On the oscillation of recurrence equations

Nonlinear Analysis 36 (1999) 231 – 268

On the oscillation of recurrence equations Ravi P. Agarwal a;∗ , Jerzy Popenda b a

Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b Institute of Mathematics, Pozna n University of Technology, Piotrowo 3A, 60-965 Poznan, Poland Received 1 April 1997; accepted 10 July 1997

Keywords: Oscillatory solutions; Recurrence equations; Orthogonal polynomials; Ordered sets; Linear spaces; Archimedean spaces; Partial recurrence equations

1. Introduction The theory of recurrence (dierence) equations, the methods used in their solutions and their wide applications have advanced beyond their adolescent stage to occupy a central position in Applicable Analysis. In fact, in the last few years, the proliferation of the subject is witnessed by hundreds of research articles and several monographs. In particular, oscillation of the solutions of the dierence equations has attracted many researchers. The purpose of this paper is to oer several new fundamental concepts in this fast developing area of research. These concept are explained through examples and supported by simple results. The plan of this paper is as follows: In Section 2, we shall consider the scalar recurrence equation and introduce concepts such as oscillation around a; oscillation around the sequence; regular oscillation, and periodic oscillation. In Section 3, we shall prove the point-wise oscillation property of several orthogonal polynomials, namely, Chebyshev polynomials of the rst and second kind, Hermite polynomials, and Legendre polynomials. Here an oscillation theorem for second-order recurrence relations is also established. In Section 4, we de ne the global oscillation of sequences of real-valued functions. Here, for a second-order nonlinear continuous-recurrence relation, sucient conditions to ensure global oscillation of solutions are provided. In Section 5, oscillation in ordered sets is considered. Here, the concept (f; R; ≤)-oscillate is introduced, and two interesting examples are given. Section 6 deals with the oscillation ∗

Corresponding author.

0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 3 7 - 6

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in linear spaces, and the ideas are elaborated in two simple theorems. In Section 7, we consider oscillation in Archimedean spaces, and prove a result for a third-order recurrence relation. In Section 8, for the partial recurrence relations we de ne oscillation across the family = { k : k ∈ N} of each other disjoint path arguments, and prove two theorems. Oscillation of system of equations is the subject matter of Section 9. Here for the system of two linear recurrence equations with constant coecients oscillation is established by using some geometric ideas. In Section 10, oscillation between sets is introduced. Here several examples illustrating the basic ideas are included. Finally, in Section 11 the oscillation of continuous-discrete recurrence relations is studied. Here two simple results dwelling upon the importance of the concepts are proved. Throughout this paper we shall use the following notations: N = {1; 2; : : :} the set of positive integers, and R = (−∞; ∞); R0 = [0; ∞); R+ = (0; ∞); and R− = (−∞; 0) the sets of real, nonnegative, positive, and negative numbers, respectively.

2. Oscillation of scalar recurrence equations Let y (= {yn }∞ n=1 ): N → R; i.e., y is a real-valued sequence. For this y the dierence operators
;
a ; a ∈ R; and
k ; k ∈ N are de ned as follows:
yn = yn+1 − yn ;


ayn = yn+1 − ayn ;

n∈N

and
kyn =
k−1yn+1 −
k−1yn ;

k¿1;


1 =
:

De nition 2.1. The sequence y is said to be oscillatory around a (a ∈ R) if there exists an increasing sequence of integers {nk }∞ k=1 such that (ynk − a)(ynk +1 − a)≤0

for all k ∈ N:

(2.1)

If in Eq. (2.1) the strict inequality holds then y is called strictly oscillatory around a. Further, if (yn+1 − a)(yn − a)¡0 for all n ∈ N then y is said to be quickly oscillatory around a. Thus, the oscillation of y is the same as the sequence {yn − a}∞ n=1 has in nite number of generalized zeros. (We recall that the sequence y has a generalized zero at n provided yn = 0 if n = 1 and if n¿1 then either yn = 0 or yn−1 yn ¡0:) Usually in literature oscillatory behavior is studied only around 0; whereas in applications mostly we need oscillation around at some other point, e.g., at stable (stationary) point. De nition 2.2. The sequence y is said to be strictly oscillatory around 0, or simply, oscillatory if for every n ∈ N there exists a k ∈ N such that yn yn+k ¡0: Further, y is said to be nonoscillatory if it is eventually of constant sign.

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233

In De nitions 2.1 and 2.2 the oscillation of y is from the phase space point of view, whereas the de nition of nonoscillation looks at the sequence as the chain of points on the enlarged space. De nition 2.3. The sequence y = {yn }∞ n=1 is called oscillatory around the sequence if there exists an increasing sequence of positive integers {nk }∞ x = {xn }∞ n=1 k=1 such that (ynk − xnk )(ynk +1 − xnk +1 )≤0

for all k ∈ N:

From the above de nition it is clear that if the sequence {yn }∞ n=1 oscillates around the ∞ ∞ sequence {xn }∞ n=1 then {xn }n=1 oscillates around the sequence {yn }n=1 : Moreover, every ∞ sequence {yn }n=1 oscillates around itself. However, the relation ‘oscillation around’ does not have the transitivity property. Further, the sequence {yn }∞ n=1 is oscillatory around the number a (the number 0) if it is oscillatory around the constant sequence x such that xn = a; n ∈ N (around the sequence  such that n = 0; n ∈ N): We also ∞ note that if the sequence {yn }∞ n=1 oscillates around the sequence {xn }n=1 ; then it also ∞ oscillates around every sequence {zn }n=1 ; where for a xed k ∈ N; zn = xn ; n ≥ k: To show the importance of the above de nitions and remarks, we need the following: Theorem 2.1. (Kocic and Ladas [10]). For the linear homogeneous dierence equation yn+k =

k−1 X

ai yn+i ;

n ∈ N;

(E1 )

i=0

where k is a positive integer and a0 ; a1 ; : : : ; ak−1 ∈ R; the following statements are equivalent: (a) Every solution of (E1 ) is oscillatory. Pk−1 (b) The characteristic equation of (E1 ); i.e.; k = i=0 ai i has no positive roots. Corollary 2.2. The equation
kyn = ayn ;

n∈N

(E2 )

oscillates; if and only if;  a¡0 in the case k is even; a¡ − 1 in the case k is odd: Further; if k is odd; and 1 ≤‘¡k the equation
kyn = ayn+‘ ;

n∈N

possesses nonoscillatory solution for any real number a; whereas if k is even it possesses nonoscillatory solution if a is positive. Now suppose that the condition (b) of Theorem 2.1 is satis ed. Since all solutions of (E1 ) are oscillatory around the null sequence  and equation (E1 ) is linear, the

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∞ ∞ dierence of any two solutions {yn }∞ n = 1 and {xn }n=1 ; i.e., the sequence {yn − xn }n=1 is also the solution of (E1 ) and oscillates around : Therefore, there exists {nk }∞ k=1 such that (ynk − xnk )(ynk +1 − xnk +1 )≤0: This means that both solutions oscillate around themselves. Hence, the set of all solutions of (E1 ) forms a bunch of sequences which oscillate around each other. Next, we shall consider the nonhomogeneous equation

yn+k =

k−1 X

ai yn+i + n ;

n∈N

(E3 )

i=0

and suppose that for the corresponding homogeneous equation (E1 ) the condition (b) ∞ of Theorem 2.1 is satis ed. Let {yn }∞ n=1 and {xn }n=1 be any two solutions of (E3 ): ∞ Then, the dierence {yn − xn }n=1 is an oscillatory solution of (E1 ) around : Hence, there exists {nk }∞ k=1 such that ((ynk − xnk ) − 0)((ynk +1 − xnk +1 ) − 0) = (ynk − xnk )(ynk +1 − xnk +1 ) ≤0; i.e., all solutions of (E3 ) oscillate around themselves, and therefore form a similar bunch as the set of solutions of (E1 ). Example 2.1. Consider the equation yn+k =

k−1 X

ai yn+i + b;

n ∈ N;

(E4 )

i=0

where b is any constant. Assume that the condition (b) of Theorem 2.1 is satis ed. Pk−1 Pk−1 Then, 1 − i=0 ai 6= 0; otherwise the characteristic equation k = i=0 ai i will have a positive root 1: It is clear that the constant sequence {yn }∞ n=1 ; where yn = b 1 −

k−1 X

!−1 ai

;

n ∈ N;

i=0

is a solution of (E4 ): Therefore, from the above remarks, every solution of (E4 ) oscilPk−1 lates around the constant b(1 − i=0 ai )−1 : As another example of the theory of oscillation we shall prove a generalization of Theorem 2 established in [16] for the dierence equation
ay(n) +

m X

ain fi (y(di (n)) = 0;

n ∈ N;

(E5 )

i=1

where a i : N → R; ain ¿0 di : N → N;

for all n ∈ N = {;  + 1; : : :};

lim di (n) = ∞;

n→∞

fi : R → R; xf(x)¿0

i = 1; 2; : : : ; m;

for x 6= 0;

i = 1; 2; : : : ; m:

i = 1; 2; : : : ; m;

(2.2) (2.3) (2.4)

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235

Theorem 2.3. If 0¡a¡1; then every solution {yn }∞ n=1 of (E5 ) is oscillatory; or limn→∞ n y(n) = 0 for some ¿1: Proof. If {yn }∞ n=1 is a nonoscillatory solution of (E5 ); say positive for n ≥ n0 ; then by Eq. (2.3) there exists an n1 ≥ n0 such that di (n) ≥ n0 for all n ≥ n1 and all i ∈ {1; 2; : : : ; m}: Hence, by Eqs. (2.2) and (2.4) we have ain fi (y(di (n)))≥ 0 for all n ≥  = max{; n1 }; i = 1; 2; : : : ; m: Therefore,
ay(n)≤ 0; n ≥ ; which is the same as y(n + 1)≤ay(n)

for n≥:

(2.5)

From Eq. (2.5), we obtain y(n + )≤ ay(n +  − 1)≤ a 2 y(n +  − 2) ≤ · · · ≤a n y():

(2.6)

Now since a ∈ (0; 1); there exists a ¿1 such that a¡1: For such ; in view of Eq. (2.6), we have n+ y(n + )≤ (a)n+ [y()a − ]; which immediately gives lim n y(n) = lim n+ y(n + )≤ lim (a)n+ [y()a − ] = 0:

n→∞

n→∞

n→∞

Now we shall study regular oscillations. For this, we recall [2] that the sequence y is called periodic (-period) if there exists an integer  ∈ N such that yn+ = yn for all n ∈ N: Thus, for the sequence y the oscillatory behavior with some regularity is the same as its periodicity. Suppose that {yn }∞ n=1 is an oscillatory sequence. To this sequence we can attribute a new sequence of signs as follows:  if y(n)¿0; 1 sgn(y; n) = −1 if y(n)¡0;  0 if y(n) = 0: De nition 2.4. A sequence y is said to be periodic oscillatory if the sequence {sgn(y; n)}∞ n=1 is periodic. It is interesting to note that the regular oscillation can have dierent forms. For example, {sgn(y; n)}∞ n=1 = 1; −1; 1; 1; −1; −1; 1; 1; 1; −1; −1; −1; : : : or {sgn(y; n)}∞ n=1 = 1; −1; 1; 1; −1; 1; 1; 1; −1; : : : : The problem connecting the distance between two consecutive zeros and regular oscillation has never been studied. We shall present a very simple result in this direction.

