Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 571935, 23 pages doi:10.1155/2011/571935

Research Article

Generalized Zeros of 2 x 2 Symplectic Difference System and of Its Reciprocal System

Ondrej Dosiy1 and Sarka Pechancova2

1 Department of Mathematics and Statistics, Masaryk University, Kotlafska 2, 611 37 Brno, Czech Republic

2 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Zizkova 17, 602 00 Brno, Czech Republic

Correspondence should be addressed to Ondrej Dosly, dosly@math.muni.cz

Received 1 November 2010; Accepted 3 January 2011

Academic Editor: R. L. Pouso

Copyright © 2011 O. Dosly and S. Pechancova. 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 x 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 x 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 x 2 symplectic difference system

'xfc+A /xk\

) = Ski ), k eZ, (S)

«k+1/ \Uk/

where Sk = (^di^) with real-valued sequences a, b, c, and d is such that detSk = akdk -bkCk = 1 for every k eh. Recall that under this condition, the matrix S is symplectic. Generally, a 2n x 2n matrix S is symplectic if

1 being the n x n identity matrix, and this conditions reduces just to the condition det S = 1 for

2 x 2 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 Sturm-Liouville difference equation, and we describe how to construct a 2 x 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 in [3, Section 4].

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 4. 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 (V) of (S) with the Casoratian w

w = xkvk - vkuk = const / 0 (1.2)

is said to be a basis of the system (S). If w = 1, it is said to be a normalized basis. An interval (m, m + 1], m eTL,is said to contain a generalized zero of a solution (X) of (S), if xm / 0 and

Xm+i = 0 or bmXmXm+1 < 0. (1.3)

A solution (X) of (S) is said to be oscillatory in Z if it has infinitely many generalized zeros in Z. In the opposite case, we say that («) is nonoscillatory in Z. System (S) is said to be nonoscillatory (offinite type) in Z if every solution of (S) is nonoscillatory in Z. A nonoscillatory system (S) is said to be 1-general in Z if it possesses two linearly independent solutions with no generalized zero, and it is said to be 1-special in Z if there is exactly one (up to the linear dependence) solution of (S) without any generalized zero in Z. 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] n Z, M,N e Z, N > M), if there exists a solution of (S) which has at least two generalized zeros in (M - 1,N + 1].

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 := rfcXfc+1 + qkXk + n^X^ = 0, (1.4)

see [7] and also [8].

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[+] = («m) with the property that limk/Xk = 0 for any solution z = (U) linearly independent of z[+]. The solution z[+] is said to be recessive at to. The recessive solution z[-] at

—<x> is defined analogously. System (S) is 1-special, respectively, 1-general if the recessive solutions z[+],z[-] have no generalized zero in Z and are linearly dependent, repectively, linearly independent. For more details concerning recessive solutions of discrete systems, we refer to [9,10].

2. Phases and Their Properties

Definition 2.1. Let (Xk) and (ykk), k e Z, form a basis of (S) with the Casoratian w. By the first phase of this basis, we understand any real-valued sequence f = (fk), k e Z, such that

fk = -

arctan — if xk f 0,

Xk (2.1) n

odd multiple of — if xk = 0,

with Afk e [0,n) if w > 0 and Afk e (-n, 0] if w < 0.

Here, by arctan, we mean a particular value of the multivalued function which is inverse to the function tangent. By the requirement Af e [0,n), respectively, Af e (-n, 0], a first phase of (X), (V) is determined uniquely up to mod n.

The first phase (and the later introduced second phase) are sometimes called zero-counting sequences, since each jump of their value over an odd multiple of n/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 (V) form a basis of (S) with the Casoratian w. Then, there exist sequences h and g, hk / 0, such that the transformation

xA / sk\

) = Rk( ), (2.2)

Uk/ \Ck/

Rk = ( hkk w/hktransforms system (S) into the so-called trigonometric system

sk+1 sk

) = Tk( ), (T)

Ck+1 Ck

where T is a symplectic matrix of the form Tk = ( -k pk) with

akhk + bkgk wbk

Vk =-j-—, qk = -7—,-• (2.3)

hk+1 hkhk+1

Sequences h, g are given by

2 2 2 Xku)

¡1 = V 4- 1/ C r. = -

hk = 4 + yl # =-JT-• (2.4)

The values of the sequence h can be chosen in such a way that wqk > 0. In particular, if bk f 0, then hk can be chosen in such a way that wqk > 0 for k e Z.

