be a generalized G B -majorized mapping such that i there exists a paracompact subset E of X such that {x ∈ X : Ax /  ∅} ⊂ E; ii for each N  {y 0 , ,y n }∈Y , there exists a compact FC-subspace L N of Y containing N such that for each x ∈ X \ K, L N  cint Ax /  ∅. Then there exists x ∈ K such that A x∅. Proof. Suppose that Ax /  ∅ for each x ∈ X. By using similar argument as in the proof of Theorem 3.5, we can show that there exists a generalized G B -mapping B : X → 2 Y such that Ax ⊂ Bx for each x ∈ X. It follows from condition ii that for each x ∈ X \ K, L N ∩ cint Bx /  ∅.ByTheorem 3.3, there exists x ∈ K such that Bx∅, and hence Ax∅ which contradicts the assumption that Ax /  ∅ for each x ∈ X. Therefore, there exists x ∈ X such that A x∅. Condition ii implies x ∈ K. This completes the proof. Remark 3.7. Theorem 3.5 generalizes 4, Theorem 2.3 in several aspects: Section 11 from G-convex space to FC-space without linear structure; Section 12 from a G B -majorized mapping to a generalized G B -majorized mapping; Section 13 condition ii of Theorem 3.5 is weaker than condition ii of 4, Theorem 2.3.IfX is compact, condition i is satisfied trivially. If X Y, ϕ N  is a compact FC-space, then by letting K  X  Y  L N for all N ∈X, conditions i and ii are satisfied automatically. Theorem 3.6 unifies and generalizes Shen’s 14, Theorem 2.1, Corollary 2.2 and Theorem 2.3 in the following ways: Section 21 from CH-convex space to FC-space without linear structure; Section 22 from H-majorized correspondences to generalized G B -majorized mapping; Section 23 condition ii of Theorem 3.6 is weaker than that in the corresponding results of Shen in 14. Theorem 3.6 also generalizes in 4, Theorem 2.4,Dingin15, Theorem 5.3, and Ding and Yuan in 16, Theorem 2.3 in several aspects. Corollary 3.8. Let X be a compact topological space, and let Y, ϕ N  be an CFC-space. Let F ∈ BY, X and A : X → 2 Y be a generalized G B -majorized mapping. Then there exists a point x ∈ X such that A x∅. Proof. The conclusion of Corollary 3.8 follows from Theorem 3.6 with E  K  X. Corollary 3.9. Let X be a topological space, and let Y, ϕ N  be an CFC-space. Let F ∈BY, X be a compact mapping and A : X → 2 Y be a generalized G B -majorized mapping. Then there exists a point x ∈ X such that Ax∅. Proof. Since F is a compact mapping, there exists a compact subset X 0 of X such that FY ⊂ X 0 . The mapping A| X 0 : X 0 → 2 Y be the restriction of A to X 0 . It is easy to see that A| X 0 is also generalized G B -majorized. By Corollary 3.8, there exists x ∈ X 0 such that A| X 0 xA x ∅. Remark 3.10. Corollary 3.8 generalizes Deguire et al. 2, Theorem 1 in the following ways: 1.1 from a convex subset of Hausdorff topological vector space to an FC-space without linear structure; 1.2 from a L S -majorized mapping to a generalized G B -majorized mapping. Corollary 3.8 also generalizes 4, Corollary 2.3 from CG-convex space to CFC-space and from a G B -majorized mapping to a generalized G B -majorized mapping. Corollary 3.9 generalizes 2, Theorem 2 and 4, Corollary 2.4 in several aspects. 