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Chapter 12 Higher-order equations
CHAPTER 12
Higher-order equations 120. Introduction
Summary o f the chapter In all preceding chapters the linear differential equations that were considered were (30. I), (30.2), i.e., (120.1)
X+Ax=O
(120.2)
a+Ax=f
These are first-order equations (although the values of x and f lie in a Banach space X) in that only the first derivative of the solution is involved. In this chapter our objects of study are the higher-order equations (120.3)
w(r!itIl
+
c A,,w(A) rri
=0
0
(1 20.4)
where m is some positive integer, and the values of w , h are in some Banach space W, while the values of each Ak are in It is usually claimed that (120.3), (120.4) may be reduced to (l20.1), (120.2) in an appropriate larger space X (namely the outer direct sum 1 copies of W ) , and that therefore the theory of the higherof m order equations can be subsumed under that of the first-order equations. T h e first part of this statement is substantially correct, and we intend to describe the claimed reduction in Section 121; the last part, however, does not follow in general. This is best illustrated in our context by the concept of admissibility of a pair of function spaces for Eq. (120.4). If B, D E b.N( W )are given, 373
m.
+
374
Ch. 12. HIGHER-ORDER EQUATIONS
it is reasonable to call the pair (B, D) admissible for (1 20.4) if for every h E B there exists a solution w E D of (120.4). If we consider the corresponding “reduced” equation (1 20.2) in the direct-sum space X = WOO... 0 W,,, , say, this condition means that for every f with the mth “component” in B and all others 0 there exists a solution x of (120.2) with its 0th “component” in D. Now the f ’ s involved do form a space in b N ( X ) , but if B were an 9 - s p a c e or a F-space this would no longer be the case for the corresponding space off’s; a more serious obstacle in the way of relating this admissibility of (B, D) with some admissibility for (120.2) and then applying the results of previous chapters is the lack of a priori restrictions on any “component” of the solution x except the 0th. Fortunately, the difficulties just described can in the main be overcome, by means of some technical devices, in the most important case of .F-pairs and similar pairs, provided certain mild assumptions are made on the A k . I t thus becomes possible to establish results substantially similar to those in Chapter 6, relating the admissibility of a 9-pair, or a similar condition, to the behavior of the solutions of the homogeneous equation (120.3) and their derivatives. We remark that, with a somewhat greater effort at technical refinement, the same methods can be applied to systems of equations with each equation of a different order, yielding essentially the same results. We do not go into this matter here. Another, and more fundamental, discrepancy between Eqs. ( 120.9, (120.4) on the one hand and ( 1 20. l), ( I 20.2) on the other is the difficulty in setting up the concept of an adjoint equation in the former case. More precisely, if we reduce (120.3), say, to the form (120.1) in the usual way and then consider the corresponding adjoint equation i* - A*x* = 0, this equation does not have the form appropriate to the reduction of a presumptive “adjoint” equation to (120.3). On the other hand, the usual definition of such an adjoint equatioipresupposes a sufficient smoothness of the A, and is not symmetrical even if the possible asymmetry of W , W* is disregarded. The bilimr functional involved in the appropriate form of Green’s Formula is iself unsymmetrical and depends on t (cf. Hartman [l], Section 12). Hartman [ I ] has studied these questions and related ntters extensively; he bases his work on a very general abstract Ethod that is adaptable to other cases besides the subject of the 86ent chapter, not necessarily involving differential equations. Our p u r p in including this chapter is merely to give the reader a glimpse of tenature of the results and of the special problems they involve; it isherefore much more restricted, as concerns both the method and t scope of the
120. INTRODUCTION
375
results. We shall be content with a bare sketch of the central theorems of the type of those of Chapter 6, as referred to above. We omit all discussion, for the higher-order equations, of the questions dealt with in Chapters 7-11. We use several proofs in Hartman's paper, but do not attempt to describe his general method. We refer to that paper for further information, especially on the adjoint equations and on certain second-order equations with different assumptions on A,, A,. See also Hartman [2], Chapter XIII. Section 121 contains a discussion of Eqs. (120.3), (120.4) and their reduction to the forms (120.1), (120.2), and an analysis of the solutions. There is an important theorem (Theorem 121.B) giving bounds for the derivatives of a solution in terms of bounds ior the solution itself and for h. Section 122 deals with the concepts of admissibility and of (B,D)-subspaces for F-pairs. Section 123 contains the main theorems.
n th primitive functions In order to speak properly of a solution of (120.3) or (120.4) we must generalize the concept of a primitive function. We define recursively an nTH P R I M I T I V E ( F U N C T I O N ) on an interval J and with values in a Banach space X:f is a first primitive iff is a primitive; f is an ( n 1)st primitive if f is a primitive and f is an nth primitive, n = 1, 2, ... . I f f is an nth primitive, we denote by f ( k ) ,k = 0, ...,n , its ~ T DHE R I V A T I V E ,defined recursively by f ( O ) = f, f ( k + l )= ( f ( k ) ) ;' each, except the last, is an ( n - k)th primitive. We prove a lemma giving estimates for the derivatives f ( O ) , ...,f ( m ) of an ( m + 1)st primitive in terms off and f ( m + l ) .
+
120.A. For every positive integer m there exists a real number c, > 0 with the following property: iff : J + X is an ( m 1)st primitive, and a A ] C J is given, then compact interval [ T , T
+
+
(120.5)
Ah I!f("(t) I/
111
< c , ,i=O1l l f ( ~
+jd/m)(1
+ Am f" l ~ f ( m + l ~ (11udu,) T
~ E [ T , T + A ] , k = O ,..., m.
Proof, It is sufficient to verify the corresponding inequality (120.6)
376
Ch. 12. HIGHER-ORDER EQUATIONS
is a real-valued ( m + 1)st primitive on [0, 13. Indeed, if + d] and k are given, there exists x * E X * such that 11 x * 1) = 1, < f ( k ) ( t l )x*) , = Ilf(k)(tl)ll.We set ~ ( s )= Re(f(7 + sd), x*), so that gP)(s) = diRe(f(i)(T + sd), x*), i = 0, ..., m + 1, and (120.6) at s =
where
y~
t , E [T,T
(tl - ~ ) / d implies (120.5) at t = t , . We proceed to prove (120.6). We consider the interpolation polynomials r,, of degree m that satisfy r,,(i/m) = 6,, , i , j = 0, ..., m ; actually, r m j ( s )= l&+j (ms - i)/(j- z]. There exists a common bound c,, > 0 for all I rg; I,j, k = 0, ..., m, on the interval [0, 13. Now y~ = # + C L o ( p ( j / m ) n m jwhere , # is an ( m 1)st primitive that vanishes at s = j / m , j = 0 , ..., m. By Rolle's Theorem there exists sk E [o, 13 such that #(k)(sk)= 0, k = 0, ..., m. Therefore
+
Using induction on m
- k we find
Since rT1) = 0 for all j , we have follows. &
#(m+l) = ~ ( ~ + land ) ,
(120.6)
Remark. A detailed analysis of the polynomials rmjand their derivatives shows that c,,, may be taken t o be m m ( g )if m is even, and mm(i(m?,J if m is odd.
121. The ( m
+ 1)st-order equation
Although we might, of course, take a more general point of view, we agree that throughout this chapter the range of t shall be J = R , . We shall always consider a positive integer m, a Banach space W, and k = 0, ..., m. If h E L(W ) , a SOLUTION of the functions Ak E L( Eq. (120.4) is an (m + 1)st primitive w : R , + Wthat, together with its derivatives, satisfies (120.4), considered as an equation in L(W). For h = 0, we obtain a SOLUTION of (120.3).
w),
121. THE( m
+ I)ST-ORDER
377
EQUATION
I n order to “reduce” Eqs. ( 1 20.3), (120.4) to the forms ( 120. I), (1 20.2), m we introduce the Banach space X = 0 W, the outer direct sum o f o m m 1 times W , provided with the supremum norm; i.e., if x = @ xj E X , 0 xi E W (superscripts will denote the components of elements in X ) , then 11 x l I x = maxi 11 xj I l w ; there is no risk in omitting the subscripts of the norms from now on. We remark that this choice of norms is so convenient that we permit it to infringe the previous agreement that the norm in a finite-dimensional space X is assumed to be euclidean (Section 15, p. 32). For given Al:€L ( r ) , k = 0, ..., m, and h E L(W) we define the Eq. (120.2) in X , given by the system
+
‘
21 - xj+I = 0 ,
(121.1)
J =
0 , ..., m
-
1
+ 1A p h = h, 911
A?‘
0
as the RED.UCED FORM of (120.4). It is easy to see that A and that, with the given norm on X ,
Iljll
=
E
L(x),f E L(X)
II h II.
Equation (120.1) with the same A is the REDUCED FORM of (120.3). T h e precise relation between (120.4) and its reduced form (120.2) is given by the following lemma, the trivial proof of which is omitted.
+
121.A. A n ( m mI)st primitive w : R, 4 W is a solution of (120.4) and only x = @ w ( j ) is a solution of the reduced form (120.2). 0 Conversely, a primitive x : R, -+ X is a solution of (120.2) if and only if xo is an ( m I)st primitive and a solution of (120.4), and xj = xo(j), j = 0, ..., m. Consequently, for any to E R, and any xo E X there exists a ) xj0 ’ j = 0, ..., m. unique solution w of (120.4) with w ( J ) ( t o=
+
T h e fundamental tool allowing us to work with solutions of (120.4) and of its reduced form is the following estimate for the derivatives of a solution of (120.4). T h e number c,,, is the constant given by 120.A.
p
121.R. THEOREM. For given to 3 0 there exists a positive integer such that
= p(tJ
Ch. 12. HIGHER-ORDER EQUATIONS
378
for any such p , and any h E L(W ) ,eoery solution w of (120.4) satisfies (121.4) 11 w Y t ) l (
< 2e!~'cn,p"C (1 4 s +j / m p ) 11 + 3e:" ma
II h ( r ) I/ dr,
["I
j=U
O
Y
K = O ,..., m,
s
[
(121.5) 11 d r ) ( t )I1 < 2 e i p (c,pnhilm
t+l+l/ml,
llw(u)ll du
+
t+1+1/nrp
Ilh(zr)ll did) ,
k = O ,..., m.
O
Proof. The existence of p = p(to) follows from the continuity of the left-hand member of (121.3) as a function of s. For given s, 0 s to there exists, on account of (l21.3), an integer i, 0 i < p , such that
< <
<
(121.6)
II A ( r ) (I dr
[:A
< 4,
=s
T
+ i/p,
A
=
l/p.