Proof. A similar statement is proved for general 2n x 2n symplectic systems in [11]. However, in contrast to [11], our transformation matrix contains the Casoratian w, and the proof for scalar 2 x 2 systems can be simplified.

Transformation (2.2) transforms the symplectic system (S) into the system

Sfc+l\ / Sk X

) = Tk( ), Tk = R-+1 Sk Rk, (2.5)

Ck+l/ \Ck

where T =: ( a 7 ) with

akhk + gkbk r wbk ctk =-;-, bk =

hk+i ' hkhk+/

Ck = — [-gk+i(akhk + bkgk) + hk+1(ckhk + dkgk)], (2.6)

7 -bkgk+1 + dkhk+i dk =-;-,

as can be verified by a direct computation. Then

detrfc = detffc-^Skfck) = det detSk detRk = - • 1 • to = 1, (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 x 2 systems where a matrix is symplectic if and only if its determinant equals 1. We have (no index means index k)

hhk+1c = — [-h(xk+iuk+i + yk+i^k+i) (ah + bg) + hh2k+l (ch + dg)] 1

= — [-(xk+i^k+i + yk+i^k+i ){ah2 + b(xu + yv+h2k+1 (ch2 + d(xu + yv)^ J 1

= — I - (xk+1uk+1 + yk+1vk+i)(x(ax + bu) + y (ay + br;))

+hk+1( x(c^ + du) + y(cy + dr))|

Advances in Difference Equations 5

= ^ [-(xfc+iWfc+1 + yk+Wk+l) (xXfc+1 + yi/fc+l) + (x2+1 + (xMfc+1 + 1/i'fc+l) j

= — [-XXfc+ll/fc+lt'fc+l - yXfc+ll/fc+lMfc+1 + yx£+1t>fc+l +Xy^.+ 1Mfc+lj 1

= — [xyfc+l(-Xfc+lt'fc+l +yfc+l«fc+l) +yXfc+l(-yfc+lMfc+l + Xfc+lt'fc+l)] = -xyfc+1 + yxk+1 = -x(ay + bu) + y(ax + bu) = -wb.

Hence, a = -(wb/hhk+1) = -a =: q. Similarly, a- d = —— [ah2 + hgb + bgk+ihk+i - dh\+jj

\ax2 + ay2 + b{xu + yv) + b(xk+1uk+1 + yk+1vk+^ - dx2k+1 - dy^j [x(ax + bu) + y(ay + bv) + xk+1(buk+1 - dxk+1) + yk+1 (bvk+1 - dyk+^] [xxk+i + yyk+i - xk+ix - yk+iy] = 0,

hhk+1 1

hhk+1 1

in the last line of this computation, we have used the fact that S 1 = (dc ~b), that is, xk = dkxk+i - bkuk+i, yk = dkyk+i - bkVk+i. Hence, a = d =: p.

Finally, concerning positivity of wq if b = 0, we fix the sign of h in a particular index,

say ho = \Jx~l + y2 and the formula coq = co2b/hhk+\ shows that the sign of /:, that is, h = ±\Jx2 + y2, at indices kf 0 can be "adjusted" in such a way that ioqk > 0 if bk f0. □

Remark 2.3. Transformation (2.2) preserves oscillatory properties of transformed systems in the following sense. If bkxkxk+1 < 0, that is, bkhkskhk+1sk+1 < 0, then since sgn(bkhkhk+1) = sgn(wqk), we have (using the positivity of the term wqk) bkhkskhk+1 sk+1 < 0 if and only if sksk+1 < 0. Note also that xk+1 = 0 if and only if sk+1 = 0, since hk/ 0 for all k.

Lemma 2.4 (see [12, Lemma 1]). Let (T) be the trigonometric system. There exists the unique (up to mod 2n) sequence yk e [0,2n) such that

sin yk = qk, cos ^ = Pk (2.10)

and the general solution (s) of (T) takes the form

sA /sin(£k + a)\

= n „ J, (2.11)

Ck J \cos(lk + a)

where k eh, a,p e R and £ is any sequence such that A£k = yk.

