Advances in Difference Equations Volume 2011, Article ID 571935,23pages doi:10.1155/2011/571935
Research Article
Generalized Zeros of
2
×
2
Symplectic Difference
System and of Its Reciprocal System
Ondˇrej Doˇsl ´y
1and ˇS ´arka Pechancov ´a
21Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
611 37 Brno, Czech Republic
2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering,
Brno University of Technology, ˇZiˇzkova 17, 602 00 Brno, Czech Republic
Correspondence should be addressed to Ondˇrej Doˇsl ´y,[email protected]
Received 1 November 2010; Accepted 3 January 2011
Academic Editor: R. L. Pouso
Copyrightq2011 O. Doˇsl ´y and ˇS. Pechancov´a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We establish a conjugacy criterion for a 2×2 symplectic difference system by means of the concept of a phase of any basis of this symplectic system. We also describe a construction of a 2×2 symplectic difference system whose recessive solution has the prescribed number of generalized zeros in .
1. Introduction
The main aim of this paper is to establish a conjugacy criterion for the 2×2 symplectic difference system
xk1
uk1
Sk
xk
uk
, k∈ , S
whereSk
akbk ck dk
with real-valued sequences a,b,c, anddis such that detSk akdk− bkck1 for everyk∈ . Recall that under this condition, the matrixSissymplectic. Generally, a 2n×2nmatrixSis symplectic if
STJSJ, J
0 I
−I 0
Ibeing then×nidentity matrix, and this conditions reduces just to the condition detS1 for 2×2 matrices. We introduce concepts of the first and second phase of any basis of systemS, and we study some of their properties. We generalize results introduced in1–4for a Sturm-Liouville difference equation, and we describe how to construct a 2×2 symplectic difference system whose recessive solution has a prescribed number of generalized zeros. This result generalizes a construction for a Sturm-Liouville difference equation and so solves an open problem posed in3, Section 4.
The paper is organized as follows. InSection 2, we introduce the definition of the first phase of any basis of the systemS, and we establish a formula for the forward difference of this phase. We apply this formula to study the relationship betweenSand its reciprocal system in Section 3, where the concept of the second phase is introduced. The forward difference of a first phase of S plays the crucial role in a conjugacy criterion for system
S, which is proved inSection 4. InSection 5, we show how to construct systemSwith prescribed oscillatory properties.
Definition of some concepts we need in our paper is now in order. A pair of linearly independent solutionsx
uandyvofSwith the Casoratianω
ω≡xkvk−ykukconst/0 1.2
is said to bea basisof the systemS. Ifω ≡ 1, it is said to bea normalized basis.An interval
m, m1,m∈ , is said to containa generalized zeroof a solutionx
uofS, ifxm/0 and
xm1 0 or bmxmxm1<0. 1.3
A solutionxuofSis said to beoscillatory in if it has infinitely many generalized zeros in . In the opposite case, we say that xu isnonoscillatory in . System Sis said to be
nonoscillatoryoffinite typein if every solution ofSis nonoscillatory in . A nonoscillatory system S is said to be 1-general in if it possesses two linearly independent solutions with no generalized zero, and it is said to be 1-specialin if there is exactly oneup to the linear dependencesolution ofSwithout any generalized zero in . The definition of these concepts via recessive solutions ofSis given later. SystemSis said to beconjugatein the intervalM, N M, Nrepresents the discrete setM, N∩ ,M, N ∈ ,N > M, if there exists a solution ofSwhich has at least two generalized zeros inM−1, N1.
Note that the terminology conjugacy/1-general/1-special equation is borrowed from the theory of differential equations, see 5, 6, and it is closely related to the concepts of supercriticality/criticality/subcriticality of the Jacobi operators associated with the three-term recurrence relation
Tx:rkxk1qkxkrk−1xk−10, 1.4
see7and also8.
At the end of this section, we recall the concept of the recessive solution of S and its relationship to conjugacy and other concepts defined above. Suppose that S is nonoscillatory. Then, there exists the uniqueup to a multiplicative factorsolutionz
x u
−∞ is defined analogously. System S is 1-special, respectively, 1-general if the recessive solutions z, z− have no generalized zero in and are linearly dependent, repectively, linearly independent. For more details concerning recessive solutions of discrete systems, we refer to9,10.