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For this, let us denote n X

pos(y; m; n) =

max(sgn(y; i); 0);

neg(y; m; n) =

i=m

n X

max(−sgn(y; i); 0):

i=m

These functions de ne the number of positive and negative terms of the sequence y(m); y(m + 1); : : : ; y(n): A simple result for the linear dierence equation y(n + k) =

k−1 X

ain y(n + i);

n ∈ N;

(E6 )

i=0

where the sequences a i : N → R; i = 0; 1; : : : ; k − 1; is stated in the following: Theorem 2.4. If the sequences {sgn(a i ; n)} are -periodic;  ≤ k and such that pos(a k−j ; 1; ) =  − s;

neg(a k−j ; 1; ) = s

for j = 2 + s;  = 0; 1; : : : ; s = 1; 2; : : : ; ; 2 + s ≤ k; pos(a k−j ; 1; ) = s;

neg(a k−j ; 1; ) =  − s

for j = (2 + 1) + s;  = 0; 1; : : : ; s = 1; 2; : : : ; ; (2 + 1) + s ≤ k; sgn(a k−j ; n) = bpn ; j ;

j = 1; 2; : : : ; ; n = 1; 2; : : : ;  − 1

where (p1 ; : : : ; p ) is some cyclic permutation of (1; 2; : : : ; );   −1 −1 · · · −1  1 −1 · · · −1    1 · · · −1  [bs;  ]× =   1 ;  ··· ··· ··· ···  1 1 · · · −1 sgn(a i ; n) = sgn(a i+ ; n);

i = 0; 1; : : : ; k −  − 1; n = 1; 2; : : : ;  − 1;

then the equation (E6 ) possesses 2-periodically oscillated solutions with  positive and  negative terms in each 2 consecutive terms. A sequence which oscillates around 0 consists of a ‘string’ of nonnegative terms followed by a string of negative terms, or vice versa, and so on. We call these strings positive and negative semicycles, respectively. When we study the oscillation around ; the semicycles are de ned relative to  and consist of strings greater than or equal to  followed by strings of terms less than ; and so on. More precisely we have the following de nitions of the semicycles. De nition 2.5. A positive semicycle of a sequence y consists of a string of terms {y(m); y(m +1); : : : ; y(n)}; all greater than or equal to ; and such that either m = 1

or

m¿1

and y(m − 1)¡

R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

237

and either n = ∞

or

n¡∞

and

y(n + 1)¡:

Similarly, a negative semicycle of a sequence y consists of a string of terms {y(m); y(m + 1); : : : ; y(n)}; all less than ; and such that either m = 1

or

m¿1

either n = ∞

or

n¡∞

and y(m − 1)≥ 

and and

y(n + 1) ≥:

It is clear that a sequence may have a nite number of semicycles or in nitely many semicycles. Further, periodic oscillation can be described in terms of semicycles. 3. Oscillation of orthogonal polynomials We shall need the following: Theorem 3.1. If a : N → R− ; and lim inf n→∞ an = ¡0; then for the dierence equation
2y(n) = an y(n);

n ∈ N(n ∈ N ∪ {0})

(E7 )

the following hold: (a) Every solution of (E7 ) oscillates. (b) For every nontrivial solution of (E7 ) the distance between successive generalized zeros is greater than 1. Proof. (a) Let us assume that y is a nonoscillatory (say eventually positive) solution of (E7 ): Let y(n)¿0 for n≥ n1 ; n1 ∈ N: Then,
y(n + 1) −
y(n) =
2y(n) = an y(n)¡0;

n ≥n1 :

Therefore,
y(n + 1)¡
y(n); and the sequence {
y(n)}∞ n=1 is decreasing for n ≥ n1 : We need now to consider the following two possible cases: (a1 ) there exists a ≥ n1 such that
y(n)¡0 for all n ≥ ; and (a2 )
y(n)¿0 for all n≥n1 : In the case (a1 ) we have
y(n)≤
y() for all n ≥ : Thus, on summing, it follows that y(n + 1) − y() =

n X


y(j)≤(n + 1 − )
y();

j=

i.e., y(n + 1) ≤y() + (n + 1 − )
y();

n≥ :

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In the above inequality the left-hand side is positive whereas the right-hand side tends to −∞ as n → ∞: This contradiction shows that the case (a1 ) is impossible. Now assume that the case (a2 ) holds. Since
y(n)¿0 and
2y(n)¡0 the sequence {
y(n)}∞ n=1 is decreasing and bounded from below by 0: Therefore, lim n→∞
y(n) = g ≥ 0 exists. But, then lim
2y(n) = lim (
y(n + 1) −
y(n)) = g − g = 0:

n→∞

n→∞

On the other hand, because
y(n)¿0; we have y(n)≥ y(n1 )¿0; and consequently an y(n) ≤an y(n1 );

n≥n1 :

The above inequality yields 0 = lim inf
2y(n) = lim inf an y(n)≤y(n1 ) lim inf an n→∞

n→∞

n→∞

¡ ( − )y(n1 )¡0 for some small ¿0 such that ( − )¡0: Thus, the case (a2 ) is also not possible. Hence, there does not exist any eventually positive solution of (E7 ): The case of eventually negative solution can be reduced, by the linearity of the equation (E7 ); to the previous one. Thus, every solution of (E7 ) is oscillatory. (b) Suppose that the statement is false. Then, there exists an increasing sequence of positive integers {nk }∞ k=1 such that for every k ∈ N one of the following cases hold   y(nk ) = y(nk + 1) = 0; y(nk ) = 0; y(nk + 1)¿0; y(nk + 2) = 0; (b1 )  y(nk ) = 0; y(nk + 1)¡0; y(nk + 2) = 0;  y(nk ) = 0; y(nk + 1)¿0; y(nk + 2)¡0; (b2 ) y(nk ) = 0; y(nk + 1)¡0; y(nk + 2)¿0;  y(nk )¡0; y(nk + 1)¿0; y(nk + 2)¡0; (b3 ) y(nk )¿0; y(nk + 1)¡0; y(nk + 2)¿0: In the case (b1 ), we get y(nk + 2) = 0; and consequently y(n) = 0 for all n ≥nk ; which is a contradiction to the nontriviality of the solution y: In the case (b2 ); we get y(nk + 2) − 2y(nk + 1) = 0; i.e., y(nk + 2) = 2y(nk + 1) which is a contradiction to the fact that y(nk + 2) and y(nk + 1) are of opposite sign. In the case (b3 ); we get the relation y(nk + 2) + y(nk ) − ank y(nk ) = 2y(nk + 1); in which the left-hand side is negative (positive) while the right-hand side is positive (negative). The above contradictions prove (b). Following [17], we note that the general solution of (E7 ) can be written as X X pi + 2n−2 y(2) pi ; n ∈ N; y(n) = 2n−3 (1 − a1 )y(1) d(n−4; 2)

d(n−3; 1)

R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

239

where pi = (−1 + ai )=4; and X

1 X

xi =

m¿1; k ≥ 0;

d1 ; : : : ; dm = 0 i=1 di di+1 = 0 i = 1; : : : ; m − 1

d(m; k)

X

m Y (xi+k )di ;

xi =

1 X

d(1; k)

X

(x1+k )d ;

d=0

X

xi =

d(0; k)

xi = 1;

d(−1; k)

X

xi = 0:

d(−2; k)

From the above representation of the solution of (E7 ) the following result is immediate. Theorem 3.2. If there exists that one of the sequences ∞    X pi ; or   d(nk −4; 2)

k=1

an increasing sequence of positive integers {nk }∞ k=1 such   X 

pi

d(nk −3; 1)

∞  

k=1

oscillates; where pi = (−1 + ai )=4; then the equation (E7 ) possesses oscillatory solutions. Remark 3.1. Theorem 3.1 holds if a is eventually negative; i.e.; a : N → R and there exists a  ∈ N such that an ¡0 for all n≥ : Also the condition lim inf n→∞ an = ¡0 can be replaced by lim inf n→∞ an = −∞: 3.1. Chebyshev polynomials Chebyshev polynomials of the rst kind {Tn (x)}∞ n=0 satisfy the recurrence relation Tn+2 (x) = 2xTn+1 (x) − Tn (x); n ∈ N ∪ {0}; x ∈ [−1; 1];

(E8 )

where T0 (x) = 1; T1 (x) = x: We arrange the equation (E8 ) as Tn+2 (x) − 2xTn+1 (x) + x 2 Tn (x) = x 2 Tn (x) − Tn (x); take any x ∈ (−1; 1)\{0}; and put x → a; Tn (a) → y(n); to obtain y(n + 2) − 2ay(n + 1) + a 2 y(n) = (a 2 − 1)y(n): Now on dividing the above equation by a n+2 and substituting z(n) = y(n)=a n ; we get
2 z(n) =

a2 − 1 z(n); a2

n ∈ N ∪ {0}:

(3.1)

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R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

Theorem 3.3. For every x ∈ [−1; 1) the sequence {Tn (x)}∞ n=1 is oscillatory. Proof. We note that (a 2 − 1)=a 2 ¡0 for a ∈ (−1; 1)\{0}; therefore for Eq. (3.1) the assumptions of Theorem 3.1 are satis ed. Let x ∈ (0; 1): Since z(n) = Tn (x)=x n ; ∞ x n ¿0; and {z(n)}∞ n=1 oscillates by Theorem 3.1, it is clear that {Tn (x)}n=1 also oscillates. For x = 0; equation (E8 ) reduces to Tn+2 (0) = −Tn (0): Now the initial condition T0 (0) = 1 yields T2 (0) = −1;

T4 (0) = 1; : : : ; T4k (0) = 1;

T4k+2 (0) = −1;

k = 0; 1; : : :

Next, let x ∈ (−1; 0): By Theorem 3.1 (b) the sequence {z(n)}∞ n=1 where z(n) = Tn (x)=x n is not quickly oscillatory, i.e., in every string of three successive, dierent from zero, terms Tk (x) ; xk

Tk+1 (x) ; x k+1

Tk+2 (x) ; x k+2

we have Tk (x) Tk+1 (x) ¿0 xk x k+1

or

Tk+1 (x) Tk+2 (x) ¿0: x k+1 x k+2

However, since for x ∈ (−1; 0) the products x k x k+1 ; x k+1 x k+2 are negative, it follows that Tk (x)Tk+1 (x)¡0; or Tk+1 (x)Tk+2 (x)¡0: This means that the sequence {Tn (x)}∞ n=1 oscillates for x ∈ (−1; 0): Finally, let x = −1: Then, the equation (E8 ) takes the form Tn+2 (−1) = −2Tn+1 (−1) − Tn (−1): Suppose that {Tn (−1)}∞ n=1 is nonoscillatory, and let Tn (−1)¡0 for all n ≥n1 : Then, Tn1 +2 (−1) = −2Tn1 +1 (−1) − Tn1 (−1)¿0; which is a contradiction to Tn1 +2 (−1)¡0: Chebyshev polynomials of the second kind {Un (x)}∞ n=0 satisfy the same recurrence relation (E8 ); however U0 (x) = 1; U1 (x) = 2x: Theorem 3.4. For every x ∈ [−1; 1) the sequence {Un (x)}∞ n=1 is oscillatory. Proof. From Theorem 3.3 the sequence {Tn (x)}∞ n=0 is oscillatory for each xed x ∈ [−1; 1): Therefore, for any a ∈ [−1; 1) the equation yn+2 = 2ayn+1 − yn possesses an oscillatory solution, namely {Tn (a)}∞ n=0 : Further, this equation can be written in self-adjoint form
(pn
yn ) = qn yn+1 ;

n ∈ N ∪ {0};

(3.2)

where pn = 1 and qn = 2a − 2 for all n ∈ N ∪ {0}: Therefore, from a well-known result [1] every solution of Eq. (3.2) is oscillatory. Since for each a ∈ [−1; 1) the sequence {Un (a)}∞ n=0 also satis es Eq. (3.2), it has to be oscillatory.

