SOME CLASSES OF EQUATIONS IN ORDERED BANACH 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 valu

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

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-Rutman’s properties of the positive eigen-pair 16

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

that there exists an increasing mapping A : K ! K such that '[f (Y )] A [' (Y )] ; 8Y 2 M:

We want to prove the mapping f is '-condensing If 9Y 2 M 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-Rutman’s 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-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

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)

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) = f g) If in E we de…ne

a partial order by x y i¤ y x2 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) = j j 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; )

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!1xn = x i¤ lim

n!1p (xn x) = in E and we call a subset A X closed

if whenever fxng A, lim

n!1xn = 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 K g

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,

1

P

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!1an = 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

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, 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 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;

(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 (Tn

z (x) Tn

z (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; 2 M and ) 2 M:

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 + p 1(W ) : W 2 ;

x = V 2 X : 9W 2 và x + p 1(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 (Tzn(x) Tzn(y)) Qn p (x y) for n2 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:

1.4 Applications to Cauchy problems in a scale of

Ba-nach spaces.

Let f(Fs;k:ks) : s2 (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)] ; t2 ; 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)] ; t2 := [0; M ] ; x (0) = x0 2 F1 (1.4) where f : F ! F satis…es following condition

(A1) if 0 < s < r 1then f is continuous from (F;k:kr) into Fs and such that

(

kf (t; u) f (t; v)ks

Cku vk r

r s 8u; v 2 Fr; t2 ;

kf (t; )ks

B

r s; where B; C are the contants and they are independent of r; s; u; v; t:

Note 4 = f(t; s) : 0 < s < 1; 0 < t < a (1 s)g for a > 0 and su¢ ciently small We call E

a space of the functions u (t; s) such that

t7! u (t; s) is continuous on [0; a (1 s)) 8s 2 (0; 1) and

kuk := supn

ju (t; s)j :ha(1 s)

t 1i

: (t; s)2 4o

<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)24kx (t)ks:h

a(1 s)

t 1i

<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

x2 {([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 T x (t) :=Rt

0 f ( ; x ( )) d

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 = x2 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!1sn = 1 The set X is equipped with a

K-normed p : X ! K de…ned by:

p (x) = sup

t2 n

kx (t)ks n

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 kskx 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)

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)

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 2 M

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 Dsuch that 2 M, f ( ) 2 M 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

(H1) ' [f ( )] A [' ( )] whenever D, 2 M, f ( ) 2 M

(H2) 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 (H1) and

(H002) 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 f g

Then f has a …xed point in D