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EIGENVALUE AN PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS Martin Bohner Department of Mathematics and Statistics, University of Missouri-Rolla 115 Rolla Building, Rolla, MO 65409-0020, USA E-mail: bohner@umr.edu Roman Hilscher Department of Mathematics, Michigan State University East Lansing, MI 48824-1027, USA E-mail: hilscher@math.msu.edu Abstract. In this paper we consider eigenvalue problems on time scales involving linear Hamiltonian dynamic systems. We give conditions that ensure that the eigenvalues of the problem are isolated and bounded below. The presented results are applicable also to Sturm Liouville dynamic equations of higher order, and further special cases of our systems are linear Hamiltonian di erential systems as well as linear Hamiltonian di erence systems. Date: February 6, 2003 Running head: Linear Hamiltonian eigenvalue problems 1991 Mathematics Subject Classi cation. 34C10, 93B60, 39A12 Key words and phrases. Time scale, Linear Hamiltonian system, Eigenvalue, Eigenfunction, Quadratic functional, Focal point, Normality. Corresponding author. Research supported by the Czech Grant Agency under grant 201/01/0079. Linear Hamiltonian eigenvalue problems 1 1. Introduction A time scale is any nonempty closed subset of R. For an introduction to the time scales calculus we refer the reader to [6, 7], see also [8, 12]. If f is a function on T, we abbreviate ft := f (t) and f := f , where is the forward jump operator. The time scale derivative ft reduces to the usual derivative f (t) if T = R and to the forward di erence ft = ft+1 ft if T = Z. The graininess function of T is t := t t. The set of rd-continuous functions is 1 denoted by Crd and the set of rd-continuously di erentiable functions by Crd . Let T := [a, b] be a time scale interval, a < b. The set T without its possible isolated (i.e., a left-scattered) maximum will be denoted by T ; thus T = T if b is left-dense. Consider the linear Hamiltonian dynamic system x = At x + Bt u, u = Ct x AT u, t t T , (H) where A, B, C : T Rn n are real rd-continuous matrices, Bt , Ct symmetric, and I t At nonsingular. Motivated by [5], we consider eigenvalue problems (with formally self-adjoint boundary conditions) involving the system (H), where the matrices At , Bt , and Ct also depend on an eigenvalue parameter R. We give conditions, among them the notion of strict controllability for system (H), that imply that the eigenvalues of (H) are isolated and bounded below, i.e., they may be arranged as < 1 2 3 . . . , counting multiplicities. An eigenvalue problem on time scales of the Sturm Liouville type has been recently studied in [1]. The setup of this paper is as follows. In the following Section 2 we recall some preliminaries on Hamiltonian systems that are needed later. Then, in Section 3, we introduce our eigenvalue problem in detail and present basic facts about this problem, e.g., how it is possible to characterize the eigenvalues. In this section we also present our main result on isolatedness and lower boundedness of eigenvalues, which we prove by using some auxiliary results that are given in detail in the last Section 4. 