R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

241

3.2. Hermite polynomials These polynomials {Hn (x)}∞ n=0 satisfy the recurrence relation Hn+2 (x) − xHn+1 (x) + (n + 1)Hn (x) = 0;

n ∈ N ∪ {0}; x ∈ R;

(E9 )

where H0 (x) = 1; H1 (x) = x: Theorem 3.5. For every x ∈ R the sequence {Hn (x)}∞ n=0 is oscillatory. Proof. For x = 0; equation (E9 ) reduces to Hn+2 (0) = −(n + 1)Hn (0) from which the oscillation follows. For x = a 6= 0; we let zn = Hn (a) so that the equation (E9 ) can be written as zn+2 − azn+1 + (n + 1)zn = 0: On multiplying the above equation by 2n+2 =an+2 and arranging the terms, we obtain 2n+1 2n 2n 4 2n 2n+2 z − 2 z + z = z − (n + 1) zn ; n+2 n+1 n n an+2 an+1 an an a2 an which is the same as   4(n + 1) 2 yn ;
yn = 1 − a2

(3.3)

where yn = (2n =an )zn : This equation is of the form (E7 ): Furthermore, if a 6= 0; then lim infn→∞ (1−4(n + 1)=a2 ) = −∞; and (1−4(n + 1)=a2 )¡0 for n¿(a2 =4)−1: Hence, by Theorem 3.1 together with Remark 3.1 it follows that every solution {yn }∞ n=0 of Eq. (3.3) is oscillatory. But, since yn = (2n =an )Hn (a); we conclude that {Hn (a)}∞ n=0 is oscillatory for a ∈ R+ : For a ∈ R− ; we have 2n 2n+1 ¡0; an an+1

n ∈ N:

(3.4)

Further, from part (b) of Theorem 3.1 we nd that for all n ∈ N in every string of three successive terms dierent from zero 2n Hn (a); an

2n+1 Hn+1 (a); an+1

2n+2 Hn+2 (a); an+2

we have 2n+1 2n H (a) Hn+1 (a)¿0 n an an+1

or

2n+1 2n+2 H (a) Hn+2 (a)¿0: n+1 an+1 an+2

The oscillatory behavior of the sequence {Hn (x)}∞ n=0 for a ∈ R− now follows from Eq. (3.4). Thus, for each x ∈ R the sequence {Hn (x)}∞ n=0 is oscillatory.

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3.3. Legendre polynomials These polynomials {Pn (x)}∞ n=0 satisfy the recurrence relation (n + 2)Pn+2 (x) = (2n + 3)xPn+1 (x) − (n + 1)Pn (x); n ∈ N ∪ {0}; x ∈ [−1; 1];

(E10 )

where P0 (x) = 1; P1 (x) = x: Theorem 3.6. For every x ∈ (−1; 1) the sequence {Pn (x)}∞ n=0 is oscillatory. Proof. For x = 0 we get the equation Pn+2 (0) = −((n + 1)=(n + 2))Pn (0) from which the oscillation follows directly. For x 6= 0; we let x = a; y0 = P0 (a); yn =

n−2 2n Y j + 2 Pn (a) an 2j + 3

for n ≥ 1;

j=0

so that the equation (E10 ) can be written as   4(n + 1)2 yn n ∈ N ∪ {0}:
2 yn = 1 − 2 a (2n + 1)(2n + 3) Since for arbitrary a ∈ (−1; 1)\{0};   1 4(n + 1)2 = 1 − 2 ¡0; lim 1 − 2 n→∞ a (2n + 1)(2n + 3) a from Theorem 3.1 and Remark 3.1 it follows that for every a ∈ (−1; 1)\{0} the sequence {Pn (a)}∞ n=0 is oscillatory. 4. Oscillation of functions recurrence equations Theorems 3.3–3.6 prove the oscillation of orthogonal polynomials in the point-wise sense. We de ne such oscillatory behavior in the following: De nition 4.1. The sequence {fn (x)}∞ n=1 of real-valued functions de ned on the set D is said to be oscillatory in the point-wise sense on the set D if for each a ∈ D the sequence {fn (a)}∞ n=1 is oscillatory, i.e., for every a ∈ D there exists an increasing sequence of positive integers (a) = {nk (a)}∞ k=1 such that fnk (a)fnk +1 (a) ≤ 0: In our next de nition we shall introduce oscillatory behavior in the global sense. De nition 4.2. The sequence {fn (x)}∞ n=1 of real-valued functions de ned on the set D is said to be oscillatory in the global sense if there exists an increasing sequence of positive integers  = {nk }∞ k=1 such that fnk (x)fnk +1 (x) ≤ 0:

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It is clear that if the sequence {fn (x)}∞ n=1 is global oscillatory on the set D; then it is point-wise oscillatory on the same set. However, generally the converse does not hold. De nition 4.3. The sequence {fn (x)}∞ n=1 of real-valued functions de ned on the set D is said to be nonoscillatory if for each a ∈ D the sequence {fn (a)}∞ n=1 is nonoscillatory (eventually of constant sign). We remark that oscillatory and nonoscillatory sequences do not exhaust all possible behavior of sequences of functions. Indeed, there exist sequences of functions de ned on the set D such that they are point-wise oscillatory on some proper subset D1 of D and nonoscillatory on the nonempty subset D2 ⊂ D; where D1 ∪ D2 = D: Example 4.1. Let D = [0; ∞): The sequence {(−1)n xn }∞ n=1 ; x ∈ [0; ∞) which is the solution of the problem fn+1 (x) = xfn (x);

f1 (x) = −x

is global oscillatory on the set D: Example 4.2. The sequence of functions de ned by   1 fn (x) = (−1)n x2 − 2 ; x ∈ D = [−1; 1]; n ∈ N n is point-wise oscillatory, but not global oscillatory on the set D: Many results known for scalar dierence equations have their analogs for dierence equations on function spaces. In fact, now we shall present such analogs whose scalar forms rst appeared in [18]. For this, let X be the set of real-valued functions de ned on D: For a ∈ X and any sequence {yn }∞ n=1 of elements of the set X the dierence operator
a is de ned exactly as in the scalar case, i.e.,
a yn = yn+1 − ayn ; n ∈ N: For u; v ∈ X we say u¿v if u(x)¿v(x) for all x ∈ D: By  we shall denote the null function of D: The dierence equation we shall consider is of the form
2a yn = F(n; yn ;
b yn );

n ∈ N;

(E11 )

where a; b are some given elements of X; and F : N × X2 → X: For the equation (E11 ) we shall consider only the nontrivial solutions, i.e., solutions for which supn≥m [|yn |]¿ for every m ∈ N: Theorem 4.1. Let a¿ and F(n; u; v) = 

 for (n; u; v) ∈ S = N × (u; v) ∈ X2 : v + (b − a)u =  ;

F(n; u; v)[v + (b − a)u] + a[v + (b − a)u]2 ¿ Then; every solution of (E11 ) is nonoscillatory.

for (n; u; v) ∈ N × (X2 \S):

(4.1)

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Proof. Condition v + (b − a)u =  which describes the set S applying to (E11 ) is equivalent to yn+1 − ayn = : We denote by W the set of all solutions of (E11 ); by W1 a subset of W such that y ∈ W1 ; if and only if, there exists a k ∈ N for which yk+1 − ayk = ; and by W2 = W\W1 the complement set of W1 with respect to W: Now let y ∈ W1 ; so that for some s ∈ N we have ys+1 − ays = : Therefore, from (E11 ) in view of Eq. (4.1), we get
2a ys = ys+2 − 2ays+1 + a2 ys = ys+2 − 2ays+1 + ays+1 = ys+2 − ays+1 = : Continuing in this way, we obtain ys+i −ays+i−1 = ; i.e., ys+i = ai ys for all i ≥ 1: Since a¿; we also have ai ¿: Therefore, sgn ys+i (x) = sgn ys (x) for every x ∈ D and this solution is nonoscillatory. Let y ∈ W2 ; and suppose that there exists a  ∈ D such that the sequence {yn ()}∞ n=1 is oscillatory. Let m ∈ N be such that ym ()¿0; ym+1 () ≤ 0: Then, it is clear that
a() ym ()¡0:

(4.2)

Now multiplying (E11 ) by
a ym ; using the relations
2a ym =
a ym+1 − a
a ym ;


a ym =
b ym + (b − a)ym ;

and Eq. (4.1), to obtain (at the point )
a() ym+1 ()
a() ym () = F(m; ym ();
b() ym ())[
b() ym () + (b() − a())ym ()] + a()[
b() ym () + (b() − a())ym ()]2 ¿0: Therefore, by Eq. (4.2) we nd
a() ym+1 ()¡0: On repeating this reasoning, we get
a() yn () ¡0 for all n ≥ m: This means that yn ()¡0 for all n¿m + 1; but this contradicts our assumption. The proof for the case ym () ≥ 0; ym+1 ()¡0 is similar. We note that for the solutions of (E11 ) which belong to the set W2 we have proved more than we have stated in the theorem. Indeed, we have proved that these solutions are nonoscillatory at each point x ∈ D: A dual result of Theorem 4.1 is the following: Theorem 4.2. Let a¡ and F(n; u; v) = 

for (n; u; v) ∈ S = N × {(u; v) ∈ X2 : v + (b − a)u = };

F(n; u; v)[v + (b − a)u] + a[v + (b − a)u]2 ¡

for (n; u; v) ∈ N × (X2 \S):

(4.3)

Then; every solution of (E11 ) is point-wise oscillatory. Proof. Let y be a solution of (E11 ) which is not point-wise oscillatory. Then, there should be at least one point  ∈ D such that the sequence {yn ()}∞ n=1 is nonoscillatory, say, eventually positive for all n ≥ m: Let m be even. The dierence
a() ym ()

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245

cannot be equal to zero, if not, then the equality ym+1 () = a()ym () leads to the contradiction that ym+1 ()¡0: Now since
a() ym () 6= 0; (ym ;
b ym ) 6∈ {(u; v) ∈ X2 : v + (b − a)u = }: On multiplying (E11 ) (taken at the point ) by
a() ym (); and using Eq. (4.3), we get
a() ym+1 ()
a() ym () =
a() ym ()F(m; ym ();
b() ym ()) + a()(
a() ym ())2 ¡0: Next, since
a ym = am+1
(ym =am ); it follows that     ym+1 () ym () m+2 m+1 (a())
¡0 (a())
(a())m+1 (a())m and hence     ym () ym+1 ()
¿0:
(a())m+1 (a())m If
(ym ()=(a())m )¿0; then (ym+1 ()=(a())m+1 )¿(ym ()=(a())m )¿0: Therefore, ym+1 ()¡0; and we obtain a contradiction. Hence, it is necessary that
(ym ()= (a())m )¡0: But, then
(ym+1 ()=(a())m+1 )¡0; i.e., (ym+2 ()=(a())m+2 )¡(ym+1 ()= (a())m+1 )¡0; and hence ym+2 ()¡0: This contradiction completes the proof. Theorem 4.3. Let a¿ and for (n; u; v) ∈ (N × X2 )\(N × {(u; v) ∈ X2 : v + bu = }) F(n; u; v)(v + bu) + a(v + bu)[v + (b − a)u]¿:

(4.4)

Then; every solution of (E11 ) is nonoscillatory. Proof. Let y be an arbitrary solution of (E11 ): Condition v+bu =  applying to (E11 ) is equivalent to yn+1 = : Since we are considering nontrivial solutions only, there exists a m ∈ N such that ym+1 6= : For this m we have
b ym + bym = ym+1 6= : Hence, by Eq. (4.4) it follows that F(m; ym ;
b ym )(
b ym + bym ) + a(
b ym + bym )[
b ym + (b − a)ym ]¿: However, since F(m; ym ;
b ym )(
b ym + bym ) + a(
b ym + bym )[
b ym + (b − a)ym ] = F(m; ym ;
b ym )ym+1 + aym+1
a ym from Eq. (4.5) it follows that F(m; ym ;
b ym )ym+1 + aym+1
a ym ¿: Now from (E11 ); we have F(m; ym ;
b ym )ym+1 + aym+1
a ym = ym+1
a ym+1

