STABILITY AND INVARIANCE 0F FUNCTIONAL : DIFFERENTIAL EQUATIONS r Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY . RENG-SONG L0 _ ' 1975 ’ .- g __.___-—.4....—. . l Ll Ill/l ll ll ll I'll l”! l l]! Mill "1le W " It E 1 Y 1293 ‘ . . .. .. C‘ 9- «in..u.gttfl u' see This is to certify that the thesis entitled "Stability and Invariance of Functional Differential Equations" presented by Mr. Reng Song Lo has been accepted towards fulfillment of the requirements for Ph. D. degree in Mathematics 'T\ ‘ C37 X MI} 5 Al-Uw \J M ' rifgssor Date June 5, 1975 0-7639 \" ‘I ABSTRACT STABILITY AND INVARIANCE OF FUNCTIONAL DIFFERENTIAL EQUATIONS By Reng-Song Lo In recent years, Liapunov's method has been successively generalized to functional differential equations of retarded type by using Liapunov functionals. However, in many cases the problems are still open. For example, we already had a complete characteriza— tion of integral stability for ordinary differential equations by a Lipschitz Liapunov function which obeys certain bounds. But the problem in functional differential equations is still open. Another fundamental problem in differential equations is the characterization of invariance sets. In ordinary differential equations, it is known that invariance of a closed set is equivalent to a notion called subtangent. But the corresponding result in functional differential equations was not known. In this thesis, we investigate the above two Open problems. First in the case of integral stability, we found the usual approach of Liapunov's method is not very useful. Although one can easily get a lower semi-continuous functional which obeys certain bounds, but "continuity" and "Lipschitz" prOperties are extremely difficult to obtain. 0n the other hand, in the case of an invariance set, the hereditary nature of the equation also prevents one from doing Reng-Song Lo a straightforward generalization to functional differential equations. For this purpose, a new Liapunov's theorem based on a class of lower semi-continuous non-Lipschitz functionals was developed. In particular, the usual Laipunov comparison principle hold true for this class of Liapunov functionals. Complete characterizations of in- tegral stability and sets of invariance are obtained using the Laipunov theory developed earlier. As an application to the in- variance characterization we give an invariance principle for a class of asymptotically autonomous systems. STABILITY AND INVARIANCE OF FUNCTIONAL DIFFERENTIAL EQUATIONS By Reng-Song Lo A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1975 ACKNOWLEDGEMENTS I would like to express my deepest appreciation to my thesis advisors, Professor P.K.Wong and'Professor S.N. Chow, for their patient guidance, constructive suggestions and numerous discussions during the preparation of this thesis. I also would like to express my hearty thanks to those members in the department for their kindness, generosity, helpful- ness that made my stay here a very pleasant one. My thanks also go to Mrs. Noralee Burkhardt for her excellent typing of this thesis. ii TABLE OF CONTENTS Page IntrOduction ......OOOOOO......OOIOOOOOOOOOIOO00.000.00.000. 1 0.1 Definition of FDE of Retarded Type and Initial Value Problems .................................. 2 0.2 Liapunov Functions ...................... ........ 5 Chapter I. THE FIRST COMPARISON THEOREM FOR FDE .. ....... . 9 1.1 Liapunov Functional and Lower Semi-Continuity ... 9 1.2 A Comparison Principle for FDE of Retarded Type . 10 1.3 C1 Locally Lipschitzian ........................ 14 Chapter II. INTEGRAL STABILITY OF FDE ... ..... . ....... .... 17 2.1 Definition of Integral Stability ................ 17 2.2 Definition and PrOperties of V Function ...... 19 2.3 The V Functional and its Speckal Properties ... 21 2.4 Characterization of Integral Stability .......... 36 Chapter III. THE SECOND COMPARISON THEOREM FOR FDE ..... .. 42 * _ 3.1 Definition of V and V ....................... 42 3.2 A Comparison Principle for FDE .................. 46 3.3 Proof of the Comparison Theorem ................. 46 Chapter IV. SEMI-INVARIANCE OF FDE OF RETARDED TYPE . ..... 57 4.1 Semi-Invariance ................................. 57 4.2 Asymptotically Autonomous Systems ............... 61 Bibliography ......OOOOOOOOOOOOOO0.00.0000.........OOOOOOOO 67 iii INTRODUCTION A relation of the form (E) x'(t) = f(t, X(t)), where x = x(t) is a d-dimensional vector value function defined on a real interval and f(t,x) is a function from a certain region of R X Rd into Rd, is called an ordinary differential equation. The function f is called a vector field and the solution of (E) are integral curves whose tangent is prescribed by the vector field f. In most classical applications, the behavior of many phenomena are assumed to be governed by such ordinary differential equations. Implicit in this assumption is that the future behavior is uniquely determined by the present state of the system alone and is independent of its past history. There is another type of differential equations, known as functional differential equations (FDE), in which the past history influences in a significant way on the future behavior of the system. It is known [see [6], [7], [19]] that such equations arise in many areas of application. The systems under study are better represented by FDE than by ordinary differential equations. Historically FDE was first encountered in the late eighteenth century, however, very little was done during the nineteenth and early twentieth century. For the last forty years and especially the last twenty years, the subject has developed into one of the most active branches of differential equations. Much of the stimulus for this was due to the work of Volterra [17], who was interested in certain ecological models, and Krasovskii [10], who was interested in the theory of control, and other mathematicians who had encountered the problem in several different fields. A good reference for FDE is Hale [8]. In this thesis we shall study three fundamental problems of functional differential equa- tions of retarded type, namely, Liapunov theory, the characterization of invariance of a set and integral stability. 