BOUNDARY VALUE PROBLEMS. AND I PERIODIC SOLUTIONS FOR CONTINGENT DIFFERENTIAL EQUATIONS r Thesis for the Degree Of Ph. D. MICHIGAN STATE ONIVERSITY WEI-HWA SHAW 1973 IIIIIIIII IIIIIIII IIIIIII ” 'fiflfll-‘xW-‘flvu‘ .14" v-9... "u. 0 a“ .‘I ‘ I , ' . LIB R21 R. 2 , 3 129 0317 I . 4 ." . n " ‘ p E I Michigan 5‘3“: 1 ° . 't .: w Um» crsx y __ This is to certify that the thesis entitled BOUNDARY VALUE PROBLEMS AND PERIODIC SOLUTIONS FOR CONTINGENT DIFFERENTIAL EQUATIONS presented by Wei -Hwa Shaw has been accepted towards fulfillment of the requirements for Ph.D. degree inMathematics QWIIM /mfimm Date 6-5‘73 0—7639 Bmomc av I“ ‘L "In“ HOAB&SUN8’ ._.. ‘flm,_ “‘— ABS TRACT BOUNDARY VALUE PROBLEMS AND PERIODIC SOLUTIONS FOR CONTINGENT DIFFERENTIAL EQUATIONS BY Wei-Hwa Shaw In this thesis, we shall investigate contingent differential equations in which the orientor fields F(t,x) satisfy the Carathéodory conditions, i.e. F(t,x) is measurable in t for each fixed x, F(t,x) is upper semi-continuous in x for each fixed t and F(t,x) is integrably bounded on every compact subset of Ran. we begin our investigation with the fundamental theory of such equations. Two similar Kamke-type convergence theorems are proved. Following from the convergence theorems are the properties of continuous dependence of solutions on initial conditions and parameters. We then study the general boundary value prOblems of contingent differential equations. An existence theorem like Fredholm's alternative is proved by using a fixed point theorem which we formulate with degree theory. As the boundary conditions require only linearity and continuity, applications Wei-Hwa Shaw can be obtained on periodic solutions, Nicoletti prdblems and aperiodic boundary value problems. We Observe also that the set of solutions is compact in the space of conti- nuous functions. Therefore, Optiomal solutions do exist with respect to any continuous (or semi-continuous) functionals. In case the orientor fields are functional and T- periodic fOr some T > O. we have contingent equations in which a finite time lag r 2:0 is involved. A T-periodic set-valued transformation is set up from the space 6% of continuous T-periodic functions into the space of non-empty. compact and convex subsets of 6% so that the previous fixed point theorem can be applied. Thus we Obtain an existence theorem of periodic solutions for contingent functional differential equations. BOUNDARY VALUE PROBLEMS AND PERIODIC SOLUTIONS FOR CONTINGENT DIFFERENTIAL EQUATIONS BY Wei-Hwa Shaw A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1973 DEDICATED TO MY MOTHER AND IN THE MEMORY OF MY FATHER ii ACKNOWLEDGEMENTS My sincere gratitude and appreciation are due to my thesis advisor, Professor Jerry D. Schuur, for his patient guidance, enthusiastic encouragement, kind criticisms and numerous discussions during the preparation of this thesis. I am deeply indebted to Professor Shui-Nee Chow not only because he taught me differential equations and gave me many valuable suggestions in this research, but also because I owe him my interest in differential equations. My thanks also go to Professor T. Yoshizawa for the many conversations we had during his visit here in 1972-73. Last but not least, I wish to express my hearty thanks to Professor wei Eihn Kuan for his friendship and encouragement throughout my graduate career and especially his sustaining mental support in those dark and lonely days when this thesis was born. iii Introduction . Chapter I: §1O §2. TABLE OF CONTENTS mappings . §3. Chapter II: §10 §2. §3. Chapter III: §1. §2. §3. Chapter IV: Bibliography . GENERAL BOUNDARY VALUE PROBLEMS FOR CONTINGENT DIFFERENTIAL EQUATIONS A fixed point theorem Some applications iv SET-VALUED MAPPINGS Set-valued mappings and upper semi-continuity Convergence properties of solutions CONTINGENT DIFFERENTIAL EQUATIONS Existence and continuation of solutions Measurability and integrals of set-valued TOpological degree of set-valued mappings Continuous dependence of solutions on initial conditions and parameters An existence theorem like Fredholm's alternative PERIODIC SOLUTIONS OF CONTINGENT FUNCTIONAL EQUATIONS 23 28 33 4O 57 63 7O 81 90 105 INTRODUCTION A relation of the form (E) 22m = f 0, there exists a 5 > 0 such that F(x) c: B€(F(x0)) for all x e A ‘with Ix-xOI < 5, where B€(D) = Ix 6 E 3 d(X,D) < 6}- F is upper semi—continuous in the sense of metric if F is upper semi-continious at every x E A in the sense of metric. Definition 1.5. A mapping F : A 4‘9(E) is said to be upper semi-continuous at x0 6 A in the sense of .. n n' . Kuratowski 1f xn 4.x yn 4 yO and yn e F(xn) imply 0' Y0 E F(xo). F is ppper semi—continuous in the sense of Kuratowski if F is upper semi-continuous at every x e A in the sense of Kuratowski. 10 Remark 1.3. It is easy to see that F is upper semi-continuous in the sense of Definition 1.5 if and only if the graph of F, INF) = {(x,y) : y e F(x)} CiAxE, is closed in AxE. Definition 1.6. A mapping F : A 4w0(E) is said to be upper semi-continuous at x0 6 A in the sense of tppology if for each open set U 3 F(xo). there exists a 5 > 0 such that Ix-xol < 6 implies F(x) c‘U. F is upper semi-continuous in the sense of topology if F is upper semi-continuous at every x 6 A in the sense of t0pology. Proposition 1.1. Let F : A 4 0(E). Assume that for x 6 A, there exists a neighborhood N(xo) of x 0 0 such that L) F(x) is relatively compact, i.e. LI F(x) x€N(xo) x€N(xo) is compact. Then, F is upper semi-continuous at X0 in the sense of Definition 1.3 implies that F is upper semi- continuous at x in the sense of Definition 1.4. 0 Proof: Let c > 0 be given. Suppose F is not upper semi-continuous at x in the sense of Definition 1.4. 0 Then, fOr each 5 > 0, there exists x 6 A such that Ix-XOI < 5 but F(x) ¢IBe(F(xO)). Hence, there exists a m n , sequence {xn}n=1 CIA. such that xn 4x0 and there ex1sts 11 n Yn 6 F(Xn) Wlth d(Yn.F(xo)) Z 6- Since xn 4.xo, xn e N(xo) for all n sufficiently large. Hence, y 6 LI F(x) for all n sufficiently large. It n xeN (x0) follows from the relative compactness of L) F(x) that x€N (x0) Ian:;1 has at least a limit point, say yo. It is clear that yO 6 lim sup F(xn). Let {yj};;1 be a subsequence of {yn]:=1 such that yj 1 yo. It follows from the triangle inequality of d that d(yo.F(xo)) 2 |d(yj.F(xo))-d(yj.yo)1 for all j. Letting j 4 m, ‘we have d(yo,F(xo))'2 6. Hence, yo A F(xo). Therefore, F is not upper semi-continuous at X0 in the sense of Definition 1.3. [3 Example 1.1. The assumption that F be locally relatively compact at x is necessary in Proposition 1.1. 0 Let R be the space of real numbers with usual metric and let 1“ ‘be the space of all bounded sequences of real numbers with the norm of each element I§n1n=l' IIIsnIII = suplenl. Let A = {0.1,%.31-....} CR. n Consider F : A 4 I" defined by 0 for all k F(O) = {51¢}:=1 with 5k F(%) = I§k};;1 with 5k 5k Oif kin and 1 if k = n. 12 As we Observe that the unit sphere of to is not compact, F is not locally relatively compact at 0. F is upper semi-continuous at 0 in the sense of Definition 1.3 since 1i? sup F(%) = ¢'c F(O). However, d(F(%),F(0)) = 1 em for all n]; 1. F is not upper semi-continuous at 0 in the sense of Definition 1.4. Proposition 1.2. Let F :IA 4 0(E). If F(xo) is closed for some xo 6 A, then F is upper semi-continuous at X0 in the sense of Definition 1.4 implies that F is upper semi-continuous at x in the sense of Definition 1.3. O Proof: Let [xn}:;1 be any sequence in A such that n x 4 x . By Definition 1.4, for each s > 0, there exists n 0 an N > 0 such that F(xn) czB€(F(xo)) for all n 2_N. Let y 5 lim sup F(xn) be arbitrary, i.e. there exists a subsequence o a . 3' {xj}j=1 of [xnIn=l with yj e F(xj) such that yj 4 y. Thus y e Be(F(xo)). Since 6 > 0 is arbitrary and F(xo) is closed, y 6 F(xo)- C] Example 1.2. The assumption that F(xo) be closed is necessary in Pr0position 1.2. Consider F : R 4 9(R) defined by F(x) = (x-1,x+l) = [t e R : x-l < t < x+l}. Let x0 6 R be arbitrary. It is clear that F is upper semi-continuous at X0 in the sense of Definition 1.4 13 1 n (take 5 — e). Let xn — x0 + 5' Then xn 4Ix0. However, lim sup F(xn) = [xo-l,xo+1] ¢:(xo-1,xo+1) = F(xo). Proposition 1.3. Let F : A 4 0(a). If F(xo) is closed for some x0 6 A, then F is upper semi-continuous at x in the sense of Definition 1.4 implies that F is 0 upper semi-continuous at X0 in the sense of Definition 1.5. n n Proof: Let xn 4Ix0. yn 4’yo By Definition 1.4, for each 6 > 0, there exists an N1(e) > 0 and yn 6 F(xn)- such that F(xn) c Be(F(xo)) for all n 2_N1. Since n . yn 4 yo, there eXists an N2(e) > 0 such that d(yo,yn) < e/2 for all n 2_N Take N = max[N1,N2]. Then 2. d(yo.F(xO)) g_d(y1oyn) + dtynaF(x)) < 6/2 + 6/2 = e for all n 2.N. Hence, yo 6 F7§37'= F(xo) since F(xo) is closed. U Example 1.3. Proposition 1.3 is not true without the assumption that F(xo) ‘be closed. Consider F : R 4 9(R) defined by F(x) = {-1} if x < 0 {1} if x > 0 14 F(O) is not closed. It is easy to see that F is upper semi-continuous at 0 in the sense of Definition 1.4 but not in the sense of Definition 1.5. Proposition 1.4. Let F : A 4 OCE). Assume that F is locally relatively compact at some xo 6 A. Then F is upper semi-continuous at X0 in the sense of Definition 1.5 implies that F is upper semi-continuous at x0 in the sense of Definition 1.4. Proof: Suppose F is not upper semi-continuous at X0 in the sense of Definition 1.4. Let N(xo) be a neighborhood of xO such that L) F(x) is compact. x€N(xO) Then there exists an e > 0 and a sequence [xn}:;1 c:N(xo) n such that xn 4 x and F(xn) ¢:B€(F(xo)). Hence, there 0 exists a sequence [yn]:=1 such that yn E F(xn) and n=1 18 d(yn,F(xo))I2 6. Since {xn}:;l c:N(x0)a {Yn} contained in a compact subset of E by our assumption on N(x). Therefore, there exists a subsequence {yna}:.=1 of a n' {Yn}n=l such that yn. 4 YO for some yO G E. Considering the inequality d(yo,F(xO)) _>_ [£(yn..f(xo))-d(yn~yoll and letting n’ 4 a, 'we have d(yO,F(xo))‘2 e > 0. Hence Y0 t F(xo). Therefore, F is not upper semi—continuous at X0 in the sense of Definition 1.5. D 15 Example 1.4. Pr0position 1.4 is not true without the assumption that F be locally relatively compact at xO even if F is single-valued. Consider the following function F : R 4 R defined by F(x)=}t if x¥0 0 if x = 0. Evidently, F is upper semi-continuous in the sense of Definition 1.5 since its graph IXF) is closed in R2. However, F is not upper semi-continuous at 0 in the sense of Definition 1.4. Prpposition 1.5. Let F : A 4 9(E) and x0 6 A be arbitrary. F is upper semi-continuous at X0 in the sense of Definition 1.6 implies that F is upper semi- continuous at x in the sense of Definition 1.4. 0 Proof: For each x0 6 A, B€(F(xo)) is an open set containing F(xo). Definition 1.4 follows immediately from Definition 1.6. [3 Proposition 1.6. Let F : A 4 QCE). Assume that F(xo) is compact for some xo 6 A. Then, F is upper semi-continuous at X0 in the sense of Definition 1.4 implies that F is upper semi-continuous at x in the sense of 0 Definition 1.6. 16 Proof: Let U be an open set containing F(xo). . 1 ' For each y 6 F(xo) et Br(y)(y) be an open ball in E centered at y 'with radius r(Y) and Br(y)(y) c‘U. since F(xo) is compact, the covering IBrI 2 (y) : y e F(xo)} 2 has a finite subcovering, say {Br1 (yl), B (y2),...,Br (yn)I. _ :2. _r1 2 2 2 Let r = minIrl,...,rn}. we claim that Br(F(xo)) c U. Let 2 I y e F(xo) be arbitrary and y’ e B (y). n the finite P- NIH subcovering, there exist yi, i g_ 3.