Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 571935, 23 pages doi:10.1155/2011/571935 Research Article Generalized Zeros of × Symplectic Difference System and of Its Reciprocal System ˇ´ Ondˇ ej Doˇ y1 and Sarka Pechancova2 r sl ´ ´ Department of Mathematics and Statistics, Masaryk University, Kotl´ rsk´ 2, aˇ a 611 37 Brno, Czech Republic Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, ˇz Brno University of Technology, Ziˇ kova 17, 602 00 Brno, Czech Republic Correspondence should be addressed to Ondˇ ej Doˇ ly, dosly@math.muni.cz r s ´ Received November 2010; Accepted January 2011 Academic Editor: R L Pouso ˇ Copyright q 2011 O Doˇ ly and S Pechancov´ This is an open access article distributed under s ´ a 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 × 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 × symplectic difference system whose recessive solution has the prescribed number of generalized zeros in Introduction The main aim of this paper is to establish a conjugacy criterion for the × symplectic difference system xk uk Sk xk uk , k∈ , S ak bk with real-valued sequences a, b, c, and d is such that det Sk ak dk − where Sk ck dk bk ck for every k ∈ Recall that under this condition, the matrix S is symplectic Generally, a 2n × 2n matrix S is symplectic if ST JS J, J I −I , 1.1 Advances in Difference Equations I being the n×n identity matrix, and this conditions reduces just to the condition det S for × matrices We introduce concepts of the first and second phase of any basis of system S , and we study some of their properties We generalize results introduced in 1–4 for a SturmLiouville difference equation, and we describe how to construct a × 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 in 3, Section The paper is organized as follows In Section 2, we introduce the definition of the first phase of any basis of the system S , and we establish a formula for the forward difference of this phase We apply this formula to study the relationship between S and 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 in Section In Section 5, we show how to construct system S with prescribed oscillatory properties Definition of some concepts we need in our paper is now in order A pair of linearly independent solutions x and y of S with the Casoratian ω u v ω ≡ xk vk − yk uk const / 1.2 is said to be a basis of the system S If ω ≡ 1, it is said to be a normalized basis An interval m, m , m ∈ , is said to contain a generalized zero of a solution x of S , if xm / and u xm or bm xm xm < 1.3 A solution x of S is said to be oscillatory in if it has infinitely many generalized zeros u in In the opposite case, we say that x is nonoscillatory in System S is said to be u nonoscillatory of finite type in if every solution of S is nonoscillatory in A nonoscillatory if it possesses two linearly independent solutions system S is said to be 1-general in with no generalized zero, and it is said to be 1-special in if there is exactly one up to the linear dependence solution of S without any generalized zero in The definition of these concepts via recessive solutions of S is given later System S is said to be conjugate in the interval M, N M, N represents the discrete set M, N ∩ , M, N ∈ , N > M , if there exists a solution of S which has at least two generalized zeros in M − 1, N Note that the terminology conjugacy/1-general/1-special equation is borrowed from the theory of differential equations, see 5, , and it is closely related to the concepts of supercriticality/criticality/subcriticality of the Jacobi operators associated with the threeterm recurrence relation Tx : rk xk qk xk rk−1 xk−1 0, 1.4 see and also 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 unique up to a multiplicative factor solution z x x with the property that limk → ∞ xk /xk for any solution z u linearly indeu pendent of z The solution z is said to be recessive at ∞ The recessive solution z − at Advances in Difference Equations −∞ 