THAI NGUYEN UNIVERSITY
UNIVERSITY OF EDUCATION
——————————————————

NGUYEN QUYNH HOA

GENERAL QUASI-EQUILIBRIUM PROBLEMS
OF THE BLUM - OETTLI TYPE AND THEIR
APPLICATIONS

Speciality: Mathematical Analysis
Code: 9460102

DISSERTATION SUMMARY

THAI NGUYEN - 2018


The dissertation was finished at:
THAI NGUYEN UNIVERSITY - UNIVERSITY OF EDUCATION

Supervisor: Prof. Dr.Sc. Nguyen Xuan Tan

Referee 1:....................................................................
Referee 2:....................................................................
Referee 3:....................................................................

The dissertation will be defended in the university committee:
THAI NGUYEN UNIVERSITY - UNIVERSITY OF EDUCATION
At.............,2018


these two theories. The theory of operator equation and optimal theory are interrelated and
interactive. In many cases, problem (1) can be taken on problem (2) and vice versa. For example,
when X is a Hilbert space, f is a convex function and has the derivative f , problem (2) is
equivalent to problem: Find x ∈ D such that
x = PD (x − f (x)),
with PD (x) is orthogoral projection of x on the set D. Or F (x) = 0, with F (x) = PD (x −
f (x)) − x. Thus, problem (1) is equivalent to problem (2).
To solve problem (2), we classify them into problem classes based on the characteristics of
function f and set D. When f is a linear function and D convex polyhedron in Euclid space
Rn , problem (2) is called linear programming problem. In 1947, G. B. Danzig was an American
mathematical scientist who found simplex algorithm to solve this problem. When set D is a
convex and closed in Rn and f is a convex function then the problem (2) is called convex programming problem. Up to the 1960 - 1970, T. Rockaffelar was an American mathematical scien-


2

tist who gave lower differential defined of convex function to construct convex analysis to solve
convex programming problems. Next, when f is a locally Lipschitz function and D is a closed
set, (2) is called Lipschitz programming problem. After the 1970s, F. H. Clarke constructed
lower differential of locally Lipschitz function to solve Lipschitz programming problems. When
f is a continuous function, D is a closed set, problem (2) is called continuous programming
problem. In the last years of the 20th century and the early years of the 21st century, D. T.
Luc and V. Jeyakumar gave approximate Jacobian theory to solve continuous programming
problems.
Up to the 1990s of last century, Stampachia gave out variational inequality problem: Let D
be a nonempty subset of space Rn , T : D → Rn . Find x ∈ D such that
T (x), x − x ≥ 0, với mọi x ∈ D.

(3)



such that
0 ∈ F (x),

(6)

in which, X, Y be Hausdorff locally convex topological vector spaces, D is subset of X. Problem
(6) is called general equilibrium problem or multivalued equation.
In fact in many cases constrain fiela D changes and is depended by a mapping, P : D → 2D .
Then, we need to consider the problem: Find x ∈ D such that
1) x ∈ P (x);

(7)

2) 0 ∈ F (x).
Problem (7) is called general quasi-equilibrium problem. Sufficient condition for the existence
of solution for this problem was studied in the case P is a continuous multivalued mapping
with nonempty compact convex values and F is a u.s.c multivalued mapping with nonempty
compact convex values.
In recent years, many mathematicians have studied the existence of the solutions to general
quasi-equilibrium problem by minimizing continuity of mappings P, F . Then, let X, Y, Z be
Hausdorff locally convex topological vector spaces, D ⊆ X, K ⊆ Z, multivalued mappings
P : D × K → 2D , Q : D × K → 2K , F : D × K → 2Y . We are interested in the problem: Find
(x, y) ∈ D × K such that
1) x ∈ P (x, y), y ∈ Q(x, y);

(8)

2) 0 ∈ F (x, y).
The multivalued mappings P, Q are called constraint mappings, F is called utility multivalued

and scalar weakly upper semi-continous mapping.
(2)Applied the obtained results in (1) to study the existence of solution of relevant problems:
generalized quasi-equilibrium problem of type I, generalized quasi-equilibrium problem of type
II and mixed generalized quasi-equilibrium problem.
For the reasons discussed above, we chose the research topic for the dissertation “General
quasi-equilibrium problems of the Blum - Oettli type and their applications”.
In addition to an introduction, a section of conclusions and a list of references, the dissertation has three chapters.
Chapter 1 collects some basic concepts needed for subsequent chapters. The main results of
this dissertation are presented in Chapter 2 and Chapter 3.
Chapter 2 studies the existence of solutions to general quasi-equilibrium problem. Theorem
2.1.1 and Theorem 2.1.2 prove the existence to solution for general quasi-equilibrium problems
with the utility multivalued mapping is a sum of scalar weakly lower semi-continous mapping
and scalar weakly upper semi-continous mapping. Theorem 2.2.1 and Theorem 2.2.2 prove
the existence to solution for general quasi-equilibrium problems with the utility multivalued
mapping is a product of scalar weakly lower semi-continous mapping and scalar weakly upper
semi-continous mapping. In this chapter, we also give some expanded results that connect Ky
Fan Theorem and Fan - Browder Theorem (Corollary 2.1.7, Corollary 2.1.8, ...).
Chapter 3 applies the obtained results in Chapter 2 to get the existence to solutions for
generalized quasi-equilibrium problem of type I (Theorem 3.1.1, Corollary 3.1.1), generalized
quasi-equilibrium problem of type II (Theorem 3.2.1, Corollary 3.2.2, Corollary 3.2.3) and mixed
generalized quasi-equilibrium problem (Theorem 3.3.1, Theorem 3.3.2).
The main content of the thesis is written based on the articles in list of research papers
published related to the dissertation.


5

Chương 1

Preliminaries

X, K ⊆ Z be nonempty subsets. Given multi-valued mappings P : D × K → 2D , Q : D × K →
2K , F : D × K → 2Y . We are interested in the problem: Find (x, y) ∈ D × K such that
1) (x, y) ∈ P (x, y) × Q(x, y);
2) 0 ∈ F (x, y).
This problem is called a general quasi-equilibrium problem. The existence of solutions to this
problem has been studied by the authors, especially, T. T. T. Duong and N. X. Tan studied
for the case in which the multivalued mapping P is continuous, the multivalued mapping Q
is u.s.c and the multivalued mapping F is u.s.c, all these mappings P, Q and F need to have
nonempty convex and closed values and the case the multivalued mapping P has open lower
section, the multivalued mapping Q is l.s.c and the multivalued mapping F is u.s.c.
In this chapter, we study the special case of general quasi-equilibrium problem:
i) The utility multivalued mapping is a sum of scalar weakly lower semi-continous mapping
and scalar weakly upper semi-continous mapping;


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ii) The utility multivalued mapping is a product of scalar weakly lower semi-continous mapping and scalar weakly upper semi-continous mapping.

2.1

General quasi-equilibrium problems of the Blum - Oettli type

In this section, we will give out some sufficient conditions for the existence of solutions of
general quasi-equilibrium problems of the Blum - Oettli type. This is general quasi-equilibrium
problem when the utility is the sum of two mappings: F = G + H in cases:
1) G : D × K → 2X is scalar weakly lower semi-continous mapping, H : D × K → 2X is a
scalar weakly upper semi-continous mapping and Y = X.
2) G : D × K → 2X×Z is a scalar weakly lower semi-continous mapping, H : D × K → 2X×Z
is a scalar weakly upper semi-continous mapping and Y = X × Z.

iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values;
v) H : D × K → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values;
vi) For all (x, y) ∈ P (x, y) × Q(x, y), ∅ = (G(x, y) − x) + (H(x, y) ∩ TP (x,y) (x)) ⊂ TP (x,y) (x).
Then there exists (x, y) ∈ D × K such that
1) (x, y) ∈ P (x, y) × Q(x, y);
2) x ∈ G(x, y) + H(x, y).
Corollary 2.1.2. We assume that the following conditions hold.
i) D, K are nonempty convex compact subsets;
ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values;
iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values;
iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values;
v) H : D × K → 2X is a scalar weakly u.s.c mapping with closed convex values;
vi) For all (x, y) ∈ P (x, y) × Q(x, y), x ∈
/ G(x, y) + H(x, y) and ∅ = (G(x, y) − x) + (H(x, y) ∩
TP (x,y) (x)) ⊂ TP (x,y) (x).
Then there exists (x, y) ∈ D × K such that
1) (x, y) ∈ P (x, y) × Q(x, y);
2) G(x, y) + H(x, y) = ∅.
Corollary 2.1.3. We assume that the following conditions hold.
i) D is nonempty convex compact subset;
ii) G0 : D → 2X is a scalar weakly l.s.c mapping with nonempty values;
iii) H0 : D → 2X is a u.s.c multivalued mapping with nonempty closed convex values;
iv) (G0 (x) − x) + (H0 (x) ∩ TD (x)) ⊂ TD (x), for all x ∈ D.


9

Then there exists x ∈ D such that x ∈ G0 (x) + H0 (x).
Up to now, there have been many studies of the existence of solutions of general quasiequilibrium problem. But there are only few of published results related to lower semi-continuous
of the utility of the problem. Based on Theorem 2.1.1, we obtain an important result on the

v) Với mỗi (x, y) ∈ P (x, y) × Q(x, y), x ∈
/ G(x, y) and G(x, y) − x ⊂ TP (x,y) (x).
Then there exists (x, y) ∈ D × K such that
1) (x, y) ∈ P (x, y) × Q(x, y);
2) G(x, y) = ∅.
Corollary 2.1.7. We assume that the following conditions hold.
i) D, K are nonempty convex compact subsets;
ii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values;
iii) F : D × K → 2D is a scalar weakly l.s.c mapping and với (x, y) ∈ D × K, y ∈ Q(x, y) nào
đó, F (x, y) = ∅;
Then there exists (x, y) ∈ D × K such that
1) y ∈ Q(x, y);
2) x ∈ F (x, y).
Corollary 2.1.8. We assume that the following conditions hold.
i) D is nonempty convex compact subset;
ii) F : D → 2D is a scalar weakly l.s.c mapping with nonempty values.
Then there exists x ∈ D such that x ∈ F (x).
Remark 2.1.1. Corollary 2.1.8 is the expansion fixed point Theorem of X. Wu. In this result,
mapping F has removed the closed and convex condition, only satisfied that the scalar weakly
l.s.c mapping. This corollary is also the expansion of fixed point Theorem of Fan - Browder.
Next, we studied general quasi-equilibrium when F = G + H with G : D × K → 2X×Z is
a scalar weakly lower semi-continous mapping, H : D × K → 2X×Z is a scalar weakly upper
semi-continous mapping and Y = X × Z.
Lemma 2.1.2. We assume that the following conditions hold.
i) D, K are nonempty convex compact subsets;
ii) P : D × K → 2D and Q : D × K → 2K are l.s.c multivalued mappings with nonempty
values;
iii) B = {(x, y) ∈ D × K|x ∈ P (x, y), y ∈ Q(x, y)} is a closed set;
iv) φ : K × K × D × D → R is a u.s.c function such that for any fixed (y, x) ∈ K × D,
φ(y, ., x, .) : K × D → R is a convex function;

2. G : D × K → 2X is a scalar weakly lower semi-continous mapping, H : D × K → 2Z is a
scalar weakly upper semi-continous mapping and Y = X × Z.
Firstly, we consider the case F = G × H with G : D × K → 2X is a scalar weakly lower semicontinous mapping and H : D × K → 2X is a scalar weakly upper semi-continous mapping
Y = X × X.
Lemma 2.2.1. We assume that the following conditions hold.


