Ky Fan Minimax Inequalities And Existence Theorems Of Generalized Games Equilibria In L-convex Space

The aim of the paper is to proof Ky Fan minimax inequilitics, existence theorem maximal elements and equilibria existence theorem for generalized game in L-convex space.In the first chapter, we introduct history setting and recent situations of problems which are ralated to Ky Fan minimax inequilitics and game theory. Moreover, we show the work what we have done in the paper improve and generalize some well-know results.In the second chapter, we study Ky Fan minimax incquilites in L-convex space. By using KKM theory, we obtain Ky Fan minimax incquilites in L-convex space. However, we get the equivalents with section theorem, fixed point theorem and existence theorem of maximal element. As applications, we get saddle point theorem in L-convex space.In the third chapter, we work on existence theorem of maximal elements and equilibria existence theorems for generalized game. First, we put forward the conceptions of Ls class mapping and Ls majorant mapping. In certain conditions, we prove Ls majorized mapping can be improved by Ls class mapping. Second, using fixed point theorem in the secondchapter, we get the existence theorems of maximal elements for Ls class mapping and Ls majorant mapping in L-convcx space. At last, we give the notions of qualitative game and generalized game. By using existence theorem of maximal elements, we get equilibria existence theorems for qualitative game and generalized game in L-convex space.The following are the main results.Theorem 2.2.1. Let X be nonempty L-convex subset of L-convex space (E, T). Let f,g:X × X→R. ∪{±∞} be such that(a) for each (x, y) ∈ X × X, f(x, y) > 0 implies g(x, y) > 0;(b) for each x ∈ X, y → f(x,y) is lower semicontinuous on each nonempty compact subset of X;(c) for each A ∈ (X), there exists a compact L-convex subset La of X including A such that for each y ∈ La, min g(x, y) < 0.Then there exists y ∈ X such that f(x, y) < 0 for all x G X.The following are a few equivalent results with Theorem 2.2.1.Theorem 2.2.1 A Let X be nonempty L-convex subset of L-convex space (E, T). Suppose that sets B, D C X x X satisfy(01) B D-(b₁) for each x ∈ X and for any nonempty compact subset K of X, the set {y ∈ K : (x, y) ∈ B} is open in K(c₁) for each A ∈ {X}, there exists a compact L-convex subset La of X including A such that for each y ∈ La, there exists x ∈ A such that (x, y) D. Then there exists y ∈ X such that {x ∈ X : (x, y) ∈ B} = .Theorem 2.2.1 B Let X be nonempty L-convex subset of L-convex space (E, T). Suppose that sets M, N X × X satisfy(a₂) N M;(B₂) for eachx ∈ X and for any nonempty compact subset K of X, the set {y ∈k (x, y) ∈ M} is closed in K;(C2) for each A ∈ (X), there exists a compact L-convex subset LA of X containing A suchthat for each y 6 LA, there exists x e A such that (x,y) G N. Then there exists y G X such that X x {yj C M.Theorem 2.2.1 C Let X be nonempty L-convex subset of L-convex space (E, F). Suppose that P, Q : X —> 2X are set-valued mappings and satisfy(a3) for each x EX, P(x) C Q{x);(63) for each y ï¿¡ X, P^y is compact open in X;(c3) for each A G (X), there exists a compact L-convex subset L^ of X containing A such that for each y G La, there exists x G A such that a; ^ Q(y). Then there exists y X such that P(y) = 0.Theorem 2.2.2. Let A" be nonempty L-convex subset of L-convex space (E, F). For each A G (X), L-co(A) is compact. Suppose that f,g : X x X —> Ru {±00} satisfy(a) for each (x, y) G X x X, f(x, y) > 0 implies g(x, y) > 0 and g(x, x) < 0;(6) for each x X, y —? /(x, ?/) is lower semicontinuous on each nonempty compact subset of X;(c) for each y G X, the set (iï¿¡l: 3(0;, y) > 0} is L-convex. Then there exists y G X such that f(x,y) < 0 for all iï¿¡l,Corollary 2.2.1. Let X be nonempty L-convex subset of L-convex space (E, F). For each A G (X), L-co(A) is compact. Suppose that f,g:XxX—>M.U {±00} satisfy(a) for each (x, y) G X x X, f(x, y) < g(x, y);(b) for each x G X, y —> f(x,y) is lower semicontinuous on each nonempty compact subset of X;(c) for each y X, the set {ieX: g(x, y) > sup ^(x, x)} is L-convex. Then there exists y X such that f(x,y) < supg(x,x) for all x X,The following result is equivalent formulation of Theorem 2.2.2.Theorem 2.2.2 A Let (E, F) be L-convex space and X be nonempty L-convex subset of {E, F). For each A G (X), L-co(A) is compact. Suppose that P, Q : X -> 2E satisfy (ax) for each x G X, P(x) C <3(:r); (6X) for each y ï¿¡ E, P^y is compact open in X;(ci) for each iï¿¡l, Q(x) is L-convex in E:(dt) for each xï¿¡ X, P(x) / 0. Then there exists x ï¿¡ X such that x ï¿¡ Q(x).Corollary 2.2.2. Let X be nonempty L-convcx subset of L-convex space (E, F). Suppose that Q : X —> 2B is set-valued mapping such that Q : ï¿¡? —> 2X satisfies one of conditions of Lemma 2.1.1. Then there exists y ï¿¡ X such that y € L-co{Q{y)).Theorem 2.3.1. Let X be nonempty L-convex subset of L-convex space (E, F). Suppose that f : X x X ->RU {±00} satisfy(a) for each x ï¿¡ X, y —> f(x,y) is lower semicontinuous on each nonempty compact subset of X;(b) for each y ï¿¡ X, x —> f(x,y) is upper semicontinuous on each nonempty compact subset of X;(c) for each M ï¿¡ {X), there exists a compact L-convex subset LM of X containing M such that for each y ï¿¡ Lm, minxeMf(x,y) < 0;(d) for each TV (X), there exists a nonempty compact L-convex subset Ln of X containing TV such that for each x Ln, min^6jv/(x, y) > 0.Then there exists (x,y) X x X such thatf{x,y) < f{ï¿¡,y) < fix,y) ix,y) e X x XIn particularly, infy6A:supx6X/(x, y) = supxeXmiyeXf{x,y) = 0.Theorem 3.2.1. Let X be a paracompact Hausdorff topological space and Y be nonempty L-convex subset of L-convex space iE,T). For each M € (Y), L-co(M) is compact. If 5 : Y —> X is single valued and continuous mapping and A : X —> 2y is Ls-majorized mapping, then there exists x ï¿¡ X such that >l(x) = 0.Theorem 3.2.2. Let X be a paracompact Hausdorff topological space and / be index set. Suppose that for each i /, Yi is nonempty L-convex subset of L-convex space (jEj, Fj) such that for each M G {Yi), L-co(M) is compact. Let Si : Fj —> X be a single valued continuous mapping. If Ai : X —> 2K> is Ls^majorizcd mapping and \J{x E X : Aiix) 7^ 0} =16/U int{x G X : A^x) =fi 0}, then there exists x E X such that Ai{x) = 0 for each i E I.Theorem 3.2.3. Let X be a compact Hausdorff topological space, and / be index set. Suppose that for each i G /, Y, is nonempty L-convex subset of L-convex space (.E^Fi) such that for each M G (Yi), L-co(M) is compact. Let Si : Y{ —> X be a continuous single valued mapping and Ai : X —* 2Yi be a set-valued mapping. Suppose the following conditions are satisfied(i) for each x G X, there exists i € / such that Ai(x) ^ 0;(ii) for each i G / and y G Yi, the set A^y is compact open in X. Then there exists i0 G / and y G Ki0 such that y G L-co>l,0(5i0(y)).Theorem 3.2.4. Let X be a Hausdorff topological space and / be any index set. Suppose that for each i G /, Y{ is nonempty L-convex subset of L-convex space (Ei, Fj) such that for each M G (Yi), L-co(M) is compact. Let Si : Y| —? X be a single valued continuous mapping and At : X —> 2Yl be class L^{ mapping. Suppose the following condition is satisfied(i) there exists a nonempty compact subset K of X and a nonempty compact subset D = Yiiei A of y = Ilie/ Yi such that for each x E X \ K, there exists i G / such that A(x) n A ^ 0-Then there exists x e K such that Ai(x) = 0 for all i G /.Theorem 3.2.5. Let X be paracompact and Hausdorff topological space and / be index set. For each i G /, F, is non-empty L-convex subset of L-convex space (ï¿¡1i,Fi). For each M G (Y^, L-co(M) is compact. Let Si . Yt —* X be a. single valued continuous mapping and Ai : X —> 2y* be Ls^majorized mapping. Suppose the following condition is satisfied(i) there exists a nonempty compact subset K of X and a nonempty compact subset D = fJig/ A of y = fJie/ ^i sucn tnat f°r cacn x e ^ \ K, there exists i G / such thatAi(x) n A / 0.Then there exists x ï¿¡ K such that >lj(x) = 0 for each z G /.Theorem 3.2.6. Let X be a Hausdorff topological space and / be index set. Suppose that for each i G /, Yt is nonempty L-convex subset of L-convex space (.E^F,). For each M G (Yi), L-co(M) is compact. Let Si : Y{ —> X be single valued continuous mapping and fi : X x Yi —+ R be functional. The following conditions are satisfied(i) for each x G X, y —> /,(x, y) is L-quasiconcavc;(ii) for each y &Yi, x —> /i(x, y) is lower scmicontinuous on each nonempty paracompact subset K of X. Then we have(A): For each A € R, at least one of the following statement is satisfied(1) there exists x E X such that supsup/j(f, y) < A;i€l yEYi(2) there exists i G / and y G Yi such that fi(Si(y),y) > A. and(B) The following minimax inequalities holds:inf sup sup fi(x,y) < sup sup fi(Si{y),y) xeX ie/ yen ieTheorem 3.3.1. let X be nonempty L-convex subset of paracompact Hausdorff L-convex space (E, P). For each M G (X), L-co(M) is compact. Let 5 : X —> X be single valued and continuous mapping. ^4, B : X —> 2X are set valued mapping and satisfy(i) for each x E X, A(x) is nonempty and L-coA(S(x)) C B(x);(ii) A : X —> 2X is compact open valued;(iii) A n P : X —> 2X is L5 majorized mapping. Then there exists x € X such that x G ï¿¡(ï¿¡), A{x) n P(i) = 0.Theorem 3.3.2. Let A = (Xi\ Pi)i^j is qualitative game. For each i 6 /, Xi is nonempty L-convex subset of paracompact Hausdorff L-convex space (Ei,Yi). For each M G (Xt), L-co(M) is compact. Let Si : Xi —> X be single valued and continuous mapping. The following conditions are satisfied(i) Pt : X —> 2X% is L5. majorized mapping ;(ii) Wi = {x G X : Pi(x) ^ 0} is open. Then A has an equilibria point x € X.Theorem 3.3.3. Let A = (Xi\ Ai\ Bi\ Pi)iei is generalized game. For each i G /, Xi is L-convex subset of paracompact Hausdorff L-convex space (Ei, Fi). For each M G (Xi), L-co(M) is compact. Let Si : Xi —> X be single valued and continuous mapping. Suppose Ai, Bi : X —> 2Xl satisfy the following conditions...