Research Article Conjugacy of Self-Adjoint Difference Equations of Even Order

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1 Abstract and Applied Analysis Volume 2011, Article ID , 16 pages doi: /2011/ Research Article Conugacy of Self-Adoint Difference Equations of Even Order Petr Hasil Department of Mathematics, Mendel University in Brno, Zemědělsá 1, Brno, Czech Republic Correspondence should be addressed to Petr Hasil, Received 31 January 2011; Revised 5 April 2011; Accepted 18 May 2011 Academic Editor: Elena Braverman Copyright q 2011 Petr Hasil. This is an open access article distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. We study oscillation properties of 2n-order Sturm-iouville difference equations. For these equations, we show a conugacy criterion using the p-criticality the existence of linear dependent recessive solutions at and. We also show the equivalent condition of p-criticality for one term 2n-order equations. 1. Introduction In this paper, we deal with 2n-order Sturm-iouville difference equations and operators y ) n Δ ν ) r ν Δν y n ν 0, ν0 r n > 0, Z, 1.1 where Δ is the forward difference operator, that is, Δy y 1 y,andr ν,ν 0,...,n,are real-valued sequences. The main result is the conugacy criterion which we formulate for the equation y q y n 0, that is viewed as a perturbation of 1.1, and we suppose that 1.1 is at least p-critical for some p {1,...,n}. The concept of p-criticality a disconugate equation is said to be p-critical if and only if it possesses p solutions that are recessive both at and,seesection 3 was introduced for second-order difference equations in 1,and later in 2 for 1.1. For the continuous counterpart of the used techniques, see 3 5 from where we get an inspiration for our research. The paper is organized as follows. In Section 2, we recall necessary preliminaries. In Section 3, we recall the concept of p-criticality, as introduced in 2, and show the first

2 2 Abstract and Applied Analysis result the equivalent condition of p-criticality for the one term difference equation Δ n r Δ n y ) Theorem 3.4. InSection 4 we show the conugacy criterion for equation Δ n r Δ n y ) q y n 0, 1.3 and Section 5 is devoted to the generalization of this criterion to the equation with the middle terms n Δ ν ) r ν Δν y n ν q y n ν0 2. Preliminaries The proof of our main result is based on equivalency of 1.1 and the linear Hamiltonian difference systems Δx Ax 1 B u, Δu C x 1 A T u, 2.1 where A, B,andC are n n matrices of which B and C are symmetric. Therefore, we start this section recalling the properties of 2.1, which we will need later. For more details, see the papers 6 11 and the boos 12, 13. The substitution n y n 1 Δ ν 1 ) r ν Δν y n ν x y Δy n 2 ν1, u y. ) Δ r n Δ n 1 Δn y y. r n 1 r n Δn y Δ n 1 y transforms 1.1 to linear Hamiltonian system 2.1 with the n n matrices A, B,andC given by A ) n a i i,1, a 1, if i 1, i 1,...,n 1, i 0, elsewhere, B diag 0,...,0, 1 { }, C diag r 0,...,rn 1. r n 2.3 Then, we say that the solution x, u of 2.1 is generated by the solution y of 1.1.