(4.5)

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and hence ym+1
a ym+1 ¿:

(4.6)

From Eq. (4.6) it follows that ym+1 () 6= 0 for every  ∈ D: Now let for some  ∈ D; ym+1 ()¿0; then by Eq. (4.6) we have
a() ym+1 ()¿0; which implies that ym+2 ()¿ a()ym+1 ()¿0: On repeating the above reasoning with ym+2 instead of ym+1 we get ym+3 ()¿0: Inductively, we obtain yn ()¿0 for all n¿m: Similarly, if ym+1 ()¡0 for some  ∈ D then yn ()¡0 for all n¿m: Therefore, for every  ∈ D the sequence {yn ()}∞ n=1 is nonoscillatory. This completes the proof. In the following result we shall change slightly the de nition of ordering in the space X: For u; v ∈ X we shall say u  v if u(x) ≥ v(x) for all x ∈ D and there exists at least one point  ∈ D such that u()¿v(): Theorem 4.4. Let a()¡0 for all  ∈ D and for (n; u; v) ∈ (N×X2 )\(N×{(u; v) ∈ X2 : v + bu = }) F(n; u; v)(v + bu) + a(v + bu)[v + (b − a)u] ≺ :

(4.7)

Then; every solution of (E11 ) is point-wise oscillatory over the set it is equal to zero. Proof. Let y be an arbitrary solution of (E11 ): As in Theorem 4.3 condition v + bu =  is equivalent to yn+1 = : For the considered nontrivial solution, there exists a m ∈ N such that ym+1 6= : For this m we have
b ym + bym = ym+1 6= ; i.e., v + bu 6= ; and hence by Eq. (4.7), we have F(m; ym ;
b ym )(
b ym + bym ) + a(
b ym + bym )[
b ym + (b − a)ym ] ≺ ; which implies that ym+1
a ym+1 = F(m; ym ;
b ym )ym+1 + aym+1
a ym = F(m; ym ;
b ym )(
b ym + bym ) + a(
b ym + bym )[
b ym + (b − a)ym ] ≺ :

(4.8)

The above inequality implies that ym+2 6= ; because ym+2 =  leads to a contradiction that −ay2m+1 ≺ : Hence, yn 6=  for all n ≥ m + 1: Next, we note that the inequality (4.8) is the same as ym+1 ym+2 − ay2m+1 ≺ :

(4.9)

We denote by D+ (n); D− (n); D0 (n) the disjoint subsets of D such that yn ()¿0 for  ∈ D+ (n); yn ()¡0 for  ∈ D− (n); and yn () = 0 for  ∈ D0 (n); respectively, furtherH;y H;y (m) = { ∈ D : Hy; m ()¿0}; D− (m) = { ∈ more by Hy; m () = ym ym+1 −ay2m ; and D+ H;y H;y D : Hy; m ()¡0}; D0 (m) = { ∈ D : Hy; m () = 0}: It is clear that D+ (m) is empty.

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247

H;y Let  ∈ D+ (m + 1) ∩ D− (m + 1); then by (4.9) we obtain ym+2 ()¡a()ym+1 ()¡0: Similarly, for  ∈ D+ (m + 1) ∩ D0H;y (m + 1); we get ym+2 () = a()ym+1 ()¡0: These two inequalities imply that D+ (m + 1) ⊂ D− (m + 2): Similarly, we have (m + 2): Therefore, the sequence {yn ()}∞ D− (m + 1) ⊂ D+S n=1 is eventually T∞ quickly os∞ cillatory for  ∈ j=1 (D+ ( j)∪D− ( j)) and is a null sequence for  ∈ j=1 (D\(D+ (j)∪ D− ( j))): This completes the proof.

From De nition 2.1 it is clear that the oscillation of the sequence y around a implies that there are in nitely many terms yn which are greater than a and there are in nitely many less than equal to a: This means that the sequence of dierences, i.e., {
yn }∞ n=1 is oscillatory around 0: In fact, if yn ¡a and yn+1 ¿a then yn+1 −yn ¿0; while if yn ¿a and yn+1 ¡a then yn+1 − yn ¡0: Thus, oscillatory behavior of the sequence {
yn }∞ n=1 is a necessary condition for the oscillation of {yn }∞ n=1 ; but not a sucient condition. For example, for the nonoscillatory sequence {yn }∞ n=1 = {1; 3; 2; 4; 3; 5; : : :}; we have {
yn }∞ n=1 = {2; −1; 2; −1; : : :} which is oscillatory. This condition is also necessary for the global oscillation of the sequences of real-valued functions. From this we can immediately deduce that no solutions of the equations yn+1 (x) − yn (x) = x2 + 1 and yn+1 (x) − yn (x) = −1=n can be oscillatory. As a consequence of this we have the following necessary condition: Theorem 4.5. If the sequence {yn }∞ n=1 is oscillatory; then the sequences of dierences {
k yn }∞ n=1 are oscillatory for all k ∈ N: 5. Oscillation in ordered sets Consider the sequence {K(n)}∞ n=1 of subsets of the plane Oxy de ned as follows: K(n) = {(x; y) : x2 + y2 = (1 − (−1)n =n)2 };

n ∈ N;

i.e., K(1) = x2 + y2 = 22 ; K(2) = x2 + y2 = (1 − 1=2)2 ; K(3) = x2 + y2 = (1 + 1=3)2 ; K(4) = x2 + y2 = (1 − 1=4)2 ; : : : : We can say that this sequence of circles oscillates around the set of points which form the circle described by the equation x2 + y2 = 1 (K(∞)): In general, the system oscillates around some element a if it is sometimes ‘greater’ and sometimes ‘less’ than a: Thus, in the system we consider, there should be a possibility to compare at least some elements. Let X be any nonempty set. We recall that a relation x ≺ y; de ned for some pairs (x; y) of elements of the set X is called an order relation (partial order relation) in X if the following conditions are satis ed: (i) x ≺ x for any x ∈ X (re exivity), (ii) x ≺ y and y ≺ z implies x ≺ z (transitivity), and (iii) x ≺ y and y ≺ x implies x = y (antisymmetry). A set X with an order relation is called an ordered set. If X is an ordered set and if for given x; y ∈ X the relations x ≺ y and y ≺ x do not hold then these elements are called incomparable. An ordered set X is called a totally ordered set if it has no incomparable elements. Any totally ordered subset Y of the ordered set X is called a chain in X:

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If in the set K of circles on the plane described by the equation x2 + y2 = 2 ;  ∈ R; for any K(r) = {(x; y): x2 +y2 = r 2 }; K(s) = {(x; y): x2 +y2 = s2 }; we say K(r) ≺ K(s) if r ≤ s; then we have endowed the set K with the order relation. It is clear that in this set, for our sequence {K(n)}∞ n=1 ; de ned earlier, we have K(2k) ≺ K(∞) and K(∞) ≺ K(2k − 1) for all k ∈ N: De nition 5.1. Let (X; ≺) be an ordered set. A function y : N → X is said to oscil∞ late around a ∈ X if there exist in nite increasing sequences {nk }∞ k=1 and {mk }k=1 of positive integers such that ynk ≺ a and a ≺ ymk for every k ∈ N: If a; b are two elements of (X; ≺); such that a ≺ b; then the set [a; b] = {x ∈ X: a ≺ x ≺ b} is called an interval, and a and b its extremities. De nition 5.1 implies that the element a belongs to in nite number of intervals with extremities are the elements of the sequence y: We also note that these de nitions allow the sequence y to contain some incomparable elements. If all elements of the sequence y belong to the same chain and the cardinal of the set of all values of the sequence y is nite, then this sequence oscillates (or is constant starting from some term) around some element of the sequence. Two ordered sets (X; ≺) and (Y; ¡) are said to be isomorphic if there exists a oneto-one mapping f of X onto Y such that x ≺ y implies f(x)¡f(y): Let (R; ≤) be the set of reals with the natural ordering, and let (Y; ≺) be any chain in the ordered set (X; ≺); isomorphic with R: Let f : R → Y be an isomorphism such that f(0) = e ∈ Y: De nition 5.2. A sequence {yn }∞ n=1 of elements of the chain Y is said to (f; R; ≤)oscillate if the sequence of reals {f−1 (yn )}∞ n=1 oscillates. Example 5.1. Let X be the set of lines on the plane Oxy not parallel to the Oy axis, and ‘ = {(x; y): y = mx + k for some xed m; k ∈ R; x; y ∈ R}: Let ‘1 = {(x; y): y = m1 x+k1 } and ‘2 = {(x; y) : y = m2 x+k2 }: We shall write ‘1 ≺ ‘2 if (m1 +k1 ) ≤ (m2 +k2 ): Since this relation does not satisfy antisymmetry property it is not an order relation on X: Let Y ⊂ X be such that ‘ ∈ Y; if and only if, there exists a ∈ R such that ‘ = {(x; y): y = a(x − 0:5); x ∈ R}: It is clear that (Y; ≺) is a totally ordered subset of (X; ≺): Next, an isomorphism f between (R; ≤) and (Y; ≺) can be de ned as follows: f(r) = {(x; y): y = r(x − 0:5)}; r ∈ R: Now let the sequence {rn }∞ n=1 be oscillatory in is oscillatory in (Y; ≺); i.e., (f; R; ≤)-oscillatory. R; then the sequence {f(rn )}∞ n=1 Example 5.2. In the space X of real, linear, and continuous functions on [0; 1]; for the initial value problem y(n + 1; t) = −y(n; t) + [1 + (−1)n ](t − 0:5); y(1; t) = t;

t ∈ [0; 1];

(5.1)

we shall consider the following problem: Is it possible to de ne on X (on some subset Y (Y ⊆ X); containing the solution {y(n; t)}∞ n=1 of Eq. (5.1)), an order relation ≺; and is (f; R; ≤)-oscillatory? nd an isomorphism such that {y(n; t)}∞ n=1

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249

The set X can be totally ordered in the following manner: Let ‘1 = {(t; y): y = m1 t + k1 }; ‘2 = {(t; y): y = m2 t + k2 }: We say ‘1 ≺ ‘2 if m1 ¡m2 ; and in the case m1 = m2 if k1 ≤ k2 : We de ne a subset Y ⊆ X as follows:   a sgn(a) for some a ∈ R: ‘ ∈ Y if and only if ‘ = (t; y): y = at − + 2 2 Since every subset of totally ordered set is totally ordered under the same order relation, (Y; ≺) is totally ordered. The solution of the initial value problem (5.1) can be written as     1 1 + (−1)n 1 + (−1)n t − (−1)n+1 n − −1 y(n; t) = (−1)n+1 n − 2 2 2 n