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 to 9, 10 Phases and Their Properties k Definition 2.1 Let xk and yk , k ∈ , form a basis of S with the Casoratian ω By the first uk v phase of this basis, we understand any real-valued sequence ψ ψk , k ∈ , such that ψk ⎧ ⎪arctan yk ⎪ ⎨ xk ⎪ ⎪odd multiple of π ⎩ if xk / 0, 2.1 if xk 0, with Δψk ∈ 0, π if ω > and Δψk ∈ −π, if ω < Here, by arctan, we mean a particular value of the multivalued function which is inverse to the function tangent By the requirement Δψ ∈ 0, π , respectively, Δψ ∈ −π, , a first phase of x , y is determined uniquely up to mod π u v The first phase and the later introduced second phase are sometimes called zerocounting sequences, since each jump of their value over an odd multiple of π/2 gives a generalized zero of a solution of S or of its reciprocal system as we will show later Lemma 2.2 Let x and y form a basis of S with the Casoratian ω Then, there exist sequences h u v and g, hk / 0, such that the transformation xk uk Rk hk gk ω/hk Rk sk , ck 2.2 , transforms system S into the so-called trigonometric system sk ck where T is a symplectic matrix of the form Tk pk Tk pk qk −qk pk ak hk bk gk , hk sk ck , T with qk ωbk hk hk 2.3 Sequences h, g are given by h2 k xk yk , gk xk uk yk vk hk 2.4 Advances in Difference Equations The values of the sequence h can be chosen in such a way that ωqk ≥ In particular, if bk / 0, then hk can be chosen in such a way that ωqk > for k ∈ Proof A similar statement is proved for general 2n × 2n symplectic systems in 11 However, in contrast to 11 , our transformation matrix contains the Casoratian ω, and the proof for scalar × systems can be simplified Transformation 2.2 transforms the symplectic system S into the system sk ck where T : a b c d sk Tk R−1 Sk Rk , k Tk , ck 2.5 with ak hk gk bk , hk ak −gk ω ck ak hk bk g k −bk gk dk ωbk , hk hk bk hk dk hk ck hk 1 hk 2.6 dk g k , , as can be verified by a direct computation Then det R−1 Sk Rk k det Tk ·1·ω ω det R−1 det Sk det Rk k 2.7 1, 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 × systems where a matrix is symplectic if and only if its determinant equals We have no index means index k hhk c −h xk uk ω − xk uk ω yk vk yk vk 1 − xk uk ω h2 k 1 x cx ah ah2 yk vk du 1 bg b xu x ax y cy dv bu hh2 k ch dg yv h2 k y ay bv ch2 d xu yv Advances in Difference Equations − xk uk ω yk vk 1 −xxk yk vk ω xyk ω −xyk −xk vk yxk xxk − yxk yk uk yk uk −x ay yyk 1 yxk vk yxk bu xk y ax yk xuk xyk uk −yk uk 1 1 yvk 1 xk vk −ωb bu 2.8 Hence, b − ωb/hhk hhk 1 hhk 1 hhk 1 hhk a−d −c : q Similarly, ah2 hgb bgk hk ax2 ay2 b xu − dh2 k yv b xk uk yk vk 1 − dxk − dyk 2.9 x ax bu xxk yyk y ay bv xk − xk x − yk y buk − dxk yk bvk − dyk 0, d −b in the last line of this computation, we have used the fact that S−1 −c a , that is, xk dk xk − bk uk , yk dk yk − bk vk Hence, a d : p Finally, concerning positivity of ωq if b / 0, we fix the sign of h in a particular index, x0 say h0 ± x2 y0 and the formula ωq ω2 b/hhk shows that the sign of h, that is, h y2 , at indices k / can be “adjusted” in such a way that ωqk > if bk / Remark 2.3 Transformation 2.2 preserves oscillatory properties of transformed systems in the following sense If bk xk xk < 0, that is, bk hk sk hk sk < 0, then since sgn bk hk hk sgn ωqk , we have using the positivity of the term ωqk bk hk sk hk sk < if and only if sk sk < Note also that xk if and only if sk 0, since hk / for all k Lemma 2.4 see 12, Lemma Let T be the trigonometric system There exists the unique (up to mod 2π) sequence ϕk ∈ 0, 2π such that sin ϕk and the general solution s c cos ϕk pk 2.10 , 2.11 of T takes the form sk ck where k ∈ qk , β sin ξk α cos ξk α , α, β ∈ Ê and ξ is any sequence such that Δξk ϕk Advances in Difference Equations Lemma 2.5 Let x and x be a basis of system S with the Casoratian ω, and let T be the u u trigonometric system associated to S as formulated in Lemma 2.2 Then, there exists a solution s c of T such that ⎛ ⎝ xk ⎞ ⎛ ⎠ ⎝ uk where k ∈ hk gk xk ⎝ ⎞ ⎛ ⎠ ⎝ hk uk