In
We consider the solution x = @ w ( j ) of the reduced form (120.2) of 0 (120.4), Using 120.A we find, for almost all t, 7 < t ,< 7 + d,
II w(n'+l)(t) II < I1 20) II < II 4 )II I1 JEWII m
< I1 4I1 (cn,P"'x II
4 7
1=0
+ II
+ j / m p ) /I
I/
+ f'"
117~(m+1)(y)ll
dr) 4- Ilh(t)ll.
T
Integrating over the interval and using (121.6) we obtain
We use 120.A again, and find
< t < s + 1, (31.5), (121.2), (121.3) imply 11 x(t)ll < 11 h(r)ll dr), and this, together with (121.7), yields (121.4). Let now t,"O < t < to - limp, be given. Consider any s, t < s < t + limp < t o , and carry out the preceding argument as far as (121.7). t < - t < (s + 1 lip) ( s - limp) < I and Now 0 < s For any
e*P(ll
t,
+
x(~)ll
s
JH+'
-
7
-
-
121. THE(m
+ <
d s $- 1 ( 121.7) imply 7
+ I)ST-ORDEREQUATION
< t + 1 + l/mp.
379
Therefore (31.5), (121.2), (121.3),
Integrating the first and last members with respect to s over the interval [t, t limp], we obtain (121.5). Q
+
A first important consequence of Theorem 121.B is a closedness theorem analogous to Theorem 31.D. Let (Ak,,), k = 0, ..., m, (h,) be sequences in 121.C. THEOREM. L( @’), L( W), respectively, and let w, be some solution of
n = 1 , 2, ... .Zf the limits = A, , k = 0, ..., m,1imnjm Lhn = h, limn+mLw, = w exist, then the function w is (except for equivalence modulo a null set) a solution of (120.4), and wkk’ -+ w ( ~uniformly ) on every compact subinterval of R+ as n 400, for k = 0, ..., m.
Proof. With Eq. (121.8) we associate its reduced form
and with (120.4) its reduced form (120.2). Obviously, limnjm LA,n = A, m L f n = f. Now x, = @ w::) is a solution of (121.9). If we can k=O
prove that limn+mLx, = x exists the conclusion will follow at once from Theorem 31.D applied to the reduced forms, and from 121.A. We proceed to prove the L-convergence of (x,). (The actual proof anticipates a part of the proof of Theorem 3 I .D, as we shall see.) We choose an arbitrary tn > 1 and fix it for the time being. Since (A.,) converges in L ( x ) , there exists a positive integer p such that J8+l 11 A.,(r)\l dr < $p for all s, 0 < s < t, and n = I , 2, ...; then the
380
Ch. 12. HIGHER-ORDER EQUATIONS
limit A satisfies (121.3). From Theorem 121.B (specifically, (121.5)) applied to the solution w, of (121.8) we find
n = l , 2 ,....
0
Since (w,), (h,) converge in L( W) and towas arbitrarily large, we conclude that (x,) is uniformly bounded on every compact subinterval of R, . Now x, is a solution of i, Ax, = ( A - A,,)x, f, ,which is the reduced form of
+
+
,Arn+" + C A,W;' m
= gn
k-0
+
where g, = (A, - A,,,)eoik) h, . T h e uniform boundedness of (xn) on every compact subinterval of R, and the convergence of (A,.,), (h,) in L(P), L(W), respectively, imply that (g,) converges in L(W) (its limit is, in fact, h). We again choose an arbitrary to > 1, and a corresponding p such that (121.3) is satisfied. Applying Theorem 121.B as before, but to w,, - wn as a solution of (120.4) with h replaced by g,. - g, , we find
II x n * ( t ) - x n ( t ) II ,< 2 e + (cmpm+'m ~
+
s'"" )I 0
g,*(u)
- g,,(u) 1) du) ,
t,+l 0
II wnc(u) - W n ( t l ) II du
0s t
< t" - 1
I
71, 71' =
1 2, 9
... .
Therefore(x,) converges uniformly on [0, to - I]. Since towas arbitrarily large, (x,) converges uniformly on every compact subinterval of R, , a fortiori in L(X). 9. Other consequences of Theorem 121.B pertain to the case in which A, E M( k = 0, ..., m ; this is equivalent, on account of (121.2), to A E M(a). T h e following lemma is a trivial consequence of Theorem 121.B.
r),
121.D. Assume that A, E M(P), k = 0, ..., m, and t h a t p is a positive integer such that p 21 A IM . Assume that h E L( W ) , and define the function H by H ( t ) = ':J 11 h(u)ll du. Then every solution w of (120.4) satisfies
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
38 I
We can use this lemma to obtain an interesting result concerning solutions of (120.4) that belong to some space D( W), where D E b y K , Such solutions are of course again called D-SOLUTIONS.
m),
121.E. THEOREM. Assume that A, E M( k = 0, ..., m, and that m D E b y K. A solution w of (120.3) is a D-solution if and only if x = @ w ( j ) 0 is a D-solution of the reduced form (120.1). If h E L(W ) has compact support, or if h E B(W ) where B E b y or b y + or by%‘+ and D is thick with respect to B, then a solution w of (120.4) is a D-solution i f and only m if x = @ w ( j ) is a D-solution of the reduced form (120.2). 0
Proof. T h e “if” parts are obvious. Let p , H be as in 121.D. In the homogeneous equation, h = 0, hence H = 0, so that w ED(W) implies X E D ( X )by (l21.10), and in fact
If h has compact support, H has compact support and is continuous, so that H ED ; the last fact also holds if h E B(W ) and D is thick with respect to B. In both cases, (121.10) implies that if w E D ( W ) then EED(X). &
122. Admissibility and (B, D)-manifolds In the Introduction we mentioned that we should be interested in the “admissibility” of a Y-pair (B, D) with respect to (120.4), i.e., the existence of a D-solution of (120.4) for every h E B(W); and also in the extension to higher-order equations of the concept of a (B, D)-manifold. I t is tempting to regard the results of the preceding section, and especially Theorem 121.E, as a means of reducing these problems for Eq. (120.4) to corresponding problems for the reduced form (120.2), for which the machinery of Chapter 5 is available. This is indeed possible, but involves a certain amount of technical juggling of function spaces that would delay our reaching the important results. We therefore prefer to follow the course of paralleling, for the higher-order equation, the essential features of the first-order theory of Chapter 5, and using the reduced form as a convenient auxiliary device. k = 0, ..., m, given throughout the section. We assume A, E L( Although our attention will later be focused on Y-pairs and the like, the first few definitions and results are meaninful in a wider context.
m),
382
Ch. 12. HIGHER-ORDER EQUATIONS
For any space D E b M ( W ) , a solution w of (120.3) or (120.4) is a D-SOLUTION if w E D , a terminology anticipated in the preceding section. m We define the set X,,,, = {@ w ( j ) ( O ): w a D-solution of (l20.3)}, a linear manifold in X. In full, i?should be denoted by X,,,,(A, , ..., A,,&). An N-pair -more precisely, an Jv(W)-pair -(B, D ) is ADMISSIBLE (FORA , , ..., A,,L, or, more loosely, FOR (120.4)) if for every h E B there exists a D-solution w of (120.4). T h e pair is REGULARLY ADMISSIBLE if, in addition, X,,,) is closed. Concerning the concept of admissibility we merely establish the analogue of Theorem 5 I .A. 122.A. THEOREM. If the pair (B, D) is admissible for A , , ..., A , , , there exists a number K > 0 such that for every h E B and every number p > I there is a D-solution w of (120.4) with I w ID pKI h .
<
In
Proof. T h e proof is quite similar to that of Theorem 51.A (the essential change is the choice of the closedness theorem), but in view of the importance of the result we give it in full. Let V be the linear manifold l)st primitives of all D-solutions of (120.4) for all h E B,i.e., of all ( m w E D that satisfy w ( ~ + ~C" ) A , w ( ~E) B. T h e mapping 17 : V + B defined by I7w = ~ ( ~ + l ) A , w ' ~ )is linear and, by assumption, surjective. Since B,D are stronger than L(W), Theorem 121.C (with fixed A , , ..., A?,J implies that the graph of I7 is closed in D x B.Since B, D are Banach spaces, the Open-Mapping Theorem implies the existence of a number k > 0 such that for every h E B there exists w E n - l ( { h } ) with I w ID kl h T h e conclusion then holds with K equal to the infimum of all possible values of k. 9,
+
+ +
<
In.
We now specialize our study to the case we are actually interested in: we assume from now on that A , E M(W),k = 0, ..., m, and that (B, D) is a Y - p a i r , a .T+-pair, or a .T%'+-pair. Since D E b y K , Theorem 121.E implies (with the usual dropping of the arguments W , X) that (122.1)
Xn(DdAo
1
..., Ant)
= &,(A).
For this reason we simply write X,, . For such pairs as were just D)-manifold as follows. mentioned, we define the concept of a (B, A linear manifold Y in X is a (B,D)-MANIFOLD (FORA , , ..., A , , or FOR (120.4)) if Y C X n D , and if there exists a number K , > 0 such that for every h E k,B( W) and every number p > 1 there exists a solution m w of (120.4) such that, if x = 0w ' j ) is the corresponding solution 0 of (120.2), we have x z ( 0 ) E Y (whence x, E D(X), whence x E D(X),
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
383
<
whence w E D (W))and I w 1, pKYlh In . A closed (B, D)-manifold is a (B,D)-SUBSPACE. We next make a quick survey of the essential theorems about (B, D)manifolds and (B, D)-subspaces, using as much as possible the analogy with the corresponding results in Section 52. 122.B. If Y is a (B, D)-manifold for A,, ..., A,, there exists a number C , 2 0 such that, i f h E k,B( W), p > 1, and w is a solution .of (120.4) m such that x = @ w(j) has xm(0)E Y and I w ID pKyI h I,, , then II x(0)ll PCYl h ;1 ’ Proof. If p is a positive integer, p 2 21 A IM, Theorem 121.B (formula 121.5)) implies, in particular,
<
<
I1 ~(0) I1 < 2eiP(cmpm+’mor(D;1
+ l/mP) Iw l o +
so that the conclusion holds with
+
C Y = 2ef”{cn,pn1+1mKya(D; 1 I&)
1
+ l/mP) Ih In),
+ a(B; 1 + l/ntp)).