2. Phases and Their Properties
Definition 2.1. Letxk ukand
yk vk
,k∈ , form a basis ofSwith the Casoratianω. By thefirst phaseof this basis, we understand any real-valued sequenceψ ψk,k∈ , such that
ψk ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
arctanyk xk
ifxk/0,
odd multiple of π
2 ifxk0,
2.1
withΔψk∈0, πifω >0 andΔψk∈−π,0ifω <0.
Here, by arctan, we mean a particular value of the multivalued function which is inverse to the function tangent. By the requirementΔψ∈0, π, respectively,Δψ∈−π,0, a first phase ofx
u,yvis determined uniquely up to modπ.
The first phase and the later introduced second phaseare sometimes called zero-counting sequences, since each jump of their value over an odd multiple of π/2 gives a generalized zero of a solution ofS or of its reciprocal systemas we will show later.
Lemma 2.2. Letxuandvyform a basis ofSwith the Casoratianω. Then, there exist sequencesh
andg,hk/0, such that the transformation
xk
uk
Rk
sk
ck
, 2.2
Rk
hk 0 gk ω/hk
, transforms systemSinto the so-called trigonometric system
sk1
ck1
Tk
sk
ck
, T
whereTis a symplectic matrix of the formTk pk qk
−qk pk
with
pk
akhkbkgk hk1
, qk ωbk hkhk1
. 2.3
Sequencesh,gare given by
h2kx2ky2k, gk
xkukykvk hk
The values of the sequencehcan be chosen in such a way thatωqk ≥ 0. In particular, ifbk/0, then hkcan be chosen in such a way thatωqk>0fork∈ .
Proof. A similar statement is proved for general 2n×2nsymplectic systems in11. However, in contrast to11, our transformation matrix contains the Casoratianω, and the proof for scalar 2×2 systems can be simplified.
Transformation2.2transforms the symplectic systemSinto the system
sk1
ck1
Tk
sk
ck
, TkR−k11SkRk, 2.5
whereT:ab cd
with
ak
akhkgkbk hk1
, bk ωbk hkhk1
,
ck
1 ω
−gk1
akhkbkgk
hk1
ckhkdkgk
,
dk
−bkgk1dkhk1 hk
,
2.6
as can be verified by a direct computation. Then
detTkdet
R−1 k1SkRk
detR−k11detSkdetRk 1
ω·1·ω1, 2.7
which means that T is a symplectic matrix, even if the transformation matric R is not generally symplectic. This is due to the fact that we consider 2×2 systems where a matrix is symplectic if and only if its determinant equals 1.
We haveno index means indexk
hhk1c 1 ω
−hxk1uk1yk1vk1
ahbghh2k1chdg
1
ω
−xk1uk1yk1vk1
ah2bxuyvhk21ch2dxuyv
1
ω
−xk1uk1yk1vk1
xaxbu yaybv
1
Finally, concerning positivity ofωqifb /0, we fix the sign ofhin a particular index, sayh0
x2
0y20 and the formulaωq ω2b/hhk1 shows that the sign ofh, that is,h
±x2y2, at indicesk /0 can be “adjusted” in such a way thatωqk>0 ifbk/0.
Lemma 2.5. Letx1 trigonometric system associated toSas formulated inLemma 2.2. Then, there exists a solutionsc
ofTsuch that
By a direct computation, we have
s1c2−c1s21 2.16
and after a few steps
whereξis an arbitrary sequence such thatΔξk ϕkandϕis given by2.10. By2.17, we haveβi1, and by2.16, we obtain
sk1ck2−ck1sk2sinξkα1
cosξkα2
−sinξkα2
cosξkα1
sinα1−α21,
2.19
that is,α1−α2 π/2mod 2π. Hence, s1 c2 andc1 −s2 what implies2.12. Sinceξk was an arbitrary sequence such thatΔξk ϕk, changing ξk toξk−α2, we get
2.13.
Notation. In the following, by Arctan and Arccot, we mean the principal branches of the multivalued functions arctan and arccot with the values in −π/2, π/2 and 0, π, respectively.
