)-'
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.|
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
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=
A
B
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,
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
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)
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)
I
XN+I(Ao) )
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
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
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