“! such that Y E Bri(yi)‘ Hence 2 d(Y'oYi) Sd(Y’OY) + d(Y:Yi) r. <12”? S.ri- Therefore, y' 6 Br (Yi) c U. Since y' is arbitrary in i Br(y), Br(y) c‘U. Also, y e F(xo) is arbitrary. It 2 2 follows that Br(F(xO)) c‘U as desired. By Definition 1.4, 2 there exists a a = 5(g) > 0 such that F(x) (2 BE(F(XO)) C U 2 for any x e A with Ix—xOI < 5. Hence, F is upper semi- continuous at X0 in the sense of Definition 1.6. I] compact if PIX in the transle for eve Then L take x Hence, 0f DefiI Definitj 17 Example 1.5. Without the assumption of the compactness of F(xo), Pr0position 1.6 is not true even if F(xo) is closed. Consider F : R .. 9(112) defined by F(x) = I- o). It is evident that F is upper semi-continuous in the sense of Definition 1.4 since F(xl) is only a translation of F(x2) for any x1,x2 e R. F(x) is closed for every x e R. Let x0 6 R be arbitrary. Define U = {(y.z) : y > x0. (y-XOIZ >0}. Then U is Open in R2 and F(xo) c.U. For each 5 > 0, take x such that 0 < x -x < 6. we find that O 1 1 l 1 (X + -(x -X). ) = (-(X +X). ---—-) 6 F(X)\~U. 2 0 x + %(xo-x)-x 2 0 %(xo-x) Hence, F is not upper semi-continuous at x0 in the sense of Definition 1.6. The following simple example also shows that Definition 1.6 does not imply Definition 1.5 in general: Example 1.6. Consider F : R 4 9(R) defined by F(x) = (0,1) for all x e R. 18 From these examples, we see that the four definitions of upper semi-continuity are quite different. However, from the previous propositions, we find also that under certain conditions they are indeed equivalent. Theorem 1.1. Let A be an Open subset of a metric space and E be a Banach space. Suppose that F : A 4 0(E) is a set-valued mapping satisfying: (i) F(x) is compact for every x e A: and (ii) for each bounded subset D :IA, F(D) is relatively compact. Then, Definitions 1.3-1.6 of upper semi-continuity are all equivalent. From now on, the set-valued mappings that we shall consider in our contingent equations satisfy the hypotheses of Theorem 1.1. Therefore, when we say upper semi-continuity, we mean any of the Definitions 1.3-1.6 with no ambiguilty. Remark 1.3. Suppose we restrict 0(E) to Comp (E) and consider mappings from A into Comp (E). It follows from PrOpositions 1.2, 1.3 and 1.6 that Definition 1.4 is the strongest among all. For the sake of completeness, we introduce the following definitions of lower semi-continuity. 19 Definition 1.7. A mapping F : A 4 Comp (E) is said to be lower semi-continuous at x0 5 A in the sense of limit inferior if F(xo) c lim inf F(xn) n4m a: n for any sequence IxnIn=1 c A such that xn I‘XO‘ F is lower semi-continuous in the sense of limit inferior if F is lower semi-continuous at every x e A in the sense of limit inferior. Definition 1.8. A mapping F : A 4 Comp (E) is said to be lower semi-continuous at x0 6 A in the sense of metric if for each s > 0, there exists a 5 > 0 such that F(XO) : B€(F(X)) for all x E A with Ix-x 5. 0| \ F is lower semi-continuous in the sense of metric if F is lower semi-continuous at every x 6 A in the sense of metric. Remark 1.4. In view of Remark 1.1, it is easy to see that Definitions 1.7 and 1.8 are equivalent. Definition 1.9. A mapping F : A 4 Comp (E) is said to be continuous at xO e A in the sense of metric if F is both upper and lower semi-continuous at X0 in the sense of metric. 20 F is continuous in the sense of metric if F is continuous at every x e A in the sense of metric. Proposition 1.7. (Hukuhara [13]). Let E be any Banach space. The operations +,-, and n have the following prOperties of continuity with respect to the product tapology of the Hausdorff topologies induced by the Hausdorff metrics of R and Comp (E): (i) the addition +: Comp (E) x Comp (E) 4 Comp (E) defined by A + B = {atb : a 6 A and b e B] is continuous: (ii) the scalar multiplication -: R x Comp (E) 4 Comp (E) defined by GA = Iaa : a e A} is continuous; (iii) the intersection 0: Comp (E) x Comp (E) 4 Comp (E) defined by A n B = [c : c e A and c e B] is upper semi-continuous. Let A be a bounded subset of a Banach space E. The norm of A, IA], is defined to be IA] = sup[|xI : x e A). Proposition 1.8. Let F be a mapping from a subset A of a metric space X into Comp (E) and let D be a 21 closed, bounded and convex subset of the Banach space E such that 0 e D. If F is upper semi-continuous in the sense of Definition 1.4, then the mapping G from A into the space of non-empty, closed, bounded and convex subsets of. E defined by G(x) = IF(x) [D for all x e A is also upper semi-continuous in the sense of Definition 1.4. Proof: Let F be upper semi-continuous in the sense of Definition 1.4 and x0 6 A be arbitrary. Since D is bounded, there exists a positive integer n such that %IDI < 1. Given 6 > 0, by Definition 1.4, there exists a 5 > 0 such that F(x) c.B€(F(xo)) for all x with n Ix-xOI < 6. For any x e D, E) . LFU‘IL x n ([11on + 5) [F(x) Ix = (IF(xO)I + I 6 since D is convex and contains 0. Hence, we have e IF(x) [D c: (|F(xO)I + fi)D c [F(xo) [D + 36(0) = Be(IF(xO)ID). Therefore, G is upper semi-continuous in the sense of Definition 1.4. C] 22 Let X,X’, and X” be complete metric spaces, F be a mapping from X into Comp X’ and F” be a mapping from Comp X’ into Comp X”. The composite function G, denoted by G = F'F, is a mapping from X into Comp X” defined by G(x) = F’(F(x)) for each x 6 X. In the following, we shall consider the continuity properties of composite functions. But first, Definition 1.10. Let X and X’ be two complete metric spaces. A mapping F : Comp (X) 4 Comp (X’) is said to be increasing if for any A, B 6 Comp (X), A c:B implies F(A) c: F(B) . Example 1.7. Let E be a Banach space. The mappings F and G from Comp (E) x Comp (E) into Comp (E) defined by F(A,B) AA + UB where A,u E R and G(A,B) A n B are all increasing. Proposition 1.9. (Hukuhara [13]). If F and F’ are upper semi-continuous (resp. lower semi-continuous) and moreover, if F’ is increasing, then the composite function G = F'F is also upper semi-continuous (resp. lower semi-continuous)- be CC m. be fc in 23 In the above prOposition, the assumption that F’ be increasing is superfluous if F is continuous. Proposition 1.10. (Hukuhara [13]). If F is continuous and F' is upper semi—continuous (resp. lower semi-continuous), then the composite function G = F'F is upper semi-continuous (resp. lower semi-continuous). §2. Measurability and integrals of set-valued mappings: The set-valued mappings that we shall consider in this section are mappings from a subset A of RI“ into Comp (Rn). It is known that for any finite dimensional Euclidean space Rm we can define a Lebesgue measure on Rm (see [33]: pp.49-53). Then with no difficulty, one can generalize the measurability of a single-valued mapping f from A into Rn to a set-valued mapping F from A into Comp (Rn). Definition 1.11. A set-valued mapping F from a measurable subset A of Rm into Comp (Rn) is said to be measurable if the set [x e A : F(x) c B] is measurable for every closed subset B of Rn. Proposition 1.11. (Hukuhara [14]). If F : A CiRm 4 Comp (Rn) is upper (resp. lower) semi-continuous in the sense of Definition 1.4 (resp. Definition 1.8), then F is measurable. 24 The following characterization of measurability of set-valued mappings is due to Plis: Theorem 1.2. (Plis [30]). Let F. be a mapping from a bounded measurable subset A of Rm into Comp (Rn). In order that F be measurable, it is necessary and sufficient that for each s > 0, there exists a closed subset A' of A such that F is continuous on A” and the measure of A‘\A’ is less than c. The measurability of set-valued mappings can also be defined in the following way: Let (5.2% u) be a finite and positive measure space. And let d denote the Hausdorff metric on Comp (E). Definition 1.12. A set-valued mapping F : S 4 Comp (Rn) is called ursimple if it assumes only a finite number of values K1,K2,...,Kr 6 Comp (Rn) and each of them on a u-measurable set. Definition 1.13. A set-valued mapping F : S 4 Comp (Rn) is called nemeasurable if and only if there exists a sequence Fk of ursimple functions converging in nemeasure to F: that is P(Ek.F) = inf(a + qus e S :d(Ek(s),F(s)) > OI). a>0 * I 'where u (D) = inf{u(E) E e Z, and D c:E}, converges to O as R400. 25 Let g = [x1.x2,...,xn} be an orthonormal basis of R“. If x e Rn, then (al,a2,...,an) denotes the coordinates of x with respect to the basis 5; that is X = (11x1 + a2x2 +...+ anxn. Definition 1.14. Let K 6 Comp (Rn). A point x0 6 K is called the lexicographic maximum of K if _ o o 0 x0 - (a1,a2,...,an) such that (i) a? = maxIa1 : x = (al,...,an) E K] .. O (11) Gk = maxIak : x = (a1....,an) G K and a - do for i < K} i - i ' We shall use e(K,§) to denote the lexicographic maximum of K ‘with respect to the basis 5. Clearly, for any compact K, e(K,§) 6 K. The following selection theorem is due to Olech: Theorem 1.3. (01ech [29]). Let (8,2, u) be a finite measure space. If F : S 4 Comp (Rn) is urmeasurable, then the mapping e(F(s),§) of S into Rn is urmeasurable in s for each fixed orthonormal basis 5 of R“. Let T be a Lebesgue measurable subset of R and F be a set-valued mapping from T into 0(Rn). we use I to denote the family of all point-valued mappings f 26 from T into Rn such that f is Lebesgue measurable over T and f(t) e F(t) for every t e T. Then the following definitions of the integral of F is a natural generalization of the integral of a point-valued mapping: Definition 1.15. Let F : T 4 0(Rn), we define the set-valued integral of F over T, (S)IE F(t)dt, by (3)]‘T F(t)dt = []‘T f(t)dt : f e .7}. The following are some fundamental theorems which will be useful in our later develoPment: Theorem 1.4. Let F be a Lebesgue measurable function from a measurable subset T CIR into Comp (R?) such that the measure of T is finite and F is integrably bounded: i.e. there exists a point-valued function f which is integrable over T and IF(t)I g_f(t) for all t e T. Then, (SIT F(t)dt :1 (5. Proof: In order to get a measurable selection of F, it suffices to show that Definition 1.11 implies Definition 1.13 when u is the Lebesgue measure m defined on T so that we can apply Theorem 1.3. 27 Let F be Lebesgue measurable. Given any positive integer k > 0, by Theorem 1.2, there is a closed subset T’ = T(k) c T such that F is Hausdorff continuous on T' and m(T‘\T') < §%a As m(T) < a, there exist a,b e R such that T c [a,b]. Since T’ is compact, F is uni— formly continuous on T'. Let a = a0 < a1 <...< am = b be a subdivision of [a,b] with aj - aj_1 = % (b-a) for j: 1,...,m. Let Ij = [aj'aj-l], Tj = T n I]. and Tj = T’ n Ij for j = 1,...,m. we choose m so large such 1 . s . that d(F(t1), F(t2 )) <-1E 1f t1,t2 e Tj for some 3. Define Ek : T 4 Comp (Rn) by Ek(t) = F(tj ) for all t e Tj if there exists a t. T. J 6 j {o} for all t e T). if T; =.¢ It is clear that Ek is a simple Lebesgue measurable function and PIFk.F) = inf(a + m [t e ‘I‘ = d(Fk(t), F(t)) > a}) a>0 55-}, +m {t 6T = dwk (t) F(t)) >2k} 1 1 __1 Sm+m-i which converges to 0 as ‘k 4 m. Hence, F is Lebesgue measurable in the sense of Definition 1.13. [3 Theorem 1.5. (Aumann [ 1]). Let F : T 4 9(Rn) such that F is integrably bounded and F(t) is closed for all t e T. Then, (S)[T F(t)dt is compact. 