122.C. Assume that Y is a (B, D)-manifold. Then: (a) Y is a (B, , D,)-manifold for any F-pair or F+-pair or Y W - p a i r (B, ,D,) (such that (kB, ,D1) is) weaker than (B, D ) ; (b) any linear manifold Z such that Y C 2 C X,, is also a (B, D)manifold; in particular, so is X,, itself; (c) if Y C X,,, , Y is a (B, kD)-manifold. Proof.
Same as for 52.B.
9,
T h e next few results allow us to restrict our attention substantially to F - p a i r s only.
A linear manifold Y in X is a 122.D. THEOREM. for A, , ..., A,, i f and only i f it is an (lcB, D)-manifold.
(B,D)-manifold
Proof. T h e proof follows the lines of that of Theorem 52.C, but moves back and forth between (120.4) and the reduced form. T h e “if” part is trivial by 122.C,(a). Assume that Y is a (B, D)-manifold; (lcB, D) is a F - p a i r or a F--pair. Let h E k,lcB(W) and p > 1 be given and set s = s(h). As in the proof of Theorem 52.C there existsg E B( W) such that II g I1 < Ilf II (whenceg E k,B(W) with s(g) < s) and I g le < I h llcs and
384
Ch. 12. HIGHER-ORDER EQUATIONS
+
By the assumption, there exists a solution v of dm+l) 1; =g m such that y = @ v ( j ) satisfies ym(0)E Y and I v 1, < pKYl gmlr < pKyl h llcB . Let k be the solution of (120.4) such that x = @ w ( j ) 0 satisfies x(s) = y(s), so that x, = y, , xm(0)E Y. I t then follows from (121.2) and (31.5) as in Theorem 52.C that I x - y ID < ( p - 1)1 h Ilea; thus
I
ID
ID
+ I w-v
ID
dI
ID
+ I "-.Y
10
d (pKY
+ (P-l))
I
llcB
.&
122.E. If Y is a (B, D)-manifold for A,, ..., A,, , then Y is a (T-lcB, D)-manifold (where the latter pair is a F - p a i r weaker than (B, D ) and T-lcB is locally closed). Proof. T h e proof is related to that of 52.J as the proof of Theorem 122.D is related to that of Theorem 52.C. T h e details are left to the reader. &
122.F. Assume that D E b y K is given. The linear manifold Y in X is a (B,D)-manifold for A,, ..., A, for some F - p a i r or Y j - p a i r or F P - p a i r (B, D ) if and only i f i t is a (T, D)-manifold. Proof. Same as for Theorem 52.K, using 122.C, 122.E instead of 52.B, 52.5. &
Our next result concerns the connection between admissibility and (B, D)-manifolds. 122.G. THEOREM. I f (B, D ) is admissible for A, , ..., A,,, , then X,, is a ( B ,D)-manifold. Proof. Same as the proof of Theorem 52.F (first part), using Theorem 122.A instead of Theorem 51.A. & Remark. We do not give the (valid) partial converse, analogous to the second part of Theorem 52.F.
Next, we obtain the fundamental theorem on (B, D)-subspaces, analogous to Theorem 52.1. I n this theorem, S , = S,,(A) for some subspace Y C X,,(A) (Theorem 33.B). 122.H. THEOREM. Assume that Y is a (B, D)-subspacefor A,, ..., A,. For every h E k,B( W )and every A > 1 [and every (Y, A)-splitting q of XI m there exists a solution w of (120.4) such that x = @ w ( j )satisfies xm(0)E Y 0 and 11 x(0)ll Ad( Y , x(0)) [q(x(O)) = x(O)]; every solution w with these
<
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
385
Proof. T h e proof is almost identical with that of Theorem 52.1, but since that proof is itself referred to the proof of a previous theorem, we give our present proof explicitly. T o prove the existence of w in the first part of the statement, we let q be a ( Y ,A)-splitting of X . By the assumption, there exists a solution w‘ m of (120.4) such that x‘ = 0w’(i) has xL(0) E Y , and I w’ I D hKYlh IB. m 0 Let w be the solution of (120.4) such that x = @ w ( j ) satisfies x(0) = q(x‘(0)). Then q ( x ( 0 ) ) = x(O), 11 x(0)ll h d( Y , l(O)), and x - x‘ = x, - x6 is a solution with x,(O) - xL(0) = x(0) - x’(0) E Y . Therefore
<
<
Y. T o prove the second part, we let the solution w (and x) be as stated, and let p > 1 be given. By the definition and by 122.B there exists a solution w‘ of (120.4) that, with the corresponding x’, satisfies x‘(0) E Y , I w’ ID pKYlh I,, , 11 x’(0)ll < pCyl h l B . Now x - x‘ = x, - x, is a solution of (120. I ) with initial value in Y , so that 11 x(0)ll < h d( Y , x(0)) = h d( Y , x’(0)) < All x’(0)lI < phCyI h In . By Theorem 33.B, I w - w‘ ID < I x - 3’ ID S y I I ~ ( 0 ) x’(0)ll p( I X ) S y C y I h l e a Thus Xm(O) E
<
I
<
Since p
>I
<
.
+
was arbitrary, the conclusion follows.
&
Finall.y, we record the almost obvious fact that admissibility of a .F-pair or related pair for A implies admissibility of the same pair for A,, ..., A,,, , and that a similar implication holds with regard to (B, D)manifolds. 122.1. THEOREM. If (B, D ) is [regularly] admissible f o r A , it is [regularly] admissible f o r A,, ..., A,,, . If Y is u (B, D)-manifold for A , Y is a (B, D)-manifold for A,, ..., A,,, .
Proof. Assume that (B, D ) is admissible for A. If h E B(W), the corresponding f in the reduced equation (12b.2) satisfies f E B(X) by ( l 2 l . l ) , (121.2). By the assumption, the reduced equation has a D-solution, say x. By 121 .A, xo is a D-solution of (120.4), so that (B, D) is admissible for A, , ..., A,),. T h e conclusion about regular admissibility follows from (122.1). If Y is a (B, D)-manifold for A , and h E k,B( and p > 1, then f E k,B(X) and the reduced equation (120.2) has a solution x with ~ ~ (E 0Y ), I x ID pKYlf Is = pKYlh I, , where
w>
<
386
Ch. 1 1. HIGHER-ORDER EQUATIONS m
K , = KYB,D(A).By 121.A, xo is a solution of (120.4), with @ xO‘j)= x, 0 and I xo ID I x ID < pK,1 h Is. Since by the assumption Y C X,D, Y is also a ( B ,D)-manifold for A,, ..., A,,, . &
<
123. The main theorems T h e results we intend to prove in this section correspond closely to those given in Chapter 6 (especially Section 64) for first-order equations. T h e same terminology about “direct” and “converse” theorems will be used: a “direct” theorem states the way in which the regular admissibility of a certain pair (B, D) or the existence of a ( B ,D)-subspace for A,, ..., A,, implies a certain type of behavior of the solutions of the homogeneous equation (120.3), or rather of the reduced form (120.1) -in our study, more precisely the existence of a dichotomy or an exponential dichotomy for A. “Converse theorem” has then the obvious meaning.
We only deal with the case in which A, E M(@’), k = 0, ..., m, an assumption we make throughout the section. (B, D ) always denotes a given 9 - p a i r or F+-pair orYV-+-pair. As we shall see, the main reason why our direct theorems do not have the neat aspect of those of Section 64, in spite of the assumption that A, E M( k = 0, ..., m, is the need to impose additional conditions on the coefficients A , , ..., A,; observe that A, is not affected by these additional conditions, and that they are satisfied anyway in the important case that A, E La( k = 1, ..., m. T h e converse theorems, by contrast, are immediate corollaries of the converse theorems of Chapter 6 as applied to the reduced forms of the homogeneous and nonhomogeneous equations.
m),
m),
123.A. THEOREM. Assume that Y is a ( B ,D)-subspace for A,, ...,A,,, (in particular, that ( B ,D ) is regularly admissible for A, , ..., A,rl and Y = XoD).Assume further that , Y [ ~ , ~ +E, ~T-lcB A , ( a space >, B ) for all t E R , , k = 1, ..., m, with (123.1)
“’tp
I X [ t . t + l ] A k ( T I C B < 00,
=
1,
...,
(in particular, that A, E L“(@), k = 1, ..., m, or that B is weaker than 0,Ll for some T E R+). Then Y induces a dichotomy for A. Proof. 1. We observe first that the parenthetical conditions that follow (123.1) do imply (123.1). If A,: E L “ ( m , then II X ~ ~ . ~I1 + , ~ A ~
<
123. THEMAIN
387
THEOREMS
I A , Ix[I,I+lIE T-lcB, since T-lcB E b y K(23.1,23.K, 24.1, 24.L,( I)), and I x [ ~ , ~ . ,IT-lcB . ~ ~ A<, ~ I A , IP(T-lcB; 1 ) and this bound does not depend
< <
on t . If B is weaker thin O,L1, then T-lcB is weaker than L', say L1 PT-lcB; and X [ l . f + l I A k E L'(W, I XrI.f+1IAkIT-lcB \< P Jf6+lII Ak(U)lldu pI A , l M , which again does not depend on t . Next, we may and do assume without loss that B E b y K and B is locally closed, since we may otherwise replace B on account of 122.E by the weaker space T-lcB, which has these properties. Once this assumpE B for all tion is made, B coincides with T-lcB, so that X[f,f+lIAk t~ R + , k = I , ..., m,and (123.1) is replaced by
Throughout the proof, p denotes a positive integer such that P 2 2 1 A l M *
2. In order to enter the proof proper, we choose a real-valued ( m 1)st primitive q~ 3 0 defined on R+ , vanishing outside [i,I], and such that 2,:J ~ ( tdt) = I . Let c > 0 be a common bound for I ~ ( I,~ k = 0, ..., m. We intend to show that statement (e) or (f) of Theorem 41.A is satisfied by Y with respect to A. Let h > I be given and let y , z be solutions of (120.1) such that y(0) E Y , 11 z(0)lI Ad( Y , ~ ( 0 ) ) Let . A 2 1 and 7 E R+ be chosen and define w by
+
<
+
are ( m I)st primitives and, by 121.A, yo(k)= Y , Now yo, z0, zk,k = 0, ..., m. Therefore
z O ( ~ )=
+ wr.(t),
K
= 0,
..., m
+ 1,
where
k
so that
= 0,
..., m
+ 1,
1
388
Ch. 12. HIGHER-ORDER EQUATIONS
implies that zrNow (123.3) , and (123.5) yields
+
w is a solution of (120.4) for h = v , , + ~
m
We set x = @ w ( j ) . Taking into account that each v k vanishes, by 7 + A ) , k = 0, ..., m, it follows from (123.3) (123.5), outside'(7 + &I, and the assumption on q that (123.7)
=
@,+AX
in particular, x, (123.8)
m+d,
= dy
X[0,7+&AlX
=
-~X[u.r+iAlz;
and x(0) = -dz(O), so that
Xm(0) E
y,
I/ x(0) I1 d M Y , x(0)).