0.1. Definition of FDE of Retarded Type and Initial Value Problems Let Rd be the real Euclidean d-space and \x\ be any nonm. Let y > 0 and, C = C[-y,0] be the Banach space of all continuous functions m: [-r,0] —'Rd’ with the usual sup norm “q“ = sup{\¢(e)‘: -y s e s 0]. Given a continuous function x: [-y + o, o + A) ... Rd, 0 6 R, A > 0, we define for each t E [0, o'+ A) an element xt E C by xt(e) = x(t + e), -y s e s O. Let D CR X C be open and f: D «Rd be continuous. A functional differential equation of retarded type is a functional relation of the form (0.1) x'(t) = f(t, xt) Let (o,q9 E D. A solution x = x(t, o, m) of (0.1) -through (o,¢) is an absolutely continuous function defined on [-y'+ a, 0'+ A) for some A >,o such that (0.2) x0 = (p , (0.3) X'(o+) = f(a. x0) . and (0.4) x'(t) = f(t, xt), o < t < A, where x'(d+) denotes the right hand derivative of x at t = 0- After defining FDE of retarded type, the immediate questions that one may ask are: (i) When does a solution exist? (ii) When do the equations have uniqueness property? (iii) Does the family of solutions have certain properties concerning convergence and continuous dependence with respect to the initial condition? The answer to the above questions may be summarized by the following theorems whose proofs are found in most standard references, c.f. [8]. Theorem 0.1 (Existence). Suppose U is an open set in R X C[-y,0] and f: U «IRd is continuous. If (o3q0 E U, then there is a solution of (0.1) passing through (o,m)- A function f(t,m) defined on R X C[-y,0] is called Lipschitzian in m on U CIR X C[-y,0], if there exists a constant L > 0 such that HUMP) - f(tml 5L ' \lqJ - Y\\ for all (t,¢), (t,Y) E U. Theorem 0.2 (Uniqueness). Suppose U is an open set in R X C[-y,0], f: U a Rd is continuous, and f(t,¢) is Lipschitzian in Q on each compact set in U. If (o,q9 E U, then there is a unique solution of (0.1) with initial value (g,¢). Definition 0.1. A continuous function x: [-y + t, b) —.Rd which is absolutely continuous for t < s <‘b is said to be non- continuable with respect to an open set D CIR X C[-y,0] if, for each -y +-t < s < b, (s, xs) 6 D and for each closed bounded set U C1D, there exists t < tU < b such that (3, x8) i U for all tU < s < b For a function x, let Dx denote the domain of this function. Then we have the following: Theorem 0.3 (Convergence). Let f: U 4 Rd be continuous on U czR X C[-y,0] and let \f(t,q9‘ be bounded on each closed and bounded subset of U. Suppose [xn(°)} is a sequence of non- continuable functions on U such that D “:3 [to - y, to + an), for some an >'0. And ¢P(-), qKo) are :ontinuous functions such that qP(-) a m(-) uniformly on [-y,0], where mn(t - to) = xn(t), to - y S t s to. Define a sequence of func- tions {Gn(-)] by Gn(t) =- xn(t) - .p“(o) - I: f(x, x:)ds for [to, co) nn H. o x Assume that for each closed, bounded set B CZU, there exists a sequence {Bn(B)], an a 0 as n a m, such that if (t, x2) 6 B for t between to and Vn, then \Gn(Vn)l S Bn(B). Then there exists a non-continuable function x(-) and a subsequence n . {x j(o)] of {xn(-)] such that n (i) xtj a xt uniformly on compact subset of D , xt and (11) x(t) = (9(0) + flares, xs)ds for t 2 to X = t W 0 Proof: See [4]. 0.2. Liapunov Functions Let f: I X D aIRd be continuous, where D is an open set in Rd, and let I denote the interval 0 s t s m. As in the usual Liapunov theory, see [22], we consider a continuous scalar function V(t,x) defined on an open set S in I X D. Further- more, we assume that V(t,x) satisfies locally a Lipschitz con- dition with respect to x. That is, for each point (to, x0) in S, there exists a neighborhood U = U(to, x0) and a positive number L(U) such that \V(t ,X) - V(t :Y)‘ S L(U)‘X ' 3" for any (t,x) 6 U, (t,Y) 6 U. We shall denote by V 6 Lipo(x) for this fact. Corresponding to V(t,x), we define the function . _ —__ .l _ (0.5) V(E)(t,x) - éfg+ h {V(t +-h, x + hf(t,x)) V(t,x)} . Let x(t) be a continuous and differentiable function defined for S 2 t, denote by V'(t, x(t)) the upper right-hand derivative of V(t, x(t)), that is, (0.6) V'(t, x(t)) = IE -1- {V(t + h, x(t + h)) - V(t,x)] had+‘h then we have the following, c.f. [22]: Lemma 0.1. Let x = x(t) be a solution of (0.1) which stays in S. Then V'(t, x(t)) = V(E)(t.X) As is well known, 11': 6(3) (t ,x) s 0 then by Lenma 0.1 V'(t, x(t)) s 0. The function V(t, x(t)) is therefore a non- increasing function of t along a solution of ( E ). Conversely, if V(t,x) is nonincreasing along a solution of ( E ), then we have V(E)(t,x) S 0 . By a Liapunov function, in usual Liapunov theory, we always mean a continuous scalardvalued function such that V 6 Lipo(x). The following is one of the simplest forms of a very general com- parison principle, c.f. [22]. Definition 0.2. For the case d = 1 in the equation (E), if qfit is a solution of (E) passing through (T,§), existing on some interval I containing 7, with the property that every other solution m of (E) passing through (T,§) and existing on I is such that cp(t) s cpM(t) ('2 E I) then fin is called a maximum solution of (E) on I passing through (T,§). Theorem 0.1. Let V(t,x) be a Liapunov function for (E). Suppose there exists a realdvalued continuous function w(t,u) defined for 0 s t < m, \ul < an where u is a scalar, such that for all (t,x) 6 I X D (0.7) V (t,x) szt, V(t,x)) (B) Let U(t, to, uo) be the maximal solution of ' = = (0-8) p V(t, u) . ”o V(to. x0) and x(t, to, x0) be a solution of (E). Then (0.9) V(t, x(t, to, xo)) s U(t, to, x0) for all