2. Preliminaries: Hamiltonian systems 1 By a solution of (H) we mean a pair (x, u) with x, u Crd (T) satisfying the system (H) on T . When referring to solutions of (H) we use a usual agreement that the vector-valued solutions of (H) are denoted by small letters and the n n-matrix-valued solutions by capital ones. By rank M , Ker M , Im M , def M , ind M , M T , M T 1 , M , M 0, and M > 0 we denote the rank, kernel, image, defect (dimension of the kernel), index (number of negative eigenvalues), transpose, inverse of the transpose, Moore Penrose generalized inverse (see [2, Chapter 1]), positive semide niteness, and positive de niteness, respectively, of the matrix M . By setting H := C AT AB , J := 0I , I 0 z := x , u z := x , u (1) the linear Hamiltonian system (H) has the form L[z]t J z + Ht z = 0, t T . (H) A solution (X, U ) of (H) is called a conjoined basis if rank(X T U T ) = n at some (and hence at any) t T, and X T U U T X 0 on T. The Wronski matrix W = X T U U T X is constant on T for any two solutions (X, U ), (X, U ) of (H). These two solutions are normalized if W = I. The (unique) solution (X, U ), resp. (X, U ), of (H) satisfying the initial conditions Xa = 0, Ua = I, 2 Martin Bohner and Roman Hilscher resp. Xa = I, Ua = 0, is called the principal, resp. associated, solution of (H) at a. Together they are called the special normalized conjoined bases of (H) at a. Lemma 1. For any s T and any conjoined basis (X, U ) of (H) there exists another conjoined basis (X, U ) such that they are normalized, and Xs is invertible. Proof. See [13, Corollary 3.3.9], [9, Remark 5]. A conjoined basis (X, U ) of (H) is said to have no focal points in the interval (a, b], provided Xt is invertible at all dense points t T \ {a}, and Ker X Ker X and D := X(X ) AB 0 on T . Recall that a point t T is dense if it is right-dense or left-dense. System (H) is called disconjugate on T if the principal solution of (H) at a has no focal points in (a, b]. A pair (x, u) is called admissible if x is piecewise rd-continuously di erentiable, denoted by 1 x Cp (T), u is piecewise rd-continuous, denoted by u Cp (T), and (x, u) satis es x = Ax + Bu on T (at points t T, where x is not continuous, this is to be read as the corresponding right/left-sided limit). Let R, S R2n 2n with S symmetric. The quadratic functional T b xa xa F(x, u) (x )T Cx + uT Bu t t + S . xb xb a is called positive de nite (F > 0), if F(x, u) > 0 for all admissible pairs (x, u) with ( xa ) xb Im RT , x 0. Following [11], system (H) is called dense-normal on [a, s] whenever s (a, b] is a dense point and the only solution of the system u = AT u, t Bt u = 0, t [a, s] , (2) (D) is the zero solution ut 0 on [a, s]. The hypothesis of dense-normality will be denoted by System (H) is dense-normal on any interval of the form [a, s] T. Moreover, we say that (H) is normal on T if whenever xt 0 on T, then ut 0 on T, i.e., system (H) is normal on T if whenever ut solves (2) with s = b (not necessarily dense), then ut 0 on T. d The di erentiation with respect to will be denoted by d z = z. We require throughout that d dX X ( ) = ( ) (3) U d d U for every conjoined basis (X, U ) of (H). This assumption is rather restrictive, but it certainly holds for any time scale which has constant graininess, in particular for T = R and T = Z. 3. The eigenvalue problem Let be given constant matrices R, R# R2n 2n such that rank(R# R) = 2n and R# RT is symmetric. In this paper, the superscript # does not mean a generalized inverse, but it is just an ordinary upper index. For R, we consider the linear Hamiltonian system x = At ( )x + Bt ( )u, xa xb