Theory of ordered Banach spaces and related equations was rst introduced by M.G.Krein
and M.A.Rutman in the 1940s. The theory was then developed and many signi cant results
were achieved in the period of time from 1950 to 1980 in the works of M.A.Krasnoselskii and his
students. Some notable names among them are E.N.Dancer, P.Rabinowitz, R.Nussbaum and
W.V.Petryshyn. The theory has been developing until today with huge range of applications
in di¤erential and integral equations, physics, chemistry, biology, control theory, optimization,
medicine, economics, linguistics,.

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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
Contents
1 Equations in K-normed spaces 4
1.1 Ordered spaces and K-normed spaces. . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Fixed point theorem of Krasnoselskii in K-normed space with K-normed has
value in Banach space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Fixed point theorem of Krasnoselskii in K-normed space with K-normed has
value in locally convex space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Locally convex space de
ned by a family of seminorms. . . . . . . . . . . 5
1.3.2 Locally convex space de
ned by a neighbord base of zero. . . . . . . . . . 6
1.4 Applications to Cauchy problems in a scale of Banach spaces. . . . . . . . . . . 7
1.4.1 In the case of problem with non perturbation. . . . . . . . . . . . . . . . 7
1.4.2 In the case of problem with perturbation. . . . . . . . . . . . . . . . . . . 8
2 Consending mapping with cone-valued measure of noncompactness 9
2.1 Measures of noncompactness, condensing mapping and
xed point theorem. . . . 9
2.2 Application for di¤erential equation with delay in the Banach space. . . . . . . . 10
3 Multivalued equation depending on parameter in ordered spaces 11
3.1 The
xed point index for class consending multivalued operator. . . . . . . . . . 11
3.1.1 The semi-continuous and compact of multivalued operator. . . . . . . . . 11
3.1.2 The
xed point index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.3 The computation of the
xed point index for some clases of multivalued
operator and applications to
xed point problem. . . . . . . . . . . . . . 11
3.2 Multivalued equation depending on parameter with monotone minorant. . . . . 13
3.2.1 The continuity of the positive solution-set. . . . . . . . . . . . . . . . . . 13
3.2.2 Eigenvalued Interval for multivalued equation. . . . . . . . . . . . . . . . 14
3.2.3 Application to a type of control problems. . . . . . . . . . . . . . . . . . 14
3.3 The positive eigen-pair problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.1 Existence of the positive eigen-pair. . . . . . . . . . . . . . . . . . . . . . 15
3.3.2 Some Krein-Rutmans properties of the positive eigen-pair. . . . . . . . . 16
1
INTRODUCTION
Theory of ordered Banach spaces and related equations was
rst introduced by M.G.Krein
and M.A.Rutman in the 1940s. The theory was then developed and many signi
cant results
were achieved in the period of time from 1950 to 1980 in the works of M.A.Krasnoselskii and his
students. Some notable names among them are E.N.Dancer, P.Rabinowitz, R.Nussbaum and
W.V.Petryshyn. The theory has been developing until today with huge range of applications
in di¤erential and integral equations, physics, chemistry, biology, control theory, optimization,
medicine, economics, linguistics,...
In the future, theory equations in ordered space probably develop in the two ways. The
rst
is that, it will continue to develop theories for the new classes of equations in ordered spaces.
The second is that, it will
nd applications to solve the problems of the area that may not be
originally related to the equations in ordered spaces.
Our thesis will present the research in two above directions. Speci
cally, in the
rst direction
we study the multivalued equations containing parameters in ordered space; in the second
direction we use cone-normed space and measure of compactness to study the equations in
space that cannot be ordered.
I. The use of cone-normed space and cone-values measure of compactness to
study the equations.
Cone-metric or cone-normed space (also called a K-metric space or K- normed space) is a
natural extension of metric spaces or normed space, where the values of the metric (res. normed)
belong to cone of an ordered space instead of R. Included in the study since 1950, these spaces
have been used in the Numerical Analysis, Di¤erential Equations, Theory Fixed Point, ... in
the researches of Kantorovich, Collatz, P.Zabreiko and other mathematicians. We can see the
usefulness of the use of space with the cone-normed in the following example. Suppose that we
have a normed vector space (X,q) and we want to
nd a
xed point of operator T : X ! X.