3. Thus far, we have let d 2 1 be arbitrary, because we shall require the preceding argument in the next proof. But now we choose d = 1. We find that (123.6) implies, with (123.2) and (31.7), that h E koB(W )with I h IB 011 y(7) z(7)11, where (T = 2m+1eh(&B; 1) ak) does not depend on h, y, x, 7 . We may now apply Theorem 122.H to w , x in view of (123.8): we find I w ID < AKll h IB hoKkll Y ( T ) z(7)ll; by (123.7) we conclude that
+
<
xy
<
+
+
(1 23.9)
4. Let now t o , t E R, be given. If t 2 to we apply Theorem 121.B to yo (on account of (121.2), limp l/p 6 in formula (121.5)). We use (31.7) and (123.9) for 7 = t o , and find
<
<
+
II ~ ( t!I )< e*" II ~ ( t 1 ) II < 2ePCmPm+lm
< 2epcmpnr+1ma(D;3) I @ t 0 + i ~ O ID < D I l ~ ( t o )+ z(to) 11, where D = D(X) = 2he%r,,pm+1ma(D;+)OK;. If t < to we apply Theorem 121.B to xo, and use (31.7), and (123.9) for 7 = to + I , and find
< e-*PDI1 A t o + 1) + 4 1 , + 1) I1 d D II rcto)+ 4 t o )!I, with the same D is proved. &
=
D(h). Therefore condition (e) or (f) of Theorem 41.A
123. THEMAIN
389
THEOREMS
123.B. THEOREM. Let the assumptions of Theorem 123.A hold. Suppose, in addition, that (B, D) is not weaker than (L1,L,"); and that, ifD is weaker than Lz, then (123.10)
lirn d-'s;p
'1b.X
I ~[,,,+~p4,IT-l,-B
k = 1, ..., m
= 0,
(this last condition holds if A, E L"( @) or A , E T-lcB( @), k Then Y induces an exponential dichotomy for A.
=
1, ..., m).
Proof. 1. We observe that the parenthetical conditions following ( 123.10) do imply ( 1 23.10) when D is weaker than L," and consequently B is not stronger than L'. This is obvious, even without this assumption on B, when A, E T-lcB(m). If A, e L m ( r ) ,we use 23.K for the locally closed space T-lcB and find 0-' supl I X [ , , ~ +IT-lcB ~~A, d-'/3( T-lcB; d)l A, I 21 A, I/&( T-lcB; 0 ) = 21 A, 1/a(B; 0) -+ 0 as d + 00 (Remark 2 to 23.S, also applicable to B E bFV'). As in the proof of Theorem 123.A, we may and do assume that B E bF' and B is locally closed, since otherwise B may be replaced, on account of 122.E, by the weaker space T-lcB. Under this assumption, ( I 23. lo), when applicable, becomes
<
<
(123.1 1)
lim +=d-I syp I ~[,,,+,p4~, lB = 0.
A
k
=
1, ...,m.
T h e positive integer p and the real-valued function q~ are as in the preceding proof. By Theorem 123.A, Y induces a dichotomy for A ; let N o , N i = Ni(h) be the parameters in conditions (Di), (Dii) of the definition of a dichotomy. In order to prove the theorem, we show that Y satisfies condition (b) of Theorem 42.A for any given A > 1 ; we need only prove (Ei), (Eii), since (Diii) is satisfied on account of the ordinary dichotomy induced by Y .
2. We assume first that D is not weaker than Lr, so that (by 23.S) lim,+mB(D; 1) = 00. T o prove (Ei), we choose 1 >, 2 so large that B(D; 1) > 3 0 e f ~ ' ~ ~ p ~ + ~ m N , u K ; ,
where CT is as in part 3 of the preceding proof. Let y be any solution of (120.1) with y(0) E Y . We carry out parts 2 and 3 of the proof of Theorem 123.A for this y , for 0 instead of x, for d = 1, and for any T E R, . Since here h > 1 is arbitrarily close to 1 , ( 1 23.9) is replaced by
( I 23.12)
I @,do ID < OK; IIAd 11.
Ch. 12. HIGHER-ORDER EQUATIONS
390
Now we use Theorem 121.B and (123.12), (31.7), as well as 23.M, and find
II Y('
/
+ 1) II d NO(1 -
T+l Tfl
I/Y W II du
< 2 e * 1 t m p m + 1 m ~ , ( 1-
11-1
/ '+' ';1 du
11 y'(s) 11 ds
T+l
+ >
>
since 1 2 implies ( ( I - 1 ) / ( 1 + &))P(D;1 i) &?(D;I). It follows from (Di) and 20.C applied to ]I y 11-l (when y # 0) that (Ei) holds with N = 2 N 0 , v = I-' log 2. To prove (Eii), we consider A > 1 given, and choose I' = l'(A) >/ 2 so large that P(D;1') 2 30Ae*~c,pm+1mN~(h)oK~. Let z be any solution of (120.1) with 11 z(O)11 A d( Y , ~(0)).We carry out parts 2 and 3 of the proof of Theorem 123.A for this h and z, for 0 instead of y, for d = 1, and for T 1' instead of T . Proceeding as above and using Theorem 121.B, (123.9), (31.7), and 23.M, we find
<
+
I1 4 ') II < %(I'
- 11-l
J
5+L'-l
T
;I z(u) II du
I t follows from (Dii) and 20.C that (Eii) holds with N'
=
2Ni(A),
v' = l'-l log 2. Observe that this value of v' depends on A; for this
reason we have proved condition (b) of Theorem 42.A, rather than condition (a). 3. We assume now that D is weaker than Lr;then B is not stronger than L1, so that (since B is locally closed) lim,,,d-'P(B;d) limA+m2 / 4 B ; d ) = 0 (23.K, Remark 2 to 23.S); further, (123.1 1) holds. T o prove (Ei), we choose d 2 I so large that
<
123. THEMAIN
39 1
THEOREMS
Let y be any solution of (120.1) with y(0)E Y. We carry out part 2 of the proof of Theorem 123.A for thisy, for 0 instead of z, for the specified d, and for any T E R+ . Now (Di) and (123.6) imply h E k,B(W) with
I
IS
+
11 Y ( T ) 11 (&D;
2ms'cN0
m
6=1
'YP
I X[t.t+dIAk
IS)
< 2m+1CNo€AIIY(T) 1).
Continuing as in part 3 of that proof, but with this A and with h > 1 arbitrarily close to I , we find I w ID ,< K;I h IS whence, using (123.7),
I @~+dy'
(1 23.13)
ID
I
d
ID
d 2n'+1cN0K&11 y ( T )11'
By Theorem 121.B and (123.13),
(1 y
+
( ~ A ) (1
< 2efpc,pnr+*m+;$/
I1Y " 4 I1du
< 2ef"r,pnb+1ma(D;2) I&@,, ID < 2my2etPc p'"+'ma(D; $)cN,K;EI I ~ ( T ) in
11
=
4 I]JJ(T) 11.
By (Di), and 20.C applied to (1 y (I-l (when y # 0), (Ei) holds with N = 2N0 ,v = d-1 log 2. T o prove (Eii), we consider h > 1 given, and choose A' = d'(h) 2 3 so large that
+c m
d'-'(fi(B; A')
I X[f.l+d']Ak
k=l
IS)
d
= (2m+3he~~c,pm+1ma(D; $)cN;(h)K;)-'.
<
Let z be any solution of (120.1) with 11 z(0)II h d(Y,z(0)).We carry out part 2 of the proof of Theorem 123.A for this A and z,for 0 instead of y, A' instead of A , and any T E R , . Now (Dii) and (123.6) imply h E k,B( W) with
Continuing as in part 3 of that proof, but with A' instead of 1, we find, using (1 23.7), (123.14)
I X [ O . . r + i)lZ0
ID
A'-1
392
Ch. 12. HIGHER-ORDER EQUATIONS
By Theorem 121.B and (123.14),
11 Z ( T ) 11
< 2et%,,,pm+lm I( z0(u)(1 du < 2etPc,pm+'ma(D; 3) I ~[~,,+ip" ID < 2m+zAet"cpm+lma(D;#)clVdK;e' I( + A') 11 < .$ )I + d') (1. m
Z(T
By (Dii), 20.C implies that (Eii) holds with N'= 2Ni, v'=
Z(T
A'-l log2.
&
As was mentioned at the beginning of this section, there is a great contrast between the preceding rather formidable direct theorems and the largely trivial converse theorems. I f the subspace Y of X induces a dichotomy for A, 123.C. THEOREM. then Y is an (Ll,L")-subspace for A,, ...,A, and (Ll, L") is admissible for A,, ..., A, . If X,, has finite codimension with respect to Y , then X,, is an (Ll,Lg)-subspace for A, , ..., A, and (Ll,L;P) is admissible for A,, ..., A, . Proof. Theorems 63.C, 63.E, and 122.1. & Combining Theorems 123.A and 123.C, we have:
A subspace Y of X induces a dichotomy for A i f 123.D. THEOREM. and only i f Y is an (Ll,La)-subspace for A , , ..., A, , If X , is closed, X , induces a dichotomy for A if and only i f (Ll,L") is (regularly) admissible for A, , ..., A,; i f X,, is closed, X,, induces a dichotomy for A i f and only if (Ll,Lg) is (regularly) admissible for A,, ..., A,, . For exponential dichotomies the converse theorem is as follows. If there exists a subspace of X inducing an exponential 123.E. THEOREM. dichotomy for A , then ( B ,D ) is (regularly) admissible for A,, ..., A,,, i f D is thick with respect to B ; and there exists a ( B ,D)-subspace for A, , ..., A , (necessarily unique and = XoD)i f D is thick with respect to kB. Proof. Theorems 64.C and 122.1. & Remark. We do not discuss the assumptions on A,, ..., A, that would ensure the necessity of these conditions. Combination of Theorems 123.B and 123.E yields the following result.
