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bohner~mr, edu Abstract--This paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict control-lability of a discrete system, that imply isolated

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D i s c r e t e Linear H a m i l t o n i a n Eigenvalue P r o b l e m sM. BOHNERDepartment of Mathematics and Statistics University of Missouri-Rolla Rolla, M O 65409-0020, U.S.A. bohner~mr, eduA b s t r a c t - - T h i s paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict controllability of a discrete system, that imply isolatedness and lower boundedness of the eigenvalues. Due to the quite general assumptions, discrete Sturrn-Liouville eigenvalue problems of higher order are included in the presented theory. (g) 1998 Elsevier Science Ltd. All rights reserved.

K e y w o r d s - - L i n e a r Hamiltonian difference system, Discrete Sturm-Liouville eigenvalue problem, Strict controllability, Reid roundabout theorem, Comparison theorem.

I. I N T R O D U C T I O N In this paper, we introduce discrete linear Hamiltonian eigenvalue problems, i.e., eigenvalue problems which consist of a linear Hamiltonian difference system depending on an eigenvalue parameter A E R subject to self-adjointboundary conditions. The m a i n result on these problems states that, under certain assumptions, the eigenvalues may be arranged as follows:--c~< A1<~ A2<: A3<= ' ' ',

i.e., that the set of eigenvalues is bounded below and that the eigenvalues are isolated in the sense that for any A E R one may pick an 6= e(A)> 0 such that the interval (A -~, A+~) contains at most one eigenvalue. The central notion connected to this isolatedness is the new concept of strict controllability of discrete systems which is also introduced in this paper. The m a i n tools on handling these eigenvalue problems and on proving the above result is a theorem that gives a useful characterization of the eigenvalues (in terms of some matrix being singular), an index theorem (which calculates the local change of the number of some matrix-valued function's negative eigenvalues), a Reid roundabout theorem (that characterizes so-called positive definiteness of discrete quadratic functionals), and a comparison theorem (which states that positive definiteness of one functional together with certain assumptions imply positive definiteness of some other functional). Finally, it should be emphasized that our general assumptions allow us to include discrete Sturm-Liouville elgenvalue problems of higher order so that these important problems may be treated with the same techniques. Let us shortly give an overview on the existing literature of the subject. Discrete SturmLiouville difference equations of order two as well as linear Hamiltonian difference systems have been an object of recent interest. Linear Hamiltonian difference systems were introduced by Erbe and Yan in[1] a

nd examined in three proceeding papers[2-4] by the same authors, however, under assumptions that only include the case of Sturm-Liouville difference equations of order two but

Typeset by .A.,~S-'~X.179

180

M. BOHNER

not of higher order. Further important results in this matter have been obtained by Ahlbrandt, Do~l~, Heifetz, Hooker, Patula, Peil, Peterson, and Ridenhour in[5-11]. In a recent series of publications by the author[12-17] (one of them is a joint work with Do~il~), linear Hamiltonian difference systems were considered under assumptions that include the important case of SturmLiouville difference equations of higher order and that give so-called Reid roundabout theorems for those problems. This work may be considered as one origin of the results proved in the present paper. The other origin is the treatment of continuous linear Hamiltonian eigenvalue problems as is done in the paper[18] by Baur and Kratz and in the monograph[19] on the subject by Kratz. This work also contains the above cited index theorem which may be successfully applied in our discrete case also. Finally, while the study of eigenvalue problems in the existing literature basically reduces to discrete Sturm-Liouville eigenvalue problems of order two (see the books by Agarwal[20, Chapter 11] and by Kelley and Peterson[21, Chapter 7]), a special Sturm-Liouville difference equation of higher order depending on an eigenvalue parameter has been considered in the recent paper[22] by Kratz; however, there is no theory for eigenvalue problems subject to general linear Hamiltonian difference systems. A brief discussion of this paper's s e t u p is in order. The following section introduces discrete linear Hamiltonian eigenvalue problems and gives some preliminaries on linear Hamiltonian difference systems. In Section 3, we present the main result of this paper and give the assumptions that are needed; among them we introduce the concept of strict controllability of discrete systems and the notion of the so-called strict controllability index, which has no obvious analogue in the"continuous theory". Section 4 contains a characterization of the eigenvalues, and this characterization is also improved in some sense if the boundary conditions under consideration are separated. While Section 5 recalls two important auxiliary results (the index theorem from[19, Theorem 3.4.1] and the Reid roundabout theorem from[14, Theorem 3]), Section 6 contains a series of lemmas that are needed for proving the isolatedness of the eigenvalues. Finally, a comparison theorem is proved in Section 7, and as an application of it we show that the eigenvalues are bounded below.