In some cases we can
nd an ordered Banach space (E;K; k:k) (K is a cone in E), a positive
continuous linear operator Q and a K-normed p : X ! K such that q (x) = kp (x)k and
p (T (x) 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 math-
ematicians 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
prove the existence of solutions of the equations. The relationship between the measures of
noncompactness and equations in ordered space is shown in the following example. Let X
be a Banach space and a mapping f : X ! X, ' : M!K is a measure of noncompactness
(M = fY X : Y is bounded in Xg;(E;K) is an ordered space, K is a cone in E). Assume
2
that there exists an increasing mapping A : K ! K such that '[f (Y )] A [' (Y )] ; 8Y 2M:
We want to prove the mapping f is '-condensing. If 9Y 2M such that ' [f (Y )] ' (Y ) then
' (Y ) A [' (Y )]. Hence, element ' (Y ) 2 K is a lower solution of the equation u = A (u) :
We can use the results of
xed point of increasing mapping A to prove ' (Y ) = 0.
In Chapter 2 of the thesis, we show some conditions with which the mapping is a '-
consending (here ' is a cone-valued measure of noncompactness) and apply this result to study
the di¤erential equation with delay of the form
x0 (t) = f [t; x (t) ; x (h (t))] ; 0 h (t) t1=:
II. Multivalued equation depending on the parameter in ordered space.
The studies of single-value equation which depends on parameter of the form x = A (; x) in
the ordered space have earned profound results, starting from Krein-Rutmans theorem about
positive eigenvalues, positive eigenvector of the strongly positive linear operator, followed by
studies of the structural solutions set of the equation in the papers of Krasnoselskii, Dancer, Ra-
binowitz, Nussbaum, Amann,... Krasnoselskii used topology degree and hypothesis of monotone
minorant to prove that the set S1 = fx j 9 : x = A (; x)g is unbounded and continuous in
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-sets structure. In Chapter 3. we present the results of some classes
multi-equations in ordered space. We proved the continuity of the equationss 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-Rutmans properties such
as the simple geometric unique of eigen-pair.
3
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).
1.1 Ordered spaces and K-normed spaces.
Let (E;K;
) be a topogical vector space (
is topology on E and K E is a cone with K is a
closed convex subset such that K K for all 0 and K \ (K) = fg). If in E we de
ne
a partial order by x y i¤ y x 2 K then the triple (E;K;
) is called an ordered space.
De
nition 1.4
Let (E;K;
) be an ordered space and X be a real linear space. A mapping p : X ! E is
called a cone norm (or K-norm) if
(i) p (x) 2 K or equivalently p (x) E 8x 2 X and p (x) = E i¤ x = X , where E,
X are the zero elements of E and X respectively,
(ii) p (x) = jj p (x) 8 2 R, 8x 2 X,
(iii) p (x+ y) p (x) + p (y) 8x; y 2 X.
If p is a cone norm in X then the pair (X; p) is called a cone normed space (or K-normed
space). The cone normed space (X; p) endowed with a topology will be denoted by (X; p; ).
1.2 Fixed point theorem of Krasnoselskii in K-normed
space with K-normed has value in Banach space.
We shall use the following two topologies on a cone normed space.
De
nition 1.5
Let (E;K) be an ordered Banach space and (X; p) be a K-normed space.
1) We de
ne lim
n!1
xn = x i¤ lim
n!1
p (xn x) = in E and we call a subset A X closed
if whenever fxng A, lim
n!1
xn = x then x 2 A. Clearly, 1 = fG X : XnG is closedg is a
topology on X:
2) We denote by 2 the topology on X, de
ned by the family of seminorms ff p : f 2 Kg.
4
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 fxng X,1P
n=1
p (xn+1 xn) converges in E then fxng converges in (X; p; ).
2) We say that (X; p; ) is complete in the sense of Kantorovich if any sequence fxng satis
es
p (xk xl) an 8k; l n, with fang K, lim
n!1
an = E (1.1)
then fxng converges in (X; p; ).
Theorem 1.1
Let (E;K) be an ordered Banach space, (X; p; ) be a complete K-normed space in the sense
of Weierstrass and = 1 or = 2. Assume that C is a convex closed subset in (X; p; ) and
S,T : C ! X are operators such that
(i) T (x) + S (y) 2 C 8x; y 2 C;
(ii) S is continuous and S (C) is compact with respect to the topology ;
(iii) there is a positive continuous linear operator Q : E ! E with the spectral radius
r (Q) < 1 such that
p (T (x) T (y)) Q [p (x y)] for all x; y 2 C:
Then the operator T + S has a
xed point in the following cases.
(C1) = 1, K is normal.
(C2) = 2.
1.3 Fixed point theorem of Krasnoselskii in K-normed
space with K-normed has value in locally convex
space.
1.3.1 Locally convex space de
ned by a family of seminorms.