3 Abstract and Applied Analysis 3 et us consider, together with system 2.1, the matrix linear Hamiltonian system ΔX AX 1 B U, ΔU C X 1 A T U, 2.4 where the matrices A, B,andC are also given by 2.3. We say that the matrix solution X, U of 2.4 is generated by the solutions y 1,...,y n of 1.1 if and only if its columns are generated by y 1,...,y n, respectively, that is, X, U x y 1,...,x y n,u y 1,...,u y n. Reversely, if we have the solution X, U of 2.4, the elements from the first line of the matrix X are exactly the solutions y 1,...,y n of 1.1. Now, we can define the oscillatory properties of 1.1 via the corresponding properties of the associated Hamiltonian system 2.1 with matrices A, B,andC given by 2.3, for example, 1.1 is disconugate if and only if the associated system 2.1 is disconugate, the system of solutions y 1,...,y n is said to be recessive if and only if it generates the recessive solution X of 2.4, and so forth. Therefore, we define the following properties ust for linear Hamiltonian systems. For system 2.4, we have an analog of the continuous Wronsian identity. etx, U and X, Ũ be two solutions of 2.4. Then, X T Ũ U T X W 2.5 holds with a constant matrix W. We say that the solution X, U of 2.4 is a conoined basis,if X T U U T X, ) X ran U n. 2.6 Two conoined bases X, U, X, Ũ of 2.4 are called normalized conoined bases of 2.4 if W I in 2.5where I denotes the identity operator. System 2.1 is said to be right disconugate in a discrete interval l, m, l, m Z, ifthe solution X U ) of 2.4 given by the initial condition Xl 0, U l I satisfies er X 1 er X, X X 1 I A 1 B 0, 2.7 for l,...,m 1, see 6. Here er,, and stand for ernel, Moore-Penrose generalized inverse, and nonnegative definiteness of the matrix indicated, respectively. Similarly, 2.1 is said to be left disconugate on l, m, if the solution given by the initial condition X m 0, U m I satisfies er X er X 1, X 1 X B I A T 1 0, l,...,m System 2.1 is disconugate on Z, if it is right disconugate, which is the same as left disconugate, see 14, Theorem 1,onl, m for every l, m Z, l<m.system2.1 is said to be nonoscillatory at nonoscillatory at, if there exists l Z such that it is right disconugate on l, m for every m>lthere exists m Z such that 2.1 is left disconugate on l, m for every l<m.

4 4 Abstract and Applied Analysis ) We call a conoined basis XŨ of 2.4 the recessive solution at, if the matrices X are nonsingular, X X 1 1 I A 1 B 0 both for large, and for any other conoined basis XU ), for which the constant matrix X T Ũ U T X is nonsingular, we have lim X 1 X The solution X, U is called the dominant solution at. The recessive solution at is determined uniquely up to a right multiple by a nonsingular constant matrix and exists whenever 2.4 is nonoscillatory and eventually controllable. System is said to be eventually controllable if there exist N, κ N such that for any m N the trivial solution x u ) 0 0 of 2.1 is the only solution for which x m ) x m1 x mκ 0. The equivalent characterization of the recessive solution XŨ of eventually controllable Hamiltonian difference systems 2.1 is ) 1 lim X 1 1 I A 1 B X T 1 0, 2.10 see 12. Similarly, we can introduce the recessive and the dominant solutions at. For related notions and results for second-order dynamic equations, see, for example, 15, 16. We say that a pair x, u is admissible for system 2.1 if and only if the first equation in 2.1 holds. The energy functional of 1.1 is given by F y ) : n r ν ν0 Δ ν y n ν ) Then, for admissible x, u, we have F y ) n r ν ν0 n 1 r ν ν0 Δ ν y n ν ) 2 Δ ν y n ν ) 2 1 r n ) 2 Δn y r n ] [x T 1 C x 1 u T B u : Fx, u To prove our main result, we use a variational approach, that is, the equivalency of disconugacy of 1.1 and positivity of Fx, u;see6. Now, we formulate some auxiliary results, which are used in the proofs of Theorems 3.4 and 4.1. The following emma describes the structure of the solution space of Δ n r Δ n y ) 0, r >

5 Abstract and Applied Analysis 5 emma 2.1 see 17, Section2. Equation 2.13 is disconugate on Z and possesses a system of solutions y, ỹ, 1,...,n, such that y 1 y n ỹ 1 ỹ n 2.14 as,wheref g as for a pair of sequences f, g means that lim f /g 0. If 2.14 holds, the solutions y form the recessive system of solutions at, while ỹ form the dominant system, 1,...,n. The analogous statement holds for the ordered system of solutions as. Now, we recall the transformation lemma. emma 2.2 see 14, Theorem 4. et h > 0, y n ν0 Δν r ν Δν y n ν and consider the transformation y h z. Then, one has where h n y ) n Δ ν R ν Δν z n ν ), 2.15 ν0 R n h n h r n, R0 h n h, 2.16 that is, y solves y 0 if and only if z solves the equation n Δ ν ) R ν Δν z n ν ν0 The next lemma is usually called the second mean value theorem of summation calculus. emma 2.3 see 17, emma 3.2. et n N and the sequence a be monotonic for K n 1, n 1 i.e., Δa does not change its sign for K n 1, n 2). Then, for any sequence b there exist n 1,n 2 K, 1 such that 1 n a n b a Kn 1 b i a n 1 b i, in 1 K ik 1 n a n b a Kn 1 b i a n 1 b i. in 2 K ik 2.18 Now, let us consider the linear difference equation y n a n 1 y n 1 a 0 y 0, 2.19 where n 0 for some n 0 N and a 0 / 0, and let us recall the main ideas of 18 and 19, Chapter IX.