]: and every element of this sequence belongs to the set Y with a = (−1)n+1 [n − 1+(−1) 2 The isomorphism from R to Y can be de ned as follows:   sgn(r) r : f(r) = (t; y): y = rt − + 2 2 Finally, it suces to observe that the sequence ∞   1 + (−1)n −1 ∞ n+1 n− {f (y(n; t))}n=1 = (−1) 2 n=1 oscillates around 0 (in fact around every real number), therefore the sequence {y(n; t)}∞ n=1 is (f; R; ≤)-oscillates around f(0): We remark that the de nition of (f; R; ≤)-oscillations in the ordered set essentially depends on the isomorphism f: It is possible that the same sequence y is (f; R; ≤)oscillatory but not (g; R; ≤)-oscillatory, where g is some other isomorphism between (R; ≤) and (X; ≺): To provide an example for this, we recall that every strictly increasing bijection h on R preserves natural ordering of the set, if f is an isomorphism between (R; ≤) and (X; ≺), then f ◦ h is an isomorphism between (R; ≤) and (X; ≺) also. Now let the sequence {yn }∞ n=1 be (f; R; ≤)-oscillatory, and the sequence −1 be bounded, i.e., −m¡f (yn )¡M for all n ∈ N and some positive {f−1 (yn )}∞ n=1 numbers m; M: Then, the function g(r + m) = f(r) is an isomorphism between R and X; but {g−1 (yn )}∞ n=1 is not oscillatory (around 0) sequence of reals. Finally, we shall illustrate here one more example which shows that the above phenomenon depends on the order relation in the set R: Example 5.3. Let X = R: The following function h is one-to-one mapping from R to R: ( S x for x ∈ (−1; 1) ∪ k∈N {[2k; 2k + 1) ∪ (−2k − 1; −2k)}; h(x) = S −x for x ∈ k∈N {(−2k; −2k + 1] ∪ [2k − 1; 2k)}: For r1 ; r2 ∈ X we shall write r1 ≺ r2 if h(r1 ) ≤ h(r2 ): The isomorphism between (R; ≤) and (X; ≺) is given by the function h−1 : We can check that for all real numbers

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∞ r1 ; r2 if r1 ≤ r2 then h−1 (r1 ) ≺ h−1 (r2 ): Let {yn }∞ n=1 = {n + 1=n}n=1 : Under this order ; n ∈ N we have the ordering relation for the sequence {n + 1=n}∞ n=1

···5

1 1 1 1 1 ≺ 3 ≺ 1 ≺ 2 ≺ 4 ···: 6 4 2 3 5

−1 This nonoscillatory sequence {n + 1=n}∞ n=1 is (h ; R; ≤)-oscillatory because the −1 −1 ∞ ∞ sequence {(h ) (yn )}n=1 = {h(yn )}n=1 is oscillatory.

6. Oscillation in linear spaces In the previous section we have used the concept of ordering. In the following we shall omit this assumption, but suppose that X is a real vector space. We recall that if a; b ∈ X and a 6= b; then the set of elements x = a + (1 − )b;  ∈ R is called the line passing through a and b: The set {x ∈ X : x = a + (1 − )b; 0 ≤  ≤ 1} is called the segment (interval) determined by a and b; and a and b are called the extremal points of the segment. De nition 6.1. A sequence {yn }∞ n=1 of elements of the real vector space X is said to oscillate around a if there exists an increasing sequence of indices {nk }∞ k=1 such that a belongs to the segments determined by ynk and ynk+1 : We shall now present two very simple results on the oscillatory behavior of solutions of dierence equations in real vector space. Our rst result is for the dierence equation
yn = n (a − yn );

n ∈ N:

(E12 )

Theorem 6.1. Let {n }∞ n=1 be a sequence of real numbers. If there exists an increasing sequence of positive integers {nk }∞ k=1 such that nk ¿1; then every solution of the equation (E12 ) oscillates around a: ∞ Proof. Let {yn }∞ n=1 be any solution of (E12 ); and j ∈ {nk }k=1 ; i.e., j ¿1: Then, from (E12 ) it follows that

yj+1 = j a + (1 − j )yj ; which implies that a=

j − 1 1 yj+1 + yj : j j

However, since 1 ¿0; j

j − 1 ¿0 j

and

j − 1 1 + = 1; j j

a is in the segment determined by yj and yj+1 :

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251

It is clear that if all n ≥ 1 then every solution of (E12 ) is quickly oscillatory around a (a is in all segments determined by yn and yn+1 for all n ∈ N): Furthermore, every solution is in the straight line passing through a and y1 : The second result is concerned with the following dierence equation: yn+2 + byn+1 + cyn = ;

n ∈ N;

(E13 )

where b; c are real constants, and  is the null element of the linear space X: Theorem 6.2. Let b ∈ R0 and c ∈ R− : Then; the equation (E13 ) possesses a family of oscillatory solutions around : Proof. Let y1 be any element of the linear space X and let {n }∞ n=1 be any solution of the scalar dierence equation n+1 n + bn + c = 0;

n ∈ N:

(6.1)

We note that every solution of Eq. (6.1) never vanishes on N: Further, Eq. (6.1) can be written as c n+1 = − b − ; n ∈ N: n Now consider the function f(x) = − b − (c=x); where b ≥ 0 and c¡0: It is clear that f(x)¡0 for x¡0: Therefore, if we take 1 ¡0; then the suitable solution of Eq. (6.1) remains negative for all n ∈ N: By direct substitution we can check that the sequence {yn }∞ n = 1 of elements of the space X de ned by the formula   n−1 Y yn =  j y1 ; n ∈ N j=1

is the solution of the equation (E13 ): Furthermore, yn+1 = n yn ; i.e., yn+1 − n yn = :

(6.2)

Now let n = 1=(1 − n ); n ∈ N: Since n ¡0; we nd that n ∈ (0; 1): Hence, from Eq. (6.2) we obtain yn+1 +((1−n )=n )yn = ; and consequently n yn+1 +(1−n )yn =  for all n ∈ N: This means that  belongs to every interval determined by yn+1 and yn ; i.e., this solution {yn }∞ n = 1 quickly oscillates around : The following generalization of Theorem 6.2 can be proved similarly. Theorem 6.3. Let k ≥ 2 and (−1) j ak−j ¡0 for j = 2; 3; : : : ; k; ak−1 ≥ 0: Then; the equation yn+k +

k−1 X

ai yn+i = ;

n∈N

i=0

possesses a family of quickly oscillatory solutions around :

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7. Oscillation in Archimedean spaces A real vector space X is said to be an ordered vector space if an order relation has been endowed such that the following conditions are satis ed (a) if x1 ; x2 ∈ X and x1 ≺ x2 ; then x1 + x ≺ x2 + x for all x ∈ X; (b) if x1 ; x2 ∈ X and x1 ≺ x2 ; then ax1 ≺ ax2 for any a ∈ R0 : Let x; x1 ; x2 ; x3 ; x4 ∈ X; a; b ∈ R; and  be the zero element of the vector space X: The following properties of the ordered vector spaces are fundamental: (i) if x1 ≺ x2 and x3 ≺ x4 ; then x1 + x3 ≺ x2 + x4 ; (ii) if  ≺ x1 and  ≺ x2 ; then  ≺ x1 + x2 ; (iii) if x1 ≺ x2 and a ≤ 0; then ax2 ≺ ax1 ; (iv) if  ≺ x and a ≤ b; then ax ≺ bx: An element x ∈ X is said to be positive if  ≺ x; and negative if x ≺ : The set of all positive (negative) elements we shall denote by X+ (X− ): Obviously, X+ ∩ X− = : An ordered vector space X we shall call an ‘Archimedean space’ if for any element x; which is not negative, the set {ax: a ∈ R+ } is not bounded from above [5]. In the Archimedean space the following properties can be proved rather easily. (P1) For any element x, which is not positive, the set {ax: a ∈ R+ } is not bounded from below. (P2) If the set {uj : j ∈ J } is not bounded from above, then the set {uj + x: j ∈ J } is also not bounded from above for any element x ∈ X: (P3) For any element x which is not negative, and any sequence {an }∞ n=1 of positive real numbers such that limn→∞ an = ∞ the set {an x: n ∈ N} is not bounded from above. De nition 7.1. A sequence {yn }∞ n=1 of elements in the ordered real vector space X is said to be nonoscillatory around  if eventually it consists of positive or negative elements. Furthermore, we suppose that card{n ∈ N : yn = }¡ℵ0 : Here we shall study the equation
3yn = an yn ;

n ∈ N;

(E14 )

in the Archimedean space. Theorem 7.1. Let a : N → (−∞; −1): Then; every solution of the equation (E14 ) is not nonoscillatory. Proof. Suppose that there exists an eventually nonoscillatory solution y = {yn }∞ n=1 of the equation (E14 ): Since (E14 ) is linear we can assume that  ≺ yn for n ≥: Now because a¡ − 1 by (iv) we have an yn ≺ −yn : Hence, by the transitivity of the order relation we nd from (E14 ) that
3yn ≺ − yn ≺ 

for n ≥ :

(7.1)

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253

Next, from (7.1) and (a) and (b), we get 1 3 yn+3

≺
yn+1

for n ≥ 

(7.2)

which implies that  ≺ 13 yn+2 ≺
yn

for n ≥  + 1:

(7.3)

This leads to y+1 ≺ y+2 ≺ y+3 ≺ · · · ≺ yn ≺ yn+1 ≺ · · · ; i.e., the sequence {yn }∞ n=+1 is positive and increasing. (in the order sense). Hence, from Eq. (7.1), we obtain
3 yn ≺ −yk for n≥k ≥+1: Taking k = +2 and summing the inequality
3 yj ≺ −y+2 (i.e., using (i) suitable number of times) from j = p to j = n − 1; we obtain
2yn −
2yp ≺ −

n−1 X

y+2 ;

(7.4)

j=p

which is in view of (a) for p =  + 2 is the same as
2yn ≺
2y+2 − (n −  − 2)y+2

for n ≥  + 2:

(7.5)

Thus,
2yn is eventually negative. (In the scalar case from Eq. (7.5) it is clear that
2yn → − ∞ which leads to the contradiction to the positivity of yn :) Now from Eqs. (7.1) – (7.3) on using the suitable properties of the order relation, we get
2y+2 ≺
2y+1 =
y+2 −
y+1 ≺
y+2 = y+3 − y+2 ≺ 3
y+1 − y+2 = 2y+2 − 3y+1 : Therefore, from Eq. (7.5), we have
2yn ≺
2y+2 − (n −  − 2)y+2 ≺ − 3y+1 − (n −  − 4)y+2 ; which yields
2 yn ≺ 

for n ≥  + 4:

Next, on taking p ≥  + 4 in Eq. (7.4), we obtain
2yn ≺
2yp − (n − p)y+2 ≺ − (n − p)y+2

for n ≥ p

which implies that
yn −
yp+1 ≺ −

n−1 X

( j − p)y+2

j=p+1

and hence   n−1 X  ( j − p) y+2 −
yp+1 ≺ −
yn j=p+1

for n ≥ p + 1:

(7.6)

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Thus, from the de nition of the Archimedean space and the properties listed in the beginning of this section, we nd that the set ∞   n−1   X  ( j − p)y+2 −
yp+1   j=p+1

n=p+1

is not bounded from above. Therefore, from Eq. (7.6) it follows that the set {−
yn }∞ n=p+1 is not bounded from above. On the other hand, by Eq. (7.3), we have −
yn ≺  for all n ≥  + 1; therefore the set {−
yn }∞ n=p+1 is bounded from above by : This contradiction completes the proof. It is interesting to note that in view of De nition 7.1, the sequence y = {yn }∞ n=1 which is not nonoscillatory can possess in nite subsequences from the sets X+ and X− ; also subsequences which are non-comparable with  (even every element of y can be of this type). So, if we call this type of sequences oscillatory, then Theorem 7.1 states that every solution of (E14 ) is oscillatory. Further, in Theorem 7.1, we did not assume that the set X is directed (to the right), if it happens then our theorem obviously holds. As a consequence of this, Theorem 7.1 remains valid for (E14 ) in the space X(D) (real-valued functions on some set D) in which the order relation is de ned in the usual way, i.e., for x; y ∈ X(D); x ≺ y if and only if x(t) ≤ y(t); t ∈ D: 8. Oscillation of partial recurrence equations Here we shall consider the real-valued sequences in two independent variables, i.e., functions y : N2 → R: ∞ De nition 8.1. A sequence y = {y(m; n)}∞ m=1; n=1 is said to be nonoscillatory around 0 if there exist positive integers ;  such that

y(m; n)¿0

(positive sequence)

for all m ≥ ; n ≥ 

y(m; n)¡0

(negative sequence)

for all m ≥ ; n ≥ :

or

Otherwise the sequence y is called oscillatory. We note that according to the above de nition eventually zero sequences are oscillatory, we shall exclude this type of sequences from our consideration. Further, for the sequence y : {1; 2; : : : ; } × N → R the above de nition can be changed to the following: De nition 8.2. A sequence y = {y(m; n)}m=1; ∞ n=1 is said to be nonoscillatory around 0 if there exists a positive integer  such that y(m; n)¿0