2. P R E L I M I N A R I E S ON D I S C R E T E E I G E N V A L U E P R O B L E M SFirst of all, let us agree upon some terminology. While KerM, I m M, defM, indM, M T, and M? denote the kernel, the image, the dimension of the kernel, the index (i.e., the number of negative eigenvalues), the transpose, and the

Moore-Penrose Inverse (see, e.g.,[23, Theorem 1.5]) of the matrix M, respectively, M> 0 and M _> 0 mean that the (symmetric) matrix M is positive definite and positive semidefinite, respectively. Let n E N, N E 1~IU (0), J:=[0, N]¢3Z, J*:=[0, N+ 1] N Z. We abbreviate a sequence (zk)keJ. by z and use the forward difference operator A defined by Az~:= zk+l - zk, k E J. Let there be given n x n-matrices Ak, Bk, Ck for all k E J so that

I - Ak is invertible and Hk= ( - C k AkThe system

A~ ) i s s y m m e t r i c f o r a l l k E J. Bk

(where xk, uk E R n for all k E J*) is then called a linear Hamiltonian difference system. If the n x n-matrix-valued functions Ak(A), Bk(A), Ck(A) depend for all k E J continuously and differentiable on a parameter )~ E R (so that the above assumptions are satisfied for I - Ak (A) and for

Hk(~)=\ Ak(~)

Bk(~)

Hamiltonian Eigenvalue Problems for each A 6 R), then we consider the systems A= Hs,(A), 0< k< N.

181

\ u~

(Hx)

Moreover, let there be given 2n× 2n-matrices R and R* with rank ( R R* )= 2n and

RR *T= R*R T.

We are interested in so-called self-conjoined boundary conditions (see[19, Definition 2.1.1 and Proposition 2.1.1])

Now, this paper deals with discrete linear Hamiltonian eigenvalue problems of the form (Hx), i.e., A R and (R), (E)

Azk=

Ak(

)xk+,+

Bk(A)uk0< k< N, (A e R), --X0

Auk= Ck(A)x~+, - A~(A)uk J '

(E)

As usual, a number A 6 R is called an eigenvalue of (E) if (Hx) has a nontrivial solution (x, u) satisfying (R), and this solution is then called an eigenfunction corresponding to the eigenvalue A. Moreover, the set of all eigenfunctions is called the eigenspace, and its dimension is referred to as being the multiplicity of the eigenvalue. We shortly s-mmarize some basic definitions and results from[14] on linear Hamiltonian difference systems that will be needed later on. DEFINITION 1. (Conjoined Basis; see[14, Definition 1].) If the n x n-matrices Xk, Uk (instead

of the vectors xk, uk) solve (H) with

r~(X:

U:)=n

and

X:U~=U~X~,

~or~k6J*,

then (X, U) is ca//ed a conjoined basis of (H). Two conjoined bases (X, U) and (X, U) are called normalized whenever X~ Ok - U~ f(k= I (the n x n-identity-matrix), holds. The conjoined bases (X, U) and (X, U) of (H) with for all k e J*

Xo=0o=0

and U o= - R o= I|

are known as the special normalized conjoined bases of (H) at O.

LEMMA 1. (See[19, Coro//ary 3.3.9] and[14, Lemma 3].) For any m E J* and any conjoined basis (X, U) of (H), there exists another conjoined basis (f(, O) o[ (H) such that (X, U) and

(fC, O) are normal~ed and such that f¢m is invextible. Farthexmore, two matrix-valued solutions (X, U) and (X, U) are normalized conjoined bas~

of(H) iffCX*, U* ) withX'=

(0,)x g

and

182

M. BOHN~.R

is a conjoined basis of the system

zt`

0 At,

0 0

\ ut`/ '

O<k<N,

Bk/

where the occurring matrix is of size 4n x 4n.