Lemma 2.5. Let (Urn ) and ) be a basis of system (S) with the Casoratian w, and let (T) be the trigonometric system associated to (S) as formulated in Lemma 2.2. Then, there exists a solution (sc) of (T) such that

Ck -Sk

hk 0 W

(2.12)

where k e Z and h, g are given by (2.4). Further, there exists a sequence £ such that

Sk = sin Ik, Ck = cos Ik,

(2.13)

Ai,k = yk, where the sequence y is given by (2.10) and yk e [0,2n) for every k e Proof. By Lemma 2.2, there exist solutions Ç ^ ^, i = 1,2, of (T) such that

^rV&V!)-

(2.14)

that is,

s[i] = h-1x[i],

c[i] =

-gx[i] + hu[i]

(2.15)

By a direct computation, we have

s[1]c[2] - c[1] s[2] = 1

(2.16)

and after a few steps

(s^)2 + (cH)2

(2.17)

By Lemma 2.4, there exist real constants a[i], ^[t] such that

slk\ . /sin( Ik + a[i]V

scos( Ik + a[i])/

(2.18)

where £ is an arbitrary sequence such that A£k = yk and y is given by (2.10). By (2.17), we have ¡3[i] = 1, and by (2.16), we obtain

s[k]c[k] - ck1] s[2 = sin £ + a[1]) cos (£k + a[2]) - sin (£k + a[2]) cos(£k + a[1])

(2.19)

= sin(a[1] - a[2]^ = 1,

that is, a[1] - a[2] = (n/2)(mod2n). Hence, s[1] = c[2] and c[1] = -s[2] what implies (2.12). Since (£k) was an arbitrary sequence such that A£k = yk, changing £k to £k - a[2], 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 (-n/2,n/2) and (0,n), respectively.

Theorem 2.6. Let z[1] = (u) and z[2] = (V) form a basis of (S) with the Casoratian w, and let f be a first phase of this basis. If bk / 0, then

xkxk+1 + ykyk+1 .r „

Arccot-r^^--if CO > 0,

xkxk+1 + ykyk+1 .r „

Arccot---jt if co < 0.

(2.20)

If bk = 0, then Afk = 0.

Proof. Let (T) be a trigonometric system associated to (S) with the basis z[1],z[2] and with p, q satisfying (2.3). Let f be a first phase of this basis. By Lemma 2.5, there exists a solution (sc) of (T) such that sk = sin £k, ck = cos £k and z[1] = (x), z[2] = (y) satisfy

xk = hk cos £k, yk = hk sin £k, (2.21)

where h is given by (2.4), A£k = yk and yk is given by (2.10). Hence, for xk / 0,

tan £k = — (2.22)

and if xk = 0, then £k is equal to an odd multiple of n/2. On the other hand, by Definition 2.1

for xk =/ 0

tanqik = ^, (2.23)

and if xk = 0, then fk is equal to an odd multiple of n/2. Consequently,

fk = £k(mod n),

(2.24)

and it implies (since the additive multiple of n to get equality in (2.24) is independent of k)

Afk = A£k. (2.25)

For w > 0, we defined in Definition 2.1 that Afk e [0,n). Suppose that bk / 0. According to Lemma 2.2, we can choose qk > 0, and then by Lemma 2.4, we can take yk e (0,n). Using (2.25), we have yk = Afk, and thus cot Afk = pk/qk, and hence

AykArccot = —. (2.26)

Let w < 0. Then, we defined Afk e (-n, 0] and based on Lemma 2.2, under the assumption bk / 0, we can choose qk < 0 and then yk e (n, 2n) defined in (2.10). Using (2.25), we have yk = Afk + 2n, and consequently cot(Afk + 2n) = cot Afk = pk/qk and

Auik = Arccot-— jt, (2 27)

in this case. Finally, if bk = 0, then qk = 0, and we put yk = 0. Hence, by (2.25), Afk = ^k(mod n) and since by Definition 2.1 Afk e (-n,n), then we have Afk = 0. Summarizing, by a direct computation

^ = (akhk + bkgk) = [afc/4 + bk(xkuk + ykvk)^. (2.28)

Since xk+1 = akxk + bkuk and yk+1 = akyk + bkvk,

XkXk+i + ykyk+1 = ah + bk (xkUk + yv), (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 f be the first phase of (S) determined by the basis (U), (V). Then, (U) has a generalized zero in (k, k + 1) if and only if y skips over an odd multiple of n/2 between k and k + 1.