u = Ct ( )x AT ( )u, t ua ub t T , (H ) subject to the (formally self-adjoint) boundary conditions R# +R = 0. (4) Linear Hamiltonian eigenvalue problems 3 We employ the following general assumption For all R, A( ), B( ), C( ) Crd (T, Rn n ), B( ), C( ) are symmetric, and I A( ) is nonsingular on T . For all t T , At ( ), Bt ( ), and Ct ( ) are continuously di erentiable with respect to . We denote At ( ) := [I t At ( )] 1 . First we derive the Lagrange identity for (H) on any time scale T. x 1 Lemma 2 (Lagrange identity). For any z, w Crd (T, R2n ), where z = ( u ) and w = ( y ), and v with notation (1), we have b wT L[z] LT [w] z a t T t = wt J zt b a . x Proof. Let z = ( u ), w = ( y ), and z = ( xu ), w = ( yv ). For brevity, we omit the argument v t in the following computation. The integration by parts in the third equality sign and the symmetry of Ht yield b b w L[z] t = a a b T y v T J x u + wT H z t = a T (y )T u v T x + wT H t z b a T b a b =y u v x + a (y )T u + (v )T x + wT H t z T = yb vb T J b a b a xb y a ub va b J xa ua T b + a y v T JT x u + wT H z t = wT J z = w Jz T + a b J w + Hw LT [w] t. z a z t + Therefore, the required identity follows. T The boundary conditions (4) are called formally self-adjoint if wt J zt a = 0 for all z, w x 1 Crd (T, R2n ) satisfying the given boundary conditions, i.e., z = ( u ) and w = ( y ) satisfy (4) and v b R# ya yb +R va vb = 0, respectively. Let us now remark that, in view of the next result, the symmetry of R# RT is a natural assumption when considering formally self-adjoint eigenvalue problems with the system (H). Lemma 3 (Formally self-adjoint boundary conditions). Let R# and R be real 2n 2n-matrices such that rank(R# R) = 2n. Then the boundary conditions (4) are formally self-adjoint i R# RT is symmetric. Proof. The proof is the same as the proof of [13, Proposition 2.1.1]. Remark 1. By [13, Remark 2.2.1], there exist matrices S, S # R2n 2n , such that S is symmetric, rank(S # R) = 2n, Im(S # )T = Ker R, and R# = RS + S # . 4 Martin Bohner and Roman Hilscher De nition 1 (Eigenvalue problem). The eigenvalue problem (H ), R, (4), (E) consists of the linear Hamiltonian dynamic system (H ) and the boundary conditions (4). A number R is called an eigenvalue of (E) if there exists a nontrivial solution (x, u) of (H ) satisfying (4). Such a solution is then called an eigenfunction corresponding to the eigenvalue . The set of all eigenfunctions corresponding to together with the zero function is called an eigenspace, and its dimension is referred to as the multiplicity of the eigenvalue . Theorem 1 (Characterization of eigenvalues). Let R and let (X, U ), (X, U ) be any normalized conjoined bases of (H ). Then is an eigenvalue of (E) i the matrix R2n 2n de ned by Xa Xa Ua Ua := R# +R Xb Xb Ub Ub is singular, and then def is the multiplicity of the eigenvalue . Proof. Let (x, u) be a nontrivial solution of (H ). We put d := x and thus ( ut ) = t Xt Xt Ut Ut Xa Xa Ua Ua 1 xa ua = T Xa T Ua T T Ua Xa xa ua = 0, d on T. Hence, +R ua ub = R# Xa Xa Ua Ua d+R d = d. Xb Xb Ub Ub R# xa xb Thus, (x, u) satis es the boundary conditions (4) i d = 0, i.e., is an eigenvalue of (E) i is singular. Corollary 1 (Separated boundary conditions). Assume that separated boundary conditions are given, i.e., # Ra 0 Ra 0 R= , R# = , # 0 Rb 0 Rb # # # # # where the n n-matrices Ra , Rb , Ra , Rb satisfy rank(Ra Ra ) = rank(Rb Rb ) = n, Ra (Ra )T = # # # T T Ra Ra , and Rb (Rb )T = Rb Rb . Let (X, U ) be the conjoined basis of (H ), R, with Xa = #T T Ra , Ua = (Ra ) . Then is an eigenvalue of (E) i the matrix Rn n given by # := Rb Xb + Rb Ub is singular. Proof. Let R. For (X, U ) there exists a conjoined basis (X, U ) of (H ) such that they are normalized, by Lemma 1. Then, Theorem 1 implies that is an eigenvalue of (E) i is singular. Since 0 I Ua Ua Xa Xa = +R = R# # , Rb Xb + Rb Ub Xb Ub Ub Xb # we have ( c1 ) = 0 i c2 = 0 and c1 + (Rb Xb + Rb Ub )c2 = 0, i.e., i c1 = 0. Hence, is c2 singular i is singular. De nition 2 (Strict dense-normality). The set of systems (HR ):= {(H ), R}, is called strictly dense-normal on T if (i) (H ) satis es (D) for all R. Linear Hamiltonian eigenvalue problems 5 (ii) For all R, for any s T \ {a}, for any solution (x, u) of (H ), if x Ht ( ) t ut then xt = ut 0 on T. Remark 2. We are particularly interested in the case when At ( ) At and Bt ( ) Bt are independent of and C depends on linearly, i.e., it is of the form Ct Ct . In this remark we discuss some features of this special case. (i) First, we note that (ii) implies (i) in De nition 2. To show this, let R and take any solution (x, u) of (H ) such that xt = 0 on [a, s], where s T is a dense point. We have x x Ct ( ) 0 t t Ht ( ) t = = Ct ( )x = Ct x = 0 t ut ut 0 0 on [a, s] , hence (ii) implies xt = ut 0 on T, so that (H ) is dense-normal on [a, s]. (ii) Next, we show that eigenvectors corresponding to di erent eigenvalues are orthogonal. More precisely, let R, S # , S, S R2n 2n be such that S, S are symmetric, rank(S # R) = 2n, #T Im(S ) = Ker R, and put S( ) := S S, R# ( ) := RS( ) + S # . Consider the eigenvalue problem x = At x + Bt u, u = (Ct Ct )x AT u, t xa ua # +R = 0. R, R ( ) xb ub t T , = 0 for all t [a, s] , (E) If (x, u) and (y, v) are eigenfunctions of (E) belonging to eigenvalues and , respectively, = , then x y with respect to C and S, i.e., b x, y := a (x )T Ct yt t xa t + xb T ya S yb = 0. To show this, we follow the proof of [13, Proposition 2.2.2]. Since (x, u) solves (H ) and (y, v) solves (H ), integration by parts implies b (x )T (C C)y + uT Bv a b t t = uT yt a , t T t = vt xt a . b b (5) (6) )T (y (C C)x + v T Bu a t By substracting (6) from (5) we obtain b ( ) a (x )T Cy t T t = yt ut b a xT vt a . t b (7) Observe that S # RT = 0 and R# ( ) + RS = RS + S # . Moreover, from [13, Proposition 2.1.2] it follows that (x, u) and (y, v) satisfy the boundary conditions in (E) i xa xb = RT c, ua ub = {R# ( )}T c, ya yb = RT d, va vb = {R# ( )}T d, 6 Martin Bohner and Roman Hilscher for some c, d R2n . Thus, from (7) we have b ( ) x, y = ( ) a (x )T Ct yt t b xT vt a t xa t + xb T T ya ( )S yb = = b T y t ut a T xa + xb T ya ( )S yb va vb xa + xb T ya ( )S yb = dT R{R# ( )}T c + cT R{R# ( )}T d + ( ) cT RSRT d ua ub = dT R(RS + S # )T c + cT R(RS + S # )T d = 0. ya yb xa xb Hence, x y and the proof is complete. (iii) If the system is strictly dense-normal and if S and Ct are all positive semide nite (which is satis ed in the present setting note that throughout this paper, with the exception of this remark, we assume S = 0 subject to conditions (V1 ) and (V2 ) given after this remark), then all eigenvalues are real. To see this, let (x, u) be an eigenfunction corresponding to an eigenvalue . Then ( , u) is x an eigenfunction corresponding to the eigenvalue , and we may use the calculation