DEFINITION 2. (Disconjugacy; see[14, Defin

ition 2].) The discrete quadratic functionalN t`=O~N+I

T~N+X

is called positive definite (we write jr> O) if~r(z, u)> 0 holds for all admissible pairs (z, u) O.e., t h a t s a t i s f Y A z t`= A t` x t`+ x+ B t` u t` f° r a l l k E J ) w i t h x# O a n d ( - Z°E I m R T ' I f i n t h i s z N+ l ) definition R= 0 and~> O, then (H) is called disconjugate on J*.|

DEFINITION 3. (Controllability; see[12, Definition 3] and[14, Definition 5].) The system (H) is called controllable on J* if there exists k E J* such that for all solutions (x, u) of (H) and for all m E J with m+ k E J*, we have thatXm ---~Xm+l -----''"= X m+ t` -~0

implies x= u= 0 on J*. The m/nimal integer t¢ E J* with this property is then called the controllability index of (H).|

3. S T R I C T C O N T R O L L A B I L I T Y A N D M A I N R E S U L T SWe open this section with the following key definition. DEFINITION 4. (Strict Controllability). The set of s y s t e m s{(HA): )t E R}=: (HR) is called strictly controllable on J* if (i) (HA) is controllable on J* for all,~ E R (see Definition 3), and if (ii) there exists k~ J such that for al/A E R, for al/solutions (z, u) of (HA), and for al/rn E J with m+ k E J

implies z= u= 0 on J*. The minimal integer tcs E J with this property is then called the strict controllability index of (Ha). I

For stating our main results, we wish to label the following assumptions. (V1) (Ha) is strictly controllable on J*. (V2)~1 _<~2 always implies Ht`(~l) _< Ht`(~2) for all k E J. (Vs) There exists~ E R such that (.;~)> 0 and such that )~ _<~ always implies for all k E J Ker Bk(~) C Ker Bk(~) andB~()~){B~(~)-B~(~)}B~(~)>_O.

Hamiltonian Eigenvalue Problems

183

Now our main result reads as follows.THEOREM 1. Assume (V~), (V~), and (Vs). Then, ff there exist eigenvalues of (E), they may be arranged by-oo<~AI_~As_~ A3_~""

More precisely, (i) (V1) and (V2) imply that the eigenvalues are/solated, and (ii) (V2) and (Vs) imply that the eigenvalues are bounded below by A (which is not an eigenvalue) provided (HA) is controllable on J* for all A E R.|The remaining sections are devoted to the proof of the above theorem. However, here we wish to make some remarks concerning the concept of strict controllability. REMARK 1. Suppose Ak(A) -: Ae and Bk(A) -: Bk are constant for all k E J. Then condition (ii) of Definition 4 (with strict controllability index s, E J) already implies condition (i), i.e., controllability of (HA) on J* for all A E R, and the controllability indices s(A) of (HA) satisfy maxAeR~(A) _< ss+ 1 E J*. To prove this, assume (ii), let there be given A E R, a solution (z, u) of (HA), and m E J with m+~,+ 1 E J* such thatXn= Xm+l= .Tm+2=

Xm+~,+l

--~ 0

holds. Therefore,0 (A)xn+l= 0+l(A)xn+ .....= 0,

andhence(note/-/k(A)=(-C0(A)

0)0 f o r k E J ),

Condition (ii)thus implies x= u= 0 on J* so that controllability of (HA) on J* with controllability

index~(A)<~a+ 1 follows.| R E M A R K 2. Suppose as in the previous remark that Ak(A) and Bk(A) are independent of A E R for all k E J. Furthermore, assume that 0k(A) is nonsingular for all k E J and all A E R. Then controllability of (HA) on J* for all A E R with controllability indices s(A) E J implies strict controllability of (HR) on J" with strict controllability index s,< s:= maxAeR s(A). To show this, let A E R, let (x, u) be a solution of (HA), let m E J with m+ s E J (this yields m+ i E J since s(A) _> 1 trivially),and assume

Then, we haveCn(A)Xn-i-1= Cn-i-l(A)Xm..b2 ..... Orn+~CA)Zrn..[-~q-1= O,

and hence,invertibility of Cn(A),Cm+I(A),..., O,n+~(A)yieldsXm+l -.Tm+2= ' "~--- X r a+~+ l= 0.