gk ⎞ sk , ω⎠ ck hk 2.12 and h, g are given by 2.4 Further, there exists a sequence ξ such that sin ξk , sk Δξk ⎛ ⎞ ck , ω⎠ −sk hk ck cos ξk , 2.13 ϕk , where the sequence ϕ is given by 2.10 and ϕk ∈ 0, 2π for every k ∈ si ci Proof By Lemma 2.2, there exist solutions ⎛ i xk ⎝ i ⎞ ⎛ ⎠ ⎝ ,i uk 1, 2, of T such that ⎞⎛ ⎞ i sk ω ⎠⎝ i ⎠, ck hk hk gk 2.14 that is, h−1 x i , si ci −gx i hu i ω 2.15 By a direct computation, we have s1 c2 −c1 s2 2.16 2.17 and after a few steps si ci By Lemma 2.4, there exist real constants α i , β i such that ⎛ i ⎜ ⎝ sk i ck ⎞ ⎟ ⎠ ⎛ βi⎝ sin ξk αi cos ξk αi ⎞ ⎠, 2.18 Advances in Difference Equations where ξ is an arbitrary sequence such that Δξk have β i 1, and by 2.16 , we obtain 2 sk ck − ck sk sin ξk α1 ϕk and ϕ is given by 2.10 By 2.17 , we − sin ξk α2 cos ξk α2 cos ξk α1 2.19 sin α − α 1, that is, α − α π/2 mod 2π Hence, s c and c −s what implies 2.12 Since ξk was an arbitrary sequence such that Δξk ϕk , changing ξk to ξk − α , 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 x Theorem 2.6 Let z u and z a first phase of this basis If bk / 0, then Δψk If bk 0, then Δψk y v form a basis of S with the Casoratian ω, and let ψ be ⎧ ⎪Arccot xk xk yk yk ⎪ ⎪ ⎨ ωbk ⎪ ⎪ ⎪Arccot xk xk ⎩ 1 yk yk if ω > 0, 2.20 ωbk −π if ω < 0 Proof Let T be a trigonometric system associated to S with the basis z , z and with p, q satisfying 2.3 Let ψ be a first phase of this basis By Lemma 2.5, there exists a solution s c y x of T such that sk sin ξk , ck cos ξk and z ,z2 satisfy u v xk where h is given by 2.4 , Δξk hk cos ξk , 2.21 yk xk 2.22 0, then ξk is equal to an odd multiple of π/2 On the other hand, by Definition 2.1 tan ψk and if xk hk sin ξk , ϕk and ϕk is given by 2.10 Hence, for xk / 0, tan ξk and if xk for xk / yk yk , xk 2.23 0, then ψk is equal to an odd multiple of π/2 Consequently, ψk ≡ ξk mod π , 2.24 Advances in Difference Equations and it implies since the additive multiple of π to get equality in 2.24 is independent of k Δψk ≡ Δξk 2.25 For ω > 0, we defined in Definition 2.1 that Δψk ∈ 0, π Suppose that bk / According to Lemma 2.2, we can choose qk > 0, and then by Lemma 2.4, we can take ϕk ∈ 0, π Using 2.25 , we have ϕk Δψk , and thus cot Δψk pk /qk , and hence Δψk Arccot pk qk 2.26 Let ω < Then, we defined Δψk ∈ −π, and based on Lemma 2.2, under the assumption bk / 0, we can choose qk < and then ϕk ∈ π, 2π defined in 2.10 Using cot Δψk pk /qk and 2.25 , we have ϕk Δψk 2π, and consequently cot Δψk 2π Δψk Arccot pk − π, qk 2.27 in this case Finally, if bk 0, then qk 0, and we put ϕk Hence, by 2.25 , Δψk ≡ ϕk mod π and since by Definition 2.1 Δψk ∈ −π, π , then we have Δψk Summarizing, by a direct computation pk qk Since xk ak xk hk ak hk ωbk bk uk and yk xk xk 1 bk g k ak yk yk yk 1 ak h2 k ωbk bk xk uk yk vk 2.28 bk vk , ak h2 k bk xk uk yk vk , 2.29 and this gives, together with 2.26 and 2.27 , the conclusion 2.20 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 S determined by the basis x , y Then, x has a u u v generalized zero in k, k if and only if ψ skips over an odd multiple of π/2 between k and k Proof Suppose that x has a generalized zero in k, k , that is, xk xk bk < By u c Lemma 2.5 xk hk ck , yk hk sk , where −s is a solution of trigonometric T with ωqk > Lemma 2.2 Suppose that ω > 0, that is, Δψk ∈ 0, π for ω < the proof is analogical c Then, −s has a generalized zero in k, k which means that ck and ck have different sign Since ck cos ξk , ck cos ξk , where ξ is a sequence with Δξk Δψk compare 2.25 , ξk and ξk lay in different intervals whose endpoints form odd multiples of π/2 Conversely, if ψ skips an odd multiple of π/2 between k and k 1, ξ does also, and reasoning in the same way as above, we see that x has a generalized zero in k, k u Advances in Difference Equations Remark 2.8 A slightly modified statement we have in the case when x has a zero at k 1, that u 2m π/2 is, xk / and xk More precisely, by the definition of the first phase ψk for some integer m, and, if ψ is increasing, then ψk ∈ 