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MINISTRY OF EDUCATION AND TRAINING
HO CHI MINH CITY UNIVERSITY OF EDUCATION

VO VIET TRI

SOME CLASSES OF EQUATIONS IN ORDERED
BANACH SPACES
Major: Analysis
CODE: 62 46 01 02

ABSTRACT

HO CHI MINH CITY, 2016

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Contents
1 Equations in K-normed spaces
1.1 Ordered spaces and K-normed spaces. . . . . . . . . . . . . . . . . . . . . .
1.2 Fixed point theorem of Krasnoselskii in K-normed space with K-normed
value in Banach space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Fixed point theorem of Krasnoselskii in K-normed space with K-normed
value in locally convex space. . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Locally convex space de…ned by a family of seminorms. . . . . . . .
1.3.2 Locally convex space de…ned by a neighbord base of zero. . . . . . .
1.4 Applications to Cauchy problems in a scale of Banach spaces. . . . . . . .

3.1.1 The semi-continuous and compact of multivalued operator. . . . . . . . .
3.1.2 The …xed point index . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 The computation of the …xed point index for some clases of multivalued
operator and applications to …xed point problem. . . . . . . . . . . . . .
3.2 Multivalued equation depending on parameter with monotone minorant. . . . .
3.2.1 The continuity of the positive solution-set. . . . . . . . . . . . . . . . . .
3.2.2 Eigenvalued Interval for multivalued equation. . . . . . . . . . . . . . . .
3.2.3 Application to a type of control problems. . . . . . . . . . . . . . . . . .
3.3 The positive eigen-pair problem. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Existence of the positive eigen-pair. . . . . . . . . . . . . . . . . . . . . .
3.3.2 Some Krein-Rutman’s properties of the positive eigen-pair. . . . . . . . .

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T (y))

Q [p (x

y)] , x; y 2 X:

(1)

From (1) implies
9k > 0 : q (T (x)

T (y))

kq (x

y) , x; y 2 X

(2)

If we only consider (X; q) with (2), we have less information than when we work with (1).
Therefore, from (1) we can use the properties of the positive linear operator found in the
theory of equation of ordered spaces.
Recently, the study of the …xed point in the cone-metric spaces has drawn a lot of mathematicians’ attention. However, the results at later period are not deep and have no new
applications compared with the studies in the previous period. In addition, these studies in the
previous and recent period only focused on the Cacciopoli-Banach principle and its extensions.
In Chapter 1 of the thesis, we present the results of …xed point theorems for mappings T + S
in the K-normed space. We applied this result to prove the existence of solutions on [0; 1) for
a Cauchy problem on the scale of Banach spaces with weak singularity.
The cone-valued measures of noncompactness are de…ned and their properties are the same
as measure of normal noncompactness (real-valued). However, they are not used much to

the sense of the following: for every bounded open subset G and G 3 then @G \ S1 6= ?.
Dancer, Rabinowitz, Nussbaum, Amann used topology degree and a separation theorem of the
compact-connected-sets to prove the existence of unbounded connected-components in the set
S2 = f( ; x) j x 6= , x = A ( ; x)g.
Naturally, we consider an inclusion x 2 A ( ; x) ; we want to establish the results of its
solutions and solution-set’s structure. In Chapter 3. we present the results of some classes
multi-equations in ordered space. We proved the continuity of the equations’s solutions set
in the sense of Krasnoselskii (The equation has a minotone minorant); we obtained a result
of parameted interval so that the equation has a solution. We applied these results to study
the Control problem and Eigevalued problem of positive homogenuous increasing multivalued
operator. For some classes of special mapping, we proved some Krein-Rutman’s properties such
as the simple geometric unique of eigen-pair.