Now controllability of (HA) on J* with controllability index~(A) _<~ together with m+ 1 E J and m+ 1+~ E J* imply z= u= 0 on J ' . Altogether, strict controllability of (HR) on J" with strict controllability index~,<_~ follows.|

184

M. BOHNER

R E M A R K 3. The purpose of this central remark is to show that Sturm-Liouville differenceequations of order 2n depending linearly on an eigenvalue parameter A R satisfy (V1) provided N> 2n - 1 holds. To start with, let r (~)ER,

We consider the n× n-matrices 0

(01/0<u<n, 1"-."'. 0, .

and

r(k )~ O, n

for a l l k E Z .

(0)... 0 1, k J,

Ak(A)= Ak=

Bk(A)~ Bk=

,r(~")

c~(A)=

_ A

0"..

A R

r (n-l)

and the corresponding systems (HA),£ E R. Note that the 2n× 2n-matrices/:/~(A) are given by

)

ilk(A)=

k J,

0 (so that (V2) is satisfied). Let N _> 2n - 1. Clearly (HA) is controllable on J* for each A R (see, e.g.,[12, Remark 20)]) with controllability index n _( N+ 1, i.e., n J*. Let A R and pick a solution (x,u) of (HA). It is very well known (see, e.g.,[4, Section 3]) that in this case Ayk_l xk=| A2Yk-2, for a l l k J *,

\ An-lyk+l-n holds with a solution yk (1 - n _< k _< N+ 1) of the linear self-~ljoint difference equation of order 2n n

n, ta--0

(SLA)

a so-called Sturm-Liouville difference equation. (In fact, (SLA) and (HA) are even equivalent in the sense that a solution of (SLA) yields a solution of (HA) and the other way around; see, e.g.,[4, Section 3].) Now we assume that for some m E J with m+ 2n - 1 E J

I'Im(Jk'(Xm'{-1) Umholds This implies

:~-Im'{'l()~,(xm'{'2\ Ura..}-I . . . . .

/~/m+2n-l(*~) ( Um-{-2n-iXm+2n) - .~ 0

Ym+l --'--Ym+2 ----""" ---- Ym+2n ---- O,and since y is a solution of a linear difference equation of order 2n being zero at 2n consecutive values, it has to be zero always, i.e., ya=0,

for aU l - n< k< N+ l,

and x= 0 on J* follows. Controllability of (HA) on J* now implies x= u= 0 on J*. Thus (Ha) is strictly controllable on J* with strict controllability index (no smaller index works)

~s=2n-lE

Hamiltonian Eigenvalue Problems

185

4. C H A R A C T E R I Z A T I O N OF EIGENVALUESTHEOREM 2. (Characterization of Eigenval

ues.) Let A 6 R, and let (X, U) and (X:, 0) be normalized conjoined bases of (H~). Then, A is an eigenvalue of (E) if and only if the 2n x 2n-

matrixA:= R*

( xo~o)+~(~o ON+I Oo)XN+I )(N+I UN+I

is singular, and then def A is the multiplicity of the eigenvalue A.PROOF. Let (x, u) be a nontrivial solution of (HA). We put

~:=

(xo~o~(~o)= (~:~~o~] (~o)~o Vo Oo/~o -Vo x: uo

(observe that Definition 1 yields the invertibility of the occurring matrix), and thus

(,,) (x,~,)uk= Uk Ok

fork J*,

since the initial value problem under consideration has a unique solution (observe that I - Ak(A) are assumed to be invertible matrices for all k E J). Now, we have

={R* (x-NX+° 1

~,,+,)+R(21 ON+l)} Oo --x~00)

d= Ad.

Thus, (x, u) solves (R), i.e., A is an eigenvalue of (E), if and only if Ad= 0 holds with d# 0, and this proves our assertion.| Next we wish to simplify this criterion in the case of so-called separated boundary conditions. By this we mean that the boundary conditions (R) may be equivalently written with 2n x 2nmatrices

~.(~

,,a~-(~I T

R~/+I

RN+I

such that the n x n-matrices Ro, R~, RN+I, R~V+l satisfy (as usual) rank(R0

P~)=rank(RN+l

R~v+l)=n,$

P~l~"= P~P~,

T RN+IRN+I= RN+IRN+I.

In this special case, we have the following result.COROLLARY i.

(Separated Boundary Conditions.) Assume that separated (and seg-conjoined)

boundary conditions are given. Let (X, U) be the conjoined basis of (HA), A 6 R, with

Xo= - P~

and

Uo= P f .