t 2 t0 for which both x(t, to, x0) and U(t, to, x0) are defined. The comparison principle has been widely used in dealing with a variety of qualitative problems. It is a very important tool as it reduces the problem of determining the behavior of solution of (E) to the solution of a scalar equation (0.8) and properties of the Liapunov function V. In this thesis, we shall first develop the Liapunov theory by using lower semi-continuous Liapunov functionals for FDE of the retarded type. In the later chapters we shall investigate the problems of integral stability and invariance of a set in which the Liapunov theory developed earlier will play an important role. In Chapter I, we shall define the notion of "derivative along a solution" for a class of functionals which are assumed only to be lower semi-continuous and prove a comparison theorem analogous to‘Theorem 0.1 for FDE. In Chapter II, we shall deal with the problem of the integral stability for FDE. In Chapter III, we shall initiate another type of derivative for the same class of functionals as in Chapter I, and we shall also prove another comparison theorem analogous to'Theorem 0.1 for FDE. In Chapter IV, we shall deal with the characterization of invariance of a set for FDE. For reading convenience, a hollow square [:1 is used to signal the end of a proof. CHAPTER I THE FIRST COMPARISON THEOREM FOR FDE 1.1. Liapunov Functional and Lower semi-continuity In this chapter, we shall give the definition of "derivative along a solution" for a Liapunov functional, which is a natural extension of (0.6), and prove a comparison principle similar to Theorem 0.1. Consider the functional differential equation (0.1). Let x(-) be a solution of (0.1) through (a,¢) and let V: R X C[-y,0] d»R* where R* denotes the extended real numbers, we shall refer to ‘V as a Liapunov functional. ‘Throughout this thesis, the term "Liapunov functional" means only a functional substantially different from the usual sense of Liapunov function in which both continuity and local Lipschitz condition are assumed. Define (1.1) we, "a’ = 111L331» g1; tv 0, such that xt = (p. We define the upper right -hand derivative of V(t,cp) along the function X(t) by V};(t,cp) = TE+ 5- {V(t + h, xt+h) - V(t,cp)] h—~o then we have the following theorem. Theorem 1.1. Let p(t) be a continuous function on [to, a], where a > to, and let V(t,cp) be a lower semi-continuous Liapunov functional on [0,co) X CA[-r,0]. If x: [to - y, a) .. R"1 is an absolutely continuous function such that (61) Kt =cp, O (b) V;(t.xt)-<-p(t), VtOSt o . n S 11243116 V(tj(n)’ xtn ) V(to, xto) J(n) 12 On the other hand we see from (1.3) and (1.4) that n n V(ti, x n) - V(ti_1, x n ) t. t. 1 1-1 stat-.11) + If] (t‘i‘ - t‘? ) Substituting into (1.7), we get (1.8) V(T, ear) -V(to.. xt > O j(n) 5 lim inf 2 (p(t:_1) + int: - 911-1) mam i=1 j(n) n n n 1 s lfln inf 2 p(ti_1)(ti - ti-l) + lim sup ;'(t T1400 i=1 n—Doo I'l j(n) - to)’ Again, from (1.3), (1.4) and (1.5), we have n limt, =T,tr}-trf <1. J01) 1 1"]. n n-m Substituting into (1.8), we conclude that V(T: xT) -V(t0’ Xt) O n—too T . 1 5 ft p(t)dt + 11m sup E-(T - to) o s f: p(t)dt . [:1 0 Remark. In the proof of Theorem 1.1, only the assumption that x is continuous was used, absolutely continuous is not required. Lemma 1.2. For the sequence {t3}:=1 that was defined in (1.4), there exists an integer j(n) such that l3 1 T s T +"'- n S tn j(n) Proof. First we claim lim t: 2 T. Suppose not. Let n 1T” n b = lfin t. so that b <'T. If T E A (b , x ), then we assert n , 1 n n b 1am n n n that T E A (ti’ x n) for infinitely many values of i. To see ti this, we note that V(T, x) - V(t’f, x ) T 1 “ t. (1.9) 1im inf n 1 i-ooo T - t. 1 V(b ) V(tn x V(w. xT> - V(bn. xb) n Sn 1 s lim inf “ + lim sup l—too T - t i—m T - t V(T: X'T') 'V(bna Xb) V(bna Kb) S Q + lim sup n T - b . n l—roo T - t. 1 n V(ti’ x n) t. - 1im inf I: i—m T - ti But, bn = 1im t2, so it follows from (1.2) and (1.9), that we have 14m V(‘T x) V(I:n ) s " ., X _ T 1 t1; V(Te XT) V(b“: fin) (1.10) lun inf n s T _ b l—coo ’T - t, n 1 l < p(bn) + n o o n Furthermore, p is a continuous function and t. a b , so that 1 n from (1.10), we see that there must be a J1 such that n 11 V e ' V a (T xr) (ti xt.) lfin inf 1 < p(t?) + %' whenever j 2 J . tn 1' -m - 1 ‘T' I 14 Therefore there exists infinitely many values of i 2 J1 such that n V(Ta XT) - V(ti’ xtn) n l + — < p(ti) n ’ and I1 n t. < r < t. + l 1 , 1 n n n Thus 7 E A (ti’ x n) for infinitely many values of i. t H. And hence there exists a t such that Z": n n T€A(tk3xn) a tk and n l l n bn - tk < 2 (T - bn) < 2 (T tk) n But this contradicts the choice of t k 1, Since n n 1 n n n - t 2 — - - . tk 1 k 2 (T tk) >»bn tk so that tk 1 > bn Consequently we have 1 n T .imtiz . D l-coo 1.3. C1 Locally Lipschitzian For (p t CA[-y, 0], define the C norm of (p by l llcollcl = IIIII + ISYIn'Ide - The following property of the functional V(t,q9 is important, especially in the study of the behavior of solutions of perturbed systems. 15 Definition 1.3. Let V(t,q9 be a functional as before. We say V is C1 locally Lipschitzian if for every (to, qh) 6 D, where q)€ CA[-y, 0], there exists a neighborhood N(to, qb) of (t,q0 and a constant L = L(N(to, qB)) 2 0 such that (1.11) w(E, cpl) - V(E, cp2)\ s Lunl - ,pZIIC 1 for all (E, (p1), (E, ((32) e N(to, To) and nycpz e cA[-y, 0]. Lemma 1.3. For two continuous functions x(t,q9, y(T,m) with the right-hand derivatives such that xt = yT = q“ we have —T_ _L , +- , + git-l- O HXt+6(t:CP) - Y1+6(T,cp)‘.