Theorem 2.6. Letz1 xuandz2 yvform a basis of Swith the Casoratianω, and letψbe
a first phase of this basis. Ifbk/0, then
Δψk
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
Arccotxkxk1ykyk1
ωbk if ω >0,
Arccotxkxk1ykyk1
ωbk −
π if ω <0.
2.20
Ifbk0, thenΔψk0.
Proof. LetTbe a trigonometric system associated toSwith the basisz1, z2and withp, qsatisfying2.3. Letψbe a first phase of this basis. ByLemma 2.5, there exists a solutions
c ofTsuch thatsksinξk,ck cosξkandz1 xu,z2 yvsatisfy
xkhkcosξk, ykhksinξk, 2.21
wherehis given by2.4,Δξkϕkandϕkis given by2.10. Hence, forxk/0,
tanξk yk
xk 2.22
and ifxk 0, thenξkis equal to an odd multiple ofπ/2. On the other hand, byDefinition 2.1 forxk/0
tanψk yk xk
, 2.23
and ifxk0, thenψkis equal to an odd multiple ofπ/2. Consequently,
and it impliessince the additive multiple ofπto get equality in2.24is independent ofk
Δψk≡Δξk. 2.25
For ω > 0, we defined in Definition 2.1 that Δψk ∈ 0, π. Suppose that bk/0. According to Lemma 2.2, we can choose qk > 0, and then by Lemma 2.4, we can take ϕk∈0, π. Using2.25, we haveϕk Δψk, and thus cotΔψkpk/qk, and hence
ΔψkArccot pk qk
. 2.26
Let ω < 0. Then, we defined Δψk ∈ −π,0 and based on Lemma 2.2, under the assumption bk/0, we can choose qk < 0 and thenϕk ∈ π,2π defined in 2.10. Using
2.25, we haveϕk Δψk2π, and consequently cotΔψk2π cotΔψkpk/qkand
ΔψkArccot pk
qk −π, 2.27
in this case. Finally, ifbk 0, then qk 0, and we put ϕk 0. Hence, by 2.25,Δψk ≡ ϕkmodπand since byDefinition 2.1Δψk∈−π, π, then we haveΔψk0.
Summarizing, by a direct computation
pk qk
hk ωbk
akhkbkgk
1
ωbk
akh2kbk
xkukykvk
. 2.28
Sincexk1 akxkbkukandyk1akykbkvk,
xkxk1ykyk1akh2kbk
xkukykvk
, 2.29
and this gives, together with2.26and2.27, the conclusion2.20.
We continue in this section with a statement which justifies why phases are sometimes called zero-counting sequences. We formulate the statement for a first phase, for a second phase the statement is similar.
Theorem 2.7. Let ψ be the first phase of Sdetermined by the basis xu,yv. Then,xuhas a
generalized zero ink, k1if and only ifψskips over an odd multiple ofπ/2betweenkandk1.
Proof. Suppose that xu has a generalized zero in k, k 1, that is, xkxk1bk < 0. By Lemma 2.5xk hkck,yk hksk, where−csis a solution of trigonometricTwithωqk >0
Remark 2.8. A slightly modified statement we have in the case whenxuhas a zero atk1, that is,xk/0 andxk10. More precisely, by the definition of the first phaseψk1 2m1π/2 for some integerm, and, ifψis increasing, thenψk∈2m−1π/2,2m1π/2.
We illustrate the above statements concerning properties of the first phase by the following example.
Example 2.9. Consider the Fibonacci recurrence relation
xk2xk1xk, k∈ , 2.30
that is,
Δ−1kΔxk
−1kxk1 0, 2.31
which can be viewed as symplectic systemSwith the matrix
Sk ⎛
⎝ 1 −1k
−1k1 0
⎞
⎠, 2.32
that is, the entry corresponding tobkchanges its sign. A basisxu,yvofScorresponding to2.31has the first components given by
xk
1−√5 2
k
, yk
1√5 2
k
, 2.33
with the positive Casoratianω√5. ByDefinition 2.1,Δψk∈0, πand
ψk ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Arctan
1√5 1−√5
k
k
2π, k even,
Arctan
1√5 1−√5
k
k1
2 π, k odd.