28 The following theorem is an analogue of Fatou's lemma : Theorem 1.6. (Aumann [l ]). If [FkI;;1 is a sequence of set-valued functions that are all defined and bounded by the same integrable point-valued function h on a measurable set T CiR, then (8) lim sup F .3 lim sup F . IT k 409 k R 4a JIT k Theorem 1.7. (Hukuhara [14]). Let F be a measurable set-valued function defined on a set T c:R with m(T) < a. If T = T1 U T2 such that T1 and T2 are disjoint and measurable, then (s)j§ F(t)dt = (3)]Tl F(t)dt + (s)]§2 F(t)dt. §3. Topological degree of set-valued mappings: One of the important theories in non-linear analysis is that of the degree of a mapping as developed by Leray and Schauder in 1934 (see [20]). Their work not only generalized the Brouwer degree to a certain class of mappings in Banach space but also made it possible toformulate more powerful fixed point theorems. As our main interest is the set—valued mappings, we shall omit the lengthy develOpment of the degree theory of point-valued mappings which was first defined on finite dimensional linear spaces and then extended to normed and locally convex linear spaces. An extensive treatment of this subject may be found in [32],[26],[27], and [19]. I 29 The extension of the topological degree from point-valued mappings to set-valued mappings has been established also by many mathematicians, first by Granas []Lfl, then by Hmkuhara [l3] and recently by Cellina and Lasota [ 4]. Each of their approaches is different from the others. The way we present here follows the approach of Granas since it is convenient for our future purposes. For simplicity, our tepological degree will be defined for a class of set-valued mappings in a Banach space E with domains solid spheres in E. The following notations are needed: Ea : an arbitrary Banach space. Pa = Ea\[0]. V (xo,p) = [x e Ea: Ix—xOI < p} where x e E a 0 a and p > 0. Sa_1(xo,p) = [x 6 Ed : Ix-XOI = p] where x0 6 Ed and p > 0. cf(Ea) = the collection of all non—empty closed convex subsets of Ed Definition 1.16. A mapping e : A c:Ea 4 cf(Ea) is said to be compact if for each subset D of A, §(D) is relatively compact in Ea’ i.e. 'ETEY is compact. Q is said to be completely continuous on A if Q is compact and upper semi-continuous in the sense of Definition 1.5 on A. f1 30 Definition 1.17. A mapping cp : A c Ea 4 cf(Ea) is said to be a completely‘continuous multi-valued vector jiglg_on A if it can be expressed in the form m(x) = x - §(x) for all x 6 A, where G(x) is completely continuous on A. Definition 1.18. we say that a completely continuous multi-valued field ¢(x) = x - 5(x), x e A, does not vanish and denote it by T : A 4»cf(Pa) if the point 0 does not belong to the set qflx) for any x e A. Let Sa-l = Sa_l(xo,p). Va = Vd(xo,p) and f,¢ : Sa-l 4.cf(Pa) be completely continuous (resp. single and multi-valued) vector fields. f is called a selection of m if f(x) 6 m(x) for each x e Sa-l' By [32], for every such f, an integer v(f,Sa_1) is defined and is called the characteristic of f on Sa-l' The following theorems are all due to Granas: Theorem 1.8. (Granas [10]). To every non-vanishing completely continuous multi-valued vector field m : Sa-l 4»cf(Pa) we can assign an integer v(¢,sa_1) called the characteristic of the field w on S such that a-l V(Cp: Sa_1) = VIfI SCI-1) for every selection f of ¢° 31 Definition 1.19. We say that two non-vanishing completely continuous multi-valued fields ¢fi'¢2 : A 4 cf(Pd), ¢u(x) = x - §1(x), ¢2(X) = x - 92(x) are homotopic and denoted by $1 a $2: if there exists an upper semi- continuous (in the sense of Definition 1.5) function y(x,t) : A x [0,1] 4Icf(Ea) such that the following conditions are satisfied: (i) the point 0 does not belong to any set ¢(x,t) = x - y(x,t) for all x e A and t 6 [0,1]: (ii) Y(Xo0) = §1(XI. Y(X.1) = 02(X) for all x e A: (iii) the set v(A,[0,l]) is relatively compact in E . a Theorem 1.9. (Granas [1£fl). If two non-vanishing completely continuous multi-valued vector fields e1,¢2 : Sa-l 4 cf(Pa) are homotopic, $1 a $2: then their characteristics are equal, i.e. ”(ml’Sa-l) = V(¢2'Sa-l)' Theorem 1.10. (Fixed point prOperty, Granas [10]). Let qfix) = x - §(x) be a completely continuous multi-valued vector field m : Va 4 Cf(Ea) defined on a full sphere Va = V;T§S:§) into cf(Ea). If the restriction of ¢ on Sa-l does not vanish, 32 mb = ¢ISG_1 : Sa-l 4»cf(Ra) and v(¢b,Sa-1) #'0, then there exists a point x e Vd such that 0 e ¢Cx), i.e. there exists x 6 Va such that x e 6(x). Although we do not include them here, there are many important fixed point theorems, e.g. the Kakutani- Ky Fan theorem (see [15] and [ 7]). that can also be Obtained without much difficulty from the view of topological degree. we shall conclude this chapter with the following extension of the well4known theorem of Borsuk on antipodes (see [ 3]): Theorem 1.11. (Granas [11]). If a non-vanishing completely continuous multi-valued vector field T : Sa-l 4Icf(Pa). defined on a sphere Sa-l c:Ea, is odd, that is m(x) = -¢(-x) for all x e Sa-l' then its characteristic v(¢,S1_l) is odd. Chapter II CONTINGENT DIFFERENTIAL EQUATIONS In an ordinary differential equation, the tangent at each point is prescribed by a point-valued function. This gives a vector field. In a contingent differential equation, the tangent is prescribed by a set-valued function. This direction field is usually called an orientor field. This more general class of equations was developed independently by A. Marchard and S.K. Zaremba in the mid 30's and has then been intensively investigated by many other mathematicians. In this chapter, we shall study the fundamental theory of such differential systems. §l. Existence and continuation of solutions: Definition 2.1. Let I be an interval in R. A mapping F from Ian into Comp (Rn) is called an orientor field. Definition 2.2. Let x(t) be a function from an interval I c:R into R“. For each to E I, let x(t)-x(to) F(t) = x-t0 33 34 The set D*x(to) defined by D*x(t ) - [c 6 Rn - there exists a s uenc [ I“ c I o ‘ ' eq 9 tk k=1 k such that tk #'to, tk 4 to and k F(tk) 4 c} is called the contingent derivative of x(t) at to. Definition 2.3. Let x(t) : I 4 R“ and F(t,x) : Ian 4 Comp (Rn). The relation * (C) D x(t) c F(t,x(t)) is called a contingent differential equation. Let J' be an interval in R and D c Ian. We shall use the following notations: Projl D = [t E I : there exists an x e R such that (t,x) e D} Proj2 D = [x e Rn : there exists a t E R such that (t,x) e D] L1(J) = the collection of Lebesgue integrable functions from J into RP C(J) = the collection of continuous functions from J into Rn AC(J) = the collection of absolutely continuous functions from J into Rn CC(Rn) = the collection of non-empty, compact and convex subsets of Rn. 35 For each x(t) E AC(J), 'we denote by x(t) the usual derivative of x(t). The abbreviation a.e. J means almost everywhere in J. Definition 2.4. A function x = x(t) defined on an interval J c I into Rn will be called a solution of (C) in the sense of Marchand if X(t) e C(J) and D*x(t) c:F(t,x(t)) a.e. J. Definition 2.5. A function x = x(t) defined on an interval J CII into Rn ‘will be called a solution of (C) in the sense of waéewski if x(t) e AC(J) and x(t) €F(t,x(t)) a.e. J. Definition 2.6. A function x = x(t) defined on an interval J c.I into Rn will be called a solution of (C) if x(t) eAC(J) and t x(t) e x(to) + (s)jt F(s,x(s))ds o for each t,tO 6 J. The contingent differential equation that we shall study in this thesis are equations in which the orientor fields are of the type defined as follow: 36 Definition gpl, We shall say that an orientor field F(t,x) from Ian into chRn) satisfies the Caratheodory conditions provided (i) F(t,x) is measurable in t for each fixed x e Rn: (ii) F(t,x) is upper semi-continuous (in the sense of Definition 1.4) in X for each fixed t e I: and (iii) for each compact subset D of Ian, there exists a function m(t) = mD(t) ‘which is integrable over Projl. D such that [F(t,x) | g m(t) for all (t,x) 6 D. A contingent differential equation is said to be of Carathéodory type if its orientor field satisfies the Carathéodory conditions. Proposition 2.1. (Plis [31]). Let F(t,x) be an orientor field from Ian into cc(Rp) such that F satisfies conditions (i) and (ii) in Definition 2.7. Then, there exists a orientor field H(t,x) such that (i) H(t,x) c:F(t,x) for every (t,x) e m“; (ii) H(t,x) is upper semi—continuous (in the sense of Definition 1.4) in x for each fixed t e I: 37 (iii) H(t,x(t)) is measurable in t for any measurable function x(t) from I into Rn: (iv) H(t,x) is measurable in (t,x): (v) there exists a countable dense subset B of Rn such that H(t.XI = F(t.X) for (t,x) 6 1x8. Remark 2.1. For an orientor field F(t,x) satisfying conditions (i) and (ii) of Definition 2.7, it may happen that F(t,x(t)) is not measurable fOr a measurable function x(t) as the following example shows. This example shows an error in Pr0position 5 of [17] by Kikuchi and also shows the need for a theorem of the Plis type. Example 2.1. Let S be a non-measurable subset of [0,1]. Define F(t,x) : [0,1] x [0,1] 4 cc(R) by F(t,x) = [1,2] if t = x e S [0.1] if t = x z’S [1] if t #'x. It is clear that F(t,x) is upper semi-continuous (hence measurable) in t for each fixed x and F(t,x) is upper semi-continuous in x for each fixed t. Moreover, F(t,x) is measurable in (t,x) since the set [(t,x) : t = x 6 [0,1]} is a set of measure 0 in [0,1] x [0,1]. 38 Let x(t) : [0,1] 4 [0,1] defined by x(t) = t which is continuous (hence measurable). However, It : F(t,x(t)) c [1,2]) = S is non-measurable. Remark 2.2. If H(t,x) c F(t,x), then every trajectory of H(t,x) is a trajectory of F(t,x). Remark 2.3. Let F(t,x) be an orientor field satisfying the Carathéodory conditions. It follows from Theorem 1.4 and (iii) of Pr0position 2.1 that (5)] F(t.x(t))dt 7‘ I2! I for any measurable function x(t) from I into Rn with m(I) < a. The following prOposition will allow us to express a contingent differential equation of Carathéodory type in the ordinary differential form as well as the integral form: Proposition 2.2. Let F(t,x) be an orientor field from Ian into cc(Rn) such that F satisfies the Carathéodory conditions. Then, Definitions 2.4-2.6 are all equivalent. Proof: The equivalence of Definition 2.5 and 2.6 is evident. It is clear that Definition 2.5 implies Definition 2.4. When F(t,x) 6 cc(Rn). Definition 2.4 implies Definition 2.5 which is due to wazewski (see [37]). [j 39 When a contingent differential system is defined, the immediate questions that one may ask are: (i) When does a solution exist? (ii) How can a solution be continued? (iii) Does the family of solutions have certain properties of convergence and continuous dependence upon initial conditions? (iv) What can we say about the family of solutions emanating from a given initial point? The next theorem which is due to Plié gives a complete answer to the first two questions. Definition 2.8. Let S be an open subset of Ran and F(t,x) be an orientor field from S into cc(Rn). A solution x(t) defined on an open interval I is called a non—continuable solution of (C) if lim (tn,x(tn)) e S implies lim t e I. In this case, I is caII;d a maximal interval :fmexistence of x(t). Remark 2.4. (i) From the existence theorem of Plis, ‘we can see that a maximal interval I exists and is unique and we shall denote it by I = (w-,m+). (ii) If (with is the maximal interval of existence of a solution, then (t,x(t)) tends to the boundary as of S as t 4 w— or w+. To say (t,x(t)) tends to as 40 as t 4 w+ (resp. t 4 w-). 