A subspace Y of X induces an exponential dichot123.F. THEOREM. omy for A i f and only i f it is an (M, L")-subspace for A,, ..., A,; and L") is (regularly) admissible for A, , ..., A,,, . i f a n d only i f Y = X , and (M,
Higher-order equations 120. Introduction
Summary o f the chapter In all preceding chapters the linear differential equations that were considered were (30. I), (30.2), i.e., (120.1)
X+Ax=O
(120.2)
a+Ax=f
These are first-order equations (although the values of x and f lie in a Banach space X) in that only the first derivative of the solution is involved. In this chapter our objects of study are the higher-order equations (120.3)
w(r!itIl
+
c A,,w(A) rri
=0
0
(1 20.4)
where m is some positive integer, and the values of w , h are in some Banach space W, while the values of each Ak are in It is usually claimed that (120.3), (120.4) may be reduced to (l20.1), (120.2) in an appropriate larger space X (namely the outer direct sum 1 copies of W ) , and that therefore the theory of the higherof m order equations can be subsumed under that of the first-order equations. T h e first part of this statement is substantially correct, and we intend to describe the claimed reduction in Section 121; the last part, however, does not follow in general. This is best illustrated in our context by the concept of admissibility of a pair of function spaces for Eq. (120.4). If B, D E b.N( W )are given, 373
m.
+
374
Ch. 12. HIGHER-ORDER EQUATIONS
it is reasonable to call the pair (B, D) admissible for (1 20.4) if for every h E B there exists a solution w E D of (120.4). If we consider the corresponding “reduced” equation (1 20.2) in the direct-sum space X = WOO... 0 W,,, , say, this condition means that for every f with the mth “component” in B and all others 0 there exists a solution x of (120.2) with its 0th “component” in D. Now the f ’ s involved do form a space in b N ( X ) , but if B were an 9 - s p a c e or a F-space this would no longer be the case for the corresponding space off’s; a more serious obstacle in the way of relating this admissibility of (B, D) with some admissibility for (120.2) and then applying the results of previous chapters is the lack of a priori restrictions on any “component” of the solution x except the 0th. Fortunately, the difficulties just described can in the main be overcome, by means of some technical devices, in the most important case of .F-pairs and similar pairs, provided certain mild assumptions are made on the A k . I t thus becomes possible to establish results substantially similar to those in Chapter 6, relating the admissibility of a 9-pair, or a similar condition, to the behavior of the solutions of the homogeneous equation (120.3) and their derivatives. We remark that, with a somewhat greater effort at technical refinement, the same methods can be applied to systems of equations with each equation of a different order, yielding essentially the same results. We do not go into this matter here. Another, and more fundamental, discrepancy between Eqs. ( 120.9, (120.4) on the one hand and ( 1 20. l), ( I 20.2) on the other is the difficulty in setting up the concept of an adjoint equation in the former case. More precisely, if we reduce (120.3), say, to the form (120.1) in the usual way and then consider the corresponding adjoint equation i* - A*x* = 0, this equation does not have the form appropriate to the reduction of a presumptive “adjoint” equation to (120.3). On the other hand, the usual definition of such an adjoint equatioipresupposes a sufficient smoothness of the A, and is not symmetrical even if the possible asymmetry of W , W* is disregarded. The bilimr functional involved in the appropriate form of Green’s Formula is iself unsymmetrical and depends on t (cf. Hartman [l], Section 12). Hartman [ I ] has studied these questions and related ntters extensively; he bases his work on a very general abstract Ethod that is adaptable to other cases besides the subject of the 86ent chapter, not necessarily involving differential equations. Our p u r p in including this chapter is merely to give the reader a glimpse of tenature of the results and of the special problems they involve; it isherefore much more restricted, as concerns both the method and t scope of the
120. INTRODUCTION
375
results. We shall be content with a bare sketch of the central theorems of the type of those of Chapter 6, as referred to above. We omit all discussion, for the higher-order equations, of the questions dealt with in Chapters 7-11. We use several proofs in Hartman's paper, but do not attempt to describe his general method. We refer to that paper for further information, especially on the adjoint equations and on certain second-order equations with different assumptions on A,, A,. See also Hartman [2], Chapter XIII. Section 121 contains a discussion of Eqs. (120.3), (120.4) and their reduction to the forms (120.1), (120.2), and an analysis of the solutions. There is an important theorem (Theorem 121.B) giving bounds for the derivatives of a solution in terms of bounds ior the solution itself and for h. Section 122 deals with the concepts of admissibility and of (B,D)-subspaces for F-pairs. Section 123 contains the main theorems.
n th primitive functions In order to speak properly of a solution of (120.3) or (120.4) we must generalize the concept of a primitive function. We define recursively an nTH P R I M I T I V E ( F U N C T I O N ) on an interval J and with values in a Banach space X:f is a first primitive iff is a primitive; f is an ( n 1)st primitive if f is a primitive and f is an nth primitive, n = 1, 2, ... . I f f is an nth primitive, we denote by f ( k ) ,k = 0, ...,n , its ~ T DHE R I V A T I V E ,defined recursively by f ( O ) = f, f ( k + l )= ( f ( k ) ) ;' each, except the last, is an ( n - k)th primitive. We prove a lemma giving estimates for the derivatives f ( O ) , ...,f ( m ) of an ( m + 1)st primitive in terms off and f ( m + l ) .
+
120.A. For every positive integer m there exists a real number c, > 0 with the following property: iff : J + X is an ( m 1)st primitive, and a A ] C J is given, then compact interval [ T , T
+
+
(120.5)
Ah I!f("(t) I/
111
< c , ,i=O1l l f ( ~
+jd/m)(1
+ Am f" l ~ f ( m + l ~ (11udu,) T
~ E [ T , T + A ] , k = O ,..., m.
Proof, It is sufficient to verify the corresponding inequality (120.6)
376
Ch. 12. HIGHER-ORDER EQUATIONS
is a real-valued ( m + 1)st primitive on [0, 13. Indeed, if + d] and k are given, there exists x * E X * such that 11 x * 1) = 1, < f ( k ) ( t l )x*) , = Ilf(k)(tl)ll.We set ~ ( s )= Re(f(7 + sd), x*), so that gP)(s) = diRe(f(i)(T + sd), x*), i = 0, ..., m + 1, and (120.6) at s =
where
y~
t , E [T,T
(tl - ~ ) / d implies (120.5) at t = t , . We proceed to prove (120.6). We consider the interpolation polynomials r,, of degree m that satisfy r,,(i/m) = 6,, , i , j = 0, ..., m ; actually, r m j ( s )= l&+j (ms - i)/(j- z]. There exists a common bound c,, > 0 for all I rg; I,j, k = 0, ..., m, on the interval [0, 13. Now y~ = # + C L o ( p ( j / m ) n m jwhere , # is an ( m 1)st primitive that vanishes at s = j / m , j = 0 , ..., m. By Rolle's Theorem there exists sk E [o, 13 such that #(k)(sk)= 0, k = 0, ..., m. Therefore
+
Using induction on m
- k we find
Since rT1) = 0 for all j , we have follows. &
#(m+l) = ~ ( ~ + land ) ,
(120.6)
Remark. A detailed analysis of the polynomials rmjand their derivatives shows that c,,, may be taken t o be m m ( g )if m is even, and mm(i(m?,J if m is odd.
121. The ( m
+ 1)st-order equation
Although we might, of course, take a more general point of view, we agree that throughout this chapter the range of t shall be J = R , . We shall always consider a positive integer m, a Banach space W, and k = 0, ..., m. If h E L(W ) , a SOLUTION of the functions Ak E L( Eq. (120.4) is an (m + 1)st primitive w : R , + Wthat, together with its derivatives, satisfies (120.4), considered as an equation in L(W). For h = 0, we obtain a SOLUTION of (120.3).
w),
121. THE( m
+ I)ST-ORDER
377
EQUATION
I n order to “reduce” Eqs. ( 1 20.3), (120.4) to the forms ( 120. I), (1 20.2), m we introduce the Banach space X = 0 W, the outer direct sum o f o m m 1 times W , provided with the supremum norm; i.e., if x = @ xj E X , 0 xi E W (superscripts will denote the components of elements in X ) , then 11 x l I x = maxi 11 xj I l w ; there is no risk in omitting the subscripts of the norms from now on. We remark that this choice of norms is so convenient that we permit it to infringe the previous agreement that the norm in a finite-dimensional space X is assumed to be euclidean (Section 15, p. 32). For given Al:€L ( r ) , k = 0, ..., m, and h E L(W) we define the Eq. (120.2) in X , given by the system
+
‘
21 - xj+I = 0 ,
(121.1)
J =
0 , ..., m
-
1
+ 1A p h = h, 911
A?‘
0
as the RED.UCED FORM of (120.4). It is easy to see that A and that, with the given norm on X ,
Iljll
=
E
L(x),f E L(X)
II h II.
Equation (120.1) with the same A is the REDUCED FORM of (120.3). T h e precise relation between (120.4) and its reduced form (120.2) is given by the following lemma, the trivial proof of which is omitted.
+
121.A. A n ( m mI)st primitive w : R, 4 W is a solution of (120.4) and only x = @ w ( j ) is a solution of the reduced form (120.2). 0 Conversely, a primitive x : R, -+ X is a solution of (120.2) if and only if xo is an ( m I)st primitive and a solution of (120.4), and xj = xo(j), j = 0, ..., m. Consequently, for any to E R, and any xo E X there exists a ) xj0 ’ j = 0, ..., m. unique solution w of (120.4) with w ( J ) ( t o=
+
T h e fundamental tool allowing us to work with solutions of (120.4) and of its reduced form is the following estimate for the derivatives of a solution of (120.4). T h e number c,,, is the constant given by 120.A.
p
121.R. THEOREM. For given to 3 0 there exists a positive integer such that
= p(tJ
Ch. 12. HIGHER-ORDER EQUATIONS
378
for any such p , and any h E L(W ) ,eoery solution w of (120.4) satisfies (121.4) 11 w Y t ) l (
< 2e!~'cn,p"C (1 4 s +j / m p ) 11 + 3e:" ma
II h ( r ) I/ dr,
["I
j=U
O
Y
K = O ,..., m,
s
[
(121.5) 11 d r ) ( t )I1 < 2 e i p (c,pnhilm
t+l+l/ml,
llw(u)ll du
+
t+1+1/nrp
Ilh(zr)ll did) ,
k = O ,..., m.
O
Proof. The existence of p = p(to) follows from the continuity of the left-hand member of (121.3) as a function of s. For given s, 0 s to there exists, on account of (l21.3), an integer i, 0 i < p , such that
< <
<
(121.6)
II A ( r ) (I dr
[:A
< 4,
=s
T
+ i/p,
A
=
l/p.