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Chapter 1
Equations in K-normed spaces
In this Chapter, we present the basic concepts of ordered space and the complete of topology
in K-normed space. In subsections 1.2, 1.3, we proved the …xed point theorem of total two
operators in the cone-normed space. We consider two cases. In the …rst case, the values of
K-normed belong to Banach spaces (Theorem 1.1). In the second one, the values of K-normed
belong to locally convex space (Theorem 1.3, Theorem 1.5).
Next, we apply these results to prove the existence of solutions on [0; 1) to a Cauchy
problem with weak singularity on the scale of Banach spaces (Theorem 1.6, Theorem 1.7).


n!1

X closed

if whenever fxn g A, lim xn = x then x 2 A. Clearly, 1 = fG X : XnG is closedg is a
n!1
topology on X:
2) We denote by 2 the topology on X, de…ned by the family of seminorms ff p : f 2 K g.

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De…nition 1.6
Let (E; K) be an ordered Banach space, (X; p) be a K-normed space, and be a topology
on X
1) We say that (X; p; ) is complete in the sense of Weierstrass if whenever fxn g
X,
1
P
p (xn+1 xn ) converges in E then fxn g converges in (X; p; ).

n=1

2) We say that (X; p; ) is complete in the sense of Kantorovich if any sequence fxn g satis…es
p (xk

xl )


Then the operator T + S has a …xed point in the following cases.
(C1 ) = 1 , K is normal.
(C2 ) = 2 .

1.3

1.3.1

Fixed point theorem of Krasnoselskii in K-normed
space with K-normed has value in locally convex
space.
Locally convex space de…ned by a family of seminorms.

Let (E; K; ) be an ordered locally convex space with the separate topology
family of seminorms such that
x

y ) ' (x)

' (y) 8' 2 .

is de…ned by a
(1.2)

Let (X; p; ) be a K-normed space with the topology is de…ned by the convergence of the
net, that is, fx g ! x i¤ p (x
x) ! E .
Theorem 1.3
Let (E; K; ) be a sequentially complete space and (X; p; ) be a K-normed space. Assume


3.1.2

The …xed point index

3.1.3

The computation of the …xed point index for some clases of
multivalued operator and applications to …xed point problem.

Let (X; K; k:k) be an ordered Banach space, we introduce some ordering relations among subsets.
De…nition 3.3
a. For subset A; B 2 2X n f?g we de…ne
(1)

1) A
(2)

2) A

B , (8x 2 A; 9y 2 B such that x

y).

B , (8y 2 B; 9x 2 A such that x

y).

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= A (u) 8u 2 K \ @

(3.1)

then iK (A; ) = 1:
2) Suppose that X = K K. If there is a u0 -positive completely continuous linear operator
L with the spectral radius r (L) 1 such that
(2)

Lu

A (u) and u 2
= A (u) 8u 2 K \ @

(3.2)

then iK (A; ) = 0.
Theorem 3.2
Let
X be an open bounded subset,
3 and T : K ! 2K n f?g be an upper semicontinuous convex compact operator with closed values such that x 2
= T (x) for all x 2 K:
Then
1) iK (T; ) = 0 if there is ( 0 ; x0 ) 2 (1; 1) K such that 0 x0 2 T (x0 );
2) iK (T; ) = 1 if x 2
= T (x) for all > 1 8x 2 K.
De…nition 3.5
Let F and ' : K ! 2K n f?g be multivalued operators. For evrery x 2 K we de…ne
kF (x)


(3.3)

in the following cases
(i) (F; ') sati…es the condition (c 0 ) for su¢ ciently small r,
(ii) (F; ') sati…es the condition (c 1 ) for su¢ ciently large r.
Theorem 3.5
Let (X; K; k:k) be an ordered Banach space, X = K K, A : K ! 2K n f?g be an upper
semi-continuous compact operator with closed convex values and P , Q : K ! K be completely

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continuous linear operators with the spectral radius r (P ), respectively r (Q). Suppose that there
exists bounded open sets 1 , 2 ( 2 1 ( 2 );such that
(i) P is u0 -positive
(ii)
(2)

Px
or

1

;

A (x)


:

(3.5)