'we mean that either m+== a (resp. w- = -m) or for any compact subset D of S, (t,x(t)) ('D 'when t is sufficiently close to w+ (resp. w-). Theorem 2.1. (Plis [31]). Let F(t,x) be an orientor field from an open subset S c Ran into cc(Rn) such that F(t,x) satisfies the Carathéodory conditions. Then, for each point (to,xo) E S, there exists at least one non- continuable solution x(t) of (C) such that x(to) = x0. Remark 2.5. Actually, Plis proves this theorem under weaker assumptions. Instead of conditions (i) and (ii) of Definition 2.7, he only requires that F(t,x) is upper semi-continuous for almost all t and that it contains an orientor field G(t,x) which is densely measurable in t on S. §2. Convergence prOperties of solutions: The convergence property is one of the most important prOperties in differential equations. As it was shown by Strauss and Yorke (see [34]), much of the fundamental theory in ordinary differential equations follows directly from a convergence theorem. Kamke has given a convergence theorem (see [12]) in which the vector fields are assumed to be continuous. In [34], this theorem was proved with vector 41 fields of Caratheodory type. For contingent differential equations, Zaremba has shown a convergence theorem (see [41]). And later, Bebernes and Schuur also established a Kamke- type convergence theorem of initial values for contingent differential equations in which the orientor fields are assumed to be upper semi-continuous (see [12]). In this section, we shall consider contingent differential equations of Carathéodory type with certain perturbation and investigate the convergence prOperties of their solutions. . . ¢ 5 Prop031tlon 2.3. Let [An]n=1 and [B ] _1 be two sequences of subsets of RI. Then, (i) if An c:Bn for all n = 1,2,..., then lim sup An c: lim sup Bn: n4m n40 0 o o m a (11) 1f [An]n=1 and [En]n=1 are bounded, then lim sup [An+Bn} c.1im sup AD + lim sup Bn' n4m n4c n4a Proof: (i) follows immediately from the definition of limit superior and (ii) follows from the Bolzanoéweierstrass prOperty of Rn. [3 For any A,B 6 Comp (Rn), the escape of A from B is defined and denoted by PIA.B) = SUP d(a.B) aéA where d(x,B) = inf |x4b|. It is Obvious that p is not beB symmetric but does satisfy the triangle inequality. 42 Definition 2.9. Let {F1}A€A orientor fields from Ian into Comp (Rn). we say be a family of {FIIAEA is an equi-upper-semi-continuous family at x e Rn if for each 6 > 0, there exists a 5 = 5(e,x0) : O 0 such that Ix-x / 5 implies oI‘ PIFl(t.X).Fx(t.XO)) < e for all t e I and for all 1 e A. If {FA}XEA is equi-upper-semi-continuous at every x 6 Rn, we say [F I 1 AEA is equi-upper-semi—continuous on Rn. Proposition 2.4. Suppose that . n n m (i) Fn(t,x) . IxR 4 Comp (R) such that [FnIn=1 is an equi-upper-semi-continuous family on Rn: (ii) there exists a Fo(t,x) : 1an 4 Comp (Rn) such that n p(Fn(t.x).Fo(t.XI) 4 O for all (t,x) 6 Ian: ... n n (iii) ¢h(t) . I 4 R such that ¢h(t) 4 ¢b(t) for all t e I. Then, 11242up Fn(t,¢h(t)) c.FO(t,go(t)) for all t e I. Proof: For any t e I, let y(t) 6 lim sup Fn(t'¢h(t)) n4m be an arbitrary element. By definition, there exists a subsequence yn (t) e Fn (t,¢h (t)) such that yn (t) 1 y(t). j j j j 43 c Q Now, consider the sequences [Fnj(t,q;k(t))}k=1 and j = 1.2,'°'. Since ek(t) 4 qo(t), it follows from the equi-upper-semi-continuity of IF that for any 6 p 0, I? n. =1 J J there exists a K = K(e,t) ) 0 such that k > K implies a (1) P(Fnj(tamk(t)). Fnj(towb(t))) < 5 j for j = l,2,'°°. It follows from yn (t) 4 y(t) and j yn (t) E Fn (t,cp:n (t)) that for any 6 > 0, there exists j j j a J = J(e,t) > 0 such that j > J implies (2) PIY.Fn (to Con (13) ) ) < E . j 3' 2 Now, for each s > 0 and t e I, we pick r > max {J,K). Then, by (l) and (2) we have d(y.F (t.qo(t)) = 9(y.F (t.oo(t))) r r n n < p(Yanr (tocpnr(t) )) + P(Fn (t.enr 0, choose N so large such that g < 5. _. 1 - Let xl — n1 where nl > N. Then le-XOI — Ix1| < 5 and t 1 1 F (t,X ) = T 2 —— = .- . n1 1 t +1 t2+1 5 Hence, p(F l _ a 1(t,x1), Fn1(t,0)) 2.§ — 3. Therefore, [Fn)n=1 n is not equi—upper-semi-continuous at 0. By a simple calculation, we have 1im sup FnIt.cpn(t)) = {é} ¢ Io] = F0(t.cp0(t)). n-OOO 45 Proposition 2.5. Suppose that (i) IFn}:el is a family of orientor fields from Ian into Comp (R9): (ii) there exists an orientor field Fo(t,x) from Ian into Comp (Rn) such that Fo(t,x) is upper semi-continuous (in the sense of Definition 1.4) in x for each fixed t E I: (iii) p(Fn(t,x),Fo(t,x)) 3.0 uniformly with respect to x: (iv) ¢h(t) : I 4 Rn such that ¢h(t) 3 ¢b(t) for each t 6 I. Then, 11:4:up Fn(t,¢h(t)) c Fo(t,¢b(t)) for any t e I. Proof: For any t e I, let y(t) 6 lim sup Fn(t'¢h(t)) n4m be arbitrary. By definition, there exists a subsequence {Fnj(t'¢hj(t))};=l 0f IFn(tomn(t))I:;l such that (1) p(y(t>.rn.(t.en.) lo. 3 3 Given 3 / 0, by (1), there exists an integer J = J(e,t) > 0 such that j > J implies (2) P(y(t).Fn (t.ch (t))) < § . J 3 By assumption (iii), there exists an integer N = N(e,t) > 0 such that n > N implies (3) PIFfi(t.X). F0(t.X)) < f for all x e Rn. NOW, let us choose k = k(e,t) SO large 46 that k > J and nk > N. By (2) and (3), we have d(y(t).FO(t.op,5((t)) = P(Y(t).FO(t.can(t))) P(Y(t).F (t. (t))) — "a. “A. + P(F (t. (t)).F (t. (t))) "k “in. 0 “*5. < f + g = 6. Hence, k (4) d(Y(t)'F°(t'¢ht(t))) 4 0. Applying (4) with assumptions (ii) and (iv), we have d(y(t) .Fo(t.cpo(t))) P(y(t) ,Fo(t,q)o(t) )) .<_ P(Y(t).Fo(t.cpr5<(t))) k + P(Fo(to¢hk(t)).Fo(t.¢b(t))) 4 o. It follows from the compactness of Fo(t,¢b(t)) that y e F0(t,¢b(t)) as desired. C] Remark 2.6. The previous example shows also that it is essential that the convergence in assumption (iii) of Proposition 2.5 must be unifOrm in x. In Example 2.2, l 1 take t—l and let xn — 5' Then, Fn(l,xn) — 5 for all n = l,2,°--. Therefore, if we choose 6 = %, it is clear 1 that we must take N > n so that p(FN(l,xn), FO(1,xn)) < 5 — e- The following two examples will show that PrOpositions 2.4 and 2.5 are generally two different sufficient conditions for the limit superior of a sequence of orientor fields to be contained in their limit function. 47 Example 2.3. Consider the orientor fields Fn(t,x) : RxR 4 cc(R) defined by _ - SE. _ Fn(t,x) - [sin 11' l] for n — 1,2,... and F0(t,x) = [0,1]. It is clear that F is upper semi—continuous for n = 0,1,2,... n and p(Fn(t,x), Fo(t,x)) 3 0 for all (t,x) E RxR. (i) (ii) IFn}:;1 is equi-upper-semi-continuous in x on R : For any (t,x) 6 RxR, since sine function is continuous, given 9 > 0, there exists a 5 = 5(e,(t,x)) >'0 such that Iy-x] < 5 implies [sin ty - sin tx] < e. For any positive integer n, Iy-x| < 5 implies IE - El < g g_5 which implies [sin %¥-- sin €§I < e. Therefore, for each s > 0 and x e R, we have p(Fn(t,y),Fn(t,x)) ~< e for all y with Iy-xI < 5 and n = l,2,°°°. p(Fn(t,x),Fb(t,x)) 3 0 is not uniform with respect to x : Given t e R, for every 3wn positive integer n, choose xn = 3? . We have II I-‘ p = p<[-1.1].[o.1]) Hence, the convergence is not uniform. Example 2.4. Consider the orientor fields Fn(t,x) : [0,1] x R 4 cc(Rn) defined by Fn(t,x) = [-1, sin nx + E sin x] for n = 1,2,... and F0(t,x) [-l,l]. 48 It is clear that Fn is continuous for all n = 0,1,2,'°°. (i) p(Fn(t,x),Fo(t,x)) g 0 uniformly with respect to x : Clearly, P(Fn(tox).Fo(t.X)) 3.; 3 0 is independent of x. (ii) [Fn}:=1 is not equi-upper-semi-continuous : Consider (t,0) 6 [0,1] x R. Let 0 < e < 1 be given. For any 5 > 0, choose n so large such that %%-< 5. we have 7T p(Fn(to§E’)an(t—00)) = p([-1, sin g + E sin g],[-1, sin 0 + 3 sin 0]) = p([—l,l + E], {-1.01) = 1 + E _>_ 1 = 5. Hence, {Fn]:;1 is not equi-upper-semi- continuous at 0. Definition 2.10. Let be a sequence of (D [¢h}n=0 functions with domains D c:R. we say “h converges compactly to go, denoted by Qn S ¢b' if for any compact subset K of D , K csz except at most a finite number n n . of Dcpn and m“ 4 90 uniformly on K. For any open subset U of Ran, let MU denote the family of all orientor fields of Caratheodory type from U into cc(Rn). we shall consider the following contingent differential equations with initial conditions: 49 X n (1) x(t) e A + Fn(t.x(t>) x(tn) (2) x(t) e A(t.x(t)) + Fo(t.X(t)) x(to) = x0 Theorem 2.2 (convergence). Suppose that (i) U and Un are open subsets of Ran for n = 0,1,2,... such that UO C’Un c U except for a finite number of Uh's (11) A 6 MD and Fn 6 mbn for n = 0,1,2,...: (iii) {F V” n n-l is an equi-upper-semi-continuous family on Proj 2 U0 such that p(Fn(t,x),Fo(t,x)) 9. o for all (t,x) 6 U0: (iv) for each compact subset Q c qo, there exists a function m(t) = mQ(t) such that m(t) is integrable over Proj 1 Q and |Fn(t,x)| g m(t) for all (t,x) 6 Q and for all n = 0,1,2,...: (v) (tn'xn) e Uh for n = 0,1,2,... such that n Then, for every sequence {¢h}:;1 of non-continuable solutions of (1), there exists a subsequence {Qh };;1 and there exists a non-continuable solution “C of (2) such that mh S wb' J' O Proof: Let f¢h}n=1 be a sequence of non- continuable solutions of (l) with interval domains {Dcp = (w. w”)} n n "1 n n— 50 (i) For any compact subset Q c UO with (t0,xo) 5 Int Q, there exists an Open interval IQ such that t0 6 IQ c (m;,m:) except for a finite number of n: By assumption (i), Q czun for all n large enough. Define t t L(t) = It L(s)ds and M(t) = It m(s)ds O 0 where L(t) = 40(t) and m(t) = mQ(t) are integrable functions that bound A(t,x) and Fn(t,x), respectively, on Q. We note that L(t), M(t) 6 AC on Projlt Q. Let a > 0 be such that the set wa wa = {(t,x) : ‘t-tO) < a. |x-xo| < % d(to.xo).aQ) is contained in Q, where 5Q denotes the boundary of Q. Let 0 < B < a be such that |t-t0| < 6 implies |L‘ + |M| < g d((to.xo).aQ). Define , 1 WE - [(t.X) . |t-to1 '\ B. \x-xol < 71 d((to.xo).aQ)}- Clearly, we have WE c:wd CiQ c U6. By assumption (v), there exists an integer N > 0 such that (tn,xn) 6 WE for all n > N. Since mh is a solution of (l), we have ‘nn(t)-xn| g_|j:n £(s)ds| + ‘Itn m(s)ds‘ lL(t)-L(tn)| + |M(t)-M(tn)| _<_ |L(t)| + |M(t)| + |L(tn)‘ + ‘M(tn)| 1 d((t0.xo).aQ) + g- d((to.xo).aQ) /\ thI-J (DIP d ( (tooxo) o 3Q) 51 for all n > N and all t such that ‘t-t0| < B. Hence, ‘cpn (t) -XO) ‘ g ‘cpn (t) -Xn‘ + ‘Xn-Xo‘ < é d((to.xo).ao) + é d((to.xo).aQ) I NIH d ( (tooxo) o 50) for all n > N and all t such that \t-to| < B. This shows that (t'¢h(t)) e Wd for all n > N and all t such that ‘t-tol < 6. Since mh is non-continuable, this shows that ¢h(t) exists at least on the interval (to-B,to+B) for all n > N. This proves (i). (ii) For any compact interval J c.I ‘with Q m 9 . to Q Int J, [¢h(t)}n=1 has a subsequence I¢hj(t)}j=1 which converges uniformly to some function ¢b(t) on J: Without loss of generality, let J = [a,b] c I and t0 6 (a,b). Q Clearly, Qn is defined on [a,b] for n > N. In part (i) of our proof, we have shown that {(t,¢h(t)) : t e IQ, n > N} C Wu c Q. Since Q is compact, [mn]n=N+l is uniformly bounded on [a,b]. Also, by Proposition 2.2, the solution mh of (1) can be expressed as t cpn(t) E xn + (S) jtn[A(S.cpn(S)) + Fn(Socpn(S)) ]ds for t 6 [a,b] and n > N. It follows from Theorem 1.7 that cpn(t) - cpn(S) e (S)j‘:[A(r,cpn(r)) + Fn(r,cpn(r))]dr 52 for any t,s 6 [a,b] and n > N. Hence, we have ‘qnn(t) -cpn(s)‘ _<_ U:[L(r) + m(r) ]dr‘ g_|L(t)-L(s)\ + ‘M(t)-M(s)| for any t,s 5 [a,b] and n > N. As L(t), M(t) e AC([a,b]) and are independent of n, {¢h}:=N+l is equi-continuous on [a,b]. It follows from Ascoli's that there exists a subsequence {gn.(t)};;l which converges uniformly to some function ¢b(t) on [a,b]. (iii) ¢b(t) is a solution of (2) on [a,b]: Consider the limit superior of both sides of the equation cpn.(t) e xn. + (my: [A(s,cpn.(s))+ Fn.(s,qgn.(s)]ds J 3 nj 3 3 3 where t 6 [a,b]. By Proposition 2.3 (i), we have lim sup qh (t) c lim sup{xn +(S)J‘tn [A(S'¢h (3)) 54¢ j j4° nj nj j + Fn (s,¢h.(s))]ds}. J 3 Since A(t,cgn (t)) and F (t,¢ (t)) are integrably bounded j n 0 nj by L(t) and m(t) respectively for all t 6 [a,b], ‘we can apply Pr0position 2.3 (ii) and get lim sup mn (t) c lim sup xn + lim sup(S)J‘tn [A(s,gh.(s)) j4w nJ' j4~ j j4w nj + Fn (s,cpn (s))]ds j j c lim sup xn. + lim sup{(S)j‘to [A(s'¢h. (3)) 34“ J qu j n . + F (S,¢h (s))]ds J j t n. + auto] [£(s)+m(S)]ds)} 53 c lim sup xn + lim sup(S)j‘to [A(s,¢h (s) jam 3 14° j + Fn_(s,¢hj(s))]ds J tn + lim sup B(jt:j[1(s)+m(s)]ds) 54w t where B(J‘t [t(s)+m(s)]ds) is the solid sphere in Rn 0 centered at O with radius ‘j: [L(s)+m(s)]ds|. Since j 0 . . 