In
We consider the solution x = @ w ( j ) of the reduced form (120.2) of 0 (120.4), Using 120.A we find, for almost all t, 7 < t ,< 7 + d,
II w(n'+l)(t) II < I1 20) II < II 4 )II I1 JEWII m
< I1 4I1 (cn,P"'x II
4 7
1=0
+ II
+ j / m p ) /I
I/
+ f'"
117~(m+1)(y)ll
dr) 4- Ilh(t)ll.
T
Integrating over the interval and using (121.6) we obtain
We use 120.A again, and find
< t < s + 1, (31.5), (121.2), (121.3) imply 11 x(t)ll < 11 h(r)ll dr), and this, together with (121.7), yields (121.4). Let now t,"O < t < to - limp, be given. Consider any s, t < s < t + limp < t o , and carry out the preceding argument as far as (121.7). t < - t < (s + 1 lip) ( s - limp) < I and Now 0 < s For any
e*P(ll
t,
+
x(~)ll
s
JH+'
-
7
-
-
121. THE(m
+ <
d s $- 1 ( 121.7) imply 7
+ I)ST-ORDEREQUATION
< t + 1 + l/mp.
379
Therefore (31.5), (121.2), (121.3),
Integrating the first and last members with respect to s over the interval [t, t limp], we obtain (121.5). Q
+
A first important consequence of Theorem 121.B is a closedness theorem analogous to Theorem 31.D. Let (Ak,,), k = 0, ..., m, (h,) be sequences in 121.C. THEOREM. L( @’), L( W), respectively, and let w, be some solution of
n = 1 , 2, ... .Zf the limits = A, , k = 0, ..., m,1imnjm Lhn = h, limn+mLw, = w exist, then the function w is (except for equivalence modulo a null set) a solution of (120.4), and wkk’ -+ w ( ~uniformly ) on every compact subinterval of R+ as n 400, for k = 0, ..., m.
Proof. With Eq. (121.8) we associate its reduced form
and with (120.4) its reduced form (120.2). Obviously, limnjm LA,n = A, m L f n = f. Now x, = @ w::) is a solution of (121.9). If we can k=O
prove that limn+mLx, = x exists the conclusion will follow at once from Theorem 31.D applied to the reduced forms, and from 121.A. We proceed to prove the L-convergence of (x,). (The actual proof anticipates a part of the proof of Theorem 3 I .D, as we shall see.) We choose an arbitrary tn > 1 and fix it for the time being. Since (A.,) converges in L ( x ) , there exists a positive integer p such that J8+l 11 A.,(r)\l dr < $p for all s, 0 < s < t, and n = I , 2, ...; then the
380
Ch. 12. HIGHER-ORDER EQUATIONS
limit A satisfies (121.3). From Theorem 121.B (specifically, (121.5)) applied to the solution w, of (121.8) we find
n = l , 2 ,....
0
Since (w,), (h,) converge in L( W) and towas arbitrarily large, we conclude that (x,) is uniformly bounded on every compact subinterval of R, . Now x, is a solution of i, Ax, = ( A - A,,)x, f, ,which is the reduced form of
+
+
,Arn+" + C A,W;' m
= gn
k-0
+
where g, = (A, - A,,,)eoik) h, . T h e uniform boundedness of (xn) on every compact subinterval of R, and the convergence of (A,.,), (h,) in L(P), L(W), respectively, imply that (g,) converges in L(W) (its limit is, in fact, h). We again choose an arbitrary to > 1, and a corresponding p such that (121.3) is satisfied. Applying Theorem 121.B as before, but to w,, - wn as a solution of (120.4) with h replaced by g,. - g, , we find
II x n * ( t ) - x n ( t ) II ,< 2 e + (cmpm+'m ~
+
s'"" )I 0
g,*(u)
- g,,(u) 1) du) ,
t,+l 0
II wnc(u) - W n ( t l ) II du
0s t
< t" - 1
I
71, 71' =
1 2, 9
... .
Therefore(x,) converges uniformly on [0, to - I]. Since towas arbitrarily large, (x,) converges uniformly on every compact subinterval of R, , a fortiori in L(X). 9. Other consequences of Theorem 121.B pertain to the case in which A, E M( k = 0, ..., m ; this is equivalent, on account of (121.2), to A E M(a). T h e following lemma is a trivial consequence of Theorem 121.B.
r),
121.D. Assume that A, E M(P), k = 0, ..., m, and t h a t p is a positive integer such that p 21 A IM . Assume that h E L( W ) , and define the function H by H ( t ) = ':J 11 h(u)ll du. Then every solution w of (120.4) satisfies
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
38 I
We can use this lemma to obtain an interesting result concerning solutions of (120.4) that belong to some space D( W), where D E b y K , Such solutions are of course again called D-SOLUTIONS.
m),
121.E. THEOREM. Assume that A, E M( k = 0, ..., m, and that m D E b y K. A solution w of (120.3) is a D-solution if and only if x = @ w ( j ) 0 is a D-solution of the reduced form (120.1). If h E L(W ) has compact support, or if h E B(W ) where B E b y or b y + or by%‘+ and D is thick with respect to B, then a solution w of (120.4) is a D-solution i f and only m if x = @ w ( j ) is a D-solution of the reduced form (120.2). 0
Proof. T h e “if” parts are obvious. Let p , H be as in 121.D. In the homogeneous equation, h = 0, hence H = 0, so that w ED(W) implies X E D ( X )by (l21.10), and in fact
If h has compact support, H has compact support and is continuous, so that H ED ; the last fact also holds if h E B(W ) and D is thick with respect to B. In both cases, (121.10) implies that if w E D ( W ) then EED(X). &
122. Admissibility and (B, D)-manifolds In the Introduction we mentioned that we should be interested in the “admissibility” of a Y-pair (B, D) with respect to (120.4), i.e., the existence of a D-solution of (120.4) for every h E B(W); and also in the extension to higher-order equations of the concept of a (B, D)-manifold. I t is tempting to regard the results of the preceding section, and especially Theorem 121.E, as a means of reducing these problems for Eq. (120.4) to corresponding problems for the reduced form (120.2), for which the machinery of Chapter 5 is available. This is indeed possible, but involves a certain amount of technical juggling of function spaces that would delay our reaching the important results. We therefore prefer to follow the course of paralleling, for the higher-order equation, the essential features of the first-order theory of Chapter 5, and using the reduced form as a convenient auxiliary device. k = 0, ..., m, given throughout the section. We assume A, E L( Although our attention will later be focused on Y-pairs and the like, the first few definitions and results are meaninful in a wider context.
m),
382
Ch. 12. HIGHER-ORDER EQUATIONS
For any space D E b M ( W ) , a solution w of (120.3) or (120.4) is a D-SOLUTION if w E D , a terminology anticipated in the preceding section. m We define the set X,,,, = {@ w ( j ) ( O ): w a D-solution of (l20.3)}, a linear manifold in X. In full, i?should be denoted by X,,,,(A, , ..., A,,&). An N-pair -more precisely, an Jv(W)-pair -(B, D ) is ADMISSIBLE (FORA , , ..., A,,L, or, more loosely, FOR (120.4)) if for every h E B there exists a D-solution w of (120.4). T h e pair is REGULARLY ADMISSIBLE if, in addition, X,,,) is closed. Concerning the concept of admissibility we merely establish the analogue of Theorem 5 I .A. 122.A. THEOREM. If the pair (B, D) is admissible for A , , ..., A , , , there exists a number K > 0 such that for every h E B and every number p > I there is a D-solution w of (120.4) with I w ID pKI h .
<
In
Proof. T h e proof is quite similar to that of Theorem 51.A (the essential change is the choice of the closedness theorem), but in view of the importance of the result we give it in full. Let V be the linear manifold l)st primitives of all D-solutions of (120.4) for all h E B,i.e., of all ( m w E D that satisfy w ( ~ + ~C" ) A , w ( ~E) B. T h e mapping 17 : V + B defined by I7w = ~ ( ~ + l ) A , w ' ~ )is linear and, by assumption, surjective. Since B,D are stronger than L(W), Theorem 121.C (with fixed A , , ..., A?,J implies that the graph of I7 is closed in D x B.Since B, D are Banach spaces, the Open-Mapping Theorem implies the existence of a number k > 0 such that for every h E B there exists w E n - l ( { h } ) with I w ID kl h T h e conclusion then holds with K equal to the infimum of all possible values of k. 9,
+
+ +
<
In.
We now specialize our study to the case we are actually interested in: we assume from now on that A , E M(W),k = 0, ..., m, and that (B, D) is a Y - p a i r , a .T+-pair, or a .T%'+-pair. Since D E b y K , Theorem 121.E implies (with the usual dropping of the arguments W , X) that (122.1)
Xn(DdAo
1
..., Ant)
= &,(A).
For this reason we simply write X,, . For such pairs as were just D)-manifold as follows. mentioned, we define the concept of a (B, A linear manifold Y in X is a (B,D)-MANIFOLD (FORA , , ..., A , , or FOR (120.4)) if Y C X n D , and if there exists a number K , > 0 such that for every h E k,B( W) and every number p > 1 there exists a solution m w of (120.4) such that, if x = 0w ' j ) is the corresponding solution 0 of (120.2), we have x z ( 0 ) E Y (whence x, E D(X), whence x E D(X),
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
383
<
whence w E D (W))and I w 1, pKYlh In . A closed (B, D)-manifold is a (B,D)-SUBSPACE. We next make a quick survey of the essential theorems about (B, D)manifolds and (B, D)-subspaces, using as much as possible the analogy with the corresponding results in Section 52. 122.B. If Y is a (B, D)-manifold for A,, ..., A,, there exists a number C , 2 0 such that, i f h E k,B( W), p > 1, and w is a solution .of (120.4) m such that x = @ w(j) has xm(0)E Y and I w ID pKyI h I,, , then II x(0)ll PCYl h ;1 ’ Proof. If p is a positive integer, p 2 21 A IM, Theorem 121.B (formula 121.5)) implies, in particular,
<
<
I1 ~(0) I1 < 2eiP(cmpm+’mor(D;1
+ l/mP) Iw l o +
so that the conclusion holds with
+
C Y = 2ef”{cn,pn1+1mKya(D; 1 I&)
1
+ l/mP) Ih In),
+ a(B; 1 + l/ntp)).
122.C. Assume that Y is a (B, D)-manifold. Then: (a) Y is a (B, , D,)-manifold for any F-pair or F+-pair or Y W - p a i r (B, ,D,) (such that (kB, ,D1) is) weaker than (B, D ) ; (b) any linear manifold Z such that Y C 2 C X,, is also a (B, D)manifold; in particular, so is X,, itself; (c) if Y C X,,, , Y is a (B, kD)-manifold. Proof.
Same as for 52.B.
9,
T h e next few results allow us to restrict our attention substantially to F - p a i r s only.
A linear manifold Y in X is a 122.D. THEOREM. for A, , ..., A,, i f and only i f it is an (lcB, D)-manifold.
(B,D)-manifold
Proof. T h e proof follows the lines of that of Theorem 52.C, but moves back and forth between (120.4) and the reduced form. T h e “if” part is trivial by 122.C,(a). Assume that Y is a (B, D)-manifold; (lcB, D) is a F - p a i r or a F--pair. Let h E k,lcB(W) and p > 1 be given and set s = s(h). As in the proof of Theorem 52.C there existsg E B( W) such that II g I1 < Ilf II (whenceg E k,B(W) with s(g) < s) and I g le < I h llcs and
384
Ch. 12. HIGHER-ORDER EQUATIONS
+
By the assumption, there exists a solution v of dm+l) 1; =g m such that y = @ v ( j ) satisfies ym(0)E Y and I v 1, < pKYl gmlr < pKyl h llcB . Let k be the solution of (120.4) such that x = @ w ( j ) 0 satisfies x(s) = y(s), so that x, = y, , xm(0)E Y. I t then follows from (121.2) and (31.5) as in Theorem 52.C that I x - y ID < ( p - 1)1 h Ilea; thus
I
ID
ID
+ I w-v
ID
dI
ID
+ I "-.Y
10
d (pKY
+ (P-l))
I
llcB
.&
122.E. If Y is a (B, D)-manifold for A,, ..., A,, , then Y is a (T-lcB, D)-manifold (where the latter pair is a F - p a i r weaker than (B, D ) and T-lcB is locally closed). Proof. T h e proof is related to that of 52.J as the proof of Theorem 122.D is related to that of Theorem 52.C. T h e details are left to the reader. &
122.F. Assume that D E b y K is given. The linear manifold Y in X is a (B,D)-manifold for A,, ..., A, for some F - p a i r or Y j - p a i r or F P - p a i r (B, D ) if and only i f i t is a (T, D)-manifold. Proof. Same as for Theorem 52.K, using 122.C, 122.E instead of 52.B, 52.5. &
Our next result concerns the connection between admissibility and (B, D)-manifolds. 122.G. THEOREM. I f (B, D ) is admissible for A, , ..., A,,, , then X,, is a ( B ,D)-manifold. Proof. Same as the proof of Theorem 52.F (first part), using Theorem 122.A instead of Theorem 51.A. & Remark. We do not give the (valid) partial converse, analogous to the second part of Theorem 52.F.
Next, we obtain the fundamental theorem on (B, D)-subspaces, analogous to Theorem 52.1. I n this theorem, S , = S,,(A) for some subspace Y C X,,(A) (Theorem 33.B). 122.H. THEOREM. Assume that Y is a (B, D)-subspacefor A,, ..., A,. For every h E k,B( W )and every A > 1 [and every (Y, A)-splitting q of XI m there exists a solution w of (120.4) such that x = @ w ( j )satisfies xm(0)E Y 0 and 11 x(0)ll Ad( Y , x(0)) [q(x(O)) = x(O)]; every solution w with these
<
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
385
Proof. T h e proof is almost identical with that of Theorem 52.1, but since that proof is itself referred to the proof of a previous theorem, we give our present proof explicitly. T o prove the existence of w in the first part of the statement, we let q be a ( Y ,A)-splitting of X . By the assumption, there exists a solution w‘ m of (120.4) such that x‘ = 0w’(i) has xL(0) E Y , and I w’ I D hKYlh IB. m 0 Let w be the solution of (120.4) such that x = @ w ( j ) satisfies x(0) = q(x‘(0)). Then q ( x ( 0 ) ) = x(O), 11 x(0)ll h d( Y , l(O)), and x - x‘ = x, - x6 is a solution with x,(O) - xL(0) = x(0) - x’(0) E Y . Therefore
<
<
Y. T o prove the second part, we let the solution w (and x) be as stated, and let p > 1 be given. By the definition and by 122.B there exists a solution w‘ of (120.4) that, with the corresponding x’, satisfies x‘(0) E Y , I w’ ID pKYlh I,, , 11 x’(0)ll < pCyl h l B . Now x - x‘ = x, - x, is a solution of (120. I ) with initial value in Y , so that 11 x(0)ll < h d( Y , x(0)) = h d( Y , x’(0)) < All x’(0)lI < phCyI h In . By Theorem 33.B, I w - w‘ ID < I x - 3’ ID S y I I ~ ( 0 ) x’(0)ll p( I X ) S y C y I h l e a Thus Xm(O) E
<
I
<
Since p
>I
<
.
+
was arbitrary, the conclusion follows.
&
Finall.y, we record the almost obvious fact that admissibility of a .F-pair or related pair for A implies admissibility of the same pair for A,, ..., A,,, , and that a similar implication holds with regard to (B, D)manifolds. 122.1. THEOREM. If (B, D ) is [regularly] admissible f o r A , it is [regularly] admissible f o r A,, ..., A,,, . If Y is u (B, D)-manifold for A , Y is a (B, D)-manifold for A,, ..., A,,, .
Proof. Assume that (B, D ) is admissible for A. If h E B(W), the corresponding f in the reduced equation (12b.2) satisfies f E B(X) by ( l 2 l . l ) , (121.2). By the assumption, the reduced equation has a D-solution, say x. By 121 .A, xo is a D-solution of (120.4), so that (B, D) is admissible for A, , ..., A,),. T h e conclusion about regular admissibility follows from (122.1). If Y is a (B, D)-manifold for A , and h E k,B( and p > 1, then f E k,B(X) and the reduced equation (120.2) has a solution x with ~ ~ (E 0Y ), I x ID pKYlf Is = pKYlh I, , where
w>
<
386
Ch. 1 1. HIGHER-ORDER EQUATIONS m
K , = KYB,D(A).By 121.A, xo is a solution of (120.4), with @ xO‘j)= x, 0 and I xo ID I x ID < pK,1 h Is. Since by the assumption Y C X,D, Y is also a ( B ,D)-manifold for A,, ..., A,,, . &
<
123. The main theorems T h e results we intend to prove in this section correspond closely to those given in Chapter 6 (especially Section 64) for first-order equations. T h e same terminology about “direct” and “converse” theorems will be used: a “direct” theorem states the way in which the regular admissibility of a certain pair (B, D) or the existence of a ( B ,D)-subspace for A,, ..., A,, implies a certain type of behavior of the solutions of the homogeneous equation (120.3), or rather of the reduced form (120.1) -in our study, more precisely the existence of a dichotomy or an exponential dichotomy for A. “Converse theorem” has then the obvious meaning.
We only deal with the case in which A, E M(@’), k = 0, ..., m, an assumption we make throughout the section. (B, D ) always denotes a given 9 - p a i r or F+-pair orYV-+-pair. As we shall see, the main reason why our direct theorems do not have the neat aspect of those of Section 64, in spite of the assumption that A, E M( k = 0, ..., m, is the need to impose additional conditions on the coefficients A , , ..., A,; observe that A, is not affected by these additional conditions, and that they are satisfied anyway in the important case that A, E La( k = 1, ..., m. T h e converse theorems, by contrast, are immediate corollaries of the converse theorems of Chapter 6 as applied to the reduced forms of the homogeneous and nonhomogeneous equations.
m),
m),
123.A. THEOREM. Assume that Y is a ( B ,D)-subspace for A,, ...,A,,, (in particular, that ( B ,D ) is regularly admissible for A, , ..., A,rl and Y = XoD).Assume further that , Y [ ~ , ~ +E, ~T-lcB A , ( a space >, B ) for all t E R , , k = 1, ..., m, with (123.1)
“’tp
I X [ t . t + l ] A k ( T I C B < 00,
=
1,
...,
(in particular, that A, E L“(@), k = 1, ..., m, or that B is weaker than 0,Ll for some T E R+). Then Y induces a dichotomy for A. Proof. 1. We observe first that the parenthetical conditions that follow (123.1) do imply (123.1). If A,: E L “ ( m , then II X ~ ~ . ~I1 + , ~ A ~
<
123. THEMAIN
387
THEOREMS
I A , Ix[I,I+lIE T-lcB, since T-lcB E b y K(23.1,23.K, 24.1, 24.L,( I)), and I x [ ~ , ~ . ,IT-lcB . ~ ~ A<, ~ I A , IP(T-lcB; 1 ) and this bound does not depend
< <
on t . If B is weaker thin O,L1, then T-lcB is weaker than L', say L1 PT-lcB; and X [ l . f + l I A k E L'(W, I XrI.f+1IAkIT-lcB \< P Jf6+lII Ak(U)lldu pI A , l M , which again does not depend on t . Next, we may and do assume without loss that B E b y K and B is locally closed, since we may otherwise replace B on account of 122.E by the weaker space T-lcB, which has these properties. Once this assumpE B for all tion is made, B coincides with T-lcB, so that X[f,f+lIAk t~ R + , k = I , ..., m,and (123.1) is replaced by
Throughout the proof, p denotes a positive integer such that P 2 2 1 A l M *
2. In order to enter the proof proper, we choose a real-valued ( m 1)st primitive q~ 3 0 defined on R+ , vanishing outside [i,I], and such that 2,:J ~ ( tdt) = I . Let c > 0 be a common bound for I ~ ( I,~ k = 0, ..., m. We intend to show that statement (e) or (f) of Theorem 41.A is satisfied by Y with respect to A. Let h > I be given and let y , z be solutions of (120.1) such that y(0) E Y , 11 z(0)lI Ad( Y , ~ ( 0 ) ) Let . A 2 1 and 7 E R+ be chosen and define w by
+
<
+
are ( m I)st primitives and, by 121.A, yo(k)= Y , Now yo, z0, zk,k = 0, ..., m. Therefore
z O ( ~ )=
+ wr.(t),
K
= 0,
..., m
+ 1,
where
k
so that
= 0,
..., m
+ 1,
1
388
Ch. 12. HIGHER-ORDER EQUATIONS
implies that zrNow (123.3) , and (123.5) yields
+
w is a solution of (120.4) for h = v , , + ~
m
We set x = @ w ( j ) . Taking into account that each v k vanishes, by 7 + A ) , k = 0, ..., m, it follows from (123.3) (123.5), outside'(7 + &I, and the assumption on q that (123.7)
=
@,+AX
in particular, x, (123.8)
m+d,
= dy
X[0,7+&AlX
=
-~X[u.r+iAlz;
and x(0) = -dz(O), so that
Xm(0) E
y,
I/ x(0) I1 d M Y , x(0)).