Proof. Suppose that (U) has a generalized zero in (k,k + 1), that is, xkxk+ibk < 0. By Lemma 2.5 xk = hkck, yk = hksk, where (-s) is a solution of trigonometric (T) with wqk > 0 (Lemma 2.2). Suppose that w > 0, that is, Afk e (0,n) (for w < 0 the proof is analogical). Then, (-s) has a generalized zero in (k,k + 1) which means that ck and ck+1 have different sign. Since ck = cos £k, ck+1 = cos £k+1, where £ is a sequence with A£k = Afk (compare (2.25)), £k and £k+1 lay in different intervals whose endpoints form odd multiples of n/2. Conversely, if f skips an odd multiple of n/2 between k and k +1, £ does also, and reasoning in the same way as above, we see that (ft) has a generalized zero in (k,k + 1). □

Remark2.8. A slightly modified statement we have in the case when (U) has a zero at k+1, that is, xk = 0andxk+i = 0. More precisely, by the definition of the first phase fk+i = (2m + 1)(n/2) for some integer m, and, if f is increasing, then fk e ((2m - 1)(n/2), (2m + 1)(n/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+2 = xk+1 + xk, k eTL, (2.30)

that is,

A ((-1)fc Axfc) + (-1)fcxfc+i = 0, (2.31)

which can be viewed as symplectic system (S) with the matrix

/ 1 HA

Sk = .. , (2.32)

\(-1)k+! 0 )

that is, the entry corresponding to bk changes its sign. A basis (U), (V) of (S) corresponding to (2.31) has the first components given by

* = (133)

with the positive Casoratian to = V5. By Definition 2.1, Aq>k e [0,.t) and

/l+v^\fc k

Arctan -- + —jt, k even,

\l-vV 2

Vk = i (2.34)

a , fc+1

Arctan ---+—-—jt, k odd.

\l-V5j 2

Notice that every jump of the value fk over an odd multiple of n/2 corresponds to a generalized zero of x in (k,k + 1). A corresponding trigonometric system (T) to symplectic system (S) has by Lemma 2.5 two linearly independent solutions (-s) and (S), where the sequences c and s are given by (2.13). Since by (2.24) £k = fk + mn for some m eTL ,the first components of a basis of (T) can be (up to the sign) uniquely determined by

ck = cos fk, sk = sin fk. (2.35)

It means that the components x, respectively, y of solutions of (S) have generalized zeros in (k,k + 1] if and only if components c, respectively, s of solutions of (T) have a generalized

zero in (k, k + 1]. By (2.21), together with (2.24), we express the first components of the basis of (S) as

xk = hk cos fk, yk = hk sin fk. (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 wqk (i.e., bkhkhk+1) 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 f and corresponding trigonometric sequences c and s in case when w < 0. Consider (2.31), that is, the corresponding symplectic system with the basis (x), (V) having the first components

1 + v^

(2.38)

with the negative Casoratian to = -V5. By Definition 2.1,

Arctan

Arctan

\l + vV

/l-v^V k-1

k even,

k odd.

(2.39)

The first components of a basis of the trigonometric system (T) corresponding to (S) is of the form

ck = cos fk, sk = sin fk. (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 3. Reciprocal System

A reciprocal system to (S) is the symplectic system

\Uk+i/ \UkJ

, /dk -cA /0 1\

Sk=JSkJ-1 = (b J J =U o)■ (31)

related to (S) by the substitution (|) = From definition of the symplectic system (S)

and its reciprocal system (Sr), it follows that if (X) is a solution of (S), then (-Ux) is a solution of its reciprocal system (Sr).

Definition 3.1. By the second phase of the basis (X), (V) of system (S), we understand any first phase of the basis (-X), (-y) of its reciprocal system (Sr), that is, any real-valued sequence q = (Qk), k eh, such that

arctan — if iik f 0,

Qk = \ Uk n (3.2)

odd multiple of — if Uk = 0,

with AQk e [0,n) if w> 0 and AQk e (-n, 0] if w < 0.

The proofs of the next statement and of its corollary are the same as those of Lemma 2.2 and Theorem 2.6, respectively.

Lemma 3.2. Let (U) and (V) form a basis of (S) with the Casoratian w, that is, (-X) and (-y) is a basis of (Sr) with the Casoratian a' = to = -iikyk + xkrk. Then, there exist sequences h and ~g, hk f 0 for k eh ,such that the transformation

transforms system (Sr) into the trigonometric system

Vfc+1/ \-qk Vk/ Vk

which is symplectic with the sequences p, q given by

dkhk - ckgk hk+i

cktv hkhk+i

hk = U\ + Pfc, gk = ~

xkUk + ykVk hk

Moreover, transformation (3.3) preserves oscillatory properties of (Sr), and the sequence (hk), k eh, can be chosen in such a way that toqk > 0 and if Ck f 0 in such a way that toqk > 0.