\ - ‘X (t ) " y (T )\ ° Proof. See [22], page 187. The following lemma is immediate. Lemma 1.4. If V(t,¢) is C1 locally Lipschitzian and x(t), y(t) are two absolutely continuous functions defined on o - y S t S aI+ o, a > 0, such that x0 = y0 = qg then I I l U + 0 + vx(O:(P) SVy(09CP) + 21“" (U) " y (C )‘9 where x'(d+), V'(of) denote the right hand derivative at t = o, and L is the constant in (1.11) at the point (g,q9. Proof. h) " V(CTCP)] 1 V'(c,¢) = lim sup - {V(o+ h, x X had+ h 6+ 1 S Miss-up h {V(o+ h, xdh) - V(O+ha yc-l-h)} ) ' V(C:CP)] 1 + lim sgp g; {Na + h. ydh ...o h 16 , . .1. . - s Vy(o.cp) + IIESgP h {L llxdlh thhllcll (1.12) 1 g V):(o,gp) + lim sup h {Luxodrh ‘ yo-l-h‘n hdo . 1 0th I . + ”£33.51" h {L 3‘0 Ix (e) - y (e)\de} . But from Lemma 1.3 we have . 1 ' + ' + (1.13) 111:ng g {L‘lxon - yflhlll = LIx (o) - y (o >\ . Substitute (1.13) into (1.12), we conclude that + + V;(o,cp) S V):(o,cp) + L\X'(o) - V'(o )\ + L‘X'(o+) - V'(o+)\ s vy'0 35>0 suchthat \f(t,Y)-f(t,cp)\ 0, any to 2 0 and any continuous function p: [to, m) a R, there exists a 6(a) > 0 such that “¢b“'< 6(6) , (pom < e for all and I: \p(t)\dt < 6(a) imply \y(t, to o t 2 to, where y(t, to, m6) denotes a solution of 19 (2.3) y'(t) = f(t, yt> + p(t) that passes through (to, ¢b>° It is the purpose of this chapter to give a necessary and sufficient condition for the zero solution of (2.2) to be integral stable. 2.2. Definition and Properties of VL Function For an open set U CZR X CA[-y,0] let V: U a R and de- note by N((T:Y)y 6) = {(taflp) e U: \t ' T! + “(P ' Y“ S O} for all (T, Y) E U and 6 > 0. Then the following is immediate. Lemma 2.1. 1im inf V(t,¢) 2 A if, and only if, for (t “PA-(T 3‘?) each s > 0 there is a 6 > 0 such that V(t,¢) 2 A - 3, whenever (tnp) 6 N(('r,\l'), 6)- Next we define the function VL: U -oR by (2.4) vL(t,¢) = lim inf V(T, ‘l’) , (Tsw)’*(t:CP) where (T,Y), (t,m) E U. Remark: Since U is an open subset of [o,m) X CA, there- a: fore for any (t,qD E U, there ex15ts a sequence {(tn, qh)}n=1 CZU such that lim V(tn, 9n) =VL(t. cp) n—m Then we have the following lemma. Lemma 2.2. The function V is a lower semi-continuous L function on U. 20 Proof. Let (t,q9 E U. We would like to prove 1im inf VL(T,Y) 2'VL(t,q9 V (t,q» E U , (T filo-'03 TC?) i.e., for any given 6 > 0 there exists a 6 > 0 such that VL(T’Y) ZVL(t3CP) - 6 Whenever (TTY) E N((t:CP): 5) It follows from (2.4) that there must exist a 61 > 0 such that V(T,Y) 2'VL(t,¢) - 3/2 whenever (T,Y) E N((t,qD, 61). NLO) Choosing 5 = we will now show that VL(T9Y) 2 VL(t9 ) " 6/2 Whenever (Ty?) E N((t9(P)a 6) Since (T,Y) 6 N((t,m), 6), we can find a 53 > 0 such that NW. 1), spawn-.111), 51) Hence v55?) 2 VL(t,cp) - 6/2 whenever (In?) 6 Nam). 53). VL(T’ Y) 2VL(t9 (P) " 6/2 Hence we have VL(T,‘Y) 2VL(t.cp) - 6 whenever (TN) 6 N((t.cp). 6), i.e., 1im inf VL(T,Y) 2.VL(t,¢) (T TY)“.(t aCP) so VL is lower semi-continuous. [j 21 Theorem 2.1. Let U CZR X CA[-y,0] be open and let W1,W2: U -+ R be continuous. Suppose V: U ... R is any arbitrary function such that (2.5) W1(t ,(p) S V(t ,(p) S W2(t ,(p) for all (t,m) E U. Then the function VL: U a‘R defined by (2.4) is lower semi-continuous and satisfies (2.6) W1(t,cp) SVL(t,(p) SW2(t,(p) for all (t,qD E U. Proof. The semi-continuity of V follows directly from L Lemma 2.2. Next we note from (2.5) that lim inf w (T,w) 5 lim inf V(T,Y) s lim inf w (7.?) 2 (T.Y)~(t.m) (T.Y)~(t.m) (T.Y)e(t,q9 V (t,cp) E U Since W1 and W2 are continuous, (2.6) follows. C] 2.3. The V Functional and its Special Properties Let c: = We CA[-y,0] : M s n} , C:(I) = [q>€ AC(I) : sup\m(t)\ S H] a tEI c“ = weer-v.01 = M snI . CH(I) = {q)6 C(I) : sup‘th)\ S H] . tEI For (t,q9 E [2y,m) x C2, we set 22 AH(t, ) = {Y E CA[-r,t] FIC2[O,t] : Y E T on [t-y, t], Y a 0 on [~y,0]} , and define t (2.7) V(t,(p) = inf \Y' - f(u, Y )‘du, . YEAH(t,cp)I° ” We have the following: Lemma 2.3. Suppose x(t, o, q», (o, q» E [0,m) X CH is a solution of (2.2). Then for t 2 2y and sup \X(S,o,¢)l S‘H, s€[o,t] V(t, xt(o,m)) is a nonincreasing function of t. Proof. For t > s 2 2y 2 a, sup \x(t,o,¢9‘ S H, we S€[o,t] want to prove V(t, xt(g,m)) S V(s, xs(o,q9). It follows from (2.7) that there exists a sequence of functions {Tn} in AH(S, Xs(o,(p)) such that V(S, Xs(o,cp)) = 1im Fn new = lhn I: \Y$(u) - f(u,Yn)\dn . nan Next for each Yn E AH(S’ xs(o,q9), we define Tlnm) Yn(u) -y s n s s x(u) s s n s t Then we see that “n E AH(t, xt(o,q9) and ng(u) = f(u. nn H) . s s n s t. so that 23 lilng - f(u, fin “lldu = l:lYé(e) - f(n, Yn,u)\du = Fn Consequently, t V(t, X (03(9)) = inf Y'(u.) - f(u,‘i’) dp, t YéA(t.xt(o,cp))f°‘ “ ‘ S 1im Fn SV(s, xs(o',cp)) . D the Lemma 2.4. Let U C [0, co) X CH be open and A , n (tn, (pm) -- (7,?) in U. If x (t, tn, cpn), x(t, 'r, Y) are solutions of (2.2) that pass through (tn, (pm), (T, Y) respectively; then for small h > 0 there exist a subsequence of [(tn, (9:1)], which we denote also by {(tn, (pn)], such that (tn+h,x )-o('T+h,x ) n tn+h T+h Proof. See [4], or [8] . D H Lemma 2.5. Suppose x(t, o, (p), (g,q)) E [o,oo) x C is a solution of (2.2). Then for t 2 2y and sup ‘x(s,g,gp)‘ s H, sE[0.t] VL(t, xt(a,(p)) is a nonincreasing function of t. Proof. Suppose t >t 22v and sup \x(s,g,cp)\ SH. 1 2 s€[o,t] n n , H Let {(t , (p )] be a sequence in [o,oo) X CA such that n n n—voo and (tn! cPm) -° (t2: xtz) 24 n Let x (t, tn, cpn) be a solution of (2.2) that passes through (tn, cp“). If Itl - tZ‘ small enough, then by Lemma 2.4, there will exist a subsequence of {(tn, $11)], which we denote also by [(tn, qP)] such that n (tn+t1-t2,x )—.