8 Journal of Inequalities and Applications Theorem 3.11. Let X be a topological space, and let I be any index set. For each i ∈ I,letY i ,ϕ N i  be an FC-space, and let Y   i∈I Y i such that Y, ϕ N  is an FC-space defined as in Lemma 2.2.Let F ∈BY, X such that for each i ∈ I, i let A i : X → 2 Y i be a generalized G B -majorized mapping; ii  i∈I {x ∈ X : A i x /  ∅}   i∈I cint{x ∈ X : A i x /  ∅}; iii there exists a paracompact subset E i of X such that {x ∈ X : A i x /  ∅} ⊂ E i ; iv there exists a nonempty set Y 0 ⊂ Y and for each N  {y 0 , ,y n }∈Y,thereexistsa compact FC-subspace L N of Y containing Y 0  N such that the set  y∈Y 0 ccl{x ∈ X : ∃i ∈ Ix,π i y / ∈ A i x} is empty or compact, where Ix{i ∈ I : A i x /  ∅}. Then there exists x ∈ X such that A i x∅ for each i ∈ I. Proof. For each x ∈ X, Ix{i ∈ I : A i x /  ∅}. Define A : X → 2 Y by A  x   ⎧ ⎪ ⎨ ⎪ ⎩  i∈Ix π −1 i  A i  x  , if I  x  /  ∅, ∅, if I  x   ∅. 3.11 Then for each x ∈ X, Ax /  ∅ if and only if Ix /  ∅.Letx ∈ X with Ax /  ∅, then there exists j 0 ∈ Ix such that A j 0 x /  ∅. By condition ii, there exists i 0 ∈ Ix such that x ∈ cint{x ∈ X : A i 0 x /  ∅}. Since A i 0 is generalized G B -majorized, there exist an open neighborhood Nx of x in X and a generalized G B -majorant A x,i 0 of A i 0 at x such that a A i 0 z ⊂ A x,i 0 z for all z ∈ Nx, b for each N  {y 0 , ,y n }∈Y  and {y r 0 , ,y r k }⊂N, F  ϕ N  Δ k    ⎛ ⎝ k  j0 cint A −1 x,i 0  π i 0  y r j  ⎞ ⎠  ∅, 3.12 c A −1 x,i 0 : Y i → 2 X is transfer compactly open in Y i , d for each N ∈{x ∈ X : A i 0 x /  ∅}, the mapping  x∈N A −1 x,i 0 is transfer compactly open in Y i . Without loss of generality, we can assume that Nx ⊂ cint{x ∈ X : A i 0 x /  ∅}. Hence, A i 0 z /  ∅ for each z ∈ Nx. Define B x,i 0 : X → 2 Y by B x,i 0  z   π −1 i 0  A x,i 0  z  , ∀z ∈ X. 3.13 We claim that B x,i 0 is a generalized G B -majorant of A at x. Indeed, we have a   for each z ∈ Nx, Az  i∈Iz π −1 i A i z ⊂ π −1 i 0 A i 0 z ⊂ π −1 i 0 A x,i 0 z  B x,i 0 z, Journal of Inequalities and Applications 9 b   for each N  {y 0 , ,y n }∈Y and M  {y r 0 , ,y r k }⊂N,ifu ∈  k j0 cint B −1 x,i 0 π i 0 y r j , then M ⊂ cintB x,i 0 u. It is easy to see that π i 0 M ⊂ cint π i 0 B x,i 0 u,sothatπ i 0 M ⊂ cint A x,i 0 u, i.e., u ∈  k j0 cint A −1 x,i 0 π i 0 y r j  and hence u / ∈ Fϕ N Δ k  by b. It follows that F  ϕ N  Δ k    ⎛ ⎝ k  j0 cint B −1 x,i 0  π i 0  y r j  ⎞ ⎠  ∅, 3.14 c   for each y ∈ Y, we have that B −1 x,i 0  y   A −1 x,i 0  π i 0  y  3.15 is transfer compactly open in Y by c. Hence B x,i 0 is a generalized G B -majorant of A at x. For each N ∈{x ∈ X : A i 0 x /  ∅} and y ∈ Y ,by3.15, we have  x∈N B −1 x,i 0  y    x∈N A −1 x,i 0  π i 0  y  . 3.16 It follows from d that  x∈N B −1 x,i 0 is transfer compactly open in Y . Hence A : X → 2 Y is generalized G B -majorized. By condition iii, we have { x ∈ X : A  x  /  ∅ } ⊂ { x ∈ X : A i 0  x  /  ∅ } ⊂ E i 0 . 3.17 By condition iv, there exists a nonempty set Y 0 ⊂ Y and for each N  {y 0 , ,y n }∈Y, there exists a compact FC-subspace L N of Y containing Y 0  N. By the definition of A,for each y ∈ Y 0 , we have A −1  y    x ∈ X : y ∈ A  x    ⎧ ⎨ ⎩ x ∈ X : y ∈  i∈Ix π −1 i  A i  x  ⎫ ⎬ ⎭  ⎧ ⎨ ⎩ x ∈ X : π i  y  ∈  i∈Ix  A i  x  ⎫ ⎬ ⎭ . 