9n (t) 4 ¢b(t) uniformly on [a,b], tn 1 t J j and by Theorem 1.6, we have ¢b(t) E x0 + (S)Ito lim sup[A(s,¢h. (s)) + Fn (s,¢h (s))]ds j4” j nj + {0}. Define Gn (t,x) = A(t,x) + Fn (t,x) for j = 1,2,... and j j Go(t,x) = A(t,x) + Fo(t,x). By assumption (iii), [Gn };;l is equi-upper-semi-continuous j . on R“ and p(Gh (t,x),Go(t,x)) 1 o. It follows from i Proposition 2.4 that lim sup Gn (t,cpn (t)) c: G0(t,cpo(t)). j+° nj ' Hence, m0(t) 6 X0 + (S)I:O[A(s.¢b(s)) + Ro(s.¢b(s))]ds. Therefore, go(t) is a solution of (2) on [a,b]. (iv) ¢b(t) can be extended to IQ = (ao,bo) defined in part (i) such that ¢b(t) is a solution of (2) on IQ m o and {Qn}n=l has a subsequence (“Rk,k)}k=l such that wank) 4 “’0 on Io: From what we have shown above, we 54 know that for every compact interval [a,b] such that t0 6 (ao,bo) and [a,b] c (ao,bo), [¢h}n=1 has a subsequence which converges compactly to a solution qb of (2) on [a,b]. Let [[ak'bk1};;l be a sequence of compact intervals such that t0 6 (ak'hk)' [ak'bk] C:[ak+l'bk+l] and (aoobo) = kE-Jl [ak'bk]. m a: Let [m(l,r)}r=l be a subsequence of {¢h}n=1 r such that m(l,r) 4 m(0.1) uniformly on [a1,b1]. If Q . . {Cp(k,r)}r=l is chosen and m(kor) 4 @(O,k) uniformly on [ak'th' we pick a subsequence [¢Wk+l,r)}r=l of an r . {m(k' r) }r=l such that “JON-1o r) 4 ‘P(o,k+l) uniformly on [ak+l'bk+1]' Clearly, w(01k+1) is a solution of (2) on [ak+1'bk+l] and “’(o,k+1)‘[ak'bk] = own” for k = 1,2,..., Define q0(t) : (ao.bo) 4 Rn by ¢b(t) = ¢(O k)(t) where k is so large that t e [ak,bk]. From our selection process. it is clear that ¢b(t) is a solution of (2) on (a0,bo) and {¢h]n=l has a subsequence {qkk,k)}k=l such that c (v) Let id be the maximal interval which is contained in Proj1 Q and to 6 Int IQ such that a solution qo(t) of (2) is defined on f; and {wh}:;l has a subsequence which converges compactly to ¢b on 10' Then, ~ ~ IQ is closed. Moreover, if 5' and b are the left and right end points of IQ to so as t 4 5' + 0 or t 4 1'5 - 0: We shall consider the N ~ right end point b only. The left end point a can be respectively, then (t,qo(t)) tends 55 proved similarly. Since “5 is a solution, for ~ §‘ + ‘M(t2)-M(t1)|. As L(t), M(t) €AC((a,b)), canal) 5%(t2) 40 as t1.t2 4'b - 0. By the Cauchy criterion for convergence, 51163-0) exists for all n = o,l,2,~~. Also lim (t,cpn(t)) 6 Q t4b-O since Q is compact and (t,¢h(t))e Q for all t 6 (a,b) and n = 0,1,2,'-°. Define ER : (a,b] 4 Proj2 Q by ¢h(t) if t G (533) :18 :9 A A n n V v I ll — ona’S-o) if t = h” where n = 0,1,2,-o-. By the existence theorem of Plis, mm can be extended so that (t,mh(t)) ('Q. And fig is a continuous extension of ¢h in Q. Hence, 3% is a solution of (l) for n 2_l and of (2) for n=0. Consider the subsequence {qu,k)};;1 defined in part (iv). By a similar proof as in part (ii), {&%k,k)];;l has a subsequence {$(r,r) :;1 such that $(r,r) 4 g uniformly on [8351, where g < S < E. Since m(r,r) 5 cpo on [3,133, 85 = $0. ~ It follows that 55 must contain b. Hence, IQ is closed. Clearly, $(r,r)(£3 = 5r 5 go = qb(b). Suppose I ~ lim (t,¢b(t)) 6 Int Q, i.e. (b,&b(b)) 5 Int Q. Then, t4b-0 l 56 following our previous proof, we can use {(b, g n)}n-O as our new sequence of initial points and extend IQ to the right which contradicts the maximality of I Q. (vi) Let I be the maximal interval of the fb's where Q is a compact subset of U0 and (t0,xo) 6 Int Q. Then I, is always Open, say, I = (w-,m+) (w-,m+) C.(lim sup w; n4m lim inf mg). Furthermore, there exists a non-continuable n4w . , .. 4. solution m0 of (2) defined on (w ,w ) and {in}:=1 has a subsequence [¢h ]j- _1 such that mnj 4 go on (w-,m+): Let {Qk}k— _1 be a sequence of compact Jsubsets of U0 such that (to,xo) eInt 0k, chok+1 and U0 =kLJ1 0k. For each Qk' we can find a maximal interval IQkk and a subsequence r ... {m(k,r)}:=1 of [¢n}n=l such that cp(k'r) m(0,k) uni- formly on fék. Choosing (“wk r)}:;l inductively and taking , _ the diagonal process as in part (iv), we get the desired result. C The following is another convergence theorem which is parallel to the previous one. The proof is analogous. The only difference is we shall apply Pr0position 2.5 instead of PrOposition 2.4. We give the statement as follow: Theorem 2.3 (convergence). Suppose that (i) U and Un are open subsets of Ran for n = 0,1,2,... such that qo c‘Un c‘U except for a finite number of Un's‘ 57 (ii) A e Wb. and FD 6 mbn for n = 0,1,2,...; (iii) p(Fn(t,x),Fo(t,x)) 3 O uniformly with respect to x for x e Proj2 ‘UO: (iv) for each compact Q th0, there exists a function m(t) = mQ(t) such that m(t) is integrable over Proj1_ Q and ‘Fn(t,x)| g m(t) for all (t,x) g Q and all n = 0,1,2,...; (v) (tn’xn) 6 Un for n = 0,1,2,... such that (tn'xn) 3 (to,x0). Then, for every sequence {¢h}:;1 of non-continuable solutions of (1), there exists a subsequence {mh ];=1 and there exists a non-continuable solution ¢b Of (2) such that Qn 3 go. j §3. Continuous dependence of solutions on initial conditions and parameters: Let U be an open subset of Ran and A ‘be a domain, A = {x : ‘x-xo‘ < c, c > O} c.R. We define an orientor field with a parameter F(t,x,x) : UxA 4 cc(Rn) by F(t,x,x) = A(t,x) + Fx(t,x) and we assume that (i) for each x e A, UK is an Open subset of n+1 R such that UK c UK c U for all x e A; 0 (ii) A E MU and Fl 6 Wb for all X E A: l 58 (iii) {FX}XEA is an equi-upper-semi-continuous family on Projz U1 such that O p(Fx(t,x),Fko(t,x)) 4 O as X 4 10 for all (t,x) E U : "0 (iv) for each compact subset Q CLUXO, there exists a function m(t) = mQ(t) such that m(t) is integrable over Proj1 Q and \Fx(t,x)\ g m(t) for all (t,x) Q Q and all 1 6 A. We shall call this family of orientor fields {F(t,x,x) : l e A] family (P) and consider the following contingent differential equation with parameter: (P1) X(t) e F(t.X.1). X(t>\) = xx“ (t o X ) E U o by the X0 x0 10 positive solution funnel through (t ,x ) we mean the set 10 10 and m is a solution of (P1 )}. "o o For any interval I, we define z+(I) = 2+ n (Ian). Definition 2.11. For each 2+ = {(t.o(t)) : t.2 t The negative solution funnel z“ and Z-(I) can be defined similarly for t g't . And we define the solution funnel xo 2 = 2+ U z" and 2(1) = Z+(I) u Z-(I). The next theorem gives some answer to question (iv) in §1. This theorem though it follows immediately from Theorem 2.2 is actually a special case of Theorem 5 of [6 ] in which the equations are considered in Banach spaces: 59 Theorem 2.4. (Chow and Schuur: [6 ]). If I is a compact interval on which all solutions of (PXO) exist, then Z+(I), Z-(I) and Z(I) are all compact. Theorem 2.5. (continuous dependence). Suppose that all solutions of (PKG) exist on [a,b]. Then, for each 6 > 0, there exists a 5 > 0 such that for any (tl'xl'X) satisfying 9((tx.xx).z([a.b])) + \1-10\ < 5. each non-continuable solution ¢x(t) of (P1) exists at least on [a,b] and there exists a solution (t) of “’1 O (P ) such that "O \¢1(t)'w1 (t)) < e for all t 6 [a,b]. 0 Proof: Suppose the first conclusion is false. Since Z([a,b]) is compact, there exists a sequence (tkn'xl ,xn) a n (To,§o,xo) such that (TO'EO) E Z([a,b]) and a sequence Of non-continuable solutions ¢h(t) of (Pxn) with maximal interval of existence (m;,m:) and an integer N > 0 such that [a,b] ¢ (g;,w:) for all n > N. By Theorem 2.2, there . CO on eXists a subsequence [mh.(t)}j=1 of {¢h(t)]n=l and a non-continuable solution m0(t) of (PKG) X(t) 6 F(t,x(t):lo) x(TO) = go with maximal interval of existence (w5,m;) such that ¢b. 3 “D on (w-.q;). Since (P' ) 15 (P ) with only M 10 the initial condition changed and this initial point (Toqu(TOH = (TO.§O) E Z([a,b]), ¢b is actually a non— continuable solution of (P ). By our assumption, exists *0 CpO 60 at least on [a,b]. Hence, [a,b] c (m; ,m; ) for j J' 3' large enough. Choosing j so large that nj > N, we get a contradiction. The second conclusion is evidently true. Suppose not. we consider the same sequence of solutions {¢h]:;1 as before and claim that there exists an integer M > 0 such that for each n > M there exists a tn 6 [a,b] with ‘¢h(tn)-¢b(tn)"2 e for some e > 0. However, by Theorem 2.2, f¢h):;1 has a subsequence [¢h.};;l which converges uniformly to ¢b on [a,b]. Hence, (ghj(t)-¢b(t)| < e for all t 6 [a,b] and for all j sufficiently large. Choosing j so large that nj > M, we get a contradiction again. [:1 Remark 2.7. Theorem 2.5 is an extension of Theorem 4 and Corollary 4.2 in [34] to contingent differential equations. Remark 2.8. It is clear that if we replace assumption (iii) of family (P) by (iii)’ p(Fx(t,x),F (t,x)) 4’0 uniformly with "o respect to x, for x e Proj2 U as xo 1 4 10. then Theorem 2.5 also holds. Next, we shall define another family called family (9) of orientor fields. Let U be an Open subset of 61 Ran, [a,b] c: Projl U and A = [x : |x-10| < c, c > 0} CR. Let F(t,x,x) : UxA 4 cc(Rn) defined by Whit. 1) = F(t.x) + 61”) and assume that (i) F e ”b’ (ii) for each x e A, Gl(t) is a continuous function from Proj1_ U into cc(Rn): (iii) I: ‘Gx(t)|dt 4 0 as l 4 xo: (iv) [G)‘(t)})‘6A is bounded on [a,b]. Then we shall consider the following contingent differential equations with parameter: (0).) x(t) e F(t,x,x) x(t)) = xx. It is easy to see that family (Q) satisfies all the conditions in family (P). Therefore, Theorem 2.5 holds true for family (0). we have the following theorem: Theorem 2.6. (continuous dependence). Let F(t,x) : U 4 cc(Rp) satisfy the Caratheodory conditions on an Open subset U of Ran such that all solutions of (s) i(t) e F(t.x(t)) x(to) = x0 exist on [a,b]. Then for every 6 > 0, there exists a 5 > 0 such that for every continuous G from [a,b] into cc(Rg) 62 which satisfies p((w.§).2([a.b])) + J” («was < a. a each solution of ¢(t) of int) 6 F(t.x(t)) + am through (T,g) can be extended to [a,b] and there exists a solution ((t) of (B) such that (m(t)-((t)) < s for all t 6 [a,b]. Remark 2.9. Theorem 2.6 extends a result of YOshizawa (see [39] p.22) and also Corollary 4.3 of Strauss and Yorke (see [34]) to contingent differential equations. Chapter III GENERAL BOUNDARY VALUE PROBLEMS FOR CONTINGENT DIFFERENTIAL EQUATIONS It is well known that fixed point theorems play a main role in the proof of existence theorems of differential equations. The papers of Granas (see [10] and [11]) have extended the notion of tOpOlogical degree to set-valued mappings and the fixed point theorems Of Rothe and Borsuk have also been successfully established for the set-valued case. In this chapter we shall prove an existence theorem of contingent differential equations by using the degree theory described in §3 of the first chapter and we shall see some of its applications. The results obtained here are motivated by Theorem 2.1 and Theorem 2.2 of [12] (see p.413) and are also generalizations of the results in [23]. §l. A fixed point theorem: Let E be a real Banach space with norm |-|. Definition 3.1. A mapping F : E 4 cc(E) is called homogeneous if F(xx) = lF(x) for every real 1 and every x E E. 63 64 Lemma 3.1. (Chow and Lasota [ 5 ]) . Let F : E 4 cc(E) be homogeneous and completely continuous with the property that x 6 F(x) azx = O. d(F) > 0 such that Then, there exists a constant a x eF(x) +b a [x] SOL‘b] for each x e E. Lemma 3.2. Let F(x) = x - V(x) and G(x) = x - W(x) be two non-vanishing completely continuous vector fields mapping a bounded subset A of E into cc (B) such that G(x) c F(x) for every x e A. Then, F and G are homotOpic on A . 