(t1,xt) n t +t1-t2 1 Thus from (2.4) and (2.8), we have VL(t1, xt ) - VL(t2, xt ) 1 2 (2.9) s 1im inf V(tn + t1 - t2. x“n ) - 1im V(tn, xnn) n—m t +1: ‘1’. n—cco t l 2 S 1im inf [V(tn + t1 - t2, xnn )- V(tn, xnn)] . n—Ioo t +t -t t 1 2 But fran Lemma 2.3 we have (2.10) V(tn + t1 - t2, x“n ) - V(t“, xnn) s o . t +t1-t2 t Substitute (2.10) into (2.9). We conclude that VL(t1, xtl) - VL(t2, xtz) S 0 . D 15313:; 2.6. For 'T > 2y and cp 6 C2, there exists a solution x(t) of (2.2) such that xo 5 0, x!r = cp and \x(t)\ SH for O S t S T if, and only if, V(‘l‘, ) = 0. Proof. First suppose x(t) is a solution of (2.2) such that x,r = cp, then by definition of V(T,tp) we have V(T, ) = 0. Next assume V(T,tp) = 0. We would like to show that there exists a solution of (2.2) such that x E 0, xT = (p and “xtll S H for O O S t S T. Now since V(T,tp) = 0, it follows from the definition 25 of V, that there exists a sequence of absolutely continuous func- tions {xk(t)] where xk(t) E AH(G,¢) and such that (2.11) 1im file; - f(n, xk )Idn = o . k4m 2” Set ¢k(t) = xk(t) - I: f(u, xk,u)du for 0 S t S T. Since letl = ‘Xk(t) ‘ l: f o, T 2 2y such that x(9) = y(e) V T - y S e S T. Then ‘V(S: XS) " V(S, yS)\ (2.12) S Hxs - ys“c1 +-M(x,y,s) V T S s < a + T 2 where M(x,y,s) is a positive number depending on x, y and s such that lim 11.05.1112). = 0. saT s T Proof. For T < s < a + T, let {¢k] be a sequence of functions, such that (1) ‘Pt 6 AH we. *3) + jjlx' - y'Idu + sup man. In, u) k ' ’ (2.13) - f(u, cpk’undu S V(s, xs) + “XS - ySHCI + M(x, y, s) where ,, = Sf ," '-£ . M(xy 8) 8:9 [fl (U- cnkju) (IL. cpk’quu It remains to show that “mutate”, s -T S—VT Since f is uniformly continuous in m for all of t, it follows that for each 6 > 0, there exists a 6 > 0 such that 2.14 f , _ ) - f , ) < 3 whenever 0 < - T < 6 ( ) l (u qk,u (u Tk,u \ u From (2.13) and (2.14), we then have 8 .— Maa = Pf: 'f9 d (x y s) sip ,T\ (u qk,u) (u ¢k,u)‘ u S (s - T) ° 6 whenever \s - T] < 6 , and the result follows. [3 Lemma 2.8. Let x(t), y(t) E C§[T - y, T +'a) where a > 0, T 2 2y such that x(9) = y(e) V T - y S 9 S.T. Then - - l + [VL(s, x5) VL(S’ ys)\ S “x8 yslcl +~M(x,y,s) V T S s S a T, where ‘M(x,y,s) is a positive number depends on x, y and s such that lim miffI-s—l = o. SdT Proof. Assume 28 VL(S. XS) = lim V(tn. on), Dan where (t T ) a (s x ) in [o m) X CH n: n 3 S a A 0 Set Me) =y(s+e) -X(s+e) V-ySeSO . and e. = «5.. + “1 Xn(t) = (p(t - tn) for tn - V St Stn =X(t)+(pn(-y) -X(tn -y) for '1' -'\(St Stn -y, yn(t) = ¢(t - tn) for tn - y S t S tn = x(t) + Tn(-Y) - x(tn - y) for T - y S t S tn - y. Then (tn: $B) “ (S, yS) and VL(S, ys) ' VL(S) XS) (2.15) S 1im inf V(tn, (pn) - 1im V(tn, (9n) n—m n—am s 1im inf [V(tn, E'pn) - V(tn, Tn” - Ham But from (2.12) we have (2°16) V(tn: qh) - V(tn: qh) tn I l S ftn-S+T‘xn - yn‘du +'M(xn’ yn’ tn) Substituting (2.16) into (2.15) we have 29 t _ n e _ I VL(s, ys) VL(s, xs) S 111:: inf {Itn-s-i-T‘xn ynldp, + M(xn, yn, tn)] lA jjlx' - y'ldu +M0, 'r 2 2y such that x(e)=y(e) VT-ySeST. then vyTo. yo) 5V};(o. x0) + 2Ix' < > “y‘TH‘ - x°+hucl V'o,y SV'g,x +1imsp Y o x o h—ooPI- h “Y -x ll (2.19) s v};(o. x0) + ligzip ”P“ h Th db I I Y " x d6 + lim sup Ia ‘ 6H1 dh‘ I + h But, we see from Lemma 1.3 that lly 'x H (2.20) 1im 33p 0““,1 “h =\X'(o+) -y'I . h—vo Substituting (2.20) into (2.19), we conclude that V'(o. y) SV'(o. x ) + 21X'(o+) - y'(o+)\ . C] Y O X o H Lemma 2.10. For fixed T 2 2y; (T, (p) 6 [o,oo) X CA’ then V(T, (p) -0 as Htpflc ...0 . 1 Proof. Set $(t) 0 -y S t S 0 linear from 0 to ¢('Y) 0 S t S T - y (p(t-T) T'YStST. Then V(T. cp) S [21301) f(llu TJHdu s TQIZE'uoI +13%“ Spud.» S T - Max(l\ 0, there exists a function x(t) with its derivative x'(t) continuous on at S t S B and x(t) y(t) on oz-yStSa suchthat \‘I‘ZIx'm - f(t, xt)\dt - §:\y'(t) f(t, yt)dt\\ < e and x - < . ll 9 yallc1 s Proof. Given a > 0, choose 0 < 6(a) < e such that 2.21 ft, -ft, —fi—— forall StS ( ) \ ( ) ( TM < 2(6-0!) a 8 whenever “(p(t) - 1’01)“ < 6(a) Since y'(t) is integrable, there exists a continuous function 32 , p(t) such that jgly'(t) - u(t)ldt < 1/2 6(a) . Set x(t) = y(a) + I: u(s)ds, for a s t s a. =y(t), for a-yStSa. Since y(t) = y(oz) + j; y'(s)ds for a s t S B we see that we» \flU-XUHSIDWG)‘Mflws<%Md.oStSes and (2 23> [a I '(t) - x'(t)Idt = B I '10 for y > 0. Suppose not, then there exists sequences {tk} and [wk] such that (2.25) \lrpkll 2 y , (ck, tpk) 6 [2y, on) x C: and V(tk, (91240 as k—em. Let 6(y/2) be the number in Definition 2.1 of integral stability that corresponds to y/Z, choose n SO large that V(tn’qh) < 6 4/ and let ”n E AH(tn’ Th) be so chosen that t 2 fonIuyt) - f(t, umtndu < 41—15 4’ . Then it follows from Lemma 2.12 that there exists a function p(t) with continuous derivative such that t (2.26) lonle'“) - f(t, ut)\dt < 502‘), and “lit - Hulk-21 . n Next we define 34 u'(t) - f(t, u ) for t E [0, t ] (2.27) p(t) = { t n 0 for t 6 un’ co) By changing p(t) slightly if necessary, we may further assume that p(t) is continuous. From (2.26), (2.27) we therefore have fjlmedt < 562‘) . On the other hand, we see from (2.27) that p(t) is a solution of x(t) = f(t, xt) + p(t) on -y S t S tH so that, by using (2.25), (2.26), we have III, II 2 He,“ - IIe, - I, II n n > He,“ ‘ W2 2 y - y/Z 2 y/2 , which contradicts the fact that zero solution of (2.2) is integral stable. Hence (2.28) b(y) > 0 for y > O . It follows from (2.24) that b(y) is a monotonic increasing function and satisfies (1). Next, from Lemma 2.6 and (2.28), we see (3) is satisfied. Finally, canbining Lanma 2.10 and (2.24) we obtain (2). [j I Lemma_2.l4. Suppose zero solution of (2.2) is integral stable. Then there exists a monotonic increasing function b: [0, on) -0 [0, 00) such that 35 (1) Mlle“) vac. 9). v (t. 9) e [29. e) x cfi ; (2) lim b(y) = 0; and who (3) b(v) = 0 if, and only if, v = O . Proof. It follows from Lemma 2.13 that there exists a monotonic increasing function a: [0, a0 a [0, m) such that (r) a(\\rp\\) SV(t, cp) , v (t, <9) 6 [2% os) x c2; (ii) 1im a(y) = 0; and YTO ' (iii) a(y) =‘0 if, and only if, y = O . Next we define the function 'b: [0, m) a [0, m) 'by b(v) = lim inf a(y) V y E [0, m) tTY Then the function b(y) is a monotonic function. Also, we see from (i) that lim inf a(\\‘l’\\) S 11111 inf V(T, Y) (T 9Y)"(t 9C?) (T 9‘?)"