3. Thus far, we have let d 2 1 be arbitrary, because we shall require the preceding argument in the next proof. But now we choose d = 1. We find that (123.6) implies, with (123.2) and (31.7), that h E koB(W )with I h IB 011 y(7) z(7)11, where (T = 2m+1eh(&B; 1) ak) does not depend on h, y, x, 7 . We may now apply Theorem 122.H to w , x in view of (123.8): we find I w ID < AKll h IB hoKkll Y ( T ) z(7)ll; by (123.7) we conclude that
+
<
xy
<
+
+
(1 23.9)
4. Let now t o , t E R, be given. If t 2 to we apply Theorem 121.B to yo (on account of (121.2), limp l/p 6 in formula (121.5)). We use (31.7) and (123.9) for 7 = t o , and find
<
<
+
II ~ ( t!I )< e*" II ~ ( t 1 ) II < 2ePCmPm+lm
< 2epcmpnr+1ma(D;3) I @ t 0 + i ~ O ID < D I l ~ ( t o )+ z(to) 11, where D = D(X) = 2he%r,,pm+1ma(D;+)OK;. If t < to we apply Theorem 121.B to xo, and use (31.7), and (123.9) for 7 = to + I , and find
< e-*PDI1 A t o + 1) + 4 1 , + 1) I1 d D II rcto)+ 4 t o )!I, with the same D is proved. &
=
D(h). Therefore condition (e) or (f) of Theorem 41.A
123. THEMAIN
389
THEOREMS
123.B. THEOREM. Let the assumptions of Theorem 123.A hold. Suppose, in addition, that (B, D) is not weaker than (L1,L,"); and that, ifD is weaker than Lz, then (123.10)
lirn d-'s;p
'1b.X
I ~[,,,+~p4,IT-l,-B
k = 1, ..., m
= 0,
(this last condition holds if A, E L"( @) or A , E T-lcB( @), k Then Y induces an exponential dichotomy for A.
=
1, ..., m).
Proof. 1. We observe that the parenthetical conditions following ( 123.10) do imply ( 1 23.10) when D is weaker than L," and consequently B is not stronger than L'. This is obvious, even without this assumption on B, when A, E T-lcB(m). If A, e L m ( r ) ,we use 23.K for the locally closed space T-lcB and find 0-' supl I X [ , , ~ +IT-lcB ~~A, d-'/3( T-lcB; d)l A, I 21 A, I/&( T-lcB; 0 ) = 21 A, 1/a(B; 0) -+ 0 as d + 00 (Remark 2 to 23.S, also applicable to B E bFV'). As in the proof of Theorem 123.A, we may and do assume that B E bF' and B is locally closed, since otherwise B may be replaced, on account of 122.E, by the weaker space T-lcB. Under this assumption, ( I 23. lo), when applicable, becomes
<
<
(123.1 1)
lim +=d-I syp I ~[,,,+,p4~, lB = 0.
A
k
=
1, ...,m.
T h e positive integer p and the real-valued function q~ are as in the preceding proof. By Theorem 123.A, Y induces a dichotomy for A ; let N o , N i = Ni(h) be the parameters in conditions (Di), (Dii) of the definition of a dichotomy. In order to prove the theorem, we show that Y satisfies condition (b) of Theorem 42.A for any given A > 1 ; we need only prove (Ei), (Eii), since (Diii) is satisfied on account of the ordinary dichotomy induced by Y .
2. We assume first that D is not weaker than Lr, so that (by 23.S) lim,+mB(D; 1) = 00. T o prove (Ei), we choose 1 >, 2 so large that B(D; 1) > 3 0 e f ~ ' ~ ~ p ~ + ~ m N , u K ; ,
where CT is as in part 3 of the preceding proof. Let y be any solution of (120.1) with y(0) E Y . We carry out parts 2 and 3 of the proof of Theorem 123.A for this y , for 0 instead of x, for d = 1, and for any T E R, . Since here h > 1 is arbitrarily close to 1 , ( 1 23.9) is replaced by
( I 23.12)
I @,do ID < OK; IIAd 11.
Ch. 12. HIGHER-ORDER EQUATIONS
390
Now we use Theorem 121.B and (123.12), (31.7), as well as 23.M, and find
II Y('
/
+ 1) II d NO(1 -
T+l Tfl
I/Y W II du
< 2 e * 1 t m p m + 1 m ~ , ( 1-
11-1
/ '+' ';1 du
11 y'(s) 11 ds
T+l
+ >
>
since 1 2 implies ( ( I - 1 ) / ( 1 + &))P(D;1 i) &?(D;I). It follows from (Di) and 20.C applied to ]I y 11-l (when y # 0) that (Ei) holds with N = 2 N 0 , v = I-' log 2. To prove (Eii), we consider A > 1 given, and choose I' = l'(A) >/ 2 so large that P(D;1') 2 30Ae*~c,pm+1mN~(h)oK~. Let z be any solution of (120.1) with 11 z(O)11 A d( Y , ~(0)).We carry out parts 2 and 3 of the proof of Theorem 123.A for this h and z, for 0 instead of y, for d = 1, and for T 1' instead of T . Proceeding as above and using Theorem 121.B, (123.9), (31.7), and 23.M, we find
<
+
I1 4 ') II < %(I'
- 11-l
J
5+L'-l
T
;I z(u) II du
I t follows from (Dii) and 20.C that (Eii) holds with N'
=
2Ni(A),
v' = l'-l log 2. Observe that this value of v' depends on A; for this
reason we have proved condition (b) of Theorem 42.A, rather than condition (a). 3. We assume now that D is weaker than Lr;then B is not stronger than L1, so that (since B is locally closed) lim,,,d-'P(B;d) limA+m2 / 4 B ; d ) = 0 (23.K, Remark 2 to 23.S); further, (123.1 1) holds. T o prove (Ei), we choose d 2 I so large that
<
123. THEMAIN
39 1
THEOREMS
Let y be any solution of (120.1) with y(0)E Y. We carry out part 2 of the proof of Theorem 123.A for thisy, for 0 instead of z, for the specified d, and for any T E R+ . Now (Di) and (123.6) imply h E k,B(W) with
I
IS
+
11 Y ( T ) 11 (&D;
2ms'cN0
m
6=1
'YP
I X[t.t+dIAk
IS)
< 2m+1CNo€AIIY(T) 1).
Continuing as in part 3 of that proof, but with this A and with h > 1 arbitrarily close to I , we find I w ID ,< K;I h IS whence, using (123.7),
I @~+dy'
(1 23.13)
ID
I
d
ID
d 2n'+1cN0K&11 y ( T )11'
By Theorem 121.B and (123.13),
(1 y
+
( ~ A ) (1
< 2efpc,pnr+*m+;$/
I1Y " 4 I1du
< 2ef"r,pnb+1ma(D;2) I&@,, ID < 2my2etPc p'"+'ma(D; $)cN,K;EI I ~ ( T ) in
11
=
4 I]JJ(T) 11.
By (Di), and 20.C applied to (1 y (I-l (when y # 0), (Ei) holds with N = 2N0 ,v = d-1 log 2. T o prove (Eii), we consider h > 1 given, and choose A' = d'(h) 2 3 so large that
+c m
d'-'(fi(B; A')
I X[f.l+d']Ak
k=l
IS)
d
= (2m+3he~~c,pm+1ma(D; $)cN;(h)K;)-'.
<
Let z be any solution of (120.1) with 11 z(0)II h d(Y,z(0)).We carry out part 2 of the proof of Theorem 123.A for this A and z,for 0 instead of y, A' instead of A , and any T E R , . Now (Dii) and (123.6) imply h E k,B( W) with
Continuing as in part 3 of that proof, but with A' instead of 1, we find, using (1 23.7), (123.14)
I X [ O . . r + i)lZ0
ID
A'-1
392
Ch. 12. HIGHER-ORDER EQUATIONS
By Theorem 121.B and (123.14),
11 Z ( T ) 11
< 2et%,,,pm+lm I( z0(u)(1 du < 2etPc,pm+'ma(D; 3) I ~[~,,+ip" ID < 2m+zAet"cpm+lma(D;#)clVdK;e' I( + A') 11 < .$ )I + d') (1. m
Z(T
By (Dii), 20.C implies that (Eii) holds with N'= 2Ni, v'=
Z(T
A'-l log2.
&
As was mentioned at the beginning of this section, there is a great contrast between the preceding rather formidable direct theorems and the largely trivial converse theorems. I f the subspace Y of X induces a dichotomy for A, 123.C. THEOREM. then Y is an (Ll,L")-subspace for A,, ...,A, and (Ll, L") is admissible for A,, ..., A, . If X,, has finite codimension with respect to Y , then X,, is an (Ll,Lg)-subspace for A, , ..., A, and (Ll,L;P) is admissible for A,, ..., A, . Proof. Theorems 63.C, 63.E, and 122.1. & Combining Theorems 123.A and 123.C, we have:
A subspace Y of X induces a dichotomy for A i f 123.D. THEOREM. and only i f Y is an (Ll,La)-subspace for A , , ..., A, , If X , is closed, X , induces a dichotomy for A if and only i f (Ll,L") is (regularly) admissible for A, , ..., A,; i f X,, is closed, X,, induces a dichotomy for A i f and only if (Ll,Lg) is (regularly) admissible for A,, ..., A,, . For exponential dichotomies the converse theorem is as follows. If there exists a subspace of X inducing an exponential 123.E. THEOREM. dichotomy for A , then ( B ,D ) is (regularly) admissible for A,, ..., A,,, i f D is thick with respect to B ; and there exists a ( B ,D)-subspace for A, , ..., A , (necessarily unique and = XoD)i f D is thick with respect to kB. Proof. Theorems 64.C and 122.1. & Remark. We do not discuss the assumptions on A,, ..., A, that would ensure the necessity of these conditions. Combination of Theorems 123.B and 123.E yields the following result.
A subspace Y of X induces an exponential dichot123.F. THEOREM. omy for A i f and only i f it is an (M, L")-subspace for A,, ..., A,; and L") is (regularly) admissible for A, , ..., A,,, . i f a n d only i f Y = X , and (M,