2522:; Define §(x,x) : [0,1] x A 4 cc(E) by 50.x) = [1w(x)+(1-1) W(x) IO] 0 VM where U is the closed unit ball centered at O in E. Then we define uphox) : [0,1] xA 4cc(E) by cp(1.x) = x - 90.x}. Clearly, 6 (hence gp) is well—defined. It follows from Propositions 1.7, 1.8 and 1.9 that Q (hence cp) is upper semi-continuous. The following prOperties of Q and q) are immediate: (i) i(1.x) c Hitx) if 1_>_ 1’: (ii) G(O.x) = V(x) and §(l,x) = W(x): 65 (iii) for each 1 6 [0,1], ¢(x,x) is non- vanishing on A : This follows from the fact that q)(x,x) c cp(0,x) = F(x) which is non-vanishing on A. (iv) O([O,l],A) is relatively compact : Since V(x) is a compact mapping, V(A) is relatively compact. It follows from 1([0.1].A) c “0.11) = v(A) that §([O,1],A) is relatively compact. Therefore, F and G are homotopic on A ‘with homotOpy cp- [3 Theorem 3.1. (fixed point property). Let F,G, H and J : E 4 cc(E) be completely continuous such that (i) F is homogeneous with the property : x e F(x) alx = O: /\ X (ii) G is bounded, i.e. “G“ = sup |G(x)] __ er for some K: (iii) there exist a o > O and an e = €(F,o) > 0 such that WK) 1 _<_ eIXl for all x ‘with ‘x‘IZ o > 0: (iv) J(x) c F(x) + G(x) + H(x) for all x e E with |x|.2 o > 0. Then, there exists at least one x e E such that x e J(x) provided 6 is small enough. 66 Proof: Consider the following completely continuous multi—valued vector fields from E into cc(E) : cp1(x) = x - F(X): $20!) = x - F(x) - H(x): m3(x) = x - F(x) - G(x) - H(x): ¢h(x) = x - J(x). (i) Let 9( XIX) §(x,x) [0,1] x E 4 cc(E) defined by x - F(X) - W(x)- Let Sp = [x : x e E and [x] = P} where p > 0. Clearly, §(x,x) is upper semi-continuous and i([0,l],Sp) is relatively compact. Moreover, i(0,x) = qd(x) and Q(1,x) = ¢2(x). It follows from assumption (i) that ¢d does not vanish on Sp' we claim that there exists a positive number r > 0 such that ]x-F(x)| 2_r > O for all x e S Suppose not. p. Then there exists a sequence [xn]:;1 CiSp such that ]xn-F(xn)| 3 0. One can see easily that there exists Y 6 F(xn) such that ]xn—yn| 3 0. It is clear that n {yn}:;1 is contained in F(Sp) which is compact. Hence, {yn]:;1 has a subsequence [yn I" such that . J j=1. ynj 1 yO 6 FTSET} Since ]xnj-ynj| 1 O, 'we have ]xnj-yo] i O, yO E Sp since [xnj};;1 c Sp which is closed. Now, yn. 6 F(xn ), yn. 1 yO and xn 1 yo. It follows 3 j J j 67 from the upper semi-continuity of F that yo 6 F(yo). However, ]y0] = p > O. This contradicts assumption (i). Choosing e < §%- and applying assumption (iii), we have ]x-F(X)-1H(X) | _>_ \x-F(X)| — [m(XH Z,r - e]x] > r - g > O for all 1 6 [0,1] and all X‘e Sp' Hence 6(l,x) does not vanish on Sp for all 1 6 [0,1]. Therefore, ¢i is homotOpic to ¢2 on Sp' (ii) Let Y : [0,1] x E 4 cc(E) defined by MLX) = X - F(x) - AGO!) - H(x). Clearly. Y(l.x) is upper semi-continuous and Y([O,l],Sp) is relatively compact. Moreover, y(0,x) = ¢2(x) and y(l,x) = ¢3(x). we claim that Y(l.x) does not vanish on Sp for all x 6 [0,1] if p is large enough. Suppose not. Then there exists an x e Sp such that x E F(x) + xG(x) + H(x) . By Lemma 3.1, there exists an a = d(F) > 0 such that ]x] _<_ a]XG(x) + H(x) ] _<_aXK + ae|x‘. Choose a < égu we have [x] g_aXK + %|x], i.e. [x] g ZaAK. This is a contradiction since ]x] = p which can be chosen arbitrarily large. Hence, we have shown that $2 and $3 are homotopic on Sp for sufficiently large p. 68 (iii) From part (i) and part (ii), qd and $3 are homotOpic on some large sphere Sp' By Lemma 3.2, $3 and ¢Z are homotopic on S . p Hence, ¢i and qh are homotOpic on S By Theorem 1.9, p' the characteristics of ¢1 and ¢z on Sp are equal. Since F is homogeneous, we have cp1(x) = x - F(X) = -(-x+F(X)) = -(-x-F(-x)) = -cp1(-x) . It follows from Theorem 1.11 that the characteristic of $1 (hence ¢Z) on Sp is Odd. By Theorem 1.10, there exists an x e E with ]x] < P such that x 6 J(x). [1 Remark 3.1. In Theorem 3.1, condition (iii) and (iv) can be replaced by (iii)’ [H(x)] g'e]x] for some 6 = e(F.P) and all x with ]x] 3.97 and (iv)’ J(x) c.F(x) + G(x) + H(x) for all x with ]x] S_P, where p >10 is sufficiently large. Remark 3.2. In Theorem 3.1, if the condition [H(x)] g.e|x] holds for all x e E, then 6 depends on F only. The following corollary is clear from the proof of Theorem 3.1: 69 Corollary 3.1. Let F,H and J : E 4 cc(E) be completely continuous and such that (i) F is homogeneous with the prOperty : x e F(x) = x = 0: (ii) fOr any positive number p > O and all x ‘with [x] < p, ]H(x)]‘g_e]x] for some a = e(F.P): (iii) J(x) c:E(x) + H(x) for all x E E with |x| < P. Then, there exists at least one x 6 E with |x| < P such that x e J(x) provided 6 is sufficiently small. Instead of a Lipschitz type condition, Corollary 3.1 still holds if |H(x)‘ is small in a neighborhood of 0 e E: Corollary 3.2. Let F,H and J : E 4 cc(E) be completely continuous such that (i) F is homogeneous with the prOperty : x 6 F(x) ~>x = 0: (ii) for some positive number p > 0, there exists an e = e(FAD) such that W(x) \ S. N? for all x e E 'with ‘x‘ g,p: (iii) J(x) c.F(x) + H(x) for all x e E with M s P- Then, there exists at least one x e E with [x] < p such that x 6 J(x) provided a is sufficiently small. 70 Proof: Let us use the same notations as in Theorem 3.1. It suffices for us to show that 5(1.x) does not vanish on Sp and the rest of the proof is clear from Theorem 3.1. Let r > 0 be a number such that ]x-F(x)| 2.r > O for all x 6 SP. Choosing e < g%- and applying assumption (ii), we have [Hind] = \x-F(X)-1H(X)] 2 \x-F(x)\ - [1H(x)| _>_r-1eP>r-§>O S for all x 6 [0,1] and all x e [j p. The next corollary is an immediate consequence of Theorem 3.1: Corollary 3.3. Let F,G,H and J : E 4 cc(E) be completely continuous such that (i) F is homogeneous with the property: x E F(x) a x = 0: (ii) G is bounded: (iii) iii-fin 4 O as ]x] 4 an; (iv) J(x) c: F(x) + G(x) + H(x) for all x e E. Then, there exists at least one x e E such that x e J(x). §2. An existence theorem like Fredholm's alternative: Let A 'be a compact interval in R and let Cn be the Banach space of all continuous functions from A into Rn ‘with the tOpology of uniform convergence. 71 Lemma 3.3 (Lasota [21]). If {ykl:=1 is a sequence of measurable functions from A into R” such that there is an integrable function m(t) from A into R and ]yk(t)| S m(t) for all k = 1,2,... and t a.e. in A, then there exist a sequence of indices am and a system of coefficients Akm (m g k 3 am, m = 1,2,...) such that 0‘11: Z) = l, a 2.m, 2_0 m >3... m >3... and that the sequence am zm(t) = RE!“ )km Yk(t) converges to a function 20(t) a.e. in A. Pr0position 3.1. Let F(t,x) : [a,b] x Rn 41cc(Rn) be an orientor field which satisfies the Caratheodory conditions. Then the set-valued mapping G(x) : Cn 4 cc(Cn) defined by G(x) = [g(t) : g(t) = I: fx(s)ds, 'where fx 6 L1([a,b]) and fx(s) e F(s,x(s)) for all s 6 [a,b]} is completely continuous. Proof: (i) G is well-defined: Let x ecn be arbitrary. It follows from PrOposition 2.1 and Theorem 1.5 that F(t,x(t)) has an integrable selection fx(t). Hence, t G(x) # ¢. Let 91,92 6 G(x), say gl(t) = I; f1(s)ds and t 1 92(t) = I; f2(s)ds where f f2 6 L and f1(s), 1' 72 f2(s) e F(s,x(s)) for s 6 [a,b]. Then for any X. 0‘3 x‘g 1, *we have iglm + (1-1)gz(t) = I: [1f1(S) + (l-l)f2(s)]ds. Clearly, )(fl + (1-1) f2 6 L1([a,b]) and 1f1(s) + (1-x)f2(s) e F(s,x(s)) for s 6 [a,b] since F(s,x(s)) is convex. Hence, xgl + (14.)92 e G(x), G(x) is convex. Let [gn}:;1 ch(x) be any sequence of functions from G(x). Since x(t) is bounded on [a,b] and F satisfies the Caratheodory conditions, we have [sum 1 _<_ j: ‘F(s,x(s)) ‘ds 3 f: m(s)ds 3 [HM - M(a) for all t 6 [a,b] and n = l,2,---. Hence {gn]n=l is uniformly bounded. Moreover, by Theorem 1.7, t gn(t) - 9n(s) e [s F(r.X(r))dr for all t,s 6 [a,b]. Hence, t \9n(t)-9n(s) l _<_ lj’s m(r)dr| _<_ [Mm-Ms) l for any t,s 6 [a,b]. Since M,e AC([a,b]). {9n}:;1 is equi-continuous. It follows from the Ascoli Lemma that e e k {gn}n=l has a subsequence {9k}k=l such that gk 2'90 uniformly on [a,b] for some go. Since the convergence is uniform, gO 6 Cn. we claim that gO e G(x). Each t 9k can be written as gk(t) = I fk(s)ds where ' a 73 fk(s) E F(s,x(s)) for k = l,2,---. [fk];;1 is integrably bounded by m(t). By Lemma 3.3, there exist a sequence of indices am and a system of coefficients )km (m _<_ k 3 am) such that a m Zlkm=la C1 k=m and that the sequence mam. 1,,20 am h (t) = Z} (t) m k=m x... f. converges to a function ho(t) a.e. in [a,b]. Since fk(t) E F(t,x(t)) which is convex, hm(t) E F(t,x(t)) and hm(t) is measurable for m = 1,2,°-°. As F(t,x(t)) is closed, ho(t) e F(t,x(t)) and ho(t) is measurable. Clearly, a 1; am t m J; hmds = kg“ >3... Ia mews = kg x... am t converges to [6 h0(s)ds by the Lebesgue dominated conver- gence theorem. Recall that [gk};;1 is picked so that {9k}k=l converges uniformly to go. Its finite convex combinations also converge to go. That is a t m _ m k=m t Hence, go(t) = I; ho(s)ds. We have 90 e G(x). G(x) is therefore compact. (ii) G is compact: Let D = [x 6 Cn : [x] g_K} be any bounded subset of C“. We want to show that G(D) has compact closure. It is equivalent for us to show that ERBT' is sequentially compact. Let [gn}:=1 c:G(D). Then t 9n(t) = I; fn(s)ds 74 where fn(s) e F(s,x(s)) for s 6 [a,b] and some x e D. Since D is bounded and F satisfies the Caratheodory conditions, [fn}:;1 is bounded by an integrable function m(t). It follows easily from Ascoli's that [gnlnel has a subsequence [gk};;1 which converges uniformly to . n . a some function gO e C . Since {9k}k=l c G(D), 90 e G( ), G(D) is therefore compact. (iii) G is upper semi-continuous: Frommwhat we have shown above, we know that G is a compact mapping from Cn 4 cc(Cn). By Theorem 1.1, all the definitions of upper semi-continuity are equivalent. Let xo 6 Cn be . a a n arbitrary. Let [xn}n=1 CCn and {gn}n=0 c.C be such n n that xn 4»x0, gn 490 and 9n 6 G(xn) for n 2.1. Then t gn(t) = Ia fn(s)ds where fn 6 L1([a,b]) and fn(s) e F(s,xn(s)) for all s 6 [a,b] and n = l,2,--°. Since xn 3x ]xn(s)| SM 0' O o for all s 6 [a,b] and n‘z 1. Hence, [fn}n=l is bounded by an integrable function m(t). By Lemma 3.3, there exist a sequence of indices am and a system of coefficients hkm (m $;k < am) such that a m 2 =1: (1 >1“: >0 k=m)...“ m- >1... and that the sequence a m we = 23 5... am k=m converges to a function ho(t) a.e. in [a,b]. Clearly, 75 t as we saw in part (ii), go(t) = I 'h0(s)ds for all a t 6 [a,b]. Since fn(t) e F(t,xn(t)), 'we have a m hm(t) e 2 1m F(t.xk(t)) k=m for all t 6 [a,b]. For each t fixed, F(t,x(t)) is upper semi-continuous. Hence, given 6 > 0, there exists a K 3 0 such that F(t,xk(t)) c F(t,xo(t)) + B€ for all k 2;K, ‘where B6 = [x e Rn : [x] g 3}. Hence, (I. (I. m m kg“ 1km F(tfifi‘H‘J) ckijm Akm(F(t,xo(t)) + Be) = F(t.xo(t)) + Be for all m 2.K. This shows that m puma. F(t.xo(t>) .. 0. As F(t,x0(t)) is closed and hm(t) 4 h0(t) a.e. in [a,b], we have ho(t) 6 F(t,xo(t)) a.e. in [a,b]. That is gO e G(xo). Therefore, G is upper semi-continuous in the sense of Definition 1.5 (hence in the sense of all others). D Lemma 3.4. Let F,G : A 4 cc(Rp) be measurable and integrably bounded and K(t) be a ball in Rp centered at 0 with radius |K(t)[ such that |K(t)] is integrable over A. If F(t) c G(t) + K(t) for all t 6 A. then (S) F c: (S) G + (S) K. IA IA J‘A 76 Proof: Let f(t) be a measurable selection of F(t). Then the function H(t) : A 4 cc(Rp) H(t) = [f(t)+K(t)] n G(t) is well-defined and measurable. that H(t) defined by It follows from Theorem 1.4 has a measurable selection h(t). is also a measurable selection of G(t). k(t) where f(t) = h(t) - k(t) = h(t) + (-k(t)) where h(t) Remark 3.3. k(t) e K(t) is measurable. e G(t) and ék(t) e H(t) Lemma 3.4, it is easy to see that (3)] (3+K) = A (3)] 3+ (3)] K A A Clearly, h(t) And h(t) = f(t) + Hence, are both measurable. If G and K are defined as in Consider the following contingent differential equations with general boundary conditions: (1) (2) Theorem 3.2. and Q: Ax‘Rn 4 cc(Rn) Suppose that (i) k(t) eA(t.X(t)) . 