(t TSP) ’ i.e. MIMI) va(t, cp) v (t, m) e [0, co) x G? . It is clear from (ii) that lim b(y) = 0. Finally since a(y) = 0 vac if, and only if, y = 0, we conclude that b(y) = 0 if, and only if, y = 0. E3 36 2.4. Characterization of Integral Stability Canbining Leruna 2.2, Lemma 2.5, Leanna 2.8, Lanma 2.11, and Lemma 2.14, we obtain the following theorem. Theorem 2.2. Suppose the zero solution of (2.2) is integral stable. Then there exists a lower semi-continuous function H VL: [2y, as) X CA -+ [0, on) Having the following properties: (1) VL(t, tp) 2 b(l\gpl\) V (t, cp) E [2% on) X Ci, where b(y) is a monotonic increasing function such that lim b(v) 'Y-+O 0 and b(y) =0 if, and only if,y=0 (2) VL(t. (p) -+ 0 as \‘tpHCI —0 0 for each fixed t 2 2y (3) For any solution x(t, o: tp) of (2.2) with (cup) 6 [0, 0°) X CH, we have for t 2 2y and sup ‘x(s, a, tp)‘ S H, V(t, xt(c, tp)) is a non- SE[o,t] increasing function of t. (4) Let x,y€C§[T-Y,T+a),where a>0,T22y such that x(e) III)’(6) V T - y S e S T+ a. Then \vL(s, x8) - VL(s. ys)l - +M , , +a, SHxs ysuc (xys)VT + 2Ix'(o*)~ y'(o+)| and Y'(o+)l - V):(o,cp) s v;(o,rp) + 2Tx'| for all t that lie in the domain of y(t, 0, 1p)- Proof. It follows fran Lemma 2.9 that (2.30) v};(t. yt(o. o» svgc. yt - y'(o+)\ Next, we see from Leanna 2.3 that 38 (2.31) V}'{(t, yt(g, (p)) s o . On the other hand, from (2.2), (2.29) we have (2.32) \X'(o+) - y'(o+>I = IpI Substituting (2.31) , (2.32) into (2.30) we have v; 0, there exists 1‘ such that “xcwucl S 6 whenever “SP“ < Tl . Proof. Since f(t, tp) is uniformly continuous in cp for all of t and f(t, 0) = 0, there exists for given 6 > 0 an 111 < 6/2 such that (2.33) \f(s, cp)‘ < 6/2v whenever “‘9“ < 111 . Also it follows from the fact that f takes a bounded set into a bounded set that there exists an 112 > 0 such that (2.34) \lxfiyll < 111 for all Hell < le . Choosing (2.35) Tl = Minflll. n2) . then for “2p“ < 'n we have from (2.33), (2.34) , (2.35) that lx'(u)\ S \fm. xnll V H- 6 [0. 0+ )1] < 6/2v . 39 Consequently ow . .1_ HT -6 I0 \x (“)l < 2v ft 6 du - 2 ’ and 6 “me S “1 < 5 Hence we conclude that ch+yHc1 S 6 whenever “TH‘< n . C] Now we are ready to state and prove the following theorem. Theorem 2.3. For the equation (2.2) suppose there exists a lower semi-continuous functional H v: [2v, ...) x CA" [0. ...) such that the following four conditions are satisfied. (1) V(t, .9) 2 MIN“) v (t, q.) 6 [2y, 00) x cl: b(y) is a monotonic increasing function such that , where lim b(y) = o and b(y) = o if, and only if, y = o. yao (2) V(t, TD A’O as “m“c .2 0 for each fixed t 2 2y. 1 (3) For any solution x(t, o, q» of (2.2), where (cup) 6 [0, on) x CH, we have for t 2 2y and SUP \X(8, Us CP)‘ 5 H) V(t, xt(09 ((3)) is a [1011‘ s€[c,t] increasing function of t. (4) Let x, y E C2[T - v, T + a), where a >'0, T 2 2y such that Ms) =y(e). V T -YS es'r+a, then 40 \VL(S. xs) - VL(S. ys)\ Sllxs -y$l\c +M(x, y, s) VT 0, and 6 >'0 there exists a continuous function p(t), to 2 0 such that I:;\P(t)\ < 6 and a qb E C[-y, 0] such that th“ < 6. for which the equation X' = f(t. xt) + P(t) will have a solution xp = xp(t, to, T6) such that (2.36) “xp(t2, to, mo)” 2 e, for some t2 > to‘+ y . It follows from (1), (2), that we can choose N >tt2 so large that and b b (2,37) \V(to+ y, (9)) < —2L§2- whenever “(PHc < Nil 1 On the other hand, in view of Lemma 2.17, we may assume 6 >'0 to be so small that 26 < b(e) and 41 111.2 (2.38) th +Y(°, to, (po)\\61 < N < 3 whenever “mo“ < 6 . 0 Furthermore, from Lemma 2.16, we have V}; (t. xt) Sz\p(t)| Vto St SI:2 . P Thus by Theorem 1.1 we have t 2 (2.39) V(t2, xtz) S'V(to + y, xtoIY) + ftofv\p(t)‘dt But then (1), (2.36), (2.37) and (2.38) together hnply b(e) s b sv v . The next theorem says that if V satisfies a local Lipschitz condition, then the usual Liapunov theorems still hold true when ‘V is replaced by V*. Theorem 3.1. Suppose that V: R x C[-r,o] + R is well- defined and for every (t,m) e D there exists a neighborhood N(t,¢) of (t,w) and a constant L = L(t,m) 2 0 such that (3.13) |V(E.wl> - V(E.¢2>l s Lle - mg“ for all (E,¢1), (E,¢2) e N(t,¢). Then 45 * _ V (t9¢) : Vx(t: xt) for any solution x(-) of (3.1) through (t,q9. Proof. By Lemma 3.1, it suffices to show that V*(t,¢) 2 V;(t, xt). Let the solution x(-) be fixed. By Defini- tion 3.2, (3.2) and (3.13), we have —- l (3.14) Vx(t, xt) - 11m inf-E [V(t+h , x h+o t+h) ‘ V(t9¢)] s lim inf 1 [V(t+h, ¢+hf(t,¢) + hw) - V(t,w)] h+o IMO) I+o + lim sup- h 1LHX - (¢+hf(t,¢) + hW)“ h+0+ t+h where w is the same as in Definition 3.2. Since ¢ + hf(t,¢0 + hw = zt+h for some z(-) e APX(t,¢,h), th+h - (w + hf(t,¢) + hp)“ - \x(c+fi) - z(c+E)\ where 0 < h s h. For each fixed h > 0, ,1; Luxt+h - (cp + hf(tup) + hm <%-L|x(t + E) - x(t) - [z(t + h) - x(t)]l . Substituting into (3.14), IA V;(t, xt) v* (t,¢) + lim sup—1|x(t+h)- x(t) - [z(t+h) - x(t)]I h+o+ v*(c,¢) + le'(t+) - z'(t+)] V*(ta¢) B because of (3.11) and (3.1). This completes the proof. 46 Remark. In [22] (p. 186-188), it is shown that if V is continuous in (t,¢) and is Lipschitzian in m, then V¥(t, xt) is independent of any particular solution through (t,¢). Theorem ._ * 3.1 says that Vx(t, xt) is in fact equal to V (t,¢). 3.2. A Comparison Principle for FDE The next theorem shows that the usual Liapunov comparison principle may be obtained for V* derivative. Theorem 3.2. Let D c R'X C[-r,o] be Open and f: D + R4 be continuous and map bounded sets into bounded sets. Suppose that W: D + R is continuous and V: D + R is lower semi-continuous. If (3.15) v*(t.cp) s wow). (mp) e D along x'(t) = f(t, xt), then for every (t,¢) e D there exists a solution x(t,¢) such that t+s (3.16) V(t+s, xt+s) - V(t, xt) S It W(u, xu)du for all s 2 0 such that x remains to be a solution. t+s Remark. In applications, it is often easier to use V to conclude that 1V¥(t,m) 5 W(t,¢). Lemma 3.1 will show that (3.15) is true. 3.3. Proof of the Comparison Theorem In this section the proof of Theorem 3.2 is given. The following lemmas are needed. For simplicity, all the assumptions in Theorem 3.2 are assumed in these lemmas. 