(positive sequence)

for all m ≤ ; n ≥ 

R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

255

or y(m; n)¡0

(negative sequence) for all m ≤ ; n ≥ :

Otherwise the sequence y is called oscillatory. Hereafter, we shall consider sequences y which are de ned on N2 : It is clear that if the sequence y is nonoscillatory then it is nonoscillatory for each ‘section’ along each arbitrary, but xed, m; m¿ as well as arbitrary, but xed, n; n¿: Further, all these sections are of the same xed sign (positive or negative). The sucient condition for oscillation can be formulated as follows: There exists an increasing to in nity sequence ∞ m = {mk }∞ k=1 such that the sections {y(mk ; n)}n=1 are oscillatory sequences for all mk ∈ m: Analogously, if there exists an increasing to in nity sequence n = {nk }∞ k=1 such that the sections {y(m; nk )}∞ m=1 are oscillatory for all nk ∈ n: This sucient condition requires essentially more than the De nition 8.1. For this, we note that the sequence de ned as  −1 for m; n ∈ N such that m = n y(m; n) = 1 otherwise is oscillatory in the sense of De nition 8.1, however, every section for a xed m as well as for a xed n is a nonoscillatory sequence. ∞ If all sequences {y(p; n)}∞ n=1 and {y(m; q)}m=1 for each xed p; q ∈ N are oscil∞ latory, then we say that the sequence y = {y(m; n)}∞ m=1; n=1 possesses a stronger oscillatory property. Nonoscillatory sequences can be oscillatory along some (but nite number) sections. For example the nonoscillatory sequence de ned as   (−1)n for m = 1; n ∈ N; y(m; n) = (−1)m for n = 1; m ∈ N;  1 for the other (m; n) ∈ N2 ∞ has quickly oscillatory sections {y(1; n)}∞ n=1 ; {y(m; 1)}m = 1 : Now we shall de ne the oscillation of the sequence y in a dierent way. For this, we need the following:

De nition 8.3. Any sequence {mk ; nk )}k = 1 where mk ; nk ∈ N for k = 1; 2; : : : ;  such that mk+1 + nk+1 = mk + nk ± 1 for all k ∈ N if  = ∞; and k = 1; 2; : : : ;  − 1; otherwise, is called a path argument. ∞ De nition 8.4. The sequence y = {y(m; n)}∞ m=1; n=1 is called oscillatory across the family = { k : k ∈ N} of each other disjoint path arguments if (y|P∈ k )(y|Q∈ k+1 )¡0 and (y|R∈ k )(y|S∈ k ) ¿0 for all P ∈ k ; Q ∈ k+1 ; R; S ∈ k ; and for all k ∈ N: ∞ max{m; n} is oscillaExample 8.1. The sequence {y(m; n)}∞ m=1; n=1 ; where y(m; n) = (−1) tory across the family of path arguments k de ned by

k = {(m; k); m = 1; 2; : : : ; k} ∪ {(k; n); n = 1; 2; · · · ; k};

k ∈ N:

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R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

∞ For the sequence y = {y(m; n)}∞ m=1; n=1 ; we de ne the partial dierence operators


1y(m; n) = y(m + 1; n) − y(m; n)

and


2 y(m; n) = y(m; n + 1) − y(m; n);

m; n ∈ N:

We shall consider the following rst-order partial dierence equation
1y(m; n) + am bn
2 y(m; n) = 0;

m; n ∈ N:

(E15 )

Theorem 8.1. Let one of the following holds: (a) the sequence {an }∞ n=1 is eventually negative and there exists a negative constant a such that am ≤ a for all m ≥  ∈ N; (b) the sequence {bn }∞ n=1 is eventually negative and there exists a negative constant b such that bn ≥ b for all n ≥  ∈ N: Then; the equation (E15 ) possesses a family of oscillatory solutions around the zero solution. Proof. To prove this result we shall use the method of separation of variables. Suppose ∞ that {y(m; n)}∞ m=1 n=1 is never vanishing solution of (E15 ) which can be represented as the product, i.e., y(m; n) = u(m)w(n)

for all m; n ∈ N:

(8.1)

We substitute Eq. (8.1) in the dierence equation (E15 ) and divide by u(m)w(n)am ; to obtain bn
2 w(n)
1 u(m) =− ; m; n ∈ N: (8.2) am u(m) w(n) The left-hand side of Eq. (8.2) depends on m only, whereas the right-hand side is independent of m: Thus, y(m; n) = u(m)w(n) is a solution of (E15 ) if and only if u and w satisfy two ordinary dierence equations
u(m) = C; am u(m)

m∈N

and

bn
w(n) = − C; w(n)

n∈N

or
u(m) = Cam u(m);

m∈N

and


w(n) = −

C w(n); bn

n∈N

(8.3)

for some constant C: Solving the system (8.3), we get u(m) = u(1)

m−1 Y

(1 + Cai );

m∈N

and w(n) = w(1)

i=1

j=1

Therefore, the sequence y(m; n) = y(1; 1)

n−1 Y

 ! n−1   Y C  (1 + Cai )  1− bj

m−1 Y i=1

j=1

C 1− bj

 ;

n ∈ N:

R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

257

where y(1; 1) = u(1)w(1) should be a solution of (E15 ): In fact, by direct substitution we can check that the sequence of the form   A 

 ! n−1   ∞ Y C  1− (1 + Cai )   bj

m−1 Y i=1

j=1

∞

(8.4)

m=1; n=1

satisfy (E15 ) for arbitrary A; C ∈ R: Now suppose that the condition (a) is satis ed. Then, on taking C¿−1=a; we obtain −1=am ¡ − 1=a¡C; and hence 0¿1 + Cam for all m ≥ : Thus, the sequence (

!)∞

m−1 Y

(1 + Cai )

i=1

m=1

quickly oscillates for m¿: Therefore, the sequences  

A



 ! s−1   ∞ Y C  1− (1 + Cai )   bj

m−1 Y i=1

j=1

m=1

oscillate for any xed A ∈ R\{0} and all xed s ∈ N: This means that the sucient conditions for the oscillation of the sequence (8.4) are satis ed. Similarly, if we take C¡b then the condition (b) ensures the oscillation of the sequence (8.4). Method similar to that used in Theorem 8.1 can be employed for the equation
21 y(m; n) + am bn
2 y(m; n) = 0;

m; n ∈ N;

(E16 )

where
21 y(m; n) = y(m + 2; n) − 2y(m + 1; n) + y(m; n); m; n ∈ N: Theorem 8.2. Let one of the following holds: (a) the sequence {an }∞ n=1 is eventually negative and there exists a negative constant a such that lim inf m→∞ am = a; (b) the sequence {an }∞ n=1 is eventually positive and there exists a positive constant a such that lim supm→∞ am = a; (c) the sequence {bn }∞ n=1 is eventually negative and there exists a negative constant b such that bn ≥ b for all n ≥  ∈ N: Then; the equation (E16 ) possesses a family of oscillatory solutions around the zero solution. Proof. Applying the same method as in the proof of Theorem 8.1, we obtain the system of ordinary dierence equations
2 u(m) = Cam u(m);

m∈N

and


w(n) = −

C w(n); bn

n ∈ N;

(8.5)

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where C is an arbitrary constant. The equation (E16 ) possesses a family of solutions of the form     ∞ ∞ n−1  Y C  ; 1− Au(m)    bj j=1

m=1; n=1

where u(m) is a solution of the rst equation in Eq. (8.5). In the case (a) ((b)) the assumptions of Theorem 3.1 are satis ed with any positive (negative) constant C: Thus, 2 every solution {u(m)}∞ m=1 of the equation
u(m) = Cam u(m); m ∈ N is oscillatory. Hence, for all s ∈ N the sequence     ∞ s−1   Y C  1− Au(m)    bj j=1

m=1

oscillates, and thus the sucient conditions for the oscillatory behavior of y(m; n) are satis ed. The case (c) reduces to the case (b) of Theorem 8.1. Finally, we remark that De nition 8.1 can be generalized to sequences of arbitrary variables k as follows: Let n = (n1 ; n2 ; : : : ; nk ) ∈ Nk denote a multi-index. A real-valued sequence y = {y(n)}(∞;:::;∞) n=(1;:::;1) is said to be nonoscillatory around 0 if there exists a positive multi-index  = (1 ; 2 ; : : : ; k ); i ¿0 such that y(n)¿0

(positive sequence)

for all ni ≥ i ; i = 1; : : : ; k

y(n)¡0

(negative sequence)

for all ni ≥ i ; i = 1; : : : ; k:

or Otherwise, the sequence y is called oscillatory. 9. Oscillation of system of equations In the system case a solution is a sequence of vectors. Thus, here oscillation can be de ned for some or all of its components. We begin with the system of two linear dierence equations with constant coecients i yn+1 = ai1 yn1 + ai2 yn2 ;

i = 1; 2; n ∈ N:

(E17 )

Theorem 9.1. Let aij ¿0 for i; j = 1; 2: Then; 2 (a) every solution of (E17 ) is nonoscillatory; i.e.; both y1 = {yn1 }∞ n=1 and y = are nonoscillatory sequences; {yn2 }∞ n=1 (b) both y1 ; y 2 are together eventually positive or eventually negative; except the case when √ a22 − a11 −  1 2 y1 = y1 where  = (a11 − a22 )2 + 4a12 a21 : (9.1) 2a12 Further; when Eq. (9.1) holds then yn1 yn2 ¡0 for all n ∈ N:

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259

Proof. From our consideration we exclude the trivial solution of (E17 ); which is obtained when y11 = y12 = 0: (i) It is clear that if yk1 ¿0 and yk2 ¿0 for some k ∈ N then yn1 ¿0 and yn2 ¿0 for all n ≥ k: Further, if yk1 ¡0 and yk2 ¡0 for some k ∈ N then yn1 ¡0 and yn2 ¡0 for all n ≥ k: (ii) If y11 = 0 or y12 = 0 then we obtain the case (i) with k = 2: (iii) In view of (i) and (ii) we need to consider only the cases y11 ¡0; y12 ¿0 and 1 y1 ¿0; y12 ¡0: We shall consider only the case y11 ¡0; y12 ¿0; as the arguments are similar for the other case. For the system (E17 ) there corresponds the square matrix   a11 a12 A= ; a21 a22 √ √ )=2; which possesses two distinct real eigenvalues √ (a11 +a22 + )=2 and (a11 +a22 − √ and two family of eigenvectors w1 = [; ( −a11 +a22 )=(2a12 )] and w2 = [; −( + a11 − a22 )=(2a12 )]; where ;  are parameters. We observe that both components of the vector w1 are of the same sign, whereas of w2 are of √ the opposite sign. For this reason (1) = [1; (a − a − )=(2a12 )]: We shall denote by our interest is in the vector w 2 22 11 √ mw = (a22 − a11 − )=(2a12 ) the direction of the vector w2 (1) in the plane Oy1 y 2 : Now let det A¿0: Then, mw ∈ (−a11 =a12 ; −a21 =a22 ): (In the case det A¡0; we have mw ∈ (−a21 =a22 ; −a11 =a12 ); while if det A = 0 then mw = − a21 =a22 = − a11 =a12 .) From the equation (E17 ) it follows that if the point (yn1 ; yn2 ) belongs to the line 2 1 2 ; yn+1 ) belongs to the line y = my1 with m 6= − a11 =a12 ; then (yn+1 y2 =

a21 + a22 m 1 y : a11 + a12 m

(9.2)