6 6 Abstract and Applied Analysis An integer m>n 0 is said to be a generalized zero of multiplicity of a nontrivial solution y of 2.19 if y m 1 / 0, y m y m1 y m 2 0, and 1 y m 1 y m 1 0. Equation 2.19 is said to be eventually disconugate if there exists N N such that no non-trivial solution of this equation has n or more generalized zeros counting multiplicity on N,. A system of sequences u 1,...,un is said to form the D-Marov system of sequences for N, if Casoratians C u 1,...,u ) u 1 u u 1 1 u 1. u 1 1. u 1, 1,...,n 2.20 are positive on N,. emma 2.4 see 19, Theorem Equation 2.19 is eventually disconugate if and only if there exist N N and solutions y 1,...,y n of 2.19 which form a D-Marov system of solutions on N,. Moreover, this system can be chosen in such a way that it satisfies the additional condition lim y i y i1 0, i 1,...,n Criticality of One-Term Equation Suppose that 1.1 is disconugate on Z and let ŷ i and ỹ i, i 1,...,n, be the recessive systems of solutions of y 0at and, respectively. We introduce the linear space { H in ŷ 1,...,ŷ n} { in ỹ 1,...,ỹ n}. 3.1 Definition 3.1 see 2. et1.1 be disconugate on Z and let dim H p {1,...,n}. Then, we say that the operator or 1.1 is p-critical on Z.IfdimH 0, we say that is subcritical on Z. If1.1 is not disconugate on Z, thatis, / 0, we say that is supercritical on Z. To prove the result in this section, we need the following statements, where we use the generalized power function 0 1, i 1 i 1, i N. 3.2 For reader s convenience, the first statement in the following lemma is slightly more general than the corresponding one used in 2 it can be verified directly or by induction.

7 Abstract and Applied Analysis 7 emma 3.2 see 2. The following statements hold. i et z be any sequence, m {0,...,n}, and 1 ) n 1z y : 1, then n 1 m 1 ) n 1 mz 1, m n 1, Δ m y 0 n 1!z, m n. 3.4 ii The generalized power function has the binomial expansion ) n n i0 1 i n i ) n i i 1 ) i. 3.5 We distinguish two types of solutions of Thepolynomial solutions i,i 0,...,n 1, for which Δ n y 0, and nonpolynomial solutions 1 ) n 1 1 i r 1, i 0,...,n 1, for which Δ n y / 0. Using emma 3.2i we obtain Δ n y n 1! i r 1. Now, we formulate one of the results of 20. Proposition 3.3 see 20, Theorem 4. If for some m {0,...,n 1} 0 [ n m 1] 2 r 1 [ n m 1] 2 r 1 0, 3.7 then { in 1,..., m} H, 3.8 that is, 2.13 is at least m 1-critical on Z. Now, we show that 3.7 is also sufficient for 2.13 to be at least m 1-critical. Theorem 3.4. et m {0,...,n 1}. Equation 2.13 is at least m 1-critical if and only if 3.7 holds.