Then, A 6 R/ s an eigenva/ue of (E) if and only if the n x n-matr/xR~+IXN+I+ RN+IUN+I

s~.

186

M. BOHNER

PROOF. Let A G R. For the above conjoined basis of (HA), there exists another conjoined basis (~:,0) of (HA) so that (X, U) and (X, 0) are normalized (seeL e m m a 1). According to Theorem 2,A is an eigenvalue of (E) iff

~--"" (x-2~N+I)+.(~N~:I ON+i)~0 0o __(_o RN+ 1 )(X~il _~0~ o o 0 RN+I= ( -~

k UN+I UN+I )

v:~o - x: Oo --I

\ R~+IXN+I+ RN+IUN+I R*~+IRN+I+ RN+ION+I]

RN+IXN+I+ RN+IUN+I

RN+lRN+,+ RN+,ON+,]

(observe Definition 1) is singular, and this happens iff R~+xXN+I+ RN+xUN+I is singular. We wish to conclude this section with the following exanlple. EXAMPLE 1. Let n= 2 and consider the eigenvMue problem (E) given by

(~ 1)0

~.~+1~(lo 0)1,

.=(0° 0)~(:~).~(~' o,~) .~+~--(~°o)1 '

According to Remark 3, (V2) is satisfied and (Vx) holds provided N _> 2n - 1= 3. However, now we let N= 2. Then, due to Corollary 1, A E R is an eigenvalue of (E) iff X3(A)=

4A-6 3-2A)4A 6 3 2ARESULTS

is singular. Therefore R is the set of eigenvalues of (E).5. T W O AUXILIARY

In this short section, we cite two results that will be needed in the proof of Theorem 1. LEMMA 2. (Index Theorem; see[19, Theorem 3.4.1].) Let ra G N, let there be given m x mmatrices R, R*, X, U with rank(R R*)=rank(X T uT)=rrt and RR*r= R*R T,

xTu=uTx,

and let X(A), U(A) be m x m-matr/x-va/ued functions on R with XT(A)U(A)= U'

r(A)X()~),x (~ ) -~ x,

A G[Ao -¢, A o+¢],u(~) -- u, as~ -~6, A0+

[or some 6> 0,

,~o,

X(A) invertible for a//A E[A0 Suppose that

6]\{Ao}.

U(A)X-I(A) decreases strictly on[Ao - 6, Ao) and on (Ao, Ao+ e], and denote for A 6[Ao - e, Ao+¢]\{Ao}M(A)=

R*R"r+ RU(A)X-I(A)R T,h= RX+ R'U.

A(~)= RX(~)+ R'U(~),

Hamiltonian Eigenvalue Problems

187

Then, ind M ( A o )=

lirnA_.x{ind M(A)} and indM(Ao+)= limA_A+{indM(A)} both exist,°

A(A) is invertiblefor al] A e[A0 - 6,A+ 6]\{A0} for some 6 e (0,~),and the formula def h= ind M ( Ao - ind M (Ao )+ def X+)

holds.

PROOF. W e referto[19,Theorem 3.4.1] (observe also[19,Corollary 3.4.4]).

LEMMA 3. (Reid Roundabout Theorem; see[14, Theorem 3].) Suppose the system (H) is contro//able on J* (see Definition 3). Let (X, U) and (X, U) be the special normalized conjoined

bases of (H) at 0 (see Definition 1). Then,~> 0 (see Definition 2) if and only if

KerX~+t c KerXk,I

XkX~+l(I - Ak)-IBk>_O,

for all/c E J,

M:=R*R T+ R

UN+I

0)(0

X N+l invertible,XN+I

XN+I

>0,

onImR

holds.

PROOF. W e referto[14,Theorem 3] and remark that R{RtR*RtR} R T= R * R T holds.