Let D1 be the region of the second quadrant of the plane Oy1 y 2 bounded by the lines y 2 = 0 and y 2 = − (a21 =a22 )y1 ; and D2 bounded by the lines y 2 = − (a11 =a12 )y1 and y1 = 0: Take any point (y11 ; y12 ) ∈ D2 ; (i.e., y12 = my11 ; m ∈ (−∞; −a11 =a12 ); y11 ¡0). 1 From the rst equation of (E17 ); i.e., yn+1 = a11 yn1 + a12 yn2 it follows that y21 = a11 y11 + 1 a12 my1 ¿0: Furthermore, the function f(m) =

a21 + a22 m a11 + a12 m

(9.3)

is positive for m ∈ (−∞; −a11 =a12 ) and hence, by Eq. (9.2), y22 ¿0: So the point (y21 ; y22 ) belongs to the rst quadrant and we are back to the case (i). Similarly, if (y11 ; y12 ) ∈ D1 ; (i.e., y12 = my11 ; m ∈ (−a21 =a22 ; 0); y11 ¡0); then we nd that (y21 ; y22 ) belongs to the third quadrant, and so here also we return to the case (i). The image of the line y 2 = − (a11 =a12 )y1 is the line y1 = 0 (y 2 -axis). The image of the line y 2 = − (a21 =a22 )y1 is the line y 2 = 0 (y1 -axis). For the rest of the proof we should show that every solution with the initial point (y11 ; y12 ) which belongs to the open region of the second quadrant bounded by the lines y 2 = − (a11 =a12 )y1 and y 2 = − (a21 =a22 )y1 ; and such that y12 6= mw y11 ; after nite number of steps comes in the rst or third quadrant (in fact by the previous remarks,

260

R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

to the set D2 ∪ { rst quadrant}; or D1 ∪ {third quadrant}): Since det A ¿0; f0 (m) = (a11 + a12 m)2 f00 (m) =

−2a12 det A ¡0 (a11 + a12 m)3

and



f(mw ) = mw

for m ∈ (−a11 =a12 ; ∞)

a21 and f − a22

(9.4) (9.5)

 = 0;

we have f(m)¿m + a for every m ∈ (mw ; −a21 =a22 ); a21 a = (mw − m) ¿0: a21 + a22 mw

where (9.6)

Now we observe that if m¿ − a11 =a12 and (yn1 ; yn2 ) belongs to the second quadrant, 1 2 ; yn+1 ) also belongs to the second quadrant as far as f(m)¡0: then (yn+1 1 Let (y1 ; y12 ) ∈ D3 = {(y1 ; y 2 ) ∈ Oy1 y 2 : y1 ¡0; y 2 = my1 , and m ∈ (mw ; −a21 =a22 )}; i.e., y11 ¡0; y12 = m1 y11 ; and m1 ∈ (mw ; −a21 =a22 ): For our purpose, suppose that all elements of the corresponding solution {yn1 ; yn2 }∞ n=1 (yn2 = mn yn1 ) belongs to the second quadrant. In fact, all the points should be in the region D3 ; because of m1 ∈ (m w ; −a21 =a22 ); all mn ¡0; and Eq. (9.4). The sequence {mn }∞ n=1 is increasing and bounded from above by −a21 =a22 : However, this is impossible, because from Eq. (9.6), we have mn+1 = f(mn )¿mn + a = f(mn−1 ) + a¿mn−1 + 2a¿ · · · ¿m1 + na → ∞ as n → ∞; where a = (m w − m1 )a21 =(a21 + a22 m w )¿0: Let D4 = {(y1 ; y 2 ) ∈ Oy1 y 2 : y1 ¡0; y 2 = my1 ; and m ∈ (−a11 =a12 ; m w )}: We note that for m ∈ (−a11 =a12 ; m w ); by Eqs. (9.4) and (9.5), we obtain det A (m − m w ) + m w ¡m: (9.7) f(m)¡ (a11 + a12 m w ) 2 Let (y11 ; y12 ) ∈ D4 and {yn1 ; yn2 }∞ n=1 be the corresponding solution of (E17 ): If for some k¿1; we have yk1 ¡0; yk2 = mk yk1 ; mk ∈ (−a11 =a12 ; m w ); but mk+1 = f(mk )¡ − a11 =a12 ; 1 2 ; yk+1 ) ∈ {third quadrant} and we are back to the case (i). Therefore, supthen (yk+1 1 pose that (yn ; yn2 ) ∈ D4 for all n ∈ N; which yields yn2 = mn yn1 with mn ∈ (−a11 =a12 ; m w ); n ∈ N: Now from (9.7) it is clear that the sequence {mn }∞ n=1 is decreasing and bounded from below by −a11 =a12 : On the other hand, again from Eq. (9.7), we obtain mn+1 = f(mn )¡mn − a = f(mn−1 ) − a¡mn−1 − 2a¡ · · · ¡m1 − na → −∞ as n → ∞; where

 a = (m1 − m w ) 1 −

which is a contradiction.

 det A ¿0; (a11 + a12 m w ) 2

R.P. Agarwal, J. Popenda / Nonlinear Analysis 36 (1999) 231 – 268

261

1 2 1 Similarly, we can verify that if yn1 ¡0; yn2 = m w yn1 then yn+1 ¡0; yn+1 = m w yn+1 : Thus, we have covered all the possible cases in the second quadrant. A similar reasoning can be applied for the points of the fourth quadrant. The case det A = 0 immediately leads to (i) or (ii). Finally, in the case det A¡0 we note that the proof of (iii) needs some modi cations because now the trajectory can have jumps between second and fourth quadrants. This completes the proof.

As a consequence of Theorem 9.1 we have the following: Theorem 9.2. Let aij ¡0 for i; j = 1; 2: Then; 2 (a) every solution of (E17 ) is quickly oscillatory; i.e.; both y1 = {yn1 }∞ n=1 and y = 2 ∞ {yn }n=1 are quickly oscillatory sequences; (b) both y1 ; y 2 are eventually of the same sign; i.e.; yn1 yn2 ¿0; except the case √ a22 − a11 +  1 2 y1 = y1 ; where  = (a11 − a22 ) 2 + 4a12 a21 2a12 when yn1 yn2 ¡0 for all n ∈ N: i = −ai1 xn1 − Proof. In (E17 ) let xni = (−1) n yni ; i = 1; 2; n ∈ N; to obtain the system xn+1 2 ai2 xn ; i = 1; 2; for which the Theorem 9.1 can be applied. Theorem 9.2 now follows from the substitution employed.

Finally, we shall state here a result which provides sucient conditions for the one component of the following system to be quickly oscillatory
yni =

m X

anij ynj ;

i = 1; 2; : : : ; m; n ∈ N:

(E18 )

j=1

Theorem 9.3. If  for i = j; i 6= k;   [−1; ∞) ij a : N → (−∞; −1] for i = j = k;   [0; ∞) for i = 6 j; i 6= k; j 6= k; and a ik ; a kj for i 6= k; j 6= k are quickly oscillatory on N with a1ik ¡0; a1kj ¿0: Then; every solution of the system (E18 ) with the initial conditions y1i ¿0; y1k ¡0 has positive components y i for i 6= k and quickly oscillatory component y k on the whole N; while every solution with the initial conditions y1i ¡0; y1k ¿0 has negative components y i for i 6= k and quickly oscillatory component y k on the whole N: 10. Oscillation between sets We shall now present another generalization of the concept of oscillation. We recall that the oscillation around zero of the real valued sequence {yn }∞ n=1 means that

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Table 1 n yn n yn

1 1 10 0.1111

2 1 11 −1:011

3 −1:5 ··· ···

4 0.3333 26 0.04

5 1.5833 27 −1:015

6 0.2 28 0.0370

7 −1:033 29 1.0728

8 0.1428 30 0.0345

9 1.2678 31 −1:001

there exist two disjoint sets R0 and R− ; and two increasing sequences of indices ∞ {nk }∞ k=1 ; {mj }j = 1 such that ynk ∈ R0 ; ymj ∈ R− for all k; j ∈ N: De nition 10.1. Let {yn }∞ n=1 be a sequence of elements of some set X and {X : ∈ } be any family of disjoint subsets of X; (i.e., X ∩ X = ∅ for all ;  ∈ ;  6= ): We say that this sequence oscillates in relation to the family X if for all ∈ there exists {n ; k }∞ k=1 such that for all k ∈ N; yn ; k ∈ X : In the above de nition we can assume that X =

S

∈

X :

Example 10.1. Consider the initial value problem  n  1 (1 + yn ) + ; n ∈ N; yn+1 =cos 2 n y1 = 1: We present the approximate solution of this problem in Table 1. We can say that this solution oscillates in relation to the family of sets {(−∞; −1]; (−1; 1); [1; ∞)}: The terms y4k+3 belong to (−∞; −1]; y4k+1 and y2 belong to [1; ∞); and the rest are in (−1; 1): In what follows we shall consider the following system of two equations: a + ayn ; xn bxn ; yn+1 = 1 + xn yn xn+1 =

n∈ N:

We shall denote by S1 = {(x; y) ∈ R 2 : x = 0 ∨y = 0}; S2 = {(x; y) ∈ R 2 : y = −1=x ∧ x 6= 0}; X = R 2 \{S1 ∪ S2 };   p ab D1 = (x; y) ∈ X: y = ∧ x ∈ (0; a(ab + 1)] ; x   p ab D2 = (x; y) ∈ X: y = ∧ x ∈ ( a(ab + 1); ∞) ; x   p ab D3 = (x; y) ∈ X: y = ∧ x ∈ (−∞; − a(ab + 1)) ; x

(E19 )

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263



 p ab ∧ x ∈ [− a(ab + 1); 0) ; D4 = (x; y) ∈ X: y = x D = {D1 ∪ D2 ∪ D3 ∪ D4 }: Theorem 10.1. Let a; b¿0: Then; every solution {(xn ; yn )}∞ n=1 of the system (E19 ) with the initial condition in the space X is oscillatory (at least for n ≥ 2) between the sets D1 and D2 ; or between the sets D3 and D4 : Proof. Let P = (x1 ; y1 ) be an arbitrary point of the phase space X: We denote by   bx a + ay; ; f(x; y) = x 1 + xy which de nes the dynamical system described by (E19 ): It can be easily checked that f(P) ∈ D: Therefore, it suces to show that for any point (x; y) ∈ D1 there is f(x; y) ∈ D2 and also if (x; y) ∈ D2 then f(x; y) ∈ D1 : Similarly, if (x; y) ∈ D3 then f(x; y) ∈ D4 and if (x; y) ∈ D4 then f(x; y) ∈ D3 : Let Pk = (xk ; yk ) ∈ D1 then from (E19 ); we obtain p a ba 2 a + ba 2 xk+1 ¿ p +p =p = a(ab + 1): a(ab + 1) a(ab + 1) a(ab + 1) Further, since yn+1 =

ab xn+1

for all n ∈ N

it is clear that Pk+1 = (xk+1 ; yk+1 ) ∈ D2 : The rest of the relations can be proved analogously. We note that in the considered system every solution (at least from the point (x2 ; y2 )) is 2-periodic. For a system (equation) which has periodic or asymptotically periodic solutions we can always nd a family of sets for which these solutions are oscillatory (at least eventually) between them. Further, in the space with a cone of positive elements and (or) a cone of negative elements we can de ne oscillations between these cones. Moreover, the concept of oscillation can also be useful in considerations of recurrence fuzzy equations and leads to a concept of fuzzy oscillations. Finally, to show how dierent forms oscillation between sets can have, we present here two examples. Example 10.2. Let X denote the set of m × m matrices. By XL (XP ) we denote subset of X consisting of all lower triangular (upper triangle) matrices. Consider the matrix dierence equation Yn+1 = An YnT ;

n ∈ N;