Corollary 3.3. Let (U) and (V) form a basis of (S) with the Casoratian w; that is, ( ux) and (-y) form the basis of (Sr) with the same Casoratian w. Let (gk) be the second phase of the basis (U), (V) of (S). If ck / 0, k eZ, then

A<?k =

Arccot

UkUk+l + VkVk+l -wck

if w > 0,

UkUk+1 + VkVk+1 ,r „

Arccot--jt if to < 0.

If ck = 0, then Agk = 0.

In the next statement, we use the relationship between the first phase f and the second phase q of the basis (x), (V) of symplectic system (S) and the fact that the behavior of the 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 = 0 and ck / 0 which do 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 e Z, 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 (m - 1, m] and (V) be a solution which together with (U) form the basis of the solution space of (S). Denote by f and q the first and second phase of this basis. Then, by Lemma 2.5,

xk = hk cos qik, uk = gk cos qik ~ sin qsk,

Vk = hk sin qsk, vk = gk sin qik + t cos

and by Lemma 3.2,

ilk = hk cos Qk, Vk = hk sin Qk.

Hence,

hk cos Çk = gk cos fk ~ sin fk,

hk sin Çk = gk sin fk + j- cos щ.

Multiplying the first equation by - sin fk, the second one by cos fk, and adding the resulting equations, we obtain

T . / \ W

hksm{Qk-([ik) =—. (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 - fk). Suppose, to be specific, that sin(^k - fk) < 0 (if this sequence is positive, the proof is similar) then there exists an odd integer n such that

nn < Çk - fk < (n + 1)n.

(3.11)

Recall that by Definition 2.1, the first phase f and the second phase q are defined as the monotone sequences on Z. In addition, by Lemma 3.2, the Casoratian w of (S) equals to the Casoratian to of (Sr), and thus, again by Definition 2.1, both phases f and q of (S) are either nondecreasing or nonincreasing. Moreover, if to = to f 0, bk f 0 and Ck f 0, k e Z, then by Theorem 2.6 and Corollary 3.3, Afk = 0 and AQk = 0 for k e%.

Suppose that the first phase fk of the basis (U), (V) of (S) given by Definition 2.1 is increasing; that is, for every integer k, we have Afk e (0,n). If we suppose a decreasing sequence f, the proof is analogous. Since the phases are determined up to mod n, without loss of generality, we may suppose that n = -1 in (3.11), that is,

0 <fk - Çk<n.

(3.12)

Moreover, we can also suppose that fl-1 e (-n/2,n/2). Since x has consecutive generalized zeros in (l - 1, l] and (m - 1,m], we have

n 3nx у,—

\ /ж 3ж\ . , „ )' l=l + 1.....m~1'

(3.13)

that is, fk skips n/2 between l -1 and l and 3n/2 between m - 1 and m and stays in the strip (n/2,3n/2) between l and m. Formula (3.12) admits the following behavior of the sequence ç (to draw a picture may help to visualize the situation).

(i) Ql-1 < -n/2, qi e [-n/2,n/2), there exists r, l <r <m, such that

Qi e (■

i = l + 1,...,r - 1, Qj e

j = r, ...,m - 1, Qm e

3 jt 5.tx

(3.14)

(ii) The sequence q has the same behavior as in (i) up to m, where Qm < 3n/2.

(iii) We have Ql-1 e [-n/2,n/2), qi e (-n/2,n/2) and for k> l the sequence q behaves as in (i).

(iv) We have

r n n\

n 3n ~2'~

/ n 3n\ , „ 9je\2'~)' -'=Z + 1.....m_1'

3jt ~2'

(3.15)

(v) Finally,

r n n\ / n n\

^e[-2>2)> ^H-2'2)' ? = Z.....''-1

2 2 j 2 2

(3.16)

n 3n\ f n 3n\ 2'-)' 2"' ]=r + 1.....

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 [I, m - 1]. □

4. A Conjugacy Criterion

In this section, we establish a conjugacy criterion for system (S) by means of the first phase f 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+1], that is, there exists a solution (U) and two intervals (l-1,l], (m,m + 1], where M < l < m < N, such that xl-1 = 0 and either x/x/-1b/-1 < 0 or xi = 0, and xm / 0 and either xmxm+1bm < 0or xm+1 = 0. 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 + 1].