5cm 6 Q(t.X(t)) . where Q(t,x) cA(t,x) + B(t,x) + F(t,x). L x(t) L x(t) (Fredholm's alternative). Let A. B, P satisfy the Caratheodory conditions. A(t,x) is homogeneous with respect to x, that is Mt. AX) for all real x: AA (to X) C] 77 (ii) B(t,x) cK(t) where K(t) is aball in RP centered at O with radius |K(t)| such that |K(t)| is integrable over A: (iii) there exists a = e(A,o,5) > 0 such that a(t,r) 3 er for any r20>0 where o > 0 is arbitrary, a(t,r) = sup |P(t,x)‘ lx L<.r and 5 = m(A) is the measure of A: (iv) L : Cn 4 R? is linear and continuous. Then, (1) has unique solutions x(t) E 0 implies that (2) has at least one solution for any a e R“, provided 6 is small enough. i? oof: without loss of generality, we may assume A = [0,T]. Define F,G,H and J : Cn 4 cc(Cn) by m ,1 35. u (I: u(s)ds + Lx + x(O) : u(s) e A(s,x(s))}, G(x) = (I: U(s)ds - a : u(s) €K(8)}. m 5. n (I: U(S)ds [11(8)] _<_ 31MB) 1} J(x) = (I: u(s)ds + Lx - a + x(O) : u(s) 6 Q(s,x(3))}, where t e [0,T] and u(t) e L1([O,T]). Then the existence of the solutions of (l) and (2) are equivalent to the existence of fixed points of F and J respectively. It follows from PrOpOsition 3.1 and the fact that the continuous linear Operator L is bounded that F,G,H and J are all completely continuous. 78 Clearly, F is homogeneous with only x(t) s O as its fixed point. G is bounded by assumption (ii). Let S : Rn 4.cc(Rn) defined by“ P(x) = [u 6 Rn : [u] g.e|x|}. Then for any x(t) 6 Cu, S(x(t)) is a ball in RP centered at O with radius e‘x(t)| which is integrable over [0,T]. From the way we define H, one sees easily that H(X) = (I: u(8)ds : 11(8) 6 3(X(s))} and ]H(x)| g ]‘A‘Sousn ]ds 3 ]A e|x(s) ]ds 3 arm. Moreover, |P(t,x) | g d(t, ]x]) _<_ e|x| for all x, ]x] = r 2 o > 0. Hence, F(t,x) c: F(t,x) for all x, ]x].2 o. It follows from Lemma 3.4 and Remark 3.3 that t J(x) = [[6 u(s)ds + Lx - a + x(O) : u(s) 6 Q(s,x(s))} c (I: u(s)ds + Lx - a + x(O) : u(s) e A(s,x(s)) + B(s,x(s)) + P(s,x(s))} c (L: u(s)ds + Lx - a + x(O) : u(s) e A(s,x(s)) + K(s) + P(x(s))} t t t = {I6 u(s)ds + Lx + x(O) + f6 v(s)ds - a + I6'w(s)ds: u(s) e A(s,x(s)). V(s) e K(s) and w(s) e S(x(s))) = F(x) + G(x) + H(x) for all x e Cn with |x]‘2 o > 0. Hence, by Theorem 3.1, there exists at least one x G Cn such that x e J(x). (j 79 Remark 3.4. As in Remark 3.1, condition (iii) of Theorem 3.2 can be replaced by (iii)’ there exists an e = g(A.P.6) > 0 such that a(t,r) g_er for all r'g p where a(t,r) = sup ]P(t,x)|, 5 = m(A) |X\SF and p > O is sufficiently large. Remark 3.5. As in Remark 3.2, in Theorem 3.2, if (iii) holds for all x e E, then a depends on A and 5 only. In (2), when B(t,x) [0} and a = O, we have the equation: (3) k(t) e Q(t.X(t)). u(t) = o 'where Q(t,x) c A(t,x) + F(t,x). In this case, G(x) a [0}. Therefore, following the same proof of the above theorem and applying Corollary 3.1 instead of Theorem 3.1, we Obtain Corollary 3.4. Let A,P and Q : Aan 4cc(Rn) satisfy the CarathéOdory conditions. Suppose that (i) A(t,x) is homogeneous with respect to x: (ii) for any P > 0, there exists an e = 6(A.P.6) > 0 such that a(t,r) g_er for all r g_p where a(t,r) = sup |P(t.x)] and 6 = m(A) ‘x‘gr is the measure of A: 80 (iii) L : Cn 4 Rn is linear and continuous. Then, (1) has unique solution x(t) 0 implies that (3) has at least one solution ¢(t) ‘with ]m] < p provided 3 is small enough. Similarly, applying Corollary 3.2, we have Corollary 3.5. Let A,P and Q : AXE“ 4 cc(Rn) satisfy the Carathébdory conditions. Suppose that (i) A(t.X) (ii) is homogeneous with respect to x: for some positive number p > 0, there exist an e = g(A.P.6) > 0 such that a(t,p) = sup |P(t,x)| g_eP Ix ISP where 5 = m(A) is the measure of A: (iii) L : Cn 4 Rn is linear and continuous. Then, (1) has unique solution x(t) 0 implies that (3) has at least one solution ¢(t) 'with ‘¢] < p provided a is small enough. From Corollary 3.3 and the way we prove Theorem 3.2, there follows immediately Corollary 3.6. Let A,B,P and Q : Aan 4 cc(Rn) satisfy the Caratheodory conditions. Suppose that (i) A(t,x) is homogeneous with respect to x: (ii) B(t,x) c:K(t) where K(t) is a ball in Rn centered at O with radius [x(t)] such that ‘K(t)‘ is integrable over A; 81 (iii) gjgégl-4 O uniformly in t as r 4 m, where a(t,r) = sup [P(t,x)]: 1X13? (iv) L : Cn 4 Rn is linear and continuous. Then, (1) has unique solution x(t) s 0 implies that (2) has at least one solution for any a e R". §3. Some applications: (a) Periodic solutions: Consider the following contingent differential equations: (4) k(t) e A(t.X(t)): (5) k(t) e n(t.x(t)) + B(t.x(t)>. and w) iu)eMmmu)+mmmu)+mmnuL Theorem 3.3. Let A,B and P : Ran 4 cc(Rn) be T-periodic in R ‘with T > 0 and satisfy the Caratheodory conditions. Suppose that (i) A(t,x) is homogeneous with respect to x: (ii) B(t,x) c:K(t) where K(t) is a ball in Rn centered at O 'with radius [K(t)] such that ]K(t)| is integrable over [s,s+T] for any 3 e R: (iii) P(t,x) is Lipschitzian at O 6 Rn with Lipschitz content 9, i.e. |P(t,x)| g 9|x] for any (t,x) e Ran. 82 Then, (4) has only trivial T-periodic solution implies that (6) has at least one T-periodic solution provided 9 is small enough. Proof: Without loss of generality, we can restrict our consideration of orientor fields on A = [0,T]. Define L : CD 4 Rn by Lx = x(O) - x(T). Clearly, L is linear and continuous. Let a(t,r) = sup ]P(t,x)\ as before. By Ix ‘53-“ assumption (iii), we have a(t,r) = sup \P(t,x)| gDsup e‘x] = or 1" \Sr 1" L9“ for any r > 0. It follows from Theorem 3.2 with a=0 and Q(t,x) = A(t,x) + B(t,x) + P(t,x) that (5) has at least one solution m(t) defined on [0,T] and satisfying ago) - qKT) = 0 provided 9 is small enough. Define g(t) : R 4 Rn by g(t) = cp(3) where s = ttkT. s e [0,T) and k is some integer. E is continuous since m is continuous and ¢(O) = m(T). 8 is T-periodic by the way we define it. Clearly, a is a solution of (5) since g is a solution of (5) on [0,T] and A,B and P are T-periodic in R. [3 Following the same way of proof of the above theorem and applying Corollary 3.5, we have 83 Corollary_3.7. Let A and P : Ran 4 cc(Rg) be T-periodic in R with T > O and satisfy the CaratheOdory conditions. Suppose that (i) A(t,x) is homogeneous with respect to x: (ii) there exists a e > 0 such that |P(t,x)| g_ep for all t E R and ]x‘ g.p where p is some positive number. Then, (4) has only trivial T-periodic solution implies that (5) has at least one T-periodic solution ¢(t) ‘with |¢| < p provided 9 is small enough. Remark 3.6. Theorem 3.3 and Corollary 3.7 generalize Theorem 2.1 and Theorem 2.2, respectively, in [12] (see p.413) from perturbed linear ordinary differential equations to perturbed homogeneous contingent differential equations. However, in Theorem 3.3, we lose the uniqueness. (b) Optimal solutions: PrOpOsition 3.2. Let A,B,P,Q and L be defined and satisfy all the conditions (i)-(iv) as in Theorem 3.2 with e sufficiently small and A having unique solution x(t) a 0. Then the set of all solutions of (2) is a non- empty compact set in Cu. Proof: The non-emptiness of the set of solutions of (2) is guaranteed by Theorem 3.2. To show it is compact, 84 it is equivalent to show that it is sequentially compact O n I 0 Since C is a metric space. Let [x ] be a sequence of solutions of (2). n n=1 Then, xn(t) 6 (S)£: Q(s,xn(s))ds + an - a + xn(0) c (8)]: A(s,xn(s))ds + an - xn(0) + (8)]: K(s)ds + (3)]: P(xn(s))ds - a ~ for all t e [0,T], where P is defined as in the proof of Theorem 3.2. By Lemma 3.1, we have ‘x“‘ = t:?8.T]‘xn(t)‘ _<_ d(J‘: [K(s) ]ds + I: |§(xn(s)) ]ds + |a]) _<_a(§+ eT|xn\ + (ap for some a > 0. Choosing e < §%-, we have |xn\ g 2a(§ + ]a]) for all n = 1,2,---. Hence, {xn}:;1 is uniformly bounded. Moreover, t Xn(t) - xn(s) e (S) j‘s Q(r.xn(r))dr for any t,s e [0,T]. Since [xn}:;l is uniformly bounded and Q satisfies the Caratheodory conditions, there exists a a function m(r) which is integrable over [0,T] such that [Q(r.xn(r)) \ 3, mm for all r e [0,T] and n = l,2,---. Hence, 85 \xn(t)-xn(s)| g l]: m(r)dr| _g ]M(t)-M(s) |. As M e AC([O,T]). [xn}:;1 is equi-continuous on [0,T]. It follows from the Ascoli lemma that [xn}:;1 a J has a subsequence {xn.}j=l such that xn. 4xo uniformly on [0,T]. Now, for any t e [0,T], x0(t) = lim sup xn (t) 1*“ j t 6 lim sup((S) Q(s,x (s))ds + Lx - a + x (0)) j... {0 “j “j "j c 1im sup(S)I6 Q(s,xn.(s))ds + on - a + xo(0) 3*: 3 c (8)]:o 1im sup Q(s,xn.(s))ds + on - a + x0(0) j-w J c: (S) I: Q(s,xo(s))ds + Lx - a + xo(0). 0 Therefore, x0 is indeed a solution of (2). [3 As we know that any real valued lower (resp. upper) semi-continuous function assumes minimum (resp. maximum) on a compact set, from PrOpOsition 3.2 there follows immediately Theorem 3.4. (existence of optimal solutions). Let A,B,P,Q and L be defined and satisfy all the conditions as in Theorem 3.2. Furthermore, a lower (resp. upper) semi- continuous functional T : Cn 4 R is given. Then, if (1) has unique solution x(t) a 0, for any a e Rn there exists a solution of (2) which minimizes (resp. maximizes) T provided 6 is small enough. 86 Remark 3.7. Similar existence theorems of Optimal solutions can be formulated corresponding to Theorem 3.3 and Corollaries 3.4-3.7. Before discussing further applications, we give Definition 3.2. An orientor field F . Aan 4 cc(Rn) is said to satisfy the strong Caratheodory conditions if (i) F(t,x) is upper semi—continuous in t in the sense of Definition 1.4 for each fixed t 6 A: (ii) F(t,x) is measurable in t for each fixed x E Rn: (iii) there exist functions m(t) and *(t) which are integrable over A such that [F(t,x)] g ¢(t) |x| + fit) for all (t,x) e Aan. Remark 3.8. It is clear that a mapping satisfying the strong Caratheodory conditions must satisfy the Caratheodory conditions. The converse is not true even when the orientor field is compact. Consider F(t,x) : AxR 4 R defined by F(t,x) = tex. F satisfies the CaratheOdory conditions and is compact but it does not satisfy the strong Caratheodory conditions. Therefore, all the results in this chapter hold true for A,B,P and Q satisfying the strong Caratheodory conditions. 87 (C) Nicoletti prOblem: Given 0 g_t1 g,t2 g... g tn 3_T and a = (a1,a2,...,an) e Rn, we shall consider the existence of a solution x(t) = (x1(t),x2(t),...,xn(t)) of (6) which satisfies the Nicoletti conditions [28]: i (7) xi(ti) = a. , i = 1,2,...,n. Also, we consider the functional T : Cn 4.R defined by (a) T(x) = I: |x(t)|dt. Lemma 3.5. (Lasota and Olech: [22]). Suppose the function m from [0,T] into R is Lebesgue integrable and non-negative. Consider the differential inequality for an n vector valued function (9) [x(t) | g m(t) |x(t)| , o g t g T and the boundary value conditions (10) X. (ti) = 0' 0 < ti _<_ T, i = 1,2,000'no Rafi )4 T If ]‘0 cp(t)dt < , then x(t) s O is the unique solution of (9) and (10). Theorem 3.5. Let A,B and P : [0,T]xRn 4 cc(Rn) satisfy the strong Caratheodory conditions. Suppose that . . T w (i) [A(t,x) | _<_ m(t) Ix] + y(t) w1th Io m(t)dt < 5; (ii) B(t,x) c:K(t) where K(t) is a ball in Rn centered at 0 ‘with radius |K(t)| such that [K(t)] is integrable over [0,T]: 88 P t x (iii) x 4 O uniformly in t as [x] 4 a. Then, (6) has a solution which satisfies the Nicoletti conditions (7) and also minimizes (or maximizes) the functional (8). Proof: Define A;B' and Q : [O,T]x‘R‘n 4 cc(Rn) by A(t,x) = [u E Rn : [u] _<_ m(t) ]x]}, r B(t,x) = B(t,x) + [u e Rn : ]u] _<_ ¢(t)}, I and Q(t,x) = A(t,x) + B(t,x) + P(t,x). Clearly, Q(t,x) cK(t,x) + B(t,x) + P(t,x). Let n n . L . C 4»R defined by Lx — (x1(t1),x2(t2),...,xn(tn)). One can see easily that A,B,P,Q and L satisfy all the assumptions of Corollary 3.6. Since T is a continuous functional, our proof follows immediately from Lemma 3.5, Theorem 3.4, and Remark 3.7. D (d) Aperiodic boundary value prOblem: Here, we shall consider the existence of a solution x(t) of (6) which satisfies the aperiodic boundary condition: (11) x(O) + xx(T) = 0 where A > O and also minimizes (or maximizes) the functional (8). Another lemma of differential inequality is needed: Lemma 3.6. (Kasprzyk and Myjak: [16]). If ¢(t) 2_O T and £0 m(t)dt < (n2 + log2 x)1/2, then x(t) s O is the unique solution of (9) and (11). 