from the second part of this remark to obtain 0 = ( ) x, x b = ( ) a ( )T Cx x t t + a x xb T xa S xb . Clearly, x, x = 0, since otherwise the positive semide niteness of S and Ct implies xa S xb =0 and Cx = 0 on [a, b] and hence x = u 0 by strict dense-normality, which is impossible. Therefore, = 0 and our claim R follows. Let us continue with the investigation of the general eigenvalue problem (E). Given the eigenvalue problem (E), we de ne the quadratic functional b F(x, u; ) := a (x ) C( )x + u B( )u T T t xa t + xb T S xa , xb where the matrix S is determined by Remark 1. We consider the following assumptions: (V1 ) (HR ) is strictly dense-normal on T. (V2 ) 1 2 always implies Ht ( 1 ) Ht ( 2 ) for all t T . (V3 ) There exists R such that F( ; ) > 0 and always imply for all t T Ker B( ) Ker B( ) and B( ) B ( ) B ( ) B( ) 0. (V4 ) (H ) is normal on T for all R. Now the main result of this paper reads as follows. Theorem 2. Assume (V1 ) (V4 ). Then, if there exist eigenvalues of (E), they may be arranged by < 1 2 3 . . . , counting multiplicities. More precisely, Linear Hamiltonian eigenvalue problems 7 (i) (V1 ) and (V2 ) imply that the eigenvalues are isolated. (ii) (V2 ) (V4 ) imply that the eigenvalues are bounded below by , provided (H ) satis es (D) for all R. Proof. Part (i) isolatedness. Let (X( ), U ( )), (X( ), U ( )) be the special normalized conjoined bases of (H ) at a for each R. Fix 0 R. Then by Lemma 6 there exists > 0 such that Xb ( ) is invertible and I 0 Ub ( ) Ub ( ) 0 I Xb ( ) Xb ( ) 1 is strictly decreasing for all U( 0 , ), where U( 0 , ) := [ 0 , 0 + ] \ { 0 } is the closed -interval around 0 without 0 . It follows from the Index Theorem (Proposition 1 in the next section) that the singular points of ( ) = R# = R# I 0 0 I b ( ) + R Ub ( ) Ub ( ) Xb ( ) X Xa ( ) Xa ( ) Ua ( ) Ua ( ) b ( ) + R Ub ( ) Ub ( ) , Xb ( ) X i.e., the eigenvalues of (E) by Theorem 1, are isolated. Furthermore, the multiplicity of an eigenvalue 0 is def ( 0 ) = ind M ( + ) ind M ( ), 0 0 0 I # since Xb = Xb ( 0 ) Xb ( 0 ) is invertible (the matrix M ( ) is de ned in Proposition 1). Hence, part (i) is proved. Part (ii) lower boundedness. Assume that (H ) satis es (D) for all R, and that (V2 ) (V4 ) hold. For R de ne M ( ) := R {S + Q# ( )} RT , b Q# ( ) := b I 0 b ( ) Ub ( ) U 0 I b ( ) Xb ( ) X 1 . We pick 0 . Then F( ; 0 ) > 0 by the Comparison Theorem (Theorem 3 in the next # section). Since (H 0 ) is normal on T, Proposition 2 implies that Xb ( 0 ) and hence Xb = 0 I Xb ( 0 ) Xb ( 0 ) # are invertible, and M ( 0 ) > 0 on Im R. It follows that Xb ( ) is invertible on some open interval J around 0 . Moreover, the matrix Q# ( ) de ned above is strictly b decreasing on J, by Lemma 5, and ind M ( + ) = 0 = ind M ( ). Now, we may apply the Index 0 0 Theorem (Proposition 1) to obtain # def ( 0 ) = ind M ( + ) ind M ( ) + def Xb = 0, 0 0 i.e., ( 0 ) is invertible. This means in view of Theorem 1 that 0 is not an eigenvalue of (E). Therefore, if there exists an eigenvalue at all, there is the smallest one 1 and satis es 1 > . The proof is complete. 4. Auxiliary results In this section we collect auxiliary results needed in our work. Recall that U( 0 , ) is the closed -interval around 0 (the center is removed). 