8 8 Abstract and Applied Analysis Proof. et V and V denote the subspaces of the solution space of 2.13 generated by the recessive system of solutions at and, respectively. Necessity of 3.7 follows directly from Proposition 3.3. To prove sufficiency, it suffices to show that if one of the sums in 3.7 is convergent, then {1,..., m } / V V. We show this statement for the sum. The other case is proved similarly, so it will be omitted. Particularly, we show [ n m 1] 2 r 1 < m / V et us denote p : n m 1, and let us consider the following nonpolynomial solutions of 2.13: y l ) n 1 pl 1 r 1 ) p n 1 1 i 1 n 1 i pl 1 i 1 ) i r 1, i i0 where l 1 p,...,m 1. By Stolz-Cesàro theorem, since using emma 3.2i Δ n y l n 1! pl 1 r 1, these solutions are ordered, that is, yi y i1,i1 p,...,m, as well as the polynomial solutions, that is, i i1,i0,...,n 2. By some simple calculation and by emma 3.2 at first, we use i, and at the end, we use ii, we have Δ m y 1 n 1! 1 ) n m 1 1 p r 1 n m 1! 0 ) p n 1 n 1 i! 1 i 1n m 1 i p i 1 ) i r 1 i n m 1 i! i0 n 1! p! ) p p r 1 p 1 i n 1!n 1 i! i0 n 1 i!i! p i ) 1p i p i 1 ) i r 1! 0 ) n 1! 1 ) p p p 1 p r 1 p! 1 i 1 p i p i 1 ) i r 1 0 i0 i 0 n 1! 1 ) p 1 p r 1 ) p 1 p r 1 p! 0 0 0

9 Abstract and Applied Analysis 9 n 1! p! 1 p1 n 1! p! 1 ) p p r 1 ) p 1 p r 1, 1 ) p p r 1 [ p] 2 r Hence, from this and by Stolz-Cesàro theorem, we get lim 1 m m! lim y 1 Δm y 1 0, 3.12 thus y 1 m. We obtained that {1,,..., m 1,y 1 p,...,y 1 } m, which means that we have n solutions less than m, therefore m / V and 2.13 is at most m-critical. 4. Conugacy of Two-Term Equation In this section, we show the conugacy criterion for two-term equation. Theorem 4.1. et n>1, q be a real-valued sequence, and let there exist an integer m {0,...,n 1} and real constants c 0,...,c m such that 2.13 is at least m 1-critical and the sequence h : c 0 c 1 c m m satisfies lim sup q h 2 n 0. K, K 4.1 If q / 0, then Δ n r Δ n y ) q y n is conugate on Z. Proof. We prove this theorem using the variational principle; that is, we find a sequence y l 2 0 Z such that the energy functional Fy r Δ n y 2 q y 2 < 0. n At first, we estimate the first term of Fy. To do this, we use the fact that this term is an energy functional of et us denote it by F that is, F y ) r Δ n ) 2. y 4.3

10 10 Abstract and Applied Analysis Using the substitution 2.2, we find out that 2.13 is equivalent to the linear Hamiltonian system 2.1 with the matrix C 0; that is, Δx A x 1 B u, Δu A T u, 4.4 and to the matrix system ΔX A X 1 B U, ΔU A T U. 4.5 Now, let us denote the recessive solutions of 4.5 at and by X,U and X,U, respectively, such that the first m 1 columns of X and X are generated by the sequences 1,,..., m.etk,, M,andN be arbitrary integers such that N M>2n, M >2n,and K>2n some additional assumptions on the choice of K,, M, N will be specified later, and let x f,u f and x g,u g be the solutions of 4.4 given by the formulas u f u g U 1 U N 1 B K B x f X 1 1 x g B K B K X N 1 N 1 B M X B X M 1 B K ) 1x h N 1 B M X ) 1x h, X ) T 1 1 X M ) 1x h M X B K ) 1x h M, ) T 1 N 1 B M X X M ) 1x h, ) 1x h M, 4.6 where B X ) 1I 1 A 1 ) B X T 1, B X ) 1I 1 A 1 ) 4.7 B X T 1, and x h,u h is the solution of 4.4 generated by h. By a direct substitution, and using the convention that 1 0, we obtain x f K 0, xf x h, xg M xh M, xg N Now, from 4.1, together with the assumption q / 0, we have that there exist Z and ε>0 such that q ε. Because the numbers K,, M, andn have been almost free so far, we may choose them such that < <M n 1.