12

i) D, K are nonempty convex compact subsets;
ii) P : D × K → 2D is a l.s.c multivalued mapping with nonempty values;
iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values;
iv) φ : K × K × D × D → R is a u.s.c function such that for any fixed (y, x) ∈ K × D,
φ(y, ., x, .) : K × D → R is a convex function;
v) φ(y, y, x, x) = 0, for all (y, x) ∈ K × D .
Then there exists (x, y) ∈ D × K such that
1) (x, y) ∈ P (x, y) × Q(x, y);
2) φ(y, z, x, t) ≥ 0, for all t ∈ P (x, y), z ∈ Q(x, y).
Theorem 2.2.1. We assume that the following conditions hold.
i) D, K are nonempty convex compact subsets;
ii) P : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values;
iii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values;
iv) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values;
v) H : D × K → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values;
vi) For all (x, y) ∈ P (x, y) × Q(x, y), G(x, y) ⊂ TP (x,y) (x), (H(x, y) ∩ TP (x,y) (y)) = ∅.
Then there exists (x, y) ∈ D × K such that
1) (x, y) ∈ P (x, y) × Q(x, y);
2) 0 ∈ G(x, y) × H(x, y).
Corollary 2.2.1. We assume that the following conditions hold.
i) D, K are nonempty convex compact subsets;

1) (x, y) ∈ P (x, y) × Q(x, y);
2) H(x, y) = ∅.
Corollary 2.2.4. We assume that the following conditions hold.
i) D, K are nonempty convex compact subsets;
ii) Q : D × K → 2K is a u.s.c multivalued mapping with nonempty closed convex values;
iii) G : D × K → 2X is a scalar weakly l.s.c mapping with nonempty values and H(x, y) − x ⊆
TD (x), for all x ∈ D, y ∈ Q(x, y).
Then there exists (x, y) ∈ D × K such that (x, y) ∈ H(x, y) × Q(x, y).
Corollary 2.2.5. We assume that the following conditions hold.
i) D is a nonempty convex compact subset;


14

ii) P : D → 2D is a continuous multivalued mapping with nonempty closed convex values;
iii) G : D → 2X is a scalar weakly l.s.c mapping with nonempty values such that G(x) ⊂
TP (x) (x), for all x ∈ P (x).
Then there exists x ∈ D such that x ∈ G(x) ∩ P (x).
Corollary 2.2.6. We assume that the following conditions hold.
i) D is nonempty convex compact subset;
ii) P : D → 2D is a continuous multivalued mapping with nonempty closed convex values;
iii) H : D → 2X is a scalar weakly u.s.c mapping with nonempty closed convex values and
H(x) ⊂ TP (x) (x), for all x ∈ P (x).
Then there exists x ∈ D such that x ∈ H(x) ∩ P (x).
Similarly, we consider general quasi-equilibrium problem with the utility function of the
form F = G × H in which G : D × K → 2X is a scalar weakly lower semi-continous mapping,
H : D × K → 2Z is a scalar weakly upper semi-continous mapping and Y = X × Z. In this
case, P and Q are l.s.c multivalued mappings.
We have a theorem.
Theorem 2.2.2. We assume that the following conditions hold.

F1 : K × D × D × D → 2Y . We are interested in the problem: Find (x, y) ∈ D × K such that
1) (x, y) ∈ S(x, y) × T (x, y);
2) 0 ∈ F1 (y, x, x, t), for all t ∈ S(x, y).
This problem is called a generalized quasi-equilibrium problem of type I, denoted by (GEP)I ,
in which the multivalued mappings S, T are called constraint mappings, F1 is called a utility
multivalued mappings.
(GEP)I and general quasi-equilibrium problem are equivalent. Indeed, we defined the multivalued mapping F : D × K → 2X by
F (x, y) = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), ∀t ∈ S(x, y)}
If exists (x, y) satisfies 1), 2) and x ∈ F (x, y) then 0 ∈ x − F (x, y).
Setting F (x, y) = x − F (x, y).
Therefore, the solutions of (GEP)I are the solutions of general quasi-equilibrium problem
and vice versa.


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3.1.2

The existence of solution theorem

In the study results of N. X. Tan and T. T. T. Duong, they used S. Park fixed point Theorem
or Lemma KKM to give the existence to solutions for generalized quasi-equilibrium problem of
type I.
In this section, we apply the obtained new results in Chapter 2 to get the existence to
solutions for generalized quasi-equilibrium problem of type I.
Let X, Y, Z be Hausdorff locally convex topological vector spaces, D ⊂ X, K ⊂ Z be
nonempty subsets. Given multi-valued mappings S, T and F1 are defined in Section 3.1.1. We
apply the Corollary 2.1.1 to have a theorem:
Theorem 3.1.1. The following conditions are sufficient for (GEP)I to have a solution:
i) D, K are nonempty convex compact subets;