(E20 )

T where {An }∞ n=1 is a sequence of m × m diagonal matrices, and Y denotes the transpose of Y: We can check that every solution of (E20 ) with the initial condition Y1 ∈ XL ∪ XP

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oscillates between sets XL and XP : We note that XL ∩ XP = XD where XD is the set of m× m diagonal matrices. This set, in our example, plays the role similar to zero element in the ordinary oscillation de nition. If for some k ∈ N we nd that Yk ∈ X D then Yn ∈ XD for all n ≥ k: For the next example, we need a dierent version of De nition 10.1. For this, suppose that we have a set X; and the set of properties {p1 ; p2 ; : : : ; pk } which are possessed by some elements of X: By Xpi we shall denote the subset of X which possess property pi : De nition 10.2. Let {xn }∞ n=1 be a sequence of elements of X: We say that this sequence is oscillatory (strictly) in relation to the set of properties (p1 ; p2 ; : : : ; pk ) if there exist (disjoint) sequences of indices n i = {nji }∞ j=1 ; i = 1; : : : ; k such that for all i ∈ {1; 2; : : : ; k} and j ∈N; xnji ∈ Xpi : Example 10.3. Consider a scalar dierence equation. We say that this equation possesses property pc if every solution of this equation converges, and property pnc if every (may be over a nite set) solution is nonconvergent. We now consider the sequence {xm }∞ m = 1 of dierence equations of the form   (−1) m yn ; n ∈ N; m ∈ N: (E21 ) xm : yn+1 = 1 + m (It is clear that (E21 ) is a particular case of the equation yn+1 = f(n; yn ; m); n ∈ N with the natural valued parameter m; and oscillation is in the sense of variation of parameters.) It is easy to check that for m odd the equation xm has the property pc ; whereas for m even property pnc : We can say that this sequence of equations oscillate in relation to the set of properties {pc ; pnc }: 11. Oscillation of continuous-discrete recurrence equations Let J be an interval (bounded or unbounded) on the real line, and let y be a realvalued function de ned on J: Usually, the function y is called oscillatory on J if it possesses an in nite number of zeros contained in J; where by a zero of the function y we mean a point x such that y(x) = 0: For continuous functions this de nition is quite satisfactory. However, the function y de ned on [0; 1]; where ( 1 for x rational; y(x) = −1 for x irrational; can be called oscillatory, but it does not vanish on [0; 1]: Thus, in this case we should talk about generalized zeros. Let (X; d) be a metric space and y : X → R be a given function. We say that at the point x the function y possesses a generalized zero, if in every suciently small

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265

neighborhood o (x) of the point x there exist two points p; q ∈ o (x) such that y(p)¿0 ∧ y(q) = 0 ∨ y(p)¡0 ∧ y(q) = 0 ∨ y(p)¿0 ∧ y(q)¡0:

(11.1)

Now let J be a subset of (X; d); by o (x)|J we shall denote the neighborhood of the point x in the set J; i.e., o (x)|J = o (x) ∩ J: The set of all generalized zeros of the function y contained in the set J we shall denote by gzer(y; J): De nition 11.1. Let J be a subset of the metric space (X; d) and y : J → R: We say that the function y oscillate on the set J if card(gzer(y; J)) ≥ ℵ0 ; and y does not oscillate if card(gzer(y; J))¡ℵ0 : The space X, for example, can be m-dimensional Euclidean space R m : We also note that in the above de nition the set R can be replaced by any ordered set (S; ≺); zero by a ∈ (S; p); y : J → S and the condition (11.1) by y(p) ≺ a ∧ a ≺ y(q) ∧ (y(p) 6= a 6= y(q)) or y(p) = a ∧ a ≺ y(q) ∧ a 6= y(q) or y(p) = a ∧ y(q) ≺ a ∧ a 6= y(q): The above remarks are enough to study equations of functions in more than one variable, one of which takes discrete values and the second continuous. De nition 11.2. Let X = N × J; where J ⊂ R m ; and y : X → R: We say that the func∞ ∞ tion y is oscillatory if there exists a sequence {xk }∞ k=1 = {(nk ; tk )}k=1 with {nk }k=1 strictly increasing sequence of positive integers such that the sequence {y(nk ; tk )}∞ k=1 is oscillatory. In the above de nition all tk ; k ∈ N can be the same, so all sections y(n; t) can be nonoscillatory as the functions of the variable t for each xed n ∈ N: A more restrictive de nition is the following: De nition 11.3. Let X = N×J; where J ⊂ R m ; and y : X → R: We say that the function y is oscillatory in relation to the set of discrete sections; if there exists a sequence {nk }∞ k=1 such that for each nk the suitable section y(nk ; t) is an oscillatory function on J: It is clear that if the function y is oscillatory in relation to the set of discrete sections, then it is oscillatory in the sense of De nition 11.2, however, the converse is not true. In our rst result here, we shall consider the equation y(n + 1; Tt) = a(t)y(n; t);

n ∈ N; t ∈ R m ;

(E22 )

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where T is some one-to-one mapping from the space R m onto R m : Let T −1 denote the inverse of T and T −j = T −1 ◦ T −1 ◦ · · · ◦ T −1 ; i.e., T −1 composed j-times. Theorem 11.1. Every solution of (E22 ) with the initial function y(1; t) such that y(1; T −nk tk )y(1; T −nk +1 tk )a(T −nk tk )¡0;

k ∈ N;

(11.2)

∞ where {nk }∞ k=1 is some strictly increasing sequence of positive integers; and {tk }k=1 is m a sequence of elements of the space R ; is oscillatory in the sense of De nition 11:2.

Proof. The solution of the equation (E22 ) can be written as " n−1 # Y −i y(n; t) = a(T t) y(1; T −n+1 t): i=1

Hence, for any  ∈ R m we have #2 " n−1 Y a(T −i ) {a(T −n )y(1; T −n )y(1; T −n+1 )} y(n + 1; )y(n; ) = i=1

and now the result follows from Eq. (11.2). The operator T can be for example Tt = t +1; T : R → R; in fact in this case we have the equation y(n + 1; t + 1) = a(t)y(n; t); n ∈ N; t ∈ R; or Tt = (c1 t1 ; c2 t2 ); c1 ; c2 ∈ R; T : R 2 → R 2 which yields the equation y(n + 1; c1 t1 ; c2 t2 ) = a(t1 ; t2 )y(n; t1 ; t2 ); n ∈ N; t1 ; t2 ∈ R: Finally, we shall consider the following dierence-dierential equation
1 y(n; t) + an

@ 2 y(n; t) = 0; @t 2

n ∈ N; t ∈ R0 :

(E23 )

Theorem 11.2. Let a : N → R be such that an ¡ for all n ∈ N and some  ∈ R+ : Then; (E23 ) possesses both oscillatory and nonoscillatory solutions in the sense of De nition 11:2. Proof. We shall seek the solution of (E23 ) in the form y(n; t) = u(n)v(t): For this, let y to be of the form   n−1 Y y(n; t) = (1 + Caj ) v(t); n ∈ N; t ∈ R0 ; (11.3) j=1

where C is an arbitrary constant. We take C such that 1 + Can 6= 0 for all n ∈ N: On substituting Eq. (11.3) in (E23 ); we obtain   n−1 Y an  (1 + Caj ) [v00 (t) + Cv(t)] = 0: (11.4) j=1

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From Eq. (11.4) it follows that for Eq. (11.3) to be a solution of (E23 ) the function v must satisfy the dierential equation v00 (t) + Cv(t) = 0:

(11.5)

Let C = −d 2 be such √ that 1 −d 2 an ¿0 for all n ≥  ∈ N: This choice is always possible because for 0¡d¡1= ; we have 1 − d 2 an ¿1 − d 2 ¿0: For such a C the sequence Q n−1 { j=1 (1 − d 2 aj )}∞ n=1 is nonoscillatory for n ≥ ; and sgn

n−1 Y

(1 − d 2 aj ) = sgn

j=1

−1

Y

(1 − d 2 aj )

j=1

for all n ≥ : Furthermore, every solution of Eq. (11.5) Q n−1is in this case nonoscillatory, so the same is for y(n; t) = u(n)v(t) with u(n) = [ j=1 (1 + Caj )]: Similarly, if we take C = d 2 then every solution of Eq. (11.5) is oscillatory on [0; ∞): Hence, for each k ∈ N the section   k−1 Y y(k; t) =  (1 + d 2 aj ) v(t) j=1

is oscillatory on [0; ∞): Therefore, the solution y(n; t) is oscillatory in the sense of De nition 11.3 and hence in the sense of De nition 11.2. References [1] R.P. Agarwal, Dierence Equations and Inequalities, Marcel Dekker, New York, 1992. [2] R.P. Agarwal, J. Popenda, On periodic solutions of rst order linear dierence equations, Math. Comput. Modelling 22 (1) (1995) 11–19. [3] R.P. Agarwal, P.J.Y. Wong, Advanced Topics in Dierence Equations, Kluwer Academic Publishers, Dordrecht, 1997. [4] C.D. Ahlbrandt, A.C. Peterson, Discrete Hamiltonian Systems: Dierence Equations, Continued Fractions and Riccati Equations, Kluwer Academic Publishers, Dordrecht, 1996. [5] R. Cristescu, Ordered Vector Spaces and Linear Operator, Abacus Press, Tunbridge Wells, Kent, 1976. [6] S. Elaydi, An Introduction to Dierence Equations, Springer, New York, 1996. [7] I. Gyori, G. Ladas, Oscillation Theory of Dierential Equations with Applications, Clarendon Press, Oxford, 1991. [8] A.J. Jerri, Linear Dierence Equations with Discrete Transform Methods, Kluwer Academic Publishers, Dordrecht, 1996. [9] W.G. Kelley, A.C. Peterson, Dierence Equations: An Introduction with Applications, Academic Press, New York, 1991. [10] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Dierence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [11] V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. [12] V. Lakshmikantham, D. Trigiante, Dierence Equations with Applications to Numerical Analysis, Academic Press, New York, 1988. [13] R.E. Mickens, Dierence Equations: Theory and Applications, 2nd ed., Van Nostrand Reinhold, New York, 1990.

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[14] R.E. Mickens, Nonstandard Finite Dierence Models of Dierential Equations, World Scienti c, Singapore, 1994. [15] J. Popenda, On the system of nite dierence equations, Fasc. Math. 15 (1984) 119 –126. [16] J. Popenda, B. Szmanda, On the oscillation of solutions of certain dierence equations, Demonstr. Math. 17 (1984) 153 –164. [17] J. Popenda, One expression for the solutions of second-order dierence equations, Proc. Amer. Math. Soc. 100 (1987) 87–93. [18] J. Popenda, Oscillation and nonoscillation theorems for second-order dierence equations, J. Math. Anal. Appl. 121 (1987) 34 –38. [19] J. Popenda, E. Schmeidel, Nonoscillatory solutions of third-order dierence equations, Portug. Math. