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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-Rutman’s 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 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. 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) = f g). 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) = 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; ). 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 xn = x i¤ lim p (xn x) = in E and we call a subset A X closed n!1 n!1 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. 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 fxn g X, P1 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 ) an 8k; l n, with fan g K, lim an = E (1.1) n!1 then fxn g 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, 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 P1 of positive continuous operators fQn : E ! Egn2N such that (2a) The series n=1 Qn ( ) 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) < + " ) ' [Qr p (x y)] < " ) (2c) For every z 2 C; then p (Tzn (x) Tzn (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 1 x = x+p (W ) : W 2 ; 1 x = V 2 X : 9W 2 và x + p (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 P1 of positive continuous operators fQn : E ! Egn2N such that (2a) the series n=1 Qn ( ) 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 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 ( Cku vkr kf (t; u) f (t; v)ks r s 8u; v 2 Fr ; t 2 ; B kf (t; )ks 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 t 7! u (t; s) is continuous on [0; a (1 s)) 8s 2 (0; 1) and n h i o kuk := sup ju (t; s)j : a(1t s) 1 : (t; s) 2 4 < 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; a(1 s)); Fs ) such that 0
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 fsn gn=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 kx (t)ksn ; n = [0; n]. t2 n n=1;2;::: 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 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 D such 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 (H 1 ) ' [f ( )] A [' ( )] whenever D, 2 M , f ( ) 2 M (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 00 (H 2 ) 1) The mapping A is increasing, the sequence fA (xn )g converges whenever fxn g 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. 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 does not depend on second variable (2.1) has been studied. Here we use the cone-valued measure of nocompactness 'c de…ned as follows. Let (Y; j:j) be a Banach space and ' be a real-valued measure of noncompactness de…ned for all bounded subsets of Y . We assume that ' satis…es the following properties: regular, semi-homogeneous, algebraic semi-additive, invariant under translations. Let X = C ([a; b] ; Y ) be the Banach space of all the continous functions on [a; b] endowed with the norm kxk = sup fjx (t)j : t 2 [a; b]g. For each bounded subset X we set (t) = fx (t) : x 2 g and de…ne a function 'c ( ) : [a; b] ! R by 'c ( ) (t) = ' [ (t)]. Let B (x0 ; r) be a ball in Y , f : [0; b] B (x0 ; r) B (x0 ; r) ! Y be a uniformly continuous bounded mapping and h : [0; b] ! R be a continuous function, satisfying (f 1 ) 9m; l > 0, 9 2 (0; 1] : ' [f (t; L; M )] l' (L) + m [' (M )] for all subsets L; M B (x0 ; r) ; (f 2 ) 0 h (t) t1= : Then, we obtain the following theorem. Theorem 2.3 Assume the hypotheses (f 1 ),(f 2 ) be satis…ed. Then there exists a number b1 2 [0; b] such that (2.1) has a solution on [0; b1 ] : 10
Chapter 3 Multivalued equation depending on parameter in ordered spaces In this Chapter (subsections 3.1.1, 3.1.2), we present the concepts of semicontinuous and topo- logical degree for compact multivalued operator. We extend the continuity of solutions set in the sense of Krasnoselskii for multivalued equa- tions depending on parameter (Theorem 3.7), we proved the existence interval of parameter’s values so that the equation has a solution (Theorem 3.8). We apply these results to prove the existence of solution of boundary value problem with a control multivalued function (Theorem 3.9). We Computate the …xed point index for some clases of multivalued operator via a linear mapping (Theorem 3.1), a convex mapping (Theorem 3.2) or its approximate mapping at (or 1) (Theorem 3.4), after that we apply it to …xed point problems (Theorem 5., Theorem 3.6). We prove the existence of a positive eigen-pair for a class of increasing positive homogenouss multivalued operator and evaluate the lower-bound for the corresponding positive eigenvalue (Theorem 3.10, Theorem 3.11, Theorem 3.12, Theorem 3.13). We also prove some Krein- Rutman’s properties of positive eigen-pair for multivalued operator in ordered space (Theorem 3.15, Theorem 3.16). 3.1 The …xed point index for class consending multival- ued operator. 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. Let (X; K; k:k) be an ordered Banach space, we introduce some ordering relations among sub- sets. De…nition 3.3 a. For subset A; B 2 2X n f?g we de…ne (1) 1) A B , (8x 2 A; 9y 2 B such that x y). (2) 2) A B , (8y 2 B; 9x 2 A such that x y). 11
(3) 3) A B , (8x 2 A; 8y 2 B then x y). b. An operator F : M X!2X n f?g is said to be (k)-increasing, k = 1; 2, if x; y 2 M; (k) x y implies F (x) F (y); moreover, it is said to be (3)-increasing if x; y 2 M; x < y implies (3) F (x) F (y) :. Theorem 3.1 Let be an open bounded subset of the ordered Banach space, 2 and A : K \ ! 2K n f?g be an upper semicontinuous compact operator with closed convex values. 1) If L is a positive continuous linear operator with spectral radius r (L) 1 such that (1) A (u) Lu and u 2 = 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 semi- continuous 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) ' (x)k0 = sup fky y 0 k : y 2 F (x) ; y 0 2 ' (x)g : kF (x) '(x)k0 1) The pair (F; ') is said to satisfy the condition (c0 ) if lim kxk = 0; x2K;kxk!0 kF (x) '(x)k0 2) The pair (F; ') is said to satisfy the condition (c1 ) if lim kxk = 0: x2K;kxk!1 Theorem 3.4 Let (X; K; k:k) be an ordered Banach space and F; ' : K ! 2K n f?g be upper semi- continuous compact operators with closed convex values. Suppose that 2 F ( ) and ' be a positively 1-homogeneous (i.e ' ( x) = ' (x) ; 8 > 0) such that x 2 = ' (x) for all x 2 K. Then iK (F; Br ( )) = iK ('; Br ( )) (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 12
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) (1) 1 2 Px A (x) 8x 2 K \ @ ; A (x) Qx 8x 2 K \ @ ; (3.4) or (2) (1) 2 1 Px A (x) 8x 2 K \ @ ; A (x) Qx 8x 2 K \ @ : (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) ; sup ? = 0; r (') = inf > 0 : 9x 2 Ksuch that x 2 ' (x) ; 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 semi- continuous 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 monotone minorant. Let X be a Banach space with an order de…ned by cone K and F : K ! 2K n f?g be a multivalued operator: In this section, we prove that the solution set of F is an unbounded continuous branch, emanating from zero. x 2 F (x) (3.6) We proved the existence interval of parameter’s values so that this equation has a solution 3.2.1 The continuity of the positive solution-set. Theorem 3.7 Let (X; K; k:k) be an ordered Banach space and F : K ! 2K n f?g be an upper semi- continuous (or lower semi-continuous) compact operator with closed convex values: Assume that there is ( 2)-increasing operator G : K ! 2K n f?g satisfying (2) (i) F (x) G (x) for x 2 K; (2) (ii) there are positive numbers a; b; and an element u 2 Kn f g such that G (tu) atu for all t 2 [0; b] : 13
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 [ (x) < b = lim inf [ (x) (3.7) r!0 x2Kr \S r!1 x2S;kxk r or a = lim sup [ (x) < b = lim+ inf [ (x) : (3.8) r!1 x2S;kxk r r!0 x2Kr \S Then the equation x 2 F (x) has a positive solution for every 2 (a; b). 3.2.3 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; (3.9) (t) 2 F (t; x (t)) ; t 2 [0; 1]: Assume that the functions f and F satisfy the following conditions: (a1) f : R+ ! R+ is a continuous function. (a2) F : [0; 1] R+ ! 2R+ n f?g is a multivalued mapping with compact convex values. In addition, it is an upper-Caratheory, it means that the multivalued mapping t !7 F (t; x) is measurable for all x 2 R+ , that is, for every y 2 R then D (t) = inf fjy zj ; z 2 F (t; x)g is a measureble function, for almost every t 2 [0; 1] ; the multivalued mapping x 7! F (t; x) is an upper semicontin- uous, for each r > 0, there exists a function 'r 2 L1 [0; 1] such that supx2[0;r] F (t; x) 'r (t) a.e on [0; 1]. The above problem is equivalent to the following equation 8 Z 1 < x (t) = G (t; s) (s) f (x (s)) ds; (3.10) : 0 (t) 2 F (t; x (t)) 8t 2 [0; 1] ; where G : [0; 1] [0; 1] ! R+ is the Green function for (3.9). 14
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 : Z 1 Au = y 2 K : 9x 2 Fu ; y (t) = 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) : (3.12) Using the result of Theorem 3.7 and Theorem 3.8, we obtain the following result. Theorem 3.9 Assume that F and f satisfy the conditions (a1), (a2). In addition, (a3) there is an increasing operator g : R+ ! R+ ; and the positive numbers s1 ; s2 ( s1 < s2 ), a; b ( a > b) such that (2) (i) F (t; s) f (s) g (s) 8s 2 R+ , g (s) as 8s 2 [0; s1 ] ; (1) (ii) F (t; s) f (s) bs 8s 2 [s2 ; 1): Then 1) The solution set S of (3.12) forms an unbounded continuous branch, emanating from : 2) For every 2 10 a ; 10 b then the problem (3.12) has a positive solution, in particular, 1 if b < 0 < a then (3.11) has a positive solution. 3.3 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) u: Then A admits a positive eigen-pair ( 0 ; x0 ) with kx0 k = 1 and 0 : 15
Theorem 3.11 Let A : K ! 2K n f?g be a positive 1-homogeneous, compact, upper semi-continuous operator with closed convex such that (i) A is (2)-increasing, (2) (ii) there exists u 2 Kn f g such that the number = inf > 0 : 9x u; A (x) x is 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 (A) = supu2K;kuk=1 inf > 0 : 9x u; A (x) x is positive. 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) (tii) 8x 9u 2 A (x) : u (or A (x) 8x ); (2) (iii) 9u 2 Kn f g ; 9 > 0 : A (u) u: Then A has a positive eigen-pair ( 0 ; x0 ) with kx0 k = 1 and 0 : In the case of the non-increasing multivalued operator. In this subsection, we prove existence of positive eigen-pair for a class of non-increasing opera- tors but, in addition, the operator is compact. Proposition 3.9 Let F : S ! 2K n f?g be an upper semi-continuous convex operator with closed values. Suppose that the following conditions satisfy (i) F (S) is relatively compact, (ii) 8p 2 S+ , 8x 2 S then (F (x) ; p) > 0, (1) (iii) There is an element u 2 S and a positive number such that u F (u) : Then 1 hp; xi 1) If 0 is de…ned by = sup inf then there exists x0 2 S such that 0 p2S+ x2S (F (x) ; p) 0 x0 2 F (x0 ) and 1 hp; x0 i = sup ; 0 p2S+ (F (x0 ) ; p) 2) If > 0 and x 2 S satisfying x 2 F (x) then 0: 3.3.2 Some Krein-Rutman’s properties of the positive eigen-pair. We extend the concepts of u0 -positive, u0 -increasing, strongly positive for multivalued operator and prove some Krein-Rutman’s properties of positive eigen-pair for multivalued operator in 16
ordered space. In what follows, we assume that (X; K; k:k) is an ordered Banach space. For every u0 2 K we denote hu0 i+ = ftu0 : t > 0g : De…nition 3.9 Let A : K ! 2K n f?g : (2) (1) 1) A is said to be u0 -positive if 8x 2 K then hu0 i+ A (x) hu0 i+ . (2) (1) 2) A is said to be strongly u0 positive if 8x 2 K then 9 ; > 0 such that u0 A (x) u0 : De…nition 3.10 An operator F : X ! 2X n f?g is said to be strongly positive if F K 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 ; 2) We de…ne r (A) = sup (x) ; r (A) = inf (x) . x2Knf g x2Knf g if intK 6= ? we denote or (A) = sup (x) ; or (A) = inf (x) : x2intK x2intK Theorem 3.14 Assume that the operator A is positively 1-homogeneous compact upper semicontinuous with closed convex values. In addition, let A be (2) increasing and r (A) > 0: Then A admits a positive eigen-pair ( 0 ; x0 ) with 0 r (A) : Moreover, 1) if A is (1) increasing then a) r (A) 0 r (A) if A is strongly u0 positive. b) x0 2intK and r (A) 0 or (A) if A is semi strong positive. 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 17
hg; x yi = 0 and hg; ui > 0 for all u 2 A (x) A (y) De…nition 3.13 1) Let ( 0 ; x0 ) be a positive eigen-pair of A. Then 0 is said to be geometrically simple if from 0 x 2 A (x) with x 2 K it follows that x 2 hx0 i+ : 2) We say that the positive eigen-pair ( 0 ; x0 ) of the operator A is unique if for any positive eigen-pair ( ; x) of A one has = 0 and x 2 hx0 i+ : Theorem 3.15 Let A : K!2K n f?g be a positively 1-homogeneous, u0 positive, u0 increasing operator and ( 0 ; x0 ) be a positive eigen-pair of A: Then 1) 0 is geometrically simple. 2) If A is (3) increasing then ( 0 ; x0 ) is unique. Theorem 3.16 Let intK 6= ?, A : K ! 2K n f?g be a positively 1-homogeneous operator and ( 0 ; x0 ) be a positive eigen-pair of A: In addition, A is semi strongly increasing. Then 1) 0 is geometrically simple, furthemore, if ( 1 ; x1 ) is a eigen-pair of A then 1 = 0 or x1 2 hx0 i+ . 2) if A is (3)-increasing the ( 0 ; x0 ) is unique. CONCLUSION We have presented the results of research in two main directions in this thesis. In the former, 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. 18
6. Extending the concepts of u0 -positive, u0 -increasing, strongly positive for multivalued operator; proving of some Krein-Rutman’s properties of positive eigen-pair for multivalued operator in ordered space. The next research directions 1. Searching for a …xed point theorem of Krasnoselskii in cone-normed space that is strong enough to be able to apply to the Cauchy problem in Banach spaces with singularities of Ovcjannikov. 2. Learning how to apply the derivatives of multi-valued mappings to extend Rabinowitz- Dancer’s theorem of branching solution set into multivalued equations. 3. Searching for conditions that are not too restrictive for mappings to obtain the unique of the positive eigen-pair and maximum eigenvalue respectively. 19