8 Martin Bohner and Roman Hilscher Proposition 1 (Index Theorem [13, Theorem 3.4.1, Corollary 3.4.4]). Let m N and let there be given matrices R, R# , X, U Rm m with rank(R# R) = rank ( X ) = m and R(R# )T = U R# RT , X T U = U T X. Let X( ), U ( ) Rm m be matrices such that X T ( )U ( ) are symmetric for all U( 0 , ), for some > 0, X( ) X, U ( ) U as 0 , and X( ) is invertible for U( 0 , ). Suppose that U ( )X 1 ( ) decreases strictly on [ 0 , 0 ) and on ( 0 , 0 + ], and denote M ( ) := R# RT + RU ( )X 1 ( )RT , Then ind M ( ) := lim ind M ( ), 0 0 ( ) := R# X( ) + RU ( ), := R# X + RU. ind M ( + ) := lim+ ind M ( ) 0 0 both exist, ( ) is invertible for all U( 0 , ) for some (0, ), and def = ind M ( + ) ind M ( ) + def X. 0 0 Proposition 2 (Jacobi Condition [10, 11]). Suppose (D) holds. Let (X, U ), (X, U ) be the special normalized conjoined bases of (H) at a. Then F > 0 i (X, U ) has no focal points in # 0I (a, b] and S + Q# > 0 on Im RT Im Xb , where X # := X X and Q# is a certain 2n 2n b U ). Moreover, if (H) is normal on T, then F > 0 implies Xb matrix built up from (X, U ), (X, # (and hence Xb ) is invertible. Lemma 4. Suppose that (X( ), U ( )) is a conjoined basis of (H ) for all R with Xa ( ) = 0 = Ua , i.e., Xa and Ua are independent of . Then t XtT ( )Ut ( ) UtT ( )Xt ( ) = a X ( ) U ( ) T X ( ) H ( ) U ( ) holds for all t T and for all R. Proof. In the computation below we skip the evaluation at t T. Compare [5, Lemma 4]. We have X ( ) [U ( ) U ( )] U ( )[X( ) X( )] = = U ( ) X( ) X ( ) U ( ) T T T = X( ) U ( ) T U ( ) X( ) X ( ) X ( ) U ( ) U ( ) T U ( ) X( ) T {H( ) H( )} X ( ) . U ( ) T Now, dividing by and letting (observe that (3) is used) yields X T ( )U ( ) U T ( )X( ) Integrating from a to t we get t = X ( ) U ( ) X ( ) H( ) . U ( ) a X ( ) U ( ) T X ( ) H ( ) U ( ) T T = X ( )U ( ) U ( )X ( ) , a t and Xa ( ) = 0 = Ua yields the result. Linear Hamiltonian eigenvalue problems 9 Lemma 5. Suppose that (X( ), U ( )), (X( ), U ( )) are normalized conjoined bases of (H ) for each R with Xa ( ) = Ua ( ) = 0 = Xa ( ) = Ua ( ). Let t T, t > a. Assume that Xt ( ) is invertible for in some open interval J. For J put Qt ( ) := I 0 t ( ) Ut ( ) U 0 I t ( ) Xt ( ) X 1 . Then (V2 ) implies that Qt ( ) decreases on J. Moreover, (V1 ) and (V2 ) imply Qt ( ) that decreases strictly on J. Proof. The proof is similar to the proof of [5, Lemma 5], so we sketch it only. Let t T, t > a, and J. We apply Lemma 4 to Xt# ( ) := 0 I , Xt ( ) Xt ( ) Ut# ( ) := I 0 . Ut ( ) Ut ( ) Then for d R2n 2n it follows that t dT Qt ( )d = a x u T H x u 0, where x u := X( ) X( ) (X # ) 1 ( )d, U ( ) U ( ) and where we used (V2 ), i.e., H( ) 0. Suppose that (V1 ) and (V2 ) hold with dT Qt ( )d = 0. Then H ( ) x = 0 for all [a, t] . Strict dense-normality implies xt = ut 0 on T, i.e., u d = 0. Thus, Qt < 0 follows. Lemma 6. Let (X( ), U ( )), (X( ), U ( )) be the special normalized conjoined bases of (H ) at a for each R. Assumptions (V1 ) and (V2 ) imply that for all 0 R there exists > 0 such that Xb ( ) is invertible and Qb ( ) de ned by Qb ( ) := I 0 Ub ( ) Ub ( ) 0 I Xb ( ) Xb ( ) 1 (8) is strictly decreasing for all U( 0 , ). Proof. Fix 0 R and let (X, U ) be the conjoined basis of (H 0 ) such that (X( 0 ), U ( 0 )) and (X, U ) are normalized and Xb is invertible, see Lemma 1. Let (X( ), U ( )) be the conjoined basis of (H ) with Xa ( ) = Xa , Ua ( ) = Ua , R. Due to continuity, X( ) is invertible on some open interval around 0 and on that interval we have, by Lemma 5 with ( X( ), U ( )) and (X( ), U ( )), that the matrix I 0 b ( ) Ub ( ) U 0 I b ( ) Xb ( ) X 1 = 1 1 Xb ( )Xb ( ) Xb ( ) T 1 Xb 1 ( ) Ub ( )Xb ( ) 1 is strictly decreasing. Consequently, Xb ( )Xb ( ) is strictly decreasing as well. It follows that Xb ( ) is invertible on U( 0 , ) for some > 0. Applying Lemma 5 again, the strict monotonicity of the matrix Qb ( ) in (8) follows. 