2m − π/2 , 2m π/2 We illustrate the above statements concerning properties of the first phase by the following example Example 2.9 Consider the Fibonacci recurrence relation xk xk xk , k∈ , 2.30 that is, Δ −1 k Δxk −1 k xk 0, 2.31 which can be viewed as symplectic system S with the matrix ⎛ ⎝ Sk −1 −1 k k ⎞ ⎠, 2.32 that is, the entry corresponding to bk changes its sign A basis to 2.31 has the first components given by √ 1− xk with the positive Casoratian ω k , yk √ , y v of S corresponding k , 2.33 √ By Definition 2.1, Δψk ∈ 0, π and ⎧ ⎪ ⎪ ⎪ ⎪Arctan ⎪ ⎪ ⎨ ψk x u ⎪ ⎪ ⎪ ⎪ ⎪Arctan ⎪ ⎩ √ √ 1− √ √ 1− k k k π, k even, 2.34 k π, k odd Notice that every jump of the value ψk over an odd multiple of π/2 corresponds to a generalized zero of x in k, k A corresponding trigonometric system T to symplectic c system S has by Lemma 2.5 two linearly independent solutions −s and s , where the c sequences c and s are given by 2.13 Since by 2.24 ξk ψk mπ for some m ∈ , the first components of a basis of T can be up to the sign uniquely determined by ck cos ψk , sk sin ψk 2.35 It means that the components x, respectively, y of solutions of S have generalized zeros in k, k if and only if components c, respectively, s of solutions of T have a generalized 10 zero in k, k of S as Advances in Difference Equations By 2.21 , together with 2.24 , we express the first components of the basis xk hk cos ψk , yk hk sin ψk 2.36 By Lemma 2.2, we choose the sign of the sequence h in such a way that , h0 > 0, h1 > 0, h2 < 0, h3 < 0, h4 > 0, , 2.37 so the term ωqk i.e., bk hk hk is positive Such a choice of the sign of the members of hk must agree with the sign of sequences xk and yk In fact, then by 2.36 , yk is positive for any k and xk is positive for every even and negative for every odd integer k Next, we describe the behavior of the phase ψ and corresponding trigonometric sequences c and s in case when ω < Consider 2.31 , that is, the corresponding symplectic system with the basis x , y having the first components u v xk with the negative Casoratian ω √ , yk k , 2.38 √ − By Definition 2.1, ⎧ ⎪ ⎪ ⎪ ⎪Arctan ⎪ ⎪ ⎨ ψk √ 1− k 1− ⎪ ⎪ ⎪ ⎪ ⎪Arctan ⎪ ⎩ √ 1− √ √ √ k − k π, k even, 2.39 k − k−1 π, k odd The first components of a basis of the trigonometric system T corresponding to S is of the form ck cos ψk , sk sin ψk 2.40 Choosing the sign of the sequence h as follows , h0 > 0, h1 > 0, h2 < 0, h3 < 0, h4 > 0, , 2.41 we get xk positive for any k and yk positive for every even and negative for every odd integer k Advances in Difference Equations 11 Reciprocal System A reciprocal system to S is the symplectic system xk uk xk r Sk uk , k∈ , J Sr , where r Sk dk −ck JSk J−1 −bk ak −1 , 3.1 J x From definition of the symplectic system S related to S by the substitution x u u r u and its reciprocal system S , it follows that if x is a solution of S , then −x is a solution u r of its reciprocal system S Definition 3.1 By the second phase of the basis x , y of system S , we understand any first u v v u phase of the basis −x , −y of its reciprocal system Sr , that is, any real-valued sequence k , k ∈ , such that k with Δ k ∈ 0, π if ω > and Δ ⎧ ⎪arctan vk ⎨ uk ⎪odd multiple of π ⎩ k if uk / 0, if uk 3.2 0, ∈ −π, if ω < The proofs of the next statement and of its corollary are the same as those of Lemma 2.2 and Theorem 2.6, respectively v u Lemma 3.2 Let x and y form a basis of S with the Casoratian ω, that is, −x and −y is a u v r basis of S with the Casoratian ω ω −uk yk xk vk Then, there exist sequences h and g, hk / for k ∈ , such that the transformation ⎛ xk ⎜ ⎝ uk hk gk ⎞ ⎟ sk , ω⎠ ck hk 3.3 transforms system Sr into the trigonometric system sk pk qk sk ck −qk pk ck Tr 12 Advances in Difference Equations which is symplectic with the sequences p, q given by pk dk hk − ck g k hk , − qk ck ω hk hk , 3.4 3.5 where hk u2 k vk , − gk xk uk yk vk hk Moreover, transformation 3.3 preserves oscillatory properties of Sr , and the sequence hk , k ∈ can be chosen in such a way that ωqk ≥ and if