Download miễn phí Luận văn The solution existence of equilibrium problems and generalized problems



Table of Contents
Foreword 1
Part 1. Equilibrium problems 4
Chapter 1. Existence conditionsfor equilibrium problems 5
Chapter 2. The solution existence of systems of quasiequilibrium problems 17
Chapter 3. Existence conditions for approximate solutions to 30
quasiequilibrium problems
Part 2. Variational inclusion problems 48
Chapter 4. Sufficient conditions for the solution existence of variational 49
inclusion problems
Chapter 5. Systems of quasivariational inclusion problems 83
List of the papers related to the thesis 97


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s proposed including, among others,
equilibrium problems, implicit variational inequalities, and quasivariational inequali-
ties involving multifunctions. Sufficient conditions for the existence of solutions with
and without relaxed pseudomonotonicity are established. Even semicontinuity may
not be imposed. These conditions improve several recent results in the literature.
Keywords Quasiequilibrium problems · Quasivariational inequalities ·
0-level-quasiconcavity · Upper semicontinuity · KKM–Fan theorem
1 Introduction
Equilibrium problems, which include as special cases various problems related to
optimization theory such as fixed point problems, coincidence point problems, Nash
equilibria problems, variational inequalities, complementarity problems, and maxi-
mization problems have been studied by many authors; see e.g., Refs. [1–6]. The
main attention has been paid to the sufficient conditions for the existence of solutions.
There has also been interest in getting such conditions for more general problem set-
tings and under weaker assumptions about continuity, monotonicity and compacity.
Communicated by S. Schaible.
This work was partially supported by the National Basic Research Program in Natural Sciences
of Vietnam. The authors are very grateful to Professor Schaible and the referees for their
valuable remarks and suggestions which helped to improve remarkably the paper.
N.X. Hai
Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology
of Vietnam, Hochiminh City, Vietnam
P.Q. Khanh ()
Department of Mathematics, International University at Hochiminh City, Hochimin City,
Vietnam
e-mail: [email protected]
J Optim Theory Appl
In the present paper, we propose a general vector quasiequilibrium problem, which
includes vector equilibrium problems, vector quasivariational inequalities, and qua-
sicomplementarity problems, etc. We establish sufficient conditions for solution ex-
istence with and without relaxed pseudomonotonicity.
In the sequel, if not otherwise specified, let X, Y and Z be real topological vector
spaces, let X be Hausdorff and let A ⊆ X be a nonempty closed convex subset. Let
C : A → 2Y , K : A → 2X and T : A → 2Z be multifunctions such that C(x) is a
closed convex cone with intC(x) = ∅ and K(x) is nonempty convex, for each x ∈
A. Let f : T (A) × A × A → Y be a single-valued mapping. The quasiequilibrium
problem under consideration is as follows:
(QEP) Find x¯ ∈ A ∩ clK(x¯) such that, for each y ∈ K(x¯), there exists t¯ ∈ T (x¯)
satisfying f (t¯, y, x¯) /∈ intC(x¯).
To motivate the problem setting, let us look at several special cases of (QEP).
(a) If K(x) ≡ A and Z = L(X,Y ), the space of linear continuous mappings of X
into Y , then (QEP) coincides with an implicit vector variational inequality studied
in Refs. [7, 8]: find x¯ ∈ A such that, for each y ∈ A, there exists t¯ ∈ T (x¯) satisfying
f (t¯, y, x¯) /∈ intC(x¯).
(b) If K(x) ≡ A and T is single-valued, then setting f (T (x), y, x) := h(y, x),
(QEP) becomes the vector equilibrium problem considered e.g. in Refs. [1–3, 5, 6]:
(EP) Find x¯ ∈ A such that, for each y ∈ A, h(y, x¯) /∈ intC(x¯).
(c) If Z = L(X,Y ), f (t, y, x) = (t, x − y), where (t, x) denotes the value of a
linear mapping t at x, then (QEP) reduces to the vector quasivariational inequality
problem investigated by many authors:
(QVI) Find x¯ ∈ A ∩ clK(x¯) such that, for each y ∈ K(x¯), there exists t¯ ∈ T (x¯)
satisfying (t¯ , y − x¯) /∈ − intC(x¯).
(d) Let X be a Banach space, let Y = R, Z = X∗, C(x) ≡ R+, let A be a closed
convex cone, T : A → 2X∗ and S : A → 2A. The quasicomplementarity problem is as
follows:
(QCP) Find x¯ ∈ A such that, ∀s¯ ∈ K ∩S(x¯),∃t¯ ∈ (−A∗)∩T (x¯) satisfying 〈t¯ , s¯〉 = 0,
where 〈t, s〉 denotes the value of a linear functional t at s.
Then, setting K(x) := x −A∩S(x)+A and f (t, y, x) := 〈t, x −y〉, (QEP) collapses
to (QCP), see Ref. [9].