89 If we set Lx = x(O) + xx(l) and a=0, a result analogous to Theorem 3.5 follows from Lemma 3.6 and Theorem 3.4. As the proof is similar, we give the statement as follows: Theorem 3.6. Let A,B and P : [03]an 4 cc(Rn) satisfy the strong Caratheodory conditions. Suppose that (i) [A(t,x)] g m(t) |x| + ((t) with T . yo smdt -< (1r2 + 1092 111/2: 4 (ii) B(t,x) c:K(t) where K(t) is a ball in Rn centered at 0 ‘with radius |K(t)] such that [K(t)] is integrable over [0,T]: (iii) lngTéLl 4 O uniformly in t as ]x] 4 o. Then, (6) has a solution which satisfies the aperiodic condition (11) and also minimizes (or maximizes) the functional (8). Chapter IV PERIODIC SOLUTIONS OF CONTINGENT FUNCTIONAL EQUATIONS In control problems, it may happen that the control system is described by a functional differential equation. Therefore, by eliminating the control term, we Obtain a contingent functional differential equation. In this chapter, we shall consider the periodic solutions of such equations and formulate an existence theorem for Fredholm's alternative analogous to Theorem 3.2. Suppose r 2.0 is a given real number, R = (~a,m) and Rn is the n-dimensional Euclidean space with norm ].]. Let Cr = C([-r,O],Rp) be the Banach space of all continuous functions from [-r,0] into RF with the norm of each element m, “T“ = sup |m(e)]. For any function x E C(R,Rn) and t e R, w:_g§2ine a function xt : [-r,O] 4 Rn by xt( e) = K(t+e) where -r g_e 3.0. Clearly, xt 6 Cr' The function xt can be considered as the segment of x(T) defined on [t-r,t] and translated to [-r,0]. 9O 91 Definition 4.1. A mapping F : RxCr 4 Comp (Rn) is called a functional orientor field. And a relation of the form (F) x(t) e F(t,xt) is called a Contingent functional differential equation. Definition 4.2. For a fixed x 6 C(R,R”), we say a function F(t,xt) : RxCr 4 Comp (R9) is measurable in t ~ if the function F : R 4 Comp (Rn) defined by F(t) = F(t,xt) is measurable. Definition 4.3. For a fixed t E R, ‘we say a I ' n I function F(t,xt) from RxCr into Comp (R) is 32%;. semi-continuous in x (in the sense of metric) if, for any a > O and any x 6 C(R,Rfi), there exists a 6 > 0 such that F(t.yt) c Be(F(t.xt)) n . for all y e C(R,R ) with “yt-xt“ < 5. Definition 4.4. we say a functional orientor field F(t,xt) : RxCr 4'Comp (Rn) satisfies the Caratheodory conditions if (i) F(t,xt) is measurable in t for each fixed x 6 C(R,Rn): (ii) F(t,xt) is upper semi-continuous in x for each fixed t e R: 92 (iii) for any closed and bounded subset D of RxCr. |F(D)] is bounded. The equation (F) is said of the Carathéodoryytype if its functional orientor field F satisfies the Caratheodory conditions. Remark 4.1. It follows from condition (iii) of Definition 4.4 that a functional orientor field satisfying the Caratheodory conditions must be compact. Definition 4.5. A function x(t) is said to be a solution of (F) in the sense of Marchand if there exist t0 6 R and A > 0 such that x(t) e C([to-r,to+A],Rg) and D*x(t) c F(t,xt) for almost every t e [to,to+A]. Definition 4.6. A function x(t) is said to be a solution ofng) in the sense of wazewski if there exist to e R and A > 0 such that n x(t) e C([to-r,to+A],R ), x(t) 6 AC([to,to+A]) and x(t) e F(t,xt) for almost every t e [to,to+A]. Definition 4.7. A function x(t) is said to be a solution of (F) if there exist t E R and A > 0 such that O 93 X(t) E C([to’ro to+A]pRn) I x(t) E AC([to,to+A]) and x(t) e x(to) + (3)]:0 F(s,xs)ds for t e [to,to+A]. The following prOposition shows that under certain conditions the solutions of (F) defined above are all equivalent: PrOposition 4.1. (Kikuchi [18]). Let P(F). Y(F) and T(F) be the collection Of all solutions of (F) with respect to Definitions 4.5, 4.6 and 4.7 respectively. Suppose that (i) F(t,xt) satisfies the Caratheodory conditions: and (ii) F(t,xt) e cc(Rn) for each (t,xt) e RxCr. Then, P(F) = Y(F) = T(F). we shall consider the periodic solution of the following contingent functional differential equations of retarded type: (1) x(t) e A(t.xt) (2) x(t) e 0(t.xt) where Q(t,xt) c A(t,xt) + B(t,xt) + P(t,xt). Theorem 4.1. (Fredholm's alternative). Let A,B, P and Q : Rxcr 4 cc(Rn) satisfy the Caratheodory conditions 94 and be T-periodic in R for some T > 0. Suppose that (i) A(t,xt) is homogeneous with respect to x: i.e. A(t.lxt) = m(t.xt) for all A e R and xt 6 Cr: (ii) B(t,xt) c:K(t) 'where K(t) is a ball in Rn centered at 0 ‘with radius [K(t)] such that [K(t)] is integrable over any T-interval [t,t+T]: (iii) there exists an e = C(A,p.T) > 0 such that a(t,m) _<_ em for all mZp > 0 ‘where p >10 is arbitrary and a(t,m) = sup [P(t,xtH. t‘Sm Then, (1) has x(t) s O as a unique T-periodic solution implies that (2) has at least one T-periodic solution, provided a is small enough. 2399;; Let 6% denote the set of all continuous T-periodic functions from R into Rn, K(Cn) denote the set of all non-empty convex subsets of CD and Li°c(R) denote the set of all functions from R into R? which are integrable over any finite interval in R. (i) Let P*(t,xt) : RxCr 41cc(Rg) be defined by ‘- 95 * * Clearly, P is well-defined and P(t,xt) : P (t,xt) for * any (t,xt) e RxCr. One can check easily that P satisfies the Caratheodory conditions. (ii) Define the Operators F,G.H and J : 9 4 T K(Cn) by F(x) = (j: e‘(t‘3)[x(s)+u(s)]ds : u e Li°°(n) and ,1 u(s) e A(s,xs) for all s e R], G(x) = []‘t e-(t-S) u(s)ds : u e Lioc(R) and -w u(s) 6 K(s) for all s 6 R}, H(x) = {]Ew e-(t-S) u(s)ds : u e Li°c(R) and u(s) e P*(s,xs) for all s e R}, and J(x) = [It e-(t-s)[x(s)+u(sX]ds: u 6 Li°c(R) and u(s) e Q(s,xs) for all s 6 R}. From our hypothesis that B(t,xt) is T-periodic in R, we can assume without loss of generality that the ball K(t) which contains B(t,xt) is also T-periodic. All the imprOper integrals defined above converge since x,A,K,P* and Q are periodic (hence bounded by (iii) of Definition 4.4). For each x e 6%, A(t,xt) is measurable in t by (i) of Definition 4.3 and is integrably bounded by (iii) of Definition 4.4 and the periodicity of A(t,xt). Hence, F(x) is not empty. The convexity follows immediately from the l fact that A is convex valued. Therefore, F is well—defined. ! Similarly, G,H and J are well-defined. } i 96 ~~~ (iii) Let us define F,G,H and 3': 6% 4 cc(éfi) by fix) = F(x) no . S(x) = G(x) neT H(x) = H(x) n er and 36:) = J(x) n 9.1,. Since for each x e 6% A(t,xt) is T-periodic, measurable and bounded, there exists a T-periodic function fx(t) E Li°c(R) such that fx(t) e A(t,xt) for all t e R. This is possible because we can have a measurable selection fx(t) on [0,T] first with fx(O) = fx(T) and then duplicate it on the intervals [kT,(k+l)T] where k is a non-zero integer. Let EX (t) : R 4 Rn defined by fx(t) = If” e-(t-S)[x(s)+f*(s)]ds. It is clear that f; E F(x) n 6%. Hence, F(x) is not empty. Since 9% is a convex subspace of Cn, F(x) is convex. By using the Ascoli lemma and Lemma 3.3 as we did in the proof of Proposition 3.1, one finds that the set F(x) “0',” = {fx(t) [[O'T] : fx 6 F(XH is compact in C([O,T],Rn). However, since Fkx) is a family of T-periodic continuous functions, a sequence of functions in F(x) converges uniformly in [0,T] implies that it converges uniformly in R and the limit function is also T-periodic. Hence, for each x e 9 , F(x) is indeed compact in 4%. Therefore, F is well—defined. Similarly, G,H' and 3' are all well-defined. ...... 4, . 97 (iv) F,G,H and J are compact: Let D = {x e 65 : [x] g_K] be a bounded subset of 6%. We want to show that F(D) is relatively compact. In a metric o O 0 w- 0 0 space, it is equivalent to show that F(D) is sequentially compact. Let {Yn};=1 C.FYD) c F(D). Then, Yn(t) E (8)1:1, e-(t-s) [xn(s)4-A(s,xn )]ds 3 where xn e D. Since D is bounded and A(t,xt) is periodic in t, it follows from Remark 4.1 that |A(s,xn )] g I? for s all s 6 R and n = l,2,---. Hence, \ynm] g If e- O. (x) For every (t,xt) e RxCr, we have Q(t,xt) c A(t,xt) + B(t,xt) + P(t,xt) * C A(t,xt) + K(t) + P (t,xt). * For each x 6 6’. K(t) and P (t,xt) are balls in Rn centered at 0 with radii ]K(t)| and eHXtH respectively. 101 Hence, by Lemma 3.4, we have J(x) : F(x) + G(x) + H(x) for any x 6 9T' It follows that for any j E J(x) n 6% x we have jx(t) = fx(t) + gx(t) + hx(t) where fx 6 F(x), g e G(x), hx € H(x) and t E R. Since X j E 6%. ‘we can restrict our consideration on [0,T]. Let x fx(t) = If“ e-(t-s)[x(s)+u(s)]ds, 9x(t) = It e-(t-s) v(s)ds, and hx(t) = ]‘t {(1513) w(s)ds, ...-Q where u,v,w e L1([O,T]) and u(s) e A(s,xs), V(s) e K(s) and 'w(s) e P*(s,xs) for all s E [0,T]. Without loss of generality, we may assume that u(O) = u(T), V(O) = V(T) and W(O) = W(T) o , w * * n Define u , v and w : R 4 R by 'k u (t) u(s) for t = kT + s 'k v (t) = v(s) for t kT + s 'k 'w (t) = w(s) for t = kT + s ' * where k is an integer and s e [0,T). Clearly, u , v* 'k and w are all T-periodic by the way we define them and * * * loc . . * u ,v ,w 6 L1 (R). It is also eVident that u (t) e A(t,xt). v*(t) E K(t) and w1(t) e P*(t,xt) for all t 6 R. Now, ~ . ~ ~ n define fx' gX and hx . R 4 R by 102 f;(t)= If” e-(t-S)[x(s)+u*(s)]ds, 3x“) " It .1“: S) v (s)ds, and fi¥(t) = It ‘(t‘3) w (s)ds. One can see esaily that f; e F(x), E; e G(x) and hx e H(x). It is also clear that Jx(t) = fx(t) + gx 0 such that a(t,m) g cm. for all m g'p where p is a sufficiently large number and a(t,m) = su |P(t,x )|. “le3“ t From Corollary 3.3 and the way we prove Theorem 4.1, there follows immediately 103 Corollary 4.1. Let A,B,P and Q : Rxcr 4 cc(Rn) satisfy the Caratheodory conditions and be T-periodic in R for some T > 0. Suppose that (i) A(t,xt) is homogeneous with respect to x: (ii) B(t,xt) c K(t) where K(t) is a ball in RF centered at O ‘with radius [K(t)] such that [K(t)‘ is integrable over any T-interval [t,t+T]: (iii) QLELEL 4 O uniformly in t as m 4 a, where a(t,m) = sup [P(t,xt)|. HXtHSF Then, (1) has unique T-periodic solution x(t) e 0 implies that (2) has at least one T-periodic solution. Remark 4.3. Corollary 4.1 is a generalization of a recent result by R. Funnel (see [9]). In (2), when B(t,xt) E O, we have the following equation: (3) x(t) e Q(t,xt) where Q(t,xt) c:A(t,xt) + P(t,xt). In this case, it is clear that G(x) s G(x) E [0]. Therefore, following the same proof of Theorem 4.1 and applying Corollary 3.5 instead of Theorem 3.1, we Obtain Corollary‘4.2. Let A,P and Q : RxCr 4 cc(Rn) satisfy the Caratheodory conditions and be T-periodic in R for some T > 0. Suppose that 104 (i) A(t,xt) is homogeneous with respect to x; (ii) there exists a e > 0 such that [P(t.xt)[ _<_ 99 for all t 6 R and “xtu gDp, where p is some positive number. Then, (1) has only trivial T-periodic solution implies that (3) has at least one T—periodic solution m(t) 'with [m[ < p provided a is small enough. Remark 4.4. 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