Theorem A (see [9,Chapter 3]). If (Uk), xk / 0, is a solution of (S) on the interval [0,N + 1],then the sequence wk = uk/xk is a solution of the Riccati difference equation

ck + dkWk

«'fc+i =-;-, (4.1)

ak + bkWk

defined for k e [0,N]. Also, if (x) has no generalized zero in the interval [0,N +1 ] and bk> 0, then ak + bkwk > 0 for k e [0, N].

Theorem B (see [9, Theorem 5.30], see also [13]). Suppose that system (S) possesses a solution with no generalized zero in [M, N +1 ]. Then, every nontrivial solution (U) of this system has at most one generalized zero in this interval.

In this section, as usual, we put ^ ™=m(-) = 0 if m> n and n\=k (■) = 1 if k > I.

Theorem 4.1. Let the sequence bk in (S) be positive. Suppose that there exist positive real numbers 61 and 62 such that

^Arccot^fc > —

^ Arccot Bk >

where M <-1 and N > 1 are arbitrary fixed integers, 2

61 bk 2

( k-1 \

1 + bkl 61 ^ Fj

\ j=Q /

/ -1 \

1 + bkl 62 ^Fj

\ j=k /.

k1 j=Q

1 + bj[ 61 ^Fi

1 + bj ^62 + EFi j

ak+1 - 1 dk - 1

Then, system (S) is conjugate in [M, N].

Proof. In the first part of the proof, we show that the solution ( U ) of (S) given by the condition

xQ = 1, xi = 1 (4.6)

has a generalized zero in (Q,N + 1]. Let ( y ) be another linearly independent solution of (S) given by the condition

yQ = 1, y1 = 1 + 61 bQ.

Since xk+1 = akxk + bkuk and yk+1 = akyk + bkvk for every integer k, this holds especially for k = 0, and hence

Mo = t(X 1 - aoxo) = —(1 - UQ),

v0 = - floyo) = ^(1 - flo) +01-

The Casoratian w satisfies

w = xQvQ - yQuQ = 61 > Û. (4.9)

Suppose, by contradiction, that ( U ) has no generalized zero in (Û,N + 1], that is, due to the fact that xQ = 1 and bk > Û, we have xk> Û for every k = 1,2,...,N. Then, by Theorem B, we get

yk > xk, (4.1Û)

for k = 1,...,N + 1, because otherwise the solution ( X ) - ( V ) has generalized zeros at k = Û and in the interval (m, m + 1], m being the integer where (4.1Û) is violated.

Let y be the first phase of solutions ( U ) and ( yv ), that is, by Definition 2.1,

= arctan —, AG [0(4.11) xk

By Theorem 2.6, we have

XkXk+1 + ykyk+1

Acpk = Arccot- / —, (4.12)

taking account that = n/4 and using (4.10), we get for k = 1,...,N + 1

„ = SA« + „ = ^ArccotJ:F"' + y/y/" + f > EArccot?ffil + * (4.13)

j=0 j=0 01 bj 4 j=0 01 bj 4

Let wk = vk/yk. Then, from the first equation in (S)

and w is a solution of the Riccati equation (4.1). Denote wk = wk + (ak - 1) /bk. Then,

flfc+i ~ 1 ck + dk(wk - (ak-l)/bk) bk+1 ak + bk(zvk - (ak - 1 )/bk)

a-k+i ~ 1 + bkck + bkdkwk - akdk + dk bk+1 bk(bkwk + 1)

(4.15)

ak+1 - 1 -1 + dk(l + - +

wk+1 =

bk+1 bk (bkw k + 1)

ak+1 - 1 rffc - 1 +

bk+1 bk 1 + bkWk

Further denote Fk = (ak+1-1)/bk+1+(dk-1)/bk• Then, since 1+bkw5k = bkwk+ak = yk+1 /yk > 0,

which means that

Aidk = Fk -

1 + bkwk

k-l k-1 idk < ido + XFj = 61 + XFj.

j=0 j=o

(4.16)

(4.17)

Hence, (4.14) implies

yk = y„n C + j = Ft jj + ^ '

j=o j=o

(4.18)

and using (4.17),

1 + + X Fi

(4.19)

2ykVk+1 2

61^k 61^k

k-1 k1

1 + j 61 + £ F,

1 + bJ 61 + £ Fj

\ j=o /.

k-1 k1

1+b^61 + X Fi) 1 + b;(61 +X Fi)

(4.2o)

Let k = N + 1 in (4.13). Then, together with assumption (4.2),

EA ^ n n

Arccot*Ak + -j > -t.