10 Martin Bohner and Roman Hilscher Lemma 7. Let m N and let be given real m m-matrices A, A, B, B, C, C such that the Hamiltonian matrices C AT C AT H := , H := AB AB are symmetric. Suppose that H H, hold. Then xT Cx + uT Bu xT Cx + uT Bu for all x, u, x, u Rm with Bu Bu = (A A)x. Moreover, there exists a matrix E Rm m such that A A = (B B)E and E T (B B)E C C. Proof. The proof is similar to the discrete case [5, Lemma 7], compare also the continuous case [13, Lemma 3.1.10]. Remark 3. Observe that Ker B Ker B from the above lemma is equivalent to B = BB B = BB B, see [3, Lemma A5, pg. 94] or [4, Remark 2(iii)]. Theorem 3 (Comparison Theorem). Suppose that (V2 ) and (V3 ) hold. Then F( ; ) > 0 for all . Proof. Suppose F( ; ) > 0 and let . From (V2 ) and (V3 ) we have Bt ( ) Bt ( ), Ker Bt ( ) Ker Bt ( ), Bt ( ) Bt ( ) Bt ( ) Bt ( ) 0. Ker B Ker B, B(B B )B 0 Let (x, u) be admissible for F( ; ), i.e., x = At ( )x + Bt ( )ut , t T , with ( xa ) Im RT , xb t t and x 0. For t T we de ne ut := Bt ( )Bt ( )ut I Bt ( )Bt ( ) Et x , t where E : T Rn n is such that At ( ) At ( ) = {Bt ( ) Bt ( )}Et , by Lemma 7. Note also that Bt ( )Bt ( )Bt ( ) = Bt ( ) by Remark 3. Then (all functions evaluated at t) B( )u B( )u = B( )u B( )B ( )B( )u + B( ) B( )B ( )B( ) Ex = {B( ) B( )} Ex = {A( ) A( )} x , so that A( )x + B( )u = A( )x + B( )u = x , i.e., (x, u) is admissible for F( ; ). Applying Lemma 7 again we get b 0 < F(x, u; ) = a b (x )T C( )x + uT B( )u t + t t + T xa xb xa xb T S xa xb a (x )T C( )x + uT B( )u t xa xb S = F(x, u; ). Hence, F( ; ) > 0 as well. Linear Hamiltonian eigenvalue problems 11 References [1] R. P. Agarwal, M. Bohner, P. J. Y. Wong, Sturm Liouville eigenvalue problems on time scales, Appl. Math. Comput. 99 (1999), no. 2-3, 153 166. [2] A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications, John Wiley & Sons, Inc., New York, 1974. [3] M. Bohner, On Positivity of Discrete Quadratic Functionals, PhD dissertation. University of Ulm, 1995. [4] M. Bohner, Linear Hamiltonian di erence systems: Disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl. 199 (1996), no. 3, 804 826. [5] M. Bohner, Discrete linear Hamiltonian eigenvalue problems, Comput. Math. Appl. 36 (1998), no. 10 12, 179 192. [6] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh user, Boston, 2001. a [7] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh user, Boston, 2003. a [8] S. Hilger, Analysis on measure chains a uni ed approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18 56. [9] R. Hilscher, Linear Hamiltonian systems on time scales: Positivity of quadratic functionals, Math. Comput. Modelling 32 (2000), no. 5-6, 507 527. [10] R. Hilscher, Inhomogeneous quadratic functionals on time scales, J. Math. Anal. Appl. 253 (2001), no. 2, 473 481. [11] R. Hilscher, Positivity of quadratic functionals on time scales: Necessity, Math. Nachr. 226 (2001), no. 1, 85 98. [12] B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Boston, 1996. [13] W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, 1995.

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