6. I S O L A T E D N E S S OF E I G E N V A L U E SIn this section,we wish to establishTheorem 1(i).Consider the followingcondition. For all Ao E R there exists e> 0 such that XN+I(A) is invertible and~N+,(A) (UNI(A) 0 ) (xNOI(A) I ) -1 isstrictlydecreasingfor A E[Ao-e, ON+,(A) Ao+~]\{Ao}, where (X(A), U(A)) and (X(A), 0(A)) are the special normalized conjoined bases of (H) at 0 for every A E R.Of course, condition (I) implies by the index theorem, L e m m a 2 (observe also L e m m a 1 and the

(I)

continuity of the Hk(A), k E J), that the singular points of

A(A)= R, IxNOI(A)=

.~N+I(A)I.jcRIuN+II(A) 0+n

(-Xo(~) k~~~ x+()

x+(~°)ku~+~(~)(uo(~)t~JO°(~) N~r(~~/ l 2~ )

~-,+,.,,

i.e.,(according to Theorem 2) the eigenvaluesof (E), are isolated.Therefore our goal is to show that (V1) and (V2) imply (1).This we will achieve by showing some lemmas. L E M M A 4. Suppose (X(A), U(A)) is a conjoined basis of (HA) for eac~ A E R with Xo(A)=

0o(~)= 0. Then,k-1/x~+1(~)~T~(~) (x~+~(~)) X~(~)0k(~)-u~C~):~k(~)= - E\ U~(~) )\ U~(~)m=0

holds for al/k E J*\{0} and for al/A E R.

188

M. BOHNER

PROOF. Let A, p R and m J. Then,

n[X~(.){U.(A) - U~(.)} - U~(.){X~(A) - X~(.)}]= n[x~(~)u~(A) - G~(~)x.(A)]= A[ t u.(~) -x.(A)

t u.(~) ) j t t-x-+~(A) )

+;/x.+,(~)

{~(u.cx)

={~(-U~(")'TI (x'+r(A))-(X'+~(")'{ T t x.(.); t u.(A) t, u.(.) ) t x.(A) ) )u.(,) )j\ u.(A) ) - t u.(v): ( x,.+,(.) ) T (x.+,(A)~= - I t U.(#){Hm(A)-H,.(#)} U.(A) )" Now, division by A -# and letting/~tend to A yields

s,_x.+,<>,)+ k u~(v) )

( -x.(A)/j (tH'(X)x )}~ X'+'(A)u.(A)

~wl

-"w6

k u.(A) )

k u.CA) )$

so that )Co(A)= Co(A)= 0 prove the validityof our assertion.

L E M M A 5. Suppose (X(A), U(A)) and ()[(A),0(A)) are norma!i~,ed conjdned

bases of (HA) for

A e R with Xo(A) -- 0o(A)=~o(A)= bo(A)= o. Let~ a'. A~meinvertible on some nontrivial open interva/2[. Put

that X~(A)

QkCA):=

Uk(A)

0kCA)

0)(0

Xk(A)

2k(A)

.),

A e2[.

Then (V2) implies that Q~(A) decreases on 2[. Moreover, (V1) and (V~) imply that Qk(A)decreases strictly on 2[ provided k>~s holds, where~, E J is the strict contro//ab///ty index of

(HR).PROOF. Let k E J*\{0} and A E 2[. We may apply Lemma 4 with the conjoined basis

(X'(A),U*CA))

x(~)

~(~)

and

U'(~)=

U(~)

0(~)

of the"big" system from Lemma 4 so that for d E R 2n

Hamiitonian Eigenvalue P r o b l e m s

189

"* JCk(~)a= J{ O;(:,)x;-' (:,) - u;(:,)x;-' (~)x~(:,)x~ * - '--

{x:'/~>¢{x~"~>~-.:~>x~>}x~-'~>~~=o× x; - ' (:,)d

=-~'x;,-'(~,)d:~./

/xa+~(:,) T\ u~(~)

>(!00 o)}-em(~)0

(A)} d

A~(~)

o o

\ u;~(~)]

X,,,+~(A)~=o× X;-'k-1 ra=O Um

U~(~)

um(~) )

um(~)

(A)dT~rn

holds provided we assume (V2) and use the solution (x, u) of (HA) defined by

\ um(~) v~(~)

x~(~)

1), R~(~,)

Now we assume (V 0 and (V2), let k> ss, and suppose dV(~(A)d= 0. This yields/:/re(A)\(x'~+lum) It follows that

= O,

for all0 _< m _< k - 1.

x'),o

x'),l .....