11 Abstract and Applied Analysis 11 et us introduce the test sequence 0,,K 1, f, y : h 1 D, K, 1,, M 1, 4.9 g, M, N 1, 0, N,, where δ>0, n, D 0, otherwise To finish the first part of the proof, we use 4.4 to estimate the contribution of the term F y ) r Δ n ) 2 y u yt B u y N 1 u yt K B u y Using the definition of the test sequence y, we can split F into three terms. Now, we estimate two of them as follows. Using 4.4, weobtain 1 u ft K 1 K 1 K x ht B u f [ Δ [ Δ U 1 [ u ft u ft K u ft ) x f ) x f Δx f Δu ft x f x ft Δu f 1 )] Ax f 1 1 uft ) X 1x h X ) T 1 1 [ K ] Ax f 1 )] A T u f 1 B K u ft u ft X Δx f x f ) 1x h K u ft ] Ax f 1 xft u f 4.12 x ht X ) T 1 1 B K X ) 1x h : G,

12 12 Abstract and Applied Analysis where we used the fact that x ht U X 1 x h 0 recall that the last n m 1 entries of x h are zeros and that the first m 1 columns of X and U are generated by the solutions 1,..., m. Similarly, N 1 u gt M B u g x gt M ug M xht M X M ) T 1 N 1 B M X M ) 1x h M : H Using property 2.10 of recessive solutions of the linear Hamiltonian difference systems, we can see that G 0asK and H 0asN. We postpone the estimation of the middle term of F to the end of the proof. To estimate the second term of Fy, we estimate at first its terms 1 q f 2 n, K N 1 q g 2 n. M 4.14 For this estimation, we use emma 2.3. To do this, we have to show the monotonicity of the sequences f h g h for K n 1, n 1, for M n 1,N n et x 1,...,x 2n be the ordered system of solutions of 2.13 in the sense of emma 2.1. Then, again by emma 2.1, there exist real numbers d 1,...,d n such that h d 1 x 1 d n x n. Because h / 0, at least one coefficient d i is nonzero. Therefore, we can denote p : max{i 1,n : d i / 0}, and we replace the solution x p by h. et us denote this new system again x 1,...,x 2n and note that this new system has the same properties as the original one. Following emma 2.2, we transform 2.13 via the transformation y h z,into n Δ ν ) R ν Δν z n ν 0, 4.16 ν0 that is, Δ n ) r h h n Δ n 1 w Δ R 1 w n 1 )

13 Abstract and Applied Analysis 13 possesses the fundamental system of solutions ) ) x w 1 1 x Δ,...,w p 1 p 1 Δ, h h ) ) x w p p1 x Δ,...,w 2n 1 2n Δ. h h 4.18 Now, let us compute the Casoratians ) C w 1) x w 1 1 Δ C x 1,h ) > 0, h h h 1 C w 1,w 2) C x 1,x 2,h ) > 0, h h 1 h 2. C w 1,...,w 2n 1) C x 1,...,x p 1,x p1,...,x 2n,h ) > 0. h h 2n Hence, w 1,...,w 2n 1 form the D-Marov system of sequences on M,,forM sufficiently large. Therefore, by emma 2.4, 4.17 is eventually disconugate; that is, it has at most 2n 2 generalized zeros counting multiplicity on M,. The sequence Δg/h is a solution of 4.17, and we have that this sequence has generalized zeros of multiplicity n 1bothatM and at N;thatis, gmi Δ h Mi ) gni 0 Δ h Ni ), i 0,...,n Moreover, g M /h M 1andg N /h N 0. Hence, Δg /h 0, M, N n 1. We can proceed similarly for the sequence f/h. Using emma 2.3, we have that there exist integers ξ 1 K, 1 and ξ 2 M, N 1 such that [ 1 1 q f 2 n q h 2 n K K [ N 1 N 1 q g 2 n q h 2 n M M fn h n gn h n ) 2 ] 1 q h 2 n, ξ 1 ) 2 ] ξ 2 1 q h 2 n. M 4.21