2.34
Notice that every jump of the value ψk over an odd multiple of π/2 corresponds to a generalized zero ofxink, k1. A corresponding trigonometric systemTto symplectic system S has by Lemma 2.5two linearly independent solutions −cs andsc, where the sequencescandsare given by2.13. Since by2.24ξk ψkmπfor somem∈ , the first components of a basis ofTcan beup to the signuniquely determined by
ckcosψk, sksinψk. 2.35
It means that the componentsx, respectively,yof solutions ofShave generalized zeros in
zero ink, k1. By2.21, together with2.24, we express the first components of the basis ofSas
xkhkcosψk, ykhksinψk. 2.36
ByLemma 2.2, we choose the sign of the sequencehin such a way that
. . . , h0>0, h1>0, h2<0, h3<0, h4 >0, . . . , 2.37
so the termωqki.e.,bkhkhk1is positive. Such a choice of the sign of the members ofhk must agree with the sign of sequencesxkandyk. In fact, then by2.36,ykis positive for anykandxkis positive for every even and negative for every odd integerk.
Next, we describe the behavior of the phase ψ and corresponding trigonometric sequencescandsin case whenω <0. Consider2.31, that is, the corresponding symplectic system with the basisxu,yvhaving the first components
xk
1√5 2
k
, yk
1−√5 2
k
, 2.38
with the negative Casoratianω−√5. ByDefinition 2.1,
ψk ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Arctan
1−√5 1√5
k
− k
2π, k even,
Arctan
1−√5 1√5
k
− k−1
2 π, k odd.
2.39
The first components of a basis of the trigonometric systemTcorresponding toSis of the form
ckcosψk, sksinψk. 2.40
Choosing the sign of the sequencehas follows
. . . , h0>0, h1>0, h2<0, h3<0, h4 >0, . . . , 2.41
3. Reciprocal System
A reciprocal systemtoSis the symplectic system
u. From definition of the symplectic systemS and its reciprocal systemSr, it follows that ifxuis a solution ofS, then−uxis a solution of its reciprocal systemSr.
Definition 3.1. By thesecond phaseof the basisx
u,yvof systemS, we understand any first phase of the basis−ux,
v
−y
of its reciprocal systemSr, that is, any real-valued sequence k,k∈ , such that
odd multiple of π
2 if uk0,
3.2
withΔ k∈0, πifω >0 andΔ k∈−π,0ifω <0.
The proofs of the next statement and of its corollary are the same as those ofLemma 2.2 andTheorem 2.6, respectively.
Lemma 3.2. Letx
uandyvform a basis of Swith the Casoratianω, that is,−uxand v
−yis a
basis ofSrwith the Casoratianωω−ukykxkvk. Then, there exist sequenceshandg,hk/0
fork∈ , such that the transformation
transforms systemSrinto the trigonometric system
which is symplectic with the sequencesp,qgiven by
pk dkhk−ckgk hk1
, qk− ckω hkhk1
, 3.4
where
h2ku2kv2k, gk−xkukykvk hk
. 3.5
Moreover, transformation3.3preserves oscillatory properties of Sr, and the sequencehk,k∈ ,
can be chosen in such a way thatωqk≥0and ifck/0in such a way thatωqk>0.
Corollary 3.3. Letx
uandyvform a basis of Swith the Casoratianω; that is,−uxand v
−y
form the basis of Srwith the same Casoratianω. Let kbe the second phase of the basisxu,yv
ofS. Ifck/0,k∈ , then
Δ k
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
Arccotukuk1vkvk1
−ωck if ω >0,
Arccotukuk1vkvk1
−ωck −
π if ω <0.
3.6
Ifck0, thenΔ k0.
In the next statement, we use the relationship between the first phaseψand the second phase of the basisxu,yvof symplectic systemSand the fact that the behavior of the first and second phases of systemSplays the crucial role in counting generalized zeros of solutions of symplectic systemSand of its reciprocal systemSr.
Theorem 3.4. If systemSwith the sequencesbk/0andck/0which do not change their sign has a
solution with two consecutive generalized zeros inl−1, l, and letm−1, m,l < m,l, m∈ , then its reciprocal systemSris either conjugate inl−1, mwith a solution having a generalized zero in l−1, lorm−1, m, or there exists a solution of Srwith exactly one generalized zero inl, m.