Let (E;K;
) be an ordered locally convex space with the separate topology
is de
ned by a
family of seminorms such that
x y ) ' (x) ' (y) 8' 2 . (1.2)
Let (X; p; ) be a K-normed space with the topology is de
ned by the convergence of the
net, that is, fxg ! 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
that (X; p; ) is complete in the sense of Weierstrass, C is a closed convex subset in (X; p; )
and S,T : C ! X are operators satisfying the follwing conditions:
(1) T is uniformly continuous, S is continuous, T (C) +C C, S (C) C and S (C) is a
relatively compact subset with respect to the topology :
(2) There is a sequence of positive continuous operators fQn : E ! Egn2N such that
(2a) The series
P1
n=1Qn () is convergent in E for every 2 K;
5
(2b) 8 ('; ") 2 (0;1) then there exists (; r) 2 (0; ")N such that if 'p (x y) < +"
then
(8x; y 2 C, 'p (x y) < + " ) ' [Qrp (x y)] < " )
(2c) For every z 2 C; then p (T nz (x) T nz (y)) Qn [p (x y)] 8n 2 N, x; y 2 C:
Then the operator T + S has a
xed point in C:
1.3.2 Locally convex space de
ned by a neighbord base of zero.
De
nition 1.8
Let (E;K;
) be an ordered locally convex space:
1) A subset M of E is called normed i¤
2 K; 2M and ) 2M:
2) We say that the ordered locally convex space (E;K;
) is normed i¤ (E;K;
) is a locally
convex topological vector space such that
(i) There exists a neighborhood base of zero which contains only convex balanced normed
sets,
(ii) if V and W are normed then V \K +W \K is normed.
De
nition 1.9
Let (E;K;
) be a normed ordered locally convex space with the neighborhood base of
zero which contains only convex balanced normed sets. Assume that X is a vector space and
p : X ! K is a K-normed on X. For every x 2 X we de
ne
x =
x+ p1 (W ) : W 2 ;
x =
V 2 X : 9W 2 và x+ p1 (W ) V :
In X; we de
ne a topology with x is a neighborhood base of x 2 X. Thus, x is family of
neighborhood of x:
The following we assume that (E;K;
) is a normed ordered locally convex space.
Theorem 1.5
Let (E;K;
) be a sequentially complete space and (X; p; ) be a K-normed space. Assume
that (X; p; ) is complete in the sense of Weierstrass (or Kantorovich). C is a closed convex
subset in (X; p; ) and S,T : C ! X are operators satisfying the follwing conditions:
(1) Tz (x) = T (x) + z 2 C for all x; z 2 C;
(2) there is a sequence of positive continuous operators fQn : E ! Egn2N such that
(2a) the series
P1
n=1Qn () is convergent in E, 8 2 K;
(2b) 8V 2 ; 9W 2 and r 2 N such that Qr (W + V ) V ,
(2c) 8z 2 C then p (T nz (x) T nz (y)) Qn p (x y) for n 2 N, x; y 2 C;
(3) S is continuous, S (C) C and S (C) is relatively compact with respect to the topology
.
Then the operator T + S has a
xed point in C:
6
1.4 Applications to Cauchy problems in a scale of Ba-
nach spaces.
Let f(Fs; k:ks) : s 2 (0; 1]g be a family of Banach spaces such that
Fr Fs; kxks kxkr 8x 2 Fr if 0 < s < r 1:
Set F = \s2(0;1)Fs. Let
R; x0 2 F1, f; g :
F ! F be mappings satisfying the follwing
condition:
For evrey pair of number r; s such that 0 < s < r 1; f and g are continuous mappings
from
(F; k:kr) to (Fs; k:ks) :
Consider the Cauchy problem of the form
x0 (t) = f [t; x (t)] + g [t; x (t)] ; t 2
; x (0) = x0 (1.3)
1.4.1 In the case of problem with non perturbation.
We consider the Cauchy problem
x0 (t) = f [t; x (t)] ; t 2
:= [0;M ] ; x (0) = x0 2 F1 (1.4)
where f :
F ! F satis
es following condition
(A1) if 0 < s < r 1 then f is continuous from
(F; k:kr) into Fs and such that(
kf (t; u) f (t; v)ks Ckuvkrrs 8u; v 2 Fr; t 2
;
kf (t; )ks Brs ;
where B;C are the contants and they are independent of r; s; u; v; t:
Note 4 = f(t; s) : 0 0 and su¢ ciently small. We call E
a space of the functions u (t; s) such that
t 7! u (t; s) is continuous on [0; a (1 s)) 8s 2 (0; 1) and
kuk := sup
n
ju (t; s)j :
h
a(1s)
t
1
i
: (t; s) 2 4
o
<1:
Then E is a Banach space. In E; we consider an order de
ned by cone K which contains
only nonnegative functions.