(iii) 0 < r (Q) < r (P ) :
Then for every 2 (r (Q) ; r (P )), the equation x 2 A (x) has a solution in Kn f g.
For every multivalued operator ' : K ! 2K n f?g we de…ne
r (') = sup

> 0 : 9x 2 Ksuch that x 2 ' (x)
> 0 : 9x 2 Ksuch that x 2 ' (x)

r (') = inf

; sup ? = 0;
; inf ? = 1:

Theorem 3.6
Let (X; K; k:k) be an ordered Banach space and A : K ! 2K n f?g be an upper semicontinuous compact operator with closed convex values. Suppose that P , Q : K ! 2K n f?g are
convex upper semi-continuous compact operator with closed values. In Addition, P and Q are
positively 1-homogeneous such that
(i) (A; P ) satis…es the condition (c 0 ) and (A; Q) satis…es the condition ( c1 );
(ii) 0 < r (P ) < r (Q) < 1 or 0 < r (Q) < r (P ) < 1:
Then
if 2 (r (P ) ; r (Q)) or 2 (r (Q) ; r (P )) then the equation x 2 A (x) has a solution
in Kn f g.

3.2

Multivalued equation depending on parameter with


13

atu for


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Then the solution set S =

x 2 K : 9 > 0; x 2 F (x)

forms an unbounded continuous

branch emanating from , that is S \ @G 6= ? for any bounded open subset G 3 .

3.2.2

Eigenvalued Interval for multivalued equation.

For x 2 Kn f g we de…ne (x) = f 2 R+ n f0g : x 2 F (x)g and Kr = K \ B r ( ) :
Theorem 3.8
Let F : K ! 2K n f?g be an upper semi-continuous compact operator with closed convex
values. Assume that the following conditions satisfy
(i) 2
= F (x) for all x 2 Kn f g ;
(ii) the set S = fx 2 Kn f g : 9 > 0; x 2 F (x)g forms an unbounded continuous branch
emanating from ;
(iii) suppose that there are numbers a, b such that either
a = lim+ sup

or
a = lim

r!1

sup

[

x2S;kxk r

r!0

Then the equation x 2 F (x) has a positive solution for every

3.2.3

[

x2Kr \S

2 (a; b).

Application to a type of control problems.

We consider the following boundary value problem
x00 (t) + (t) f (x (t)) = 0; t 2 [0; 1] ; x (0) = x (1) = 0;
(t) 2 F (t; x (t)) ; t 2 [0; 1]:

(3.9)

(3.10)


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We denote = [0; 1]: Let X = C ( ) be the Banach space of all the continous real-value
functions on with the norm kxk = maxt2 jx (t)j. In X we de…ne a partial order by cone
K = fx 2 X : x (t) 0 for all t 2 g.
For u 2 K we denote
Fu = x 2 L1 ( ) : x (t) 2 F (t; u (t)) a.e on
Au =

y 2 K : 9x 2 Fu ; y (t) =

Z

:

1

G (t; s) x (s) f [u (s)] ds .

0

We need to prove that the following equation has a solution
u 2 A (u) :

(3.11)

Together with (3.11) we also consider the following equation which depends on parameter:
u 2 A (u) :


0

< a then (3.11) has a positive solution.

The positive eigen-pair problem.

In what follows, we consider an ordered Banach space (X; K) : The pair ( 0 ; x0 ) is called a
positive eigen-pair of the operator A : K!2K n f?g if x0 2 Kn f g ; 0 > 0 and 0 x0 2 A (x0 ) :

3.3.1

Existence of the positive eigen-pair.