(4.21)

On the other hand, since ( U ) has no generalized zero in (o,N + 1], it follows that fk < n/2 for every k = o,...,N + 1, a contradiction with (4.21). It means that the solution (U) has a generalized zero in (o,N + 1].

In the second part of the proof, we show that the solution ( X ) of (S) given by condition (4.6) has also a generalized zero in (M-1, o].SinceS-1 = (dc ~b) ,wehave xk = dkxk+1 -bkuk+1 and uk = -ckxk+1 + akuk+1, in particular,

x-i = d- 1X0 - b-iuo = d-1 —r—(1 - flo)-

18 Advances in Difference Equations

Let (f ) be another linearly independent solution of (S) given by the condition

y0 = 1, y_j = 62*7-1 +d-i - ^-(1 - a0), (4.23)

with the corresponding second component vq expressed by

1 _ _ 1

po = j^-(d-iyo-y-i) = ^(l-flo)-62. (4.24)

The Casoratian To of (ft), (ji^ satisfies

To = xovo - y0«o = -62 < 0. (4-25)

Suppose, by contradiction, that the solution (U) has no generalized zero in the interval (M-1, 0], that is, xk > 0 for k = M,...,0. Then, by Theorem B (using the same argument as in the first part of the proof)

yk > xk, (4.26)

for every k = M,... ,-1. Let (js be the first phase of (*) and with the Casoratian To < 0. By Definition 2.1,

7pk = arctan Ayk e (~jt,0], (4.27) xk

and by Theorem 2.6

AqJk = ArccotXkXk+1^k+1 - jr. (4.28)

-62 bk

Taking into account that q>0 = .t/4, (4.26), and that the function Arccot(-) - zr/2 is odd, we get for every k = M,...,-1

Arccot

Xj+lXj + y; + 1y;- Jf Jf

= - y Arccot

Xj+iXj + y-+1y • -1 2y-+1y •

--- < -¿jArccot-

Hence,

Vk> 4 +EArccot^-- (4-30)

j=k 2 j

Let us estimate the term (2yk+1yk)/bk by means of the Riccati equation. Let wk = vk/ yk. Then, yk = dkyk+1 - bkvk+1/ that is,

Jl*-=dk-bkwk+1, (4.31)

and from the backward Riccati equation for w (which follows from (4.1)),

— ~ck + akwk+1

wk = —-—=-. (4.32)

dk - bkWk+i

Put wk = wk - (dk-1 - 1) /bk—\. Then, substituting into (4.32), we have (no index mean the index k here and also in later computations)

d^1 - 1 -c + a{wk+1 + (d - 1)/b) awk+1 + (-cb + ad - a)/b

vu +-—-—-

bk-i d - b(wk+i + (d - 1)/b) 1 - bwk+i

_ l/b + (a/b)(bwk+1 -1) _ _a + l/b-Wfc+i + ^ ^

1 - bwk+1 b 1 - bwk+1

a - 1 w k+1 —--;- +

1 - bw k+1'

and hence,

w k - Tv k+1 = -

ak - 1 dk-1 - 1 bkwk+1

bk bk-1 1 - bkWk+1'

(4.34)

Since,

1 - bwk+1 = 1 - M Wic+1 - —— - ) = d - bwk+1

- d bC + _ cb _ 1 _ Vk > q

a + bzZJ a + bw a + yfc+1

(4.35)

we have -Awk- > Fk-1, where Fk is given by (4.5). Summing the last inequality from k + 1 to -1, we obtain

-wk+1 + wo < ^ Fj-1 (4.36)

and this means that

1 - bkwk+1 < 1 + bkl -w0 + ^ Fj-1 !.