Proof. Let xube the solution of S having consecutive generalized zeros in l−1, land
m−1, mandyvbe a solution which together withx
uform the basis of the solution space ofS. Denote byψand the first and second phase of this basis. Then, byLemma 2.5,
xkhkcosψk, ukgkcosψk− ω hk
sinψk,
ykhksinψk, vkgksinψk ω hkcosψk,
3.7
and byLemma 3.2,
Hence,
hkcos kgkcosψk− ω hksinψk,
hksin kgksinψk ω hk
cosψk.
3.9
Multiplying the first equation by−sinψk, the second one by cosψk, and adding the resulting equations, we obtain
hksin
k−ψk
ω
hk
. 3.10
Since we assume that the sequences b, c are of constant sign, the last part of Lemma 2.2 together with the second formulas in2.3,3.4imply that handhhave constant sign as well and by3.10the same holds for the sequence sin k−ψk. Suppose, to be specific, that sin k −ψk < 0 if this sequence is positive, the proof is similarthen there exists an odd integernsuch that
nπ < k−ψk<n1π. 3.11
Recall that byDefinition 2.1, the first phaseψ and the second phase are defined as the monotone sequences on . In addition, byLemma 3.2, the CasoratianωofSequals to the Casoratianω of Sr, and thus, again by Definition 2.1, both phasesψ and ofSare either nondecreasing or nonincreasing. Moreover, ifω ω /0,bk/0 andck/0,k ∈ , then byTheorem 2.6andCorollary 3.3,Δψk/0 andΔ k/0 fork∈ .
Suppose that the first phaseψk of the basisxu,yvofSgiven byDefinition 2.1is increasing; that is, for every integerk, we haveΔψk ∈ 0, π. If we suppose a decreasing sequenceψ, the proof is analogous. Since the phases are determined up to modπ, without loss of generality, we may suppose thatn−1 in3.11, that is,
0< ψk− k< π. 3.12
Moreover, we can also suppose thatψl−1 ∈−π/2, π/2. Sincexhas consecutive generalized zeros inl−1, landm−1, m, we have
ψl∈
π 2,
3π 2
, ψj∈
π
2, 3π
2
, j l1, . . . , m−1, ψm∈
3π 2 ,
5π 2
, 3.13
that is,ψkskipsπ/2 betweenl−1 andland 3π/2 betweenm−1 andmand stays in the strip
π/2,3π/2betweenlandm. Formula3.12admits the following behavior of the sequence
i l−1<−π/2, l∈−π/2, π/2, there existsr,l < r < m, such that existence of a solution with exactly one generalized zero inl, m−1.
4. A Conjugacy Criterion
In this section, we establish a conjugacy criterion for systemSby means of the first phaseψ and the associated Riccati equation.
Theorem Bsee9, Theorem 5.30, see also13. Suppose that systemSpossesses a solution with no generalized zero inM, N1. Then, every nontrivial solutionxuof this system has at most
one generalized zero in this interval.
In this section, as usual, we putnim· 0 ifm > nandlik· 1 ifk > l.
Theorem 4.1. Let the sequencebkinSbe positive. Suppose that there exist positive real numbers δ1andδ2such that given by the condition
The Casoratianωsatisfies
ωx0v0−y0u0 δ1>0. 4.9
Suppose, by contradiction, thatxuhas no generalized zero in0, N1, that is, due to the fact thatx01 andbk>0, we havexk>0 for everyk 1,2, . . . , N. Then, by Theorem B, we get
yk> xk, 4.10
fork 1, . . . , N1, because otherwise the solutionxu−yvhas generalized zeros atk0 and in the intervalm, m1,mbeing the integer where4.10is violated.