3.2
3.2.1

Generalized quasi-equilibrium problem of type II
Put the problem

Let X, Y, Z be Hausdorff locally convex topological vector spaces, given multi-valued mappings P1 : D → 2D , P2 : D → 2D , Q : K × D → 2K and F : K × D × D → 2Y . We are interested
in the problem: Find x ∈ D such that
1) x ∈ P1 (x);
2) 0 ∈ F (y, x, t), for all t ∈ P2 (x) and y ∈ Q(x, t).
This problem is called a generalized quasi-equilibrium problem of type II, denoted by (GEP)II .
The multivalued mappings P1 , P2 , Q are called constraint mappings and F is called a utility
multivalued mapping.
Extension of constraint conditions of (GEP)II with mappings S, P0 : D × K → 2D , T :
D × K → 2K , Q0 : K × D × D → 2K and F2 : K × K × D × D → 2Y . Then we have a problem:
Find (x, y) ∈ D × K such that
1) (x, y) ∈ S(x, y) × T (x, y);
2) 0 ∈ F2 (y, v, x, t), for all t ∈ P0 (x, y) and v ∈ Q0 (y, x, t).
The problem is called general (GEP)II . General (GEP)II and general quasi-equilibrium are
equivalent problems. Indeed, we define multivalued mappings: H : D × K → 2X :
H(x, y) = {z ∈ S(x, y)|0 ∈ F2 (y, v, x, t), ∀t ∈ P0 (x, y), v ∈ Q0 (y, x, t)}.
If, exists (x, y) such that 1), 2) are satisfied and x ∈ H(x, y) then 0 ∈ x − H(x, y).
Setting F (x, y) = x − H(x, y).
Then, we see the solutions of general (GEP)II are the solutions of general quasi-equilibrium
and vice versa.


18


i) D is nonempty convex compact subset;
ii) S : D × K → 2D is a multivalued mapping with a nonempty closed fixed set B = {(x, y) ∈
D × K|x ∈ S(x, y), y ∈ T (x, y)};
iii) P0 : D × K → 2D is a multivalued mapping having open lower sections and co(P0 (x, y)) ⊂
S(x, y) for each (x, y) ∈ D × K;


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iv) for any fixed t ∈ D, the set
A = {(x, y) ∈ D × K|0 ∈
/ F2 (y, v, x, t), for some v ∈ Q0 (x, t, y)}
is open in D.
Then, the multivalued mapping G : D × K → 2D defined by
G(x, y) = {t ∈ P0 (x, y)|0 ∈
/ F2 (y, v, x, t), for some v ∈ Q0 (x, t, y)},
where (x, y) ∈ D × K, is l.s.c on D × K.
We assume that for any fixed (x, y) ∈ D × K, Q0 (x, ., t) : D → 2K is a l.s.c multivalued
mapping and F2 (., ., ., t) : K × K × D → 2Y is a closed mapping. We have a lemma.
Lemma 3.2.2. We assume that the following conditions hold.
i) D is a nonempty convex compact subset;
ii) S : D × K → 2D is a multivalued mapping with nonempty closed values fixed set B =
{(x, y) ∈ D × K|x ∈ S(x, y), y ∈ T (x, y)};
iii) P0 : D × K → 2D is a multivalued mapping having open lower sections and co(P0 (x, y)) ⊆
S(x, y) for each (x, y) ∈ D × K;
iv) for any fixed t ∈ D, Q0 (., t, .) : D × K → 2K is a l.s.c multivalued mapping and F2 (., ., ., t) :
K × K × D → 2Y is a closed multivalued mapping.
Then G : D × K → 2D :
G(x, y) = {t ∈ P0 (x, y)|0 ∈
/ F2 (y, v, x, t), với v ∈ Q0 (x, t, y) nào đó},