We call X a set of functions x 2 \
0<s<1
{([0; a(1 s)); Fs) such that
q (x) = sup
(t;s)24
kx (t)ks :
h
a(1s)
t
1
i
<1
The set X is equipped with a K-normed p : X ! K de
ned by p (x) (t; s) = kx (t)ks. Then
q (x) = kp (x)k ; x 2 X.
Theorem 1.6
Assume that f satis
es the condition (A1). Then the problem (1.4) has a unique solution
x 2 {([0; a(1 s)); Fs) 8s 2 (0; 1) for su¢ ciently small a. Furthermore, the operator (I T )1
is continuous on (X; q), where Tx (t) :=
R t
0
f ( ; x ()) d .
7
1.4.2 In the case of problem with perturbation.
Consider Cauchy problem (1.3) with
= [0;1): Suppose that the mapping f :
Fs ! Fs
satis
es Lipschitz condition.
Let (E;K;
) be a locally convex space de
ned by
E =
x =
x(1); x(2); ::::
: x(j) 2 R; j 2 N ;K = x 2 E : x(j) 0; j 2 N
and the topology
de
ned by a family of seminorms = f'n : E ! Rgn=1;2;:::, 'n (x) =
x(n).
We call X a set of the mappings x from
to F satisfying the following condition: For every
s 2 (0; 1); the mapping x :
! (Fs; k:ks) is continuous. Choose the sequence fsngn=1;2;:::
(0; 1) such that s1 < s2 < ::: < sn < ::: and limn!1 sn = 1. The set X is equipped with a
K-normed p : X ! K de
ned by:
p (x) =
sup
t2
n
kx (t)ksn
n=1;2;:::
;
n = [0; n].
Assume that f and g : [0;1) F ! F satisfy the owing conditions:
(A1): For every s 2 (0; 1), f is continuous from (F; k:ks) to (Fs; k:ks) and there is a positive
numeric ks such that
kf (t; x) f (t; y)ks ks kx yks , for x; y 2 X; (1.5)
(A2): for every pair (r; s) 2 4; the mapping g is continuous from (F; k:kr) to (F; k:ks)
and the set g (I F ) is relative compact in (Fs; k:ks) for every segment I [0;1), where
4 = f(r; s) 2 (0; 1) (0; 1) : r > sg.
By ussing Theorem 1.3 we obtain the following theorem.
Theorem 1.7
Assume that the conditions (A1-A2) hold. Then equation (1.3) has a solution on [0;1).
8
Chapter 2
Consending mapping with cone-valued
measure of noncompactness
In this Chapter, we prove the existence of the conditions so that the mapping is '-consending,
where ' is a cone-valued measures of noncompactness (Theorem 2.2).
We use this result and a cone-value measure of noncompactness appropriately to prove the
existence of solutions for a class of Cauchy problem with delay (Theorem 2.3).
2.1 Measures of noncompactness, condensing mapping
and
xed point theorem.
De
nition 2.1
Let (E;K) be an ordered Banach space, X be a Banach space, M be a family of bounded
subsets of X such that: if
2 M then co (
) 2 M. A mapping ' : M ! K is called a
measure of noncompactness if ' [co (
)] = ' (
) 8
2M.
De
nition 2.2
Let (E;K) be an ordered Banach space, X be a Banach space and ' :M 2X ! K be
a cone-valued measure of noncompactness. A continuous mapping f : D X ! X is called
condensing if for
D such that
2M, f (
) 2M and ' [f (
)] ' (
) then
is relatively
compact:
Theorem 2.2
Let (E;K) be an ordered Banach space, X be a Banach space and ' : M 2X ! K
be a regular measure of noncompactness having property ' (fxn : n 1g) = ' (fxn : n 2g).
Assume that D X is a nonempty closed convex subset and f : D ! D is a continuous
mapping such that there exists a mapping A : K ! K satisfying
(H 1) ' [f (
)] A [' (
)] whenever
D,
2M , f (
) 2M
(H 2) if x0 2 K, x0 A (x0) then x0 = :
Then f has a
xed point in D.
Corollary 2.2
Suppose that the measure of noncompact ' is regular and the mapping f satis
es hypothesis
(H 1) and
(H
00
2) 1) The mapping A is increasing, the sequence fA (xn)g converges whenever fxng is
an increasing sequence in D,
2) A does not have
xed points in Kn fg.
Then f has a
xed point in D.
9
2.2 Application for di¤erential equation with delay in
the Banach space.
Let us consider the Cauchy problem
x= (t) = f [t; x (t) ; x (h (t))] ; x (0) = u0: (2.1)
In the case that f do