Letψbe the first phase of solutionsx
uandyv, that is, byDefinition 2.1,
ψkarctan yk xk
, Δψk∈0, π. 4.11
ByTheorem 2.6, we have
ΔψkArccot
xkxk1ykyk1 δ1bk
, 4.12
taking account thatψ0π/4 and using4.10, we get fork1, . . . , N1
ψk k−1
j0
Δψjψ0
k−1
j0
Arccotxjxj1yjyj1 δ1bj
π 4 >
k−1
j0
Arccot2yjyj1 δ1bj
π
4. 4.13
Letwkvk/yk. Then, from the first equation inS
wk 1 bk
yk1
yk −ak
, 4.14
andwis a solution of the Riccati equation4.1. Denotewkwk ak−1/bk. Then,
wk1 ak1−1 bk1
ckdkwk−ak−1/bk akbkwk−ak−1/bk
ak1−1 bk1
bkckbkdkwk−akdkdk bkbkwk1
ak1−1 bk1
−1dk1bkwk−bkwkbkwk bkbkwk1
ak1−1 bk1
dk−1 bk
wk 1bkwk
.
Further denoteFk ak1−1/bk1dk−1/bk. Then, since 1bkwkbkwkakyk1/yk>0,
Hence,4.14implies
Letyvbe another linearly independent solution ofSgiven by the condition
y01, y−1 δ2b−1d−1−b−1 b0
1−a0, 4.23
with the corresponding second componentv0expressed by
v0 1 b−1
d−1y0−y−1
1
b0
1−a0−δ2. 4.24
The Casoratianωofx u,
y v
satisfies
ωx0v0−y0u0 −δ2<0. 4.25
Suppose, by contradiction, that the solutionxuhas no generalized zero in the intervalM−1, 0, that is,xk >0 fork M, . . . ,0. Then, by Theorem Busing the same argument as in the first part of the proof
yk> xk, 4.26
for everyk M, . . . ,−1. Letψbe the first phase ofxuand
y v
with the Casoratianω < 0. ByDefinition 2.1,
ψkarctanyk xk
, Δψk∈−π,0, 4.27
and byTheorem 2.6
ΔψkArccotxkxk1ykyk1
−δ2bk −
π. 4.28
Taking into account thatψ0 π/4,4.26, and that the function Arccot·−π/2 is odd, we get for everykM, . . . ,−1
−ψkψ0
−1
jk
Δψj
−1
jk #
Arccotxj1xjyj1yj
−δ2bj − π 2 −
π 2 $
−−1
jk
Arccotxj1xjyj1yj δ2bj
<−
−1
jk
Arccot2yj1yj δ2bj
.
Hence,
ψk> π 4
−1
jk
Arccot2yjyj1 δ2bj
. 4.30
Let us estimate the term2yk1yk/bkby means of the Riccati equation. Letwkvk/ yk. Then,ykdkyk1−bkvk1, that is,
yk
yk1 dk−bkwk1, 4.31
and from the backward Riccati equation forwwhich follows from4.1,
wk −
ckakwk1 dk−bkwk1
. 4.32
Putwk wk−dk−1−1/bk−1. Then, substituting into4.32, we haveno index mean the
indexkhere and also in later computations
wdk−1−1 bk−1
−cawk1 d−1/b d−bwk1 d−1/b
awk1 −cbad−a/b 1−bwk1
1/b a/bbwk1−1
1−bwk1 − a b
1/b−wk1wk1 1−bwk1
−a−1
b
wk1 1−bwk1,
4.33
and hence,
wk−wk1−ak−1 bk −
dk−1−1 bk−1
bkw2k1 1−bkwk1.
4.34
Since,
1−bwk11−b
wk1−d−1 b
d−bwk1
d−bcdw
abw
ad−cb abw
1 abw
yk yk1 >0,
4.35
we have−Δwk− ≥Fk−1, whereFkis given by4.5. Summing the last inequality fromk1 to
−1, we obtain
−wk1w0≤
−1
jk1
and this means that
Substituting from4.30,
ψM> π
Remark 4.2. The conjugacy criterion for the Sturm-Liouville equation
ΔrkΔxk qkxk1 0, rk>0, 4.44
formulated in1, Theorem 2is the corollary of the above criterion forak 1, bk 1/rk, ck−qkanddk1−qk/rk.Theorem 4.1also extends the results proved in1,2,12,14.
5. Systems with Prescribed Oscillatory Properties
In this concluding section, we present a method of constructing a symplectic system S whose recessive solution has the prescribed number of generalized zeros in .