Put the problem

Let X, Y, Y1 , Y2 , Z be Hausdorff locally convex topological vector spaces, D ⊂ X, K ⊂ Z be
nonempty subsets. Given multi-valued mappings S : D × D → 2D , T : D × K → 2K , P : D →
2D , Q : K × D → 2K and F1 : K × D × D × D → 2Y1 , F : K × D × D → 2Y2 . We have a
problem: Find (x, y) ∈ D × K such that
1) x ∈ S(x, y);
2) y ∈ T (x, y);
3) 0 ∈ F1 (y, x, x, t), for all t ∈ S(x, y);
4) 0 ∈ F (y, x, t), for all t ∈ P (x) and y ∈ Q(x, t).
The problem is called mixed generalized quasi-equilibrium problem, denoted by (M GQEP ), in
which the multivalued mappings S, T, P, Q are call constraint mappings, F1 , F are called utility
multivalued mappings.
Besides, it is the combination of generalized quasi-equilibrium problem of type I and general
generalized quasi-equilibrium problem of type II to have a problem: Find (x, y) ∈ D × K such
that
1) x ∈ S(x, y);
2) y ∈ T (x, y);
3) 0 ∈ F1 (y, x, v, x), for all v ∈ T (x, y);
4) 0 ∈ F2 (y, v, x, t), for all t ∈ P0 (x, y) and v ∈ Q0 (y, x, t),


21

where, the multivalued mappings: S : D × D → 2D , T : D × K → 2K , P0 : D × K → 2D , Q0 :
K × D × D → 2K and F1 : K × D × D × D → 2Y1 , F2 : K × K × D × D → 2Y2 .
The problem is called general mixed generalized quasi-equilibrium problem, in which the
multivalued mappings S, T, P0 , Q0 are call constraint mappings, F1 , F2 are called utility multivalued mappings.
3.3.2



a) for any fixed t ∈ D, the function ψ(., ., ., t) : K × K × D → R are u.s.c;
b) ψ(y, v, x, x) ≥ 0, for all y, v ∈ K, x ∈ D.
Then, there exists (x, y) ∈ D × K such that (x, y) ∈ S(x, y) × T (x, y) and
ψ(y, v, x, t) ≥ 0, for all (t, v) ∈ S(x, y) × T (x, y).
Next, we apply Corollary 2.1.8 to prove the existence of solutions to general mixed generalized
quasi-equilibrium problem.
Theorem 3.3.2. We assume that the following conditions hold.
i) D, K are nonempty convex compact sets;
ii) S : D × K → 2D is a continuous multivalued mapping with nonempty closed convex values;
iii) T : D × K → 2K is a u.s.c continuous multivalued mapping with nonempty closed convex
values;
iv) The set A = {(y, x, z, t) ∈ K × D × D × D|0 ∈ F1 (y, x, z, t)} is closed;
v) for any fixed (y, x) ∈ K × D, the set
B = {z ∈ S(x, y)|0 ∈ F1 (y, x, z, t), for all t ∈ S(x, y)}
is nonempty and convex;
vi) P0 : D × K → 2D is a multivalued mapping having open lower sections and for each
(x, y) ∈ P0 (x, y) × T (x, y), on has
0 ∈ F1 (y, x, x, t), for all t ∈ S(x, y);
vii) Q0 : D × D × K → 2K , and F2 : K × K × D × D → 2Y are multivalued mappings such
that for any fixed t ∈ D, the set
C = {(x, y) ∈ D × K|0 ∈
/ F2 (y, v, x, t), với v ∈ Q0 (x, t, y) nào đó}
is open in D;
viii) 0 ∈ F2 (y, v, x, x), for all (x, y) ∈ S(x, y) × T (x, y), v ∈ Q0 (x, x, y).
Then there exists (x, y) ∈ D × K such that
1) (x, y) ∈ S(x, y) × T (x, y);
2) 0 ∈ F1 (y, x, x, t), for all t ∈ S(x, y);
3) 0 ∈ F2 (y, v, x, t), for all t ∈ P0 (x, y), v ∈ Q0 (x, t, y).


satisfied.
2) In the case when Q0 : D × D × K → 2K is a l.s.c multivalued mapping and φ2 : K × K × D ×
D → 2Y is a function such that for any fixed t ∈ D, the function φ2 (., ., ., t) : K ×D ×D → R
is u.s.c, then condition vii) is satisfied.