14 14 Abstract and Applied Analysis Finally, we estimate the remaining term of Fy.By4.9, we have M 1 [ r Δ n ) ] 2 y q y 2 n M 1 {r Δ n h Δ n h D 2 q h n h n D n 2} M 1 { } r Δ n h D 2 q h 2 n 2q h 2 n D n q h 2 n D2 n n { r Δ n h D 2} M 1 [ ] q h 2 n 2q h 2 n D n q h 2 n D 2 n [ n r 1 n ) h n δ ]2 M 1 [ ] q h 2 n 2δq h 2 n δ 2 q h 2 n )2 n n M 1 [ ] δ 2 h 2 n r q h 2 n 2δεh 2 n δ 2 εh 2 n )2 n n M 1 [ ] <δ 2 h 2 n r q h 2 n 2δεh 2 n Altogether, we have F y ) )2 n n M 1 [ ] 1 ξ <δ 2 h 2 n r q h 2 n 2δεh 2 n G H q h 2 n 2 1 q h 2 n ξ 1 M )2 n n ξ2 1 δ 2 h 2 n r 2δεh 2 n G H q h 2 n, ξ where for K sufficiently small is G <δ 2 /3, for N sufficiently large is H <δ 2 /3, and, from 4.1, ξ 2 1 ξ 1 q h 2 n <δ2 /3forξ 1 <and ξ 2 >M. Therefore, F y ) )2 n n <δ 2 δ 2 h 2 n r 2δεh 2 n )2 δ δ n n 1 h 2 n r εh 2 n, which means that Fy < 0forδ sufficiently small, and 4.2 is conugate on Z. 4.24

15 Abstract and Applied Analysis Equation with the Middle Terms Under the additional condition q 0forlarge, and by combining of the proof of Theorem 4.1 with the proof of 2, emma 1, we can establish the following criterion for the full 2n-order equation. Theorem 5.1. et n>1, q be a real-valued sequence, and let there exist an integer m {0,...,n 1} and real constants c 0,...,c m such that 1.1 is at least m 1-critical and the sequence h : c 0 c 1 c m m satisfies lim sup q h 2 n 0. K, K 5.1 If q 0 for large and q / 0, then y ) q y n n Δ ν ) r ν Δν y n ν q y n ν0 is conugate on Z. Remar 5.2. Using Theorem 3.4, we can see that the statement of Theorem 4.1 holds if and only if 3.7 holds. Finding a criterion similar to Theorem 3.4 for 1.1 is still an open question. Remar 5.3. In the view of the matrix operator associated to 1.1 in the sense of 21, we can see that the perturbations in Theorem 4.1 affect the diagonal elements of the associated matrix operator. A description of behavior of 1.1, with regard to perturbations of limited part of the associated matrix operator but not only of the diagonal elements, is given in 2. Acnowledgment The research was supported by the Czech Science Foundation under Grant no. P201/ 10/1032. References 1 F. Gesztesy and Z. Zhao, Critical and subcritical Jacobi operators defined as Friedrichs extensions, Journal of Differential Equations, vol. 103, no. 1, pp , O. Došlý and P. Hasil, Critical higher order Sturm-iouville difference operators, to appear in Journal of Difference Equations and Applications. 3 O. Došlý, Existence of conugate points for selfadoint linear differential equations, Proceedings of the Royal Society of Edinburgh. Section A, vol. 113, no. 1-2, pp , O. Došlý, Oscillation criteria and the discreteness of the spectrum of selfadoint, even order, differential operators, Proceedings of the Royal Society of Edinburgh. Section A, vol. 119, no. 3-4, pp , O. Došlý and J. Komenda, Conugacy criteria and principal solutions of self-adoint differential equations, Archivum Mathematicum, vol. 31, no. 3, pp , M. Bohner, inear Hamiltonian difference systems: disconugacy and Jacobi-type conditions, Journal of Mathematical Analysis and Applications, vol. 199, no. 3, pp , 1996.

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