(e) Consider the following maximization problem:
(MP) Find the Pareto maximizer of a mapping J : A → Y , where Y is ordered by a
convex cone C.
Then setting C(x) ≡ C,K(x) ≡ A, T (x) = {x} and f (T (x), y, x) := J (y) − J (x),
we see that (QEP) is equivalent to (MP).
Our aim now is to develop sufficient conditions for the existence of solutions
to (QEP) under weak assumptions and to derive as consequences several improve-
ments of known results for vector equilibrium problems and vector quasivariational
inequalities.
J Optim Theory Appl
2 Preliminaries
We recall first some definitions needed in the sequel. Let X and Y be topological
spaces. A multifunction F : X → 2Y is said to be upper semicontinuous (usc) at
x0 ∈ domF := {x ∈ X : F(x) = ∅} if, for each neighborhood U of F(x0), there is a
neighborhood N of x0 such that F(N) ⊆ U . F is called usc if F is usc at each point
of domF . In the sequel, all properties defined at a point will be extended to domains
in this way. F is called lower semicontinuous (lsc) at x0 ∈ domF if for each open
subset U satisfying U ∩ F(x0) = ∅ there exists a neighborhood N of x0 such that,
for all x ∈ N,U ∩ F(x) = ∅. F is said to be continuous at x ∈ domF if F is both
usc and lsc at x. F is termed closed at x ∈ domF if ∀xα → x, ∀yα ∈ F(xα) such that
yα → y, then y ∈ F(x). It known that, if F is usc and has closed values, then F is
closed.
A multifunction H of a subset A of a topological vector space X into X is said
to be a KKM mapping in A if, for each {x1, . . . , xn} ⊆ A, one has co{x1, . . . , xn} ⊆⋃n
i=1 H(xi), where co{} stands for the convex hull.
The main machinery for proving our results is the following well-known KKM-
Fan theorem (Ref. [10]).
Theorem 2.1 Assume that X is a topological vector space, A ⊆ X is nonempty
and H : A → 2X is a KKM mapping with closed values. If there is a subset X0
contained in a compact convex subset of A such that ⋂x∈X0 H(x) is compact, then⋂
x∈A H(x) = ∅.
The following fixed-point theorem is a slightly weaker phiên bản (suitable for our
use) of the Tarafdar theorem (Ref. [11]), which is equivalent to Theorem 2.1.
Theorem 2.2 Assume that X is a Hausdorff topological vector space, A ⊆ X is non-
empty and convex and ϕ : A → 2A is a multifunction with nonempty convex values.
Assume that:
(i) ϕ−1(y) is open in A for each y ∈ A;
(ii) there exists a nonempty subset X0 contained in a compact convex set of A such
that A \ ⋃y∈X0 ϕ−1(y) is compact or empty.
Then, there exists xˆ ∈ A such that xˆ ∈ ϕ(xˆ).
The next theorem on fixed points is modified (for our use) from a theorem in
Ref. [12].
Theorem 2.3 Assume that V is a convex set in a Hausdorff topological vector space
and that f : V → 2V is a multifunction with convex values. Assume that:
(i) V = ⋃x∈V intf −1(x);
(ii) there exists a nonempty compact subset D ⊆ V such that, for all finite subsets
M ⊆ V , there is a compact convex subset LM of V , containing M , such that
LM \ D ⊆ ⋃x∈LM f −1(x).
Then, there is a fixed point of f in V .
J Optim Theory Appl
Using Theorem 2.3, we derive the following modification of Theorem 2.1.
Theorem 2.4 Assume that V is a convex set in a Hausdorff topological vector space
and H : V → 2V is a KKM mapping in V with closed values. Assume further that
there exists a nonempty compact subset D ⊆ V such that, for all finite subsets M ⊆ V ,
there is a compact convex subset LM of V , containing M , such that
LM \ D ⊆
⋃
x∈LM
(V \ H(x)). (1)
Then,
⋂
x∈V H(x) = ∅.
Proof Suppose that ⋂x∈V H(x) = ∅. Define the multifunction g : V → 2V by
g(y) = {x ∈ V : y /∈ H(x)}. Then g(y) = ∅, ∀y ∈ V , and g−1(x) = V \ H(x).
Hence, g−1(x) is open and V = ⋃x∈V g−1(x). Define further f : V → 2V by
f (x) = cog(x), where co means the convex hull. One has V = ⋃x∈V f −1(x). More-
over, LM \ D ⊆ ⋃x∈LM g−1(x) ⊆ ⋃x∈LM f −1(x).
By Theorem 2.3 there is x0 ∈ V such that x0 ∈ f (x0). Therefore, one can find
xj ∈ g(x0) and λj ≥ 0, j = 1, . . . ,m, ∑mj=1 λj = 1 such that x0 = ∑mj=1 λjxj . By
the definition of g, x0 /∈ H(xj ), j = 1, . . . ,m. Thus, x0 = ∑mj=1 λjxj /∈ ⋃mj=1 H(xj ),
which is impossible, since H is KKM. 
3 Main Results
We propose first a very relaxed quasiconcavity. Let Z, A, C, T and f be as for prob-
lem (QEP). For x ∈ A, the mapping f is said to be 0-level-quasiconcave with respect
to T (x) if, for any finite subsets {y1, . . . , yn} ⊆ A and any αi ≥ 0, i = 1, . . . , n, with∑n
i=1 αi = 1, there exists t ∈ T (x) such that
[f (T (x), yi, x) ⊆ intC(x), i = 1, . . . , n] ⇒
[
f
(
t,...

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