47 Lemma 3.2. Let (t,¢) e D be given and An(t,¢) denote the set of (7.1) e D such that (3.17) t 0 and $1 6 C[—r,o] such that (3.21) h + Iw (0)] < l;- and 1 2n ’ l - l (3.22) 1h [V(t+h, mwhf(t,¢) + hwl) - V(t,¢)] < W(t,¢) +-; Let zl(-) e APX(t,¢,h) be such that (3.5) is satisfied. Let T = t+h and w = It is easy to see that (3.17), (3.19) z1,t;+h' and (3.20) are satisfied for this choice of (T,¢). Moreover, by (3.5) 42(0) - cp(0) = 7‘1,t+h(°) - cp(0) = hf(tyco) + hw1(0) Hence - 1 |‘1(°?r_ag°)- f(t,:p)| s ”1“” < a . This proves that (t,w) e An(t,¢) . D 48 For each n sufficiently large, we now construct an approxi- mate solution xn(-) of (3.1) through (t,m). Since An(t,m) is non-empty, sup{h : (t + h,w) e An(t,@)} > 0. We may therefore find (t1, m1) 5 An(t,m) such that t1 - t >-% sup{h : (t+h, W) e An(t,m)} . Now, for each i = 1,2,3,... there exists inductively (ti+1’ ¢i+l) e An(ti, mi) such that 1 (3.23) t1+1 - ti >-§ sup {h . (ti + h,¢) e An(ti, mi)}. Let bn = sup ti . 121 d Define xn: [-r + t, bn) + R by (3.24) x = mi. 1 = 1,2,3,... . The following is immediate from the definition of An(t,¢) . Lemma 3.3. xn: [-r + t, bn) + Rd is continuous and is absolutely continuous for t s s < bn' Moreover, (s, xn,s) e D for all t s s < bn' Lemma 3.4. For each n, xn(-) is non-continuable with respect to D. 49 Proof. We may assume that bn.<.n. If xn(-) is not non-continuable with respect to D, then there exists a sequence t -> b as k +00 such that k n (tk, x t ) e U for all k = 1,2,... “’1: where U c D is some closed bounded subset. This implies that xn(s), -r + t s s 1 0 denotes the bound of If(1,w)l for (r,w) in the closure of {(s, xn,s) : t s s < bn}’ then it follows from (3.20) and (3.10) that Ix'(s)| S M +-l a e t s s < b n n , . ., n . Thus, xn is uniformly continuous on [t - r, bn)' This implies {(s, xn S) : t S s < bn} belongs to a compact set in D. Hence, 9 is well-defined and (bn’ x ) e U. We now claim that xn,b n,b n n (bn’ xn b ) f U. This contradiction will prove the lemma. 9 n Proof of claim. (bn’ xn,bn) l U. If (t,¢) e An(bn’ xn,bn)’ and if (T,w) e A (t ,x ) for all sufficiently large 1, then n i n,t1 for all sufficiently large 1 1 1 t114-1":1l s e. d Then there exists a non-continuable function x: [t - r, b) + R and a subsequence {x (-)} such that (3.33) xnk(-)‘+ x(-) uniformly on compact subsets of [t - r, b) as nk'+ m . (3.34) x(s) - ¢(o) + ;: f(u, xu)du xt = m . 52 Proof. See [4]. We remark that in [4] it is assumed that the projection of D onto C[-r, o] is bounded. However, the same proof may be used in our case with almost no changes. D Lemma 3.6. xn(ti+l) - xn(ti) _ ' f(t ,X )1 ' (t - t.) t1+1 ti 1 .n,ti 1+1 1 (3.35) [ t =IS+1[ t1+1 1 It x;(s) - f(s, xn’s)]ds - [f(ti’ x ) - n,ti f(s, xn,s)]ds . Lemma 3.7. Let U C D be closed and bounded and (3.36) Ul = {(s, xn,s) : n = 1,2,...; (u, xn’u) e U for all t S u S s} . Then U1 is relatively compact. Proof. For each n, we obtain from (3.10) that if (S,X )EU n,s 1’ (3.37) Ixr"(s)l s M +¥1§5 (1 + M), a.e., where M = sup{|f(s,w)| : (S.W) e U}. We also note that xn t = m 9 for all n. For each xn,s such that (s, xn,s) 6 U1, x e C[—r, 0] may be broken into two parts. Namely, one part is n,s some portion of m and the other part is absolutely continuous and satisfies (3.37). Since w is uniformly continuous and the bound in (3.37) is independent of n, the set {xn : (s, Xu 8) 6 U1} ’ ,S is equi—continuous. Now, an argument using Ascoli theorem will complete the proof. D 53 Proof of Theorem 3.2. Let U, U1 and M be as in Lemma 3.7, and L = sup{s - t : (s,¢) e U for some w e C[-r, 0]}. Let an(U) = suplf(sl, $1) - f(sz, wz)l where sup 18 taken over the set of ($1, $1), (32, wz) e Ul such that 1 13 -321 <1; 1 “1’1 ' 1’2“ < max{(M-n+1)/n2. 1(a)} where y(n) is determined from the uniform continuity of xn on 1 l [t - r, t +-H], i.e., if Isl - 82' ('33 then Ixn(sl) - xn(sz)] < y(n). It follows from the uniform continuity of Q and (3.37) that y(n) + o as n + w. Let (U) - ( (u) +-1)L +-1(2M + 1) 8n - 0"n n n ° Since U1 has compact closure, on(U) < m. Moreover, the uniform continuity of f on U1 yields that on(U) + o and 8n(U) + o as n + w. Let Gn(t) be as in Lemma 3.5. We claim that the condition (3.32) is satisfied by the above choice of an. Let v > t and (s, x ) e U for all t s s s v . We have n n,s n V (3.38) IGn(vn)| Iftnlx;(s) - f(s, xn,s)]dsl t 1 0 1ft [xn(s) - f(s, xn’8)]ds +...+ 1’. 1+1 , ft1 [xn(s) - f(s, xn,s)]ds +...+ V ft:[xé(s) - f(s, xn,s)]dsl 54 where ti's are from the definition of xn(°) and tj is such that tj S vn < tj+l° It follows from Lemma 3.6 and (3.18) that t1+1 1 ' - — .- (3.39) Ifti [xn(s) f(s, xn,s)]dsl S n (t1+1 ti) ' t1+1 + It |£(ci, xn,t ) - f(s, xn,S)|ds . i 1 . l — < < — Since for r _ 6 _ 0, ti 5 s s t1+1 < ti + n and ti + 6 2 t I"? p. A CD V II lxn(s + 6) - xn(ti + 6)| IA 9+6 , ' fti+e|xn(u)ldu, we obtain from (3.10) and the uniform continuity of xn(-) on 1 [t-r, t+a], s max{(Mn + 1)/n2, y(n)} . This inequality, the definition an(U) and (3.39) yield 1'. 1 1 . 1 liti+ [xn(s) - f(s, xn’s)]dsl s [an(U) + 31(t1+1 - ti) . Substituting this into (3.38), 1 vn , [an(U) +3“:j - t] + ftj Ixn(s) - f(s, xn,s)lds IA IGn t, through (t,¢) which is non-continuable with respect to D and there exists a subsequence xn (.) such that j 55 xn (-) +-x(-) uniformly on compact subsets of j [t - r, b) as n-+ w . We now prove that this solution x(-) satisfies (3.16). For simplicity, denote the subsequence xn (.) by xj(-) and the sequence {ti} used to define x (-) by ti(j). Let sEE [t,b). J For each large j let i = i(j) be so chosen that l (j) < :16) +3; . Thus, ti(j) + s as j + w. Since xj(-) + x(') uniformly on ti(j) s s S t1+1 [t - r, s], “x - xu“ + 0 uniformly for t S u S s as j + m. Lu Let Us = {(u, xj u) : t s u s s, j - 1,2,...