Theorem 5.1. Suppose that x
u,yv ∈
2,k ∈ , are sequences such that the Casoratianω detxkyk
ukvk
1for anyk∈ . Then, these sequences form a normalized basis of symplectic systemS with
akxk1vk−yk1uk,
bk−xk1ykyk1xk,
ckuk1vk−vk1uk,
dk−uk1ykvk1xk.
5.1
Moreover, ifbk/0fork∈ ,
lim k→ ±∞
xk
yk 0, 5.2
andx
uhasm−1generalized zeros in , then the first phase determined byxu,yvsatisfies
lim
k→ ∞ψk−klim→ −∞ψk
k∈ ,bk/0
Arccotxkxk1ykyk1
bk
mπ. 5.3
Proof. Letxuandyvbe sequences with the Casoratian equal to 1, and leta, b, c, anddbe given by5.1. Then, by the Cramer rule, we obtain
xk1akxkbkuk,
uk1ckxkdkuk,
yk1akykbkvk,
vk1ckykdkvk,
5.4
assumptions of the second part of the theorem are satisfied. Then,Sis nonoscillatory in
We finish the paper with an example illustrating the previous theorem.
Example 5.2. Consider a pair of two-dimensional sequencesxu,yvwith
xk
fork∈ , the solutionxuhas no generalized zero in . Consequently,5.3readsas can be again verified by a direct computation
k∈
Arccot
k2 4−2nkn2−4n19 4
%
π. 5.9
By a similar method, one can find the explicit formula for the sum of various infinite series involving the function Arccot.
Acknowledgments
Research supported by the Grants nos. 201/09/J009 and P201/10/1032 of the Czech Grant Agency and by the Research Project no. MSM0021622409 of the Ministry of Education of the Czech Government.
References
1 Z. Doˇsl´a and ˇS. Pechancov´a, “Conjugacy and phases for second order linear difference equation,”
Computers & Mathematics with Applications, vol. 53, no. 7, pp. 1129–1139, 2007.
2 I. Kumari and S. Umamaheswaram, “Conjugacy criteria for a linear second order difference equation,”Dynamic Systems and Applications, vol. 8, no. 3-4, pp. 533–546, 1999.
3 ˇS. Pechancov´a,Phases and oscillation theory of second order difference equations, Ph.D. thesis, Masaryk University, Brno, Czech Republic, 2007.
4 ˇS. Ryz´ı, “On the first and second phases of 2×2 symplectic difference systems,”Studies of the University of ˇZilina. Mathematical Series, vol. 17, no. 1, pp. 129–136, 2003.
5 O. Bor ˚uvka, Linear Differential Transformations of the Second Order, The English Universities Press, London, UK, 1971.
6 F. Neuman, Global Properties of Linear Ordinary Differential Equations, vol. 52 ofMathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Nethrlands, 1991.
7 F. Gesztesy and Z. Zhao, “Critical and subcritical Jacobi operators defined as Friedrichs extensions,”
Journal of Differential Equations, vol. 103, no. 1, pp. 68–93, 1993.
8 O. Doˇsl ´y and P. Hasil, “Critical higher order Sturm-Liouville difference operators,” to appear in
Journal of Difference Equations and Applications.
9 C. D. Ahlbrandt and A. C. Peterson,Discrete Hamiltonian Systems. Difference Equations, vol. 16 ofKluwer Texts in the Mathematical Sciences, Kluwer Academic Publishers, Dordrecht, The Nethrlands, 1996.
10 M. Bohner, O. Doˇsl ´y, and W. Kratz, “A Sturmian theorem for recessive solutions of linear Hamiltonian difference systems,”Applied Mathematics Letters, vol. 12, no. 2, pp. 101–106, 1999.
11 M. Bohner and O. Doˇsl ´y, “Trigonometric transformations of symplectic difference systems,”Journal of Differential Equations, vol. 163, no. 1, pp. 113–129, 2000.
12 Z. Doˇsl´a and D. ˇSkrab´akov´a, “Phases of linear difference equations and symplectic systems,”
Mathematica Bohemica, vol. 128, no. 3, pp. 293–308, 2003.
13 M. Bohner, O. Doˇsl ´y, and W. Kratz, “Sturmian and spectral theory for discrete symplectic systems,”
Transactions of the American Mathematical Society, vol. 361, no. 6, pp. 3109–3123, 2009.