In the case of the increasing operators.
In this subsection, we apply the Theorem 3.7 to prove the existence of the positive eigen-pair
for the positive homogenuous increasing operators or convex processes.
Theorem 3.10
Assume that (X; K; k:k) is an ordered Banach space. Let A : K ! 2K n f?g be a (2)increasing, compact, upper semi-continuous operator with closed convex, such that
(2)

(i) A (tx)

tA (x) 8 (t; x) 2 (0; 1)

K;

(2)

(ii) 9u 2 K; 9 > 0 : A (u)


positive:
Then A has a positive eigen-pair ( 0 ; x0 ) with 0
and kx0 k = 1:
Theorem 3.12.
Let A : K ! 2K n f?g be a positive 1-homogeneous compact upper semicontinuous operator
with closed convex such that
i) A is ( 2)-increasing,
(2)

ii) the number

> 0 : 9x

(A) = supu2K;kuk=1 inf

u; A (x)

is positive.

x

Then A has a positive eigen-pair ( 0 ; x0 ) with 0
(A). Moreover, if A is ( 3)-increasing
then 0 = (A) :
Theorem 3.13
Assume that (X; K; k:k) is an ordered Banach space. Let A : X ! 2X n f?g be an upper
semi-continuous compact operator, such that
(i) A is a convex process,
(2)

(iii) There is an element u 2 S and a positive number such that u F (u) :
Then
hp; xi
1
= sup inf
then there exists x0 2 S such that
1) If 0 is de…ned by
p2S+ x2S (F (x) ; p)
0
0 x0 2 F (x0 ) and
1
0

2) If

3.3.2

= sup
p2S+

hp; x0 i
;
(F (x0 ) ; p)

> 0 and x 2 S satisfying x 2 F (x) then

0:

Some Krein-Rutman’s properties of the positive eigen-pair.



> 0 such that u0

! 2X n f?g is said to be strongly positive if F

K

(1)

A (x)

int(K) and

(2)

it is said to be semi strongly positive if 9g 2 K such that hg; F (x)i > 0 = hg; xi for all
x 2 Knint(K) :
De…nition 3.11
Let A : K ! 2K n f?g :
1) For x 2 K, we denote
K (x) = ff 2 K : hf; xi > 0g ; S (x) = ff 2 K : hf; xi = 1g
and
(x) = inf fhf; zi : (f; z) 2 S (x)

A (x)g ;

(x) = sup fhf; zi : (f; z) 2 S (x)

A (x)g ;


1) if A is (1) increasing then
a) r (A)
r (A) if A is strongly u0 positive.
0
b) x0 2intK and r (A)
or (A) if A is semi strong positive.
0
2) If A is lower semicontinuous, semi strong positive and is (3) increasing then r (A) =
0 = r (A).
De…nition 3.12
Given A : K !2K n f?g ; u0 2 K.
(2)

1) The operator A is said to be u0 -increasing if x

y implies hu0 i+

[A (y)

A (x)] \ K .

2) An operator A is said to be semi strongly increasing if 9g 2 K such that if x y 2 KnintK
then

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we use cone-normed space and cone-valued measure of compactness to study the existence of
…xed point for operator and the application of abstract results to some classes of di¤erential
equations. In the second direction, we use topological degree combined with an reasonable
order to prove some global results of eigenvalue problem for multivalued operator depending
on parameter in ordered space. Main results of the thesis include:
1. Proving the …xed point theorem of total two operators in the cone-normed space with
the values of K-normed belonging to Banach spaces or to locally convex space.
Applying the received results to prove the existence of solutions on [0; 1) of a problem on
Cauchy with weak singularity on the scale of Banach spaces.
2. Applying a result of the …xed point theory for increasing mapping in ordered space to
prove the existence of the …xed point for a class of consending operators by the cone-valued
measures of noncompactness.
Using this result and a cone-value measure of noncompactness appropriately to prove the
existence of solutions for a class of Cauchy problems with delay.
3. Extending of solutions set’s continuity in the sense of Krasnoselskii for multivalued
equations containing parameters with monotone minorant and proving the existence interval of
parameter’s values so that the equation has a solution.
Applying these …ndings to prove the existence of solution of boundary value problem with
a control multivalued function.
4. The computation of the …xed point index for some clases of multivalued operator via
linear mapping, convex mapping or its approximate mapping at (or 1) and applications to
…xed point problems.
5. Proving the existence of a positive eigen-pair for a class of increasing positive homogenous
multivalued operators.

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