.. J

holds. Strict controllability of (HR) on J* with strict controllability index~s E J now forces x= u= 0 on J* so that d= 0 and hence Qk(A)< 0 follows.| LEMMA 6. (V1) and (V2) imply (I). PROOF. For every A E R, we denote the special normalized conjoined bases of (HA) at 0 by (X(A), U(A)) and (X(A), 0(A)). Let Ao E R. We pick a conjoined basis (.~, 0) of (HAo) such that (X(Ao), U(A0)) and (X, 0) are normalized and such that )(N+I is invertible (observe Lemma 1). Let (X(A), 0(A)) be the conjoined basis of (HA) with Xo(A)= )(o and 0o(A) -= 0o, A e R. Due to continuity, XN+I(A) is invertible on some nontrivial open interval that contains Ao, and on this interval we have strict monotonicity of I (-0N+I(A) 0 0 U~+~(,x)) (-RN+~(,X)XN+I(A),

)-'

by L e m m a 5 so that .~ I ( A ) X N+ I ( A ) is strictlydecreasing on this interval also. Thus, there exists~> 0 such that XN+I(A) is invertibleon[Ao-6, Ao+6]\{Ao}. W e now may apply L e m m a 5 once again to obtain that

o)(0

decreases strictly on[Ao - e, Ao+ e]\{Ao}. This shows that (I) holds and hence the proof of Theorem l(i) is done.|

190

M. BOHNZR

7. L O W E R

BOUNDEDNESS

OF EIGENVALUES

The purpose of this section is to provide a proof of Theorem l(ii). We need the following auxiliary result. LEMMA 7. Let there be given m x m-matrices A, A, B, B, C, C such that

H= are symmetric. Suppose that

and

H_>~ H,hold. Then, we have

KerBCKerB~

and

B(Bt-Bt)

B>O

xTc2, nc u T B ufor a/l z, u, u e R m with B u - B u= (A - A)z.

>_ z T C z+ uT B u

PROOF. By[19, Lemma 3.1.10], H> H implies B> B and the existence of a matrix D with A- A= (B- B)D

and

D T ( B - B ) D< C - C.

According to[14, Remark 2(iii)], Ker B C Ker B is equivalent toB= B

BtB=

BBtB.

Let x,u,u 6 R m with B u - B u= ( A - A)z= (B - B)Dz. Then,

- B)D

+,:B,,-{B,,+ (B -

= x'rD'r(B _ B B t B ) D z+ uT(B - B B t B ) u+ 2zTDT(B _ BBtB)u= (mx+ u)V(B - B B t B ) ( D z+ u)= (Dz+ u)rB(B t -

Bt)B(Dx+

u)> o.

T H E O R E M 3. (Comparison Theorem.) Suppose that conditions (%'2) and (Vs) hold.~'(.;A)> 0 for all A< A.

Then,

HamiltoniaaEigenvalue ProblemsPROOF. Suppose~'(.;A)> 0 and let A< A. By (V2) and (Vs), we have for all k J Hk(A)>_ Hk(A), Ker Bk(A) C Ker Bk(A),

191

Bk( )> O.

Let (x,u) be such that ( -x0~ i m R T, x~ 0, and Axk= Ak()t)X,k+l+ Sk()t)~l,k, k J.\ XN+I/ Define Uk:= Bt(A)Bk(A)Uk -{ I - Btk(A)Bk(A)} DkXk+l, k J, where Ak(A) - Ak(A)={Bk(A) - Bk(A)} Dk according to the proof of Lemma 7. Then,

Bk()t)~tk- Bk(~)~k~--(Bk(~)- Bk()t)} DkXk+l= (Ak(~)- Ak()t)} . T k+ land thus Axk= Ak(A)xk+l+ Bk(~)Uk for all k J, so that an application of Lemma 7 yields

o< y(z,~; !)k----O N k----0 XN+I~~ XN+I T X XN+I

= Y'(z, u;~).Hence~r(.;A)> 0 also.|

Now we are able to finish the proof of Theorem l(ii)--and hence, of Theorem 1--as follows. Assume (V2), (Vs), and controllabilityof (H),) on J* for all A R. For A R, let (X(A), U(A)) and ()[(A),0(A)) be the special normalized conjoined bases of (HA) at 0 and define

M(A):=R'RT+R

(

UN+I(A) UN+I(A)