\ j=k+i /

(4.37)

From (4.31),

= dk - bk (ivk+1 + —) = 1 - bkiok+i,

(4.38)

hence yk = (1 - bkic>k+i)yk+1, that is, from (4.37),

y^yon^-^+O ^n

j=k j=k

1 + bj{ -W0 + ^ Fi-1 ! \ i=j+! /

(4.39)

Finally,

_ - 1 c 1 - ao d-i - 1 WO = Wo---- = -C>2 + —------ = -02 ~ t-1,

b-1 b0 b-1

(4.40)

this implies

1 + bj 62 + Fi

(4.41)

Substituting from (4.30),

<Pm > f + I>rccot

1 + bk (62 + X Fj

1 + bj 62 + Fi

(4.42)

and hence

— ^ A — n n

Vm > Zj ArccotBfc + - > -.

k=M 4 2

(4.43)

On the other hand, since we suppose that (U) has no generalized zero in (M - 1,0], it holds qJM < jt/2, a contradiction with (4.43).

Summarizing, we have proved that the solution (U) has at least one generalized zero in (M - 1,0] and one generalized zero in (0,N + 1]. The proof is complete. □

Remark 4.2. The conjugacy criterion for the Sturm-Liouville equation

A(rk Axk) + qkXk+1 = 0, rk> 0, (4.44)

formulated in [1, Theorem 2] is the corollary of the above criterion for ak = 1, bk = 1/rk, ck = -qk and dk = 1 - qk/rk. Theorem 4.1 also extends the results proved in [1, 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 (U), (y) e R2 / k eh, are sequences such that the Casoratian w = det( Uk yD = 1 for any k eh. Then, these sequences form a normalized basis of symplectic system (S) with

ak = Xk+1^k - yk+1«k, bk = -Xk+1yk + yk+1Xk,

Ck = Uk+lVk - Vk+lUk, dk = -Uk+1yk + Vk+1Xk.

Moreover/ if bk = 0 for k eh,

lim ^ = 0, (5.2)

and (X) has (m - 1) generalized zeros inh, then the first phase determined by (U), (V) satisfies

V A ,XkXk+1 + ykyk+1 iim qsk - iim qsk = 2, Arccot---= nm. /c;

k ^^ k keZ ,bk = 0 bk

Proof. Let (X) and (y) be sequences with the Casoratian equal to 1, and let a, b, c, and d be given by (5.1). Then, by the Cramer rule, we obtain

Xk+1 = akXk + bkUk, Uk+1 = CkXk + dkUk,

yk+1 = akyk + bkVk, Vk+1 = Ckyk + dkVk,

so (U) and (V) are solutions of (S) with a,b,c,d given by (5.1). It is easy to verify that akdk - bkck = 1 holds for any integer k; that is, system (S) is symplectic. Now, suppose that

assumptions of the second part of the theorem are satisfied. Then, (S) is nonoscillatory in Z and ( U ) is its recessive solution both in oo and -o. Since

A _ Aykxk - Axkyk yk+1xk - xk+1yk

XkXk+i xkxk+i

(akyk + bkvk)xk - (akxk + bkuk)yk _ bk ^ XkXk+1 XkXk+l

for large and small k, the limits limk^±oyk/xk exist and by (5.2) limk^oyk/xk = oo limk^-oyk/xk = -o. It follows, by the definition of the first phase, that limits limk k > limk also exist and equal to odd multiples of n/2. This, coupled with the fact that (U)

has exactly m - 1 generalized zeros in Z;that is, yk equals (m - 1) times an odd multiple of

jt/2 or skips over this multiple, gives (5.3). □

We finish the paper with an example illustrating the previous theorem. Example 5.2. Consider a pair of two-dimensional sequences (U), (VV) with

/ 1\(n-1) n -1

Xk = \ 21 ' Uk =

(k+3/2)

2) ' k (k + 3/2)(n-1)/

where n > 1 and (k + a)(n) := (k + a) ••• (k + a - n +1). By a direct computation, one can verify that xkvk - ykuk = 1 and that (5.1) read

ak = 1,

__-w(w-l)__<5'7)

Ck ~ (k + 5/2 )(fc + 3/2)2 • • • (Jfc - n + 7/2)2(k -n + 5/2) '

_ (k + 5/2)(fc + 3/2) - n(n - 1) fc_ (fc + 5/2)(fc + 3/2)

Obviously, limk^±oxk/yk = 0, so the assumptions of the previous theorem are satisfied, and since

xkxk+1bk = (k + (k + i)4 • • • (k - n + 04(fc - n + > 0, (5.8)

for k e Z ,the solution (U) has no generalized zero in Z .Consequently, (5.3) reads (as can be again verified by a direct computation)

Arccot

2 2 19

kz + (4 - 2n)k + nz -4n + —

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.

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