}. It is shown by ’ the same proof of Lemma 3.7 that U8 is relatively compact. The uniform continuity of W on Us yields W(u, x ) + W(u, x ) uniformly on [t,s] as j + w. j,u u Thus, if wj = sup{IW(u, xj u) - W(u, xu)l : t s u s s}, then w + O as j + m. Now, 3 (3.40) V(s, x8) - V(t,¢) 5 lim+inf V(t1(j), xj,ti(j)) - V(t,¢) 1(1)-11 ( ( ) ) - lim inf 2 V t j , X 1.... k-O 1‘“ j”141(1) where to(j) = t for all j = 1,2,... . By the definition of xn(°) and by (3.19), we have 56 V(t (j). x . ) k+1 j.tk+1(J) s [W(ck(1). xj Substituting into (3.40), V(s, x8) - V(t,¢) g lim inf j—MD [tk+1(j) S lim inf j-mo [tk+1(j) 5 lim inf j-xx: + lim sup j+m - V(tk(j) , X, .tk<1)’ + J.tk(j)) ll. tk+1 ° i<1>~1 . 1 'z=o [W(tk(3), xj’tk(j)) +-;j] - tk(j)] i(j)-1 z [W(t (j), x. k=0 k 3 )1 .tk(j) - tk(j)1 i(j)-1 Z W(tk(j), x k=o tk(j)>[tm(1) - t (1)] i(j)-1 k X [W(t (j). X . ) - W(t (j). k=0 k tk(3) k xj,tk(j)][tk+1(j) - ck| : w eqfih} . If Qt+h is empty, we define do(¢ + hf(t,¢), Q ) = +0° . t+h Theorem 4.1. Let Q c D be closed. Q is semi-invariant with respect to (3.1) if and only if for each (t,¢) 5 Q, f(t,¢) is subtangential to Q at (t,¢) . Proof. Let Q be semi-invariant. For each (I,¢) 6 Q, let x(-) be the solution of (3.1) through (t,w) such that (3, x8) 5 Q for all s 2 t and x(s) is defined. It is clear that x(°) e APX(t,¢, h) and x for every small h > 0. t+h 5 Qt+h 59 Moreover, 1; [4(a) + new - xt+h(o)] = f(t,:p) - “1+“; “1) But the right hand side tends to zero as h + 0+. Hence, by definition f(t,¢) is subtangential to Q at (t,¢). Conversely, first define 0 (Cd?) 6 Q (402) V (ttfip) = Q ~1 (t.¢) { Q The closeness of Q implies V is lower semi-continuous. By Q Definition 4.2, for each a > 0 there exists h, 0 < h < 0, such that l 1h |¢(o) + hf(t,¢) - w(o)| < e where w = zt+h for some z(-) e APX(t,¢,h) such that (t + h, zt+h) e Qt+h' Let $1 "h'Izt+h - m - hf(t,m)] - We have m + hf(t,¢) + hwl = z and |w1(o)| < 5. Hence t+h VQ(t + h, m + hf(t,¢) + hwl) - VQ(t,¢) = 0 . Letting e + 0, we have from the definition * VQ(t,¢) = 0 . By Theorem 3.2, there exists a non-continuable solution x(-) of (3.1) through (t,m) such that 60 V (3, XS) - V Q (t, xt) s 0, s 2 t and x(s) is defined. Q From (4.2), (s, xs)i€ Q for s 2 t and x(s) is defined. D Corollary. Let Q c D be closed. Suppose that for each (t,¢) 5 D there exists a unique solution of (3.1) through (t,w). Then Q is invariant if and only if f(t,m) is subtangential to Q at every (t,¢) 6 Q. Remark. In applications, it is often that (3.1) is auto- nomous. For this reason, we will state Theorem 4.1 separately for the autonomous case. Let E c C[—r, o] be open and g : E + Rd be continuous and map bounded sets into bounded sets. Consider the autonomous system (4.3) X'(t) = 8(xt) . Definition 4.3. Let P c E. P is said to be semi—in- variant with respect to (4.3) if for each m 5 P there exists a solution x(°) of (4.3) through (ng) such that xt e P for all t 2 0 and x(t) is defined. We say that g(¢) is sub- tangential to P if 1 - (4.4) lim inf-E doom + hg(¢), Pb) = 0 h+o where Ph denotes the set of all w e P such that w = 2b for some z(-) e APX(0.w and for every continuous function z : [-r,e°) + Rd we have 62 T l IfT O h(t, 2t)dt| S p(To) for all 0 S To S Tl S To + 1. This condition on h is slightly more general than that given by Miller [13]. Theorem 4.3. If h satisfies condition (H) and if L is the limit set of a solution y(')‘ of (4.6), then L is semi- invariant with respect to (4.5). We shall assume that L is non-empty. Let g) e L be fixed and 6 > 0 be so small that if Hm-—¢4\S 6, then w e E. Let M = sup{|g(w)| : \hp-¢H S 6}. By definition, there exists a sequence {tn} such that tn + w as n + w and yt -+ m as n n + w. The following lemmas are needed. Lemma 4.1. If a = min{l, 6/3M} and n is large, then (4.7) Hm - ytn+tH S 6, ’0 S t S a . Proof. For large tn, tn+t+9 tn+t+9 y(tn + t +6) = y(tn + 6) +-f£n+e g(ys)ds + ftn+6 h(s,ys)ds . We assume that tn is so large that “w - yt\\ < 6/3 and n p(tn - r) S 6/3. Let tn be fixed. As long as 0 S t S l and 11ytd+t - 6][: 6 , we have for -r §_6 §_0 t +t+9 |¢(e) - y(tn + t +6)| S “W - yt\[ + tM + If n h(s,ys)dsl ‘ ' n tn+6 IA 6/3 + tM + u(tn + e) g 6/3 + tM + p(tn - r) 25 S ——- . 3 + tM argument yields (4.7). rj A "suppose not " 63 Lemma 4.2. The set Y = {yt +t : n n is relatively compact. Proof. It follows from (4.7) that Y is uniformly bounded in C[-r, o]. If —r S 6 < 8 < 6 +1, 6 1 2 1 < 0, and 0 S t S a, 2 then tn+t+62 (4.8) |y(tn + t + 62) - y(tn + t + el)| S ftn+t+61 |g(ys)|ds t +t+6 Iftn+t+62 h(s’ ys)dSI n 1 + S (e - 91)M+u(tn+ t+61) 2 IA (02 - 81)M + p(tn - r) Let a > 0. Choose tN = tN(e) so large that p(tn - r) < e/2 for all t 2 t . Let n N Mn = M(e) = sup{|g(¢)| + Ih(t,¢)l : w e B and < < co 0 _ t _ tN + a} < where B<: D is a bounded set such that yt +t e B for all tn n and 0 S t S a. (The existence of the set B is a consequence of Lemma 4.1.) We note that Mn depends only on e > 0. Thus ly'(tn + t + 6)| S Mn, a.e., for all 2r S tn S t 0 S t S a, N, and -r S 6 S 0. If I6 - all < minfl, e/MN, e/ZM}, we have 2 ly- wu s p} . Then 8(p) decreases monotonically to zero as p + 0. Corollary 2. For each t, O S t S a, {yt +t} has a n limit point 1). Such that “(p - 1])“ S 5 . Lemma 4.3. Let w be as in Corollary 2 of Lemma 4.2. Define a function 2 : [-r, t] + Rd by 20 = w, 2t = w. Then Z(°) e APX(O.m.tS 8) - Proof. Since y(tn + 6) + w(e) uniformly in 6 as n + m and y(tn + t + 6)1+ w(e) uniformly in 6 as tn + w j (where {tn } is a subsequence of {tn})’ 2 is well-defined. 3 For '0 S 51’ 52 S t, by (4.8) (4.9) l2(sl) - 2(32)| = lw(s1 - t) - w(s2 - t)| S |y(tnj + $1) - w(sl - t)| + Iy(tnj + $2) - w(s2 - t)| + Iy0° new - 8(yt +8)]ds| + lim sup p(tn) S tB(O) n n*w where B is from Lemma 4.2 and - 0 S s S t} . p = lim sup {Hm - yt‘+A\ . n 11"“ 66 For 0 S s S t, by using (4.8) [A lytn+s(e) - ¢(9)| [y(tn + e) - ¢(e)| + Iy(tn + e) — y(tn + s + e)| IA ly(tn + e) - ¢(9)| + tM + p(tn - r), -r S e S o . Hence, p S tM. From (4.10) 1“ +c'éc) L)