0)(0

XN+I(A)

whenever the inverseexists. Now we pick A0 _< A. Thus,~'(.;Ao)> 0 according to the above comparison result,Theorem 3. Our Reid roundabout theorem, L e m m a 3, now yieldsthat XN+I (Ao) is invertibleand that M(Ao)> 0 holds on Im R. Of course, XN+I(A) is invertiblein some nontrivial open interval containing Ao, UN+I(A)0N+I(A)

o)(o:~+I(A) XN+~(A)0

is strictly decreasing there

due to Lemma 5, and ind M(Ao= ind M(Ao)= 0 so that we may apply the index theorem,+) Lemma 2, to obtain def A(Ao)= ind M(A+) - ind M(Ao)+ def= def XN+I(Ao)= 0. Thus, the crucial matrix A(Ao) from our result on characterization of eigenvalues, Theorem 2, is nonsingular, and hence A0 is not an eigenvalue. Therefore there exists a smallest eigenvalue Ax--if there exists an eigenvalue at all--and it satisfies the inequality A1> A.

XN+I(Ao)

XN+I(Ao) )

192

M. BOHNER

REFERENCES1. L. Erbe and P. Yan, Disconjugacy for linear Harniltonian difference systems, J. Math. Anal. AppL 167, 355-367 (1992). 2. L. Erbe and P. Yah, Qualitative properties of Hamiltonian difference systems, J. Math. Anal. Appl. 171, 334-345 (1992). 3. L. Erbe and P. Yan, Oscillation criteria for Hamiltonian matrix difference systems, Proc. Amer. Math. Soc. 119 (2), 525-533 (1993). 4. L. Erbe and P. Yan, On the discrete Riccati equation and its applications to discrete Hamlltonian systems, Rocky Mountain J. Math. 25, 167-178 (1995). 5. C.D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems, J. Math. Anal. Appl. 180, 498-517 (1993). 6. C.D. Ahlbrandt, M. Heifetz, J.W. Hooker and W.T. Pat

ula, Asymptotics of discrete time Riccati equations, robust control, and discrete linear Hamiltonian systems, PanAmerican Mathematical Journal 5, 1-39 (1996). 7. C.D. Ahlbrandt and A. Peterson, The (n, n)-disconjugacy of a 2n th order linear difference equation, Computers Math. Applic. 28 (1-3), 1-9 (1994). 8. O. Do~ly, Transformations of linear Hamlltonian difference systems and some of their applications, J. Math. Anal. Appl. 191, 250-265 (1995). 9. T. Peil and A. Peterson, Criteria for C-disfocality of a self-adjoint vector difference equation, J. Math. Anal. Appl. 179 (2), 512-524 (1993). 10. A. Peterson, C-disfocality for linear Hamiltonian difference systems, J. Differential Equations 110 (1), 53-66 (1994). 11. A. Peterson and J. Ridenhour, The (2, 2)-disconjugacy of a fourth order difference equation, J. Difference Equations 1 (1), 87-93 (1995). 12. M. Bohner, Controllability and disconjugacy for linear Hamiltonian difference systems, In Conference Proccedings of the First International Conference on Difference Equations, pp. 65-77, Gordon and Breach, (1994). 13. M. Bohner, Inhomogeneons discrete variational problems, In Conference Proceedings o] the Second International Conference on Difference Equations, pp. 89-97, Gordon and Breach, (1995). 14. M. Bohner, Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions, J. Math. Anal. AppL 199, 804-826 (1996). 15. M. Bohner, On disconjugacy for Sturm-Liouville difference equations, J. Difference Equations 2, 227-237

(1996).16. M. Bohner, Riccati matrix difference equations and linear Hamiltonian difference systems, Dynamics of Continuous, Discrete and Impulsive Systems 2, 147-159 (1996). 17. M. Bohner and O. Doi137, Disconjugacy and transformations for symplectic systems, Rocky Mountain J. Math. 27 (3), 707-743 (1997). 18. G. Baur and W. Kratz, A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functional, Rend. C~rc. Math. Palermo 88 (2), 329-370 (1989). 19. W. Kratz, Quadratic P'~nctionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, (1995). 20. Ft. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, (1992). 21. W.G. Kelley and A.C. Peterson, Difference Equations: An Introduction~ith Applications, Academic Press, San Diego, CA, (1991). 22. W. Kratz, An inequality for finite differences via asymptotics of Riccati matrix difference equations, J. Difference Equations (to appear). 23. A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, John Wiley& Sons, New York, (1974).

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