ck / in such a way that ωqk > , v u Corollary 3.3 Let x and y form a basis of S with the Casoratian ω; that is, −x and −y u v r x form the basis of S with the same Casoratian ω Let k be the second phase of the basis u , y v of S If ck / 0, k ∈ , then Δ If ck 0, then Δ k ⎧ ⎪Arccot uk uk vk vk ⎪ ⎪ ⎨ −ωc ⎪ ⎪ ⎪Arccot uk uk ⎩ if ω > 0, k k vk vk −ωck 3.6 −π if ω < 0 In the next statement, we use the relationship between the first phase ψ and the second phase of the basis x , y of symplectic system S and the fact that the behavior of the u v first and second phases of system S plays the crucial role in counting generalized zeros of solutions of symplectic system S and of its reciprocal system Sr Theorem 3.4 If system S with the sequences bk / and ck / which not change their sign has a solution with two consecutive generalized zeros in l − 1, l , and let m − 1, m , l < m, l, m ∈ , then its reciprocal system Sr is either conjugate in l − 1, m with a solution having a generalized zero in l − 1, l or m − 1, m , or there exists a solution of Sr with exactly one generalized zero in l, m Proof Let x be the solution of S having consecutive generalized zeros in l − 1, l and u m − 1, m and y be a solution which together with x form the basis of the solution space u v of S Denote by ψ and the first and second phase of this basis Then, by Lemma 2.5, xk hk cos ψk , uk gk cos ψk − ω sin ψk , hk yk hk sin ψk , vk gk sin ψk ω cos ψk , hk 3.7 and by Lemma 3.2, uk hk cos k, vk hk sin k 3.8 Advances in Difference Equations 13 Hence, hk cos hk sin gk cos ψk − k ω sin ψk , hk gk sin ψk k ω cos ψk hk 3.9 Multiplying the first equation by − sin ψk , the second one by cos ψk , and adding the resulting equations, we obtain hk sin k ω hk − ψk 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 in 2.3 , 3.4 imply that h and h have constant sign as well and by 3.10 the same holds for the sequence sin k − ψk Suppose, to be specific, that sin k − ψk < if this sequence is positive, the proof is similar then there exists an odd integer n such that nπ < k − ψk < n π 3.11 Recall that by Definition 2.1, the first phase ψ and the second phase are defined as the monotone sequences on In addition, by Lemma 3.2, the Casoratian ω of S equals to the Casoratian ω of Sr , and thus, again by Definition 2.1, both phases ψ and of S are either nondecreasing or nonincreasing Moreover, if ω ω / 0, bk / and ck / 0, k ∈ , then by Theorem 2.6 and Corollary 3.3, Δψk / and Δ k / for k ∈ Suppose that the first phase ψk of the basis x , y of S given by Definition 2.1 is u v increasing; that is, for every integer k, we have Δψk ∈ 0, π If we suppose a decreasing sequence ψ, the proof is analogous Since the phases are determined up to mod π, without loss of generality, we may suppose that n −1 in 3.11 , that is, < ψk − k < π 3.12 Moreover, we can also suppose that ψl−1 ∈ −π/2, π/2 Since x has consecutive generalized zeros in l − 1, l and m − 1, m , we have ψl ∈ π 3π , , 2 ψj ∈ π 3π , , 2 j l 1, , m − 1, ψm ∈ 3π 5π , , 2 3.13 that is, ψk skips π/2 between l − and l and 3π/2 between m − and m and stays in the strip π/2, 3π/2 between l and m Formula 3.12 admits the following behavior of the sequence to draw a picture may help to visualize the situation 14 Advances in Difference Equations i i l−1 < −π/2, π π , ∈ − , 2 i l ∈ −π/2, π/2 , there exists r, l < r < m, such that 1, , r − 1, l ii The sequence iii We have as in i l−1 j ∈ π 3π , , 2 r, , m − 1, j has the same behavior as in i up to m, where ∈ −π/2, π/2 , m m 3π 5π , 2 3.14 ∈ < 3π/2 l ∈ −π/2, π/2 and for k > l the sequence j ∈ behaves iv We have l−1 π π ∈ − , , 2 l ∈ π 3π , , 2 π 3π , , 2 j l 1, , m − 1, m ≥ 3π 3.15 v Finally, l−1 π π ∈ − , , 2 j π π ∈ − , , 2 l, , r − 1, j 3.16 r π 3π , , ∈ 2 j ∈ π 3π , , 2 j r 1, , m The cases i – iv correspond to conjugacy of Sr , while the last case corresponds to the existence of a solution with exactly one generalized zero in l, m − A Conjugacy Criterion In this section, we establish a conjugacy criterion for