3.18 It follows from condition iv that K   y∈Y 0 cint A −1 y c   y∈Y 0 ccl{x ∈ X : ∃i ∈ Ix,π i y / ∈ A i x} is empty or compact and hence all conditions of Theorem 3.5 are satisfied. By Theorem 3.5, there exists x ∈ X such that Ax∅ which implies Ix∅,thatis, A i x∅ for each i ∈ I. 10 Journal of Inequalities and Applications Theorem 3.12. Let X be a topological space, and let I be any index set. For each i ∈ I,letY i ,ϕ N i  be an CFC-space, and let Y   i∈I Y i .LetF ∈By, x be a compact mapping such that for each i ∈ I, i let A i : X → 2 Y i be a generalized G B -majorized mapping; ii  i∈I {x ∈ X : A i x /  ∅}   i∈I cint{x ∈ X : A i x /  ∅}. Then there exists x ∈ X such that A i x∅ for each i ∈ I. Proof. Since for each i ∈ I,letY i ,ϕ N i  be an CFC-space, then for each N i ∈Y i , there exists a compact FC-subspace L N i of Y i containing N i .LetL N   i∈I L N i and N   i∈I N i ∈Y , then L N is a compact FC-subspace of Y for each N ∈Y , L N is a compact FC-subspace of Y containing N. Hence Y, ϕ N  is also an CFC-space. For each x ∈ X, Ix{i ∈ I : A i x /  ∅}. Define A : X → 2 Y A  x   ⎧ ⎪ ⎨ ⎪ ⎩  i∈Ix π −1 i  A i  x  , if I  x  /  ∅, ∅, if I  x   ∅. 3.19 Then for each x ∈ X, Ax /  ∅ if and only if Ix /  ∅. By using similar argument as in the proof of Theorem 3.11, we can show that A : X → 2 Y is a generalized G B -majorized mapping. By Corollary 3.9, there exists x ∈ X such that Ax∅,andsoIx∅. Hence, we have A i x∅ for each i ∈ I. Theorem 3.13. Let X be a topological space, let K be a nonempty compact subset of X, and let I be any index set. For each i ∈ I,letY i ,ϕ N i  be an FC-space, and let Y   i∈I Y i such that Y, ϕ N  is an FC-space defined as in Lemma 2.2.LetF ∈BY, X such that for each i ∈ I, A i : X → 2 Y i be a generalized G B -mapping such that i for each i ∈ I and N i ∈Y i , there exists a compact FC-subspace L N i of Y i containing N i and for each x ∈ X \ K,thereexistsi ∈ I satisfying L N i  cint A i x /  ∅. Then there exists x ∈ K such that A i x∅ for each i ∈ I. Proof. Suppose that the conclusion is not true, then for each x ∈ K, there exists i ∈ I such that A i x /  ∅. Since A i is a generalized G B -mapping, A −1 i is transfer compactly open valued. By Lemma 3.1, we have K ⊂  i∈I  y i ∈Y i  cint A −1 i  y i   . 3.20 Since K is compact, there exists a finite set J ⊂ I such that for each j ∈ J, there exists N j  {y 1 j ,y 2 j , ,y m j j }⊂Y j with K ⊂  j∈J  m j k1 cint A −1 j y k j . It follows that for each x ∈ K, there exists a j ∈ J ⊂ I such that N j  cint A j x /  ∅. We may take any fixed y 0 y 0 i  i∈I ∈ Y . For each i ∈ I \J,letN i  {y 0 i }. By condition i, for each i ∈ I, there exists a compact FC-subspace L N i of Y i containing N i and for each x ∈ X \K, there exists i ∈ I satisfying L N i  cint A i x /  ∅. Hence for each x ∈ X, there exists i ∈ I such that L N i  cint A i x /  ∅.LetL N   i∈I L N i , then L N is a compact FC-subspace of Y and hence it is also a compact CFC-space. 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Mechanics, vol 24, no 9, pp 899–905, 2003 6 X.-P Ding and F Q Xia, “Equilibria of nonparacompact generalized games with LFc -majorized correspondence in G-convex spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol 56, no 6, pp 831–849, 2004 7 X.-P Ding, J.-C Yao, and L.-J Lin, “Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces,” Journal . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 905605, 12 pages doi:10.1155/2009/905605 Research Article Some Maximal Elements’ Theorems in. mappings unifies and generalizes the corresponding notions of G B -majorized mappings in 4; L S -majorized mappings in 2, 13; H-majorized mappings in 14. Some new existence theorems of maximal. a generalized G B -mapping resp., G B -majorant mapping. The inverse is not true in general. 3. Maximal Elements In order to obtain our main results, we need the following lemmas. Lemma 3.1