system S by means of the first phase ψ and the associated Riccati equation The conjugacy of S in M, N means that there exists a solution of S with at least two generalized zeros in M−1, N , that is, there exists a solution x and two intervals l−1, l , u m, m , where M ≤ l < m ≤ N, such that xl−1 / and either xl xl−1 bl−1 < or xl 0, and xm / and either xm xm bm < or xm Conversely, we say that system S is disconjugate in M, N if every solution of S has at most one generalized zero in M − 1, N Theorem A see 9, Chapter If xk , xk / 0, is a solution of S on the interval 0, N , then uk the sequence wk uk /xk is a solution of the Riccati difference equation wk defined for k ∈ 0, N Also, if ak bk wk > for k ∈ 0, N x u ck ak dk w k , bk w k has no generalized zero in the interval 0, N 4.1 and bk > 0, then Advances in Difference Equations 15 Theorem B see 9, Theorem 5.30 , see also 13 Suppose that system S possesses a solution with no generalized zero in M, N Then, every nontrivial solution x of this system has at most u one generalized zero in this interval n i m In this section, as usual, we put · l i k if m > n and · if k > l Theorem 4.1 Let the sequence bk in S be positive Suppose that there exist positive real numbers δ1 and δ2 such that N Arccot Ak ≥ k 0 π , Arccot Bk ≥ k M 4.2 π , 4.3 where M ≤ −1 and N ≥ are arbitrary fixed integers, ⎡ Ak ⎛ ⎣ δ bk bk ⎝δ1 ⎛ ⎣ δ bk ⎞⎤ Fj ⎠⎦ j ⎡ Bk k−1 bk ⎝δ2 Fj ⎠⎦ j k bj −1 Fi δ1 , i ⎡ ⎛ ⎣1 bj ⎝δ2 −1 ⎞⎤2 4.4 Fi ⎠⎦ , i j j k ak − bk Fk j ⎞⎤ −1 j−1 k−1 dk − bk 4.5 Then, system S is conjugate in M, N Proof In the first part of the proof, we show that the solution x0 has a generalized zero in 0, N given by the condition Let y0 Since xk ak xk k 0, and hence bk uk and yk u0 1, ak yk 1, y v x1 x u of S given by the condition 4.6 be another linearly independent solution of S y1 δ b0 4.7 bk vk for every integer k, this holds especially for x1 − a0 x0 b0 1 − a0 , b0 4.8 v0 y1 − a0 y0 b0 1 − a0 b0 δ1 16 Advances in Difference Equations The Casoratian ω satisfies x0 v0 − y0 u0 ω δ1 > 4.9 Suppose, by contradiction, that x has no generalized zero in 0, N , that is, due u to the fact that x0 and bk > 0, we have xk > for every k 1, 2, , N Then, by Theorem B, we get yk > xk , 4.10 for k 1, , N 1, because otherwise the solution x − y has generalized zeros at k u v and in the interval m, m , m being the integer where 4.10 is violated Let ψ be the first phase of solutions x and y , that is, by Definition 2.1, u v ψk arctan yk , xk Δψk ∈ 0, π 4.11 By Theorem 2.6, we have Δψk taking account that ψ0 k−1 ψk Δψj xk xk yk yk 1 , δ bk π/4 and using 4.10 , we get for k k−1 ψ0 Arccot j Let wk Arccot xj xj j yj yj δ bj 1 π > 4.12 1, , N k−1 Arccot j 2yj yj δ bj π 4.13 vk /yk Then, from the first equation in S yk − ak , yk bk wk and w is a solution of the Riccati equation 4.1 Denote wk wk ak − bk ck ak ak − bk bk ck ak − bk −1 ak − bk dk − bk 4.14 wk ak − /bk Then, dk wk − ak − /bk bk wk − ak − /bk bk dk wk − ak dk bk bk w k dk dk bk w k − bk w k bk bk w k wk bk w k bk w k 4.15 Advances in Difference Equations ak −1 /bk Further denote Fk 17 dk −1 /bk Then, since bk wk Δ wk Fk − bk w k bk wk ak yk /yk > 0, ≤ Fk , bk w k 4.16 which means that k−1 wk ≤ w0 k−1 Fj δ1 Fj j 4.17 j Hence, 4.14 implies k−1 k−1 yk w j bj y0 bj w j aj j , 4.18 j and using 4.17 , yk ≤ j−1 k−1 bj Fi δ1 j 4.19 i Now, 2yk yk δ bk ≤ δ bk j−1 k−1 bj δ1 i j ⎡ ⎣ δ bk ⎛ bk ⎝δ1 j−1 k Fi k−1 bj δ1 ⎞⎤ Fj ⎠⎦ j Fi i j j−1 k−1 bj δ1 j Fi 4.20 i Ak Let k N in 4.13 Then, together with assumption 4.2 , N ψN > Arccot Ak k π π ≥ 4.21 On the other hand, since x has no generalized zero in 0, N , it follows that ψk < π/2 u for every k 0, , N 1, a contradiction with 4.21 It means that the solution x has a u generalized zero in 0, N In the second part of the proof, we show that the solution x of S given by condition u d −b , we have x dk xk −bk uk 4.6 has also a generalized zero in M−1, Since S−1 k −c a and uk −ck xk ak uk , in particular, x−1 d−1 x0 − b−1 u0 d−1 − b−1 − a0 b0 4.22 18 Let Advances in Difference Equations y v be another linearly independent solution of S given by the condition y0 1, y−1 d−1 − δ2 b−1 b−1 − a0 , b0 4.23 with the corresponding second component v0 expressed by d−1 y − y −1 b−1 v0 The Casoratian ω of x u , y v 1 − a0 − δ2 b0 4.24 −δ2 < 4.25 satisfies ω x0 v0 − y0 u0 Suppose, by contradiction, that the solution x has no generalized zero in the interval M−1, u , that is, xk > for k M, , Then, by Theorem B using the same argument as in the first part of the proof yk > xk , for every k M, , −1 Let ψ be the first phase of By Definition 2.1, ψk arctan yk , xk 4.26 x u and y v with the Casoratian ω < Δψ k ∈ −π, , 4.27 y k yk −δ2 bk 4.28 and by Theorem 2.6 Δψ k xk xk 1 − π π/4, 4.26 , and that the function Arccot · − π/2 is odd, we Taking into account that ψ get for every k M, , −1 −ψ k Arccot −1 ψ0 Δψ j j k −1 Arccot j k xj xj yj y j −δ2 bj − π π − 2 4.29 − −1 Arccot j k xj xj yj yj δ bj −1 π Arccot 2yj y j δ bj j k 4.30 Let us estimate the term 2yk yk /bk by means of the Riccati equation Let wk y k Then, y k dk y k − bk v k , that is, yk yk d k − bk w k , vk / 4.31 and from the backward Riccati equation for w which follows from 4.1 , −ck ak wk d k − bk w k wk 4.32 wk − dk−1 − /bk−1 Then, substituting into 4.32 , we have no index mean the Put wk index k here and also in later computations w dk−1 − bk−1 d − /b d − /b −c a wk d − b wk 1/b − a/b bwk − b wk a−1 b awk −1 − a b −cb ad − a /b − b wk 1 1/b − wk wk − b wk 1 4.33 wk , − b wk and hence, wk − wk − bk w k ak − dk−1 − − bk bk−1 1 − bk w k 4.34 Since, − b wk 1 − b wk − c dw d−b a bw d−1 b d − bwk ad − cb a bw 4.35 a bw yk > 0, yk we have −Δwk − ≥ Fk−1 , where Fk is given by 4.5 Summing the last inequality from k −1, we obtain −wk w0 ≤ to −1 Fj−1 j k 4.36 20 Advances in Difference Equations and this means that ⎛ − bk w k bk ⎝−w0 ≤1 ⎞ −1 Fj−1 ⎠ 4.37 j k From 4.31 , yk yk 1 − bk w k hence y k dk − bk 1 − bk w k , 4.38 y k , that is, from 4.37 , −1 yk d k − bk w k y0 − bj w j ≤ j k ⎡ −1 ⎛ bj ⎝−w0 ⎣1 −1 ⎞⎤ Fi−1 ⎠⎦ 4.39 i j j k Finally, w0 w0 − d−1 − b−1 −δ2 − a0 d−1 − − b0 b−1 −δ2 − F−1 , 4.40 this implies yk ≤ −1 ⎡ ⎛ ⎣1 bj ⎝δ2 −1 ⎞⎤ Fi ⎠⎦ 4.41 i j j k Substituting from 4.30 , ψM > π ⎡ −1 Arccot k M ⎣ δ bk ⎛ bk ⎝δ2 −1 ⎞⎤ Fj ⎠⎦ j k −1 ⎡ ⎛ ⎣1 bj ⎝δ2 j k −1 ⎞⎤2 Fi ⎠⎦ , 4.42 i j and hence −1 ψM > k M Arccot Bk π π ≥ 4.43 On the other hand, since we suppose that x has no generalized zero in M − 1, , it holds u ψ M < π/2, a contradiction with 4.43 Summarizing, we have proved that the solution x has at least one generalized zero u in M − 1, and one generalized zero in 0, N The proof is complete Advances in Difference Equations 21 Remark 4.2 The conjugacy criterion for the Sturm-Liouville equation Δ rk Δxk qk xk 0, rk > 0, 4.44 formulated in 1, Theorem is the corollary of the above criterion for ak 1, bk 1/rk , ck −qk and dk − qk /rk Theorem 4.1 also extends the results proved in 1, 2, 12, 14 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 , y ∈ Ê2 , k ∈ , are sequences such that the Casoratian ω u v k for any k ∈ Then, these sequences form a normalized basis of symplectic system S det xk yk uk v with xk vk − yk uk , ak −xk yk bk ck Moreover, if bk / for k ∈ xk lim x u vk xk , k → ±∞ yk and 5.1 uk vk − vk uk , −uk yk dk yk xk , 0, 5.2 has m − generalized zeros in , then the first phase determined by lim ψk − lim ψk k→∞ k → −∞ Arccot xk xk k∈ ,bk / yk yk bk x u mπ , y v satisfies 5.3 Proof Let x and y be sequences with the Casoratian equal to 1, and let a, b, c, and d be u v given by 5.1 Then, by the Cramer rule, we obtain xk bk u k , ck xk dk uk , yk ak yk bk vk , vk y v ak xk uk so x and u ak dk − bk ck 1 ck yk dk vk , 5.4 are solutions of S with a, b, c, d given by 5.1 It is easy to verify that holds for any integer k; that is, system S is symplectic Now, suppose that 22 Advances in Difference Equations assumptions of the second part of the theorem are satisfied Then, S is nonoscillatory in and x is its recessive solution both in ∞ and −∞ Since u Δ Δyk xk − Δxk yk xk xk yk xk yk xk − xk yk xk xk bk vk xk − ak xk xk xk ak yk 5.5 bk uk yk bk xk xk > 0, for large and small k, the limits limk → ±∞ yk /xk exist and by 5.2 limk → ∞ yk /xk ∞ limk → −∞ yk /xk −∞ It follows, by the definition of the first phase, that limits limk → ∞ ψk > limk → −∞ ψk also exist and equal to odd multiples of π/2 This, coupled with the fact that x u has exactly m − generalized zeros in ; that is, ψk equals m − times an odd multiple of π/2 or skips over this multiple, gives 5.3 We finish the paper with an example illustrating the previous theorem x u Example 5.2 Consider a pair of two-dimensional sequences n k yk n−1 k xk , n−1 uk , vk y v , k 3/2 n n k 3/2 n−1 with , 5.6 , where n ≥ and k α n : k α · · · k α − n By a direct computation, one can verify that xk vk − yk uk and that 5.1 read ak k bk ck k ··· k −n k−n , 5.7 −n n − k 3/2 · · · k − n 5/2 k k dk Obviously, limk → ±∞ xk /yk since xk xk bk 2 1, 7/2 k−n 5/2 , 5/2 k 3/2 − n n − k 5/2 k 3/2 0, so the assumptions of the previous theorem are satisfied, and k 2 k ··· k − n k−n 2 > 0, 5.8 Advances in Difference Equations 23 for k ∈ , the solution x has no generalized zero in u again verified by a direct computation Arccot k2 k∈ − 2n k Consequently, 5.3 reads as can be n2 − 4n 19 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 ˇ Z Doˇ l´ and S Pechancov´ , “Conjugacy and phases for second order linear difference equation,” sa a Computers & Mathematics with Applications, vol 53, no 7, pp 1129–1139, 2007 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 ˇ S Pechancov´ , Phases and oscillation theory of second order difference equations, Ph.D thesis, Masaryk a University, Brno, Czech Republic, 2007 ˇ S Ryz´, “On the first and second phases of 2×2 symplectic difference systems,” Studies of the University ı ˇ of Zilina Mathematical Series, vol 17, no 1, pp 129–136, 2003 O Boruvka, Linear Differential Transformations of the Second Order, The English Universities Press, ˚ London, UK, 1971 F Neuman, Global Properties of Linear Ordinary Differential Equations, vol 52 of Mathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Nethrlands, 1991 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 O Doˇ ly and P Hasil, “Critical higher order Sturm-Liouville difference operators,” to appear in s ´ Journal of Difference Equations and Applications C D Ahlbrandt and A C Peterson, Discrete Hamiltonian Systems Difference Equations, vol 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic Publishers, Dordrecht, The Nethrlands, 1996 10 M Bohner, O Doˇ ly, and W Kratz, “A Sturmian theorem for recessive solutions of linear Hamiltonian s ´ difference systems,” Applied Mathematics Letters, vol 12, no 2, pp 101–106, 1999 11 M Bohner and O Doˇ ly, “Trigonometric transformations of symplectic difference systems,” Journal s ´ of Differential Equations, vol 163, no 1, pp 113–129, 2000 ˇ 12 Z Doˇ l´ and D Skrab´ kov´ , “Phases of linear difference equations and symplectic systems,” sa a a Mathematica Bohemica, vol 128, no 3, pp 293–308, 2003 13 M Bohner, O Doˇ ly, and W Kratz, “Sturmian and spectral theory for discrete symplectic systems,” s ´ Transactions of the American Mathematical Society, vol 361, no 6, pp 3109–3123, 2009 ˇ a 14 O Doˇ ly and P Reh´ k, “Conjugacy criteria for second-order linear difference equations,” Archivum s ´ Mathematicum (Brno), vol 34, no 2, pp 301–310, 1998  ... xk 1 bk g k ak yk yk yk 1 ak h2 k ωbk bk xk uk yk vk 2. 28 bk vk , ak h2 k bk xk uk yk vk , 2. 29 and this gives, together with 2. 26 and 2. 27 , the conclusion 2. 20 We continue in this section... multiple of π ⎩ k if uk / 0, if uk 3 .2 0, ∈ −π, if ω < The proofs of the next statement and of its corollary are the same as those of Lemma 2. 2 and Theorem 2. 6, respectively v u Lemma 3 .2 Let x and. .. Acknowledgments Research supported by the Grants nos 20 1/09/J009 and P201/10/10 32 of the Czech Grant Agency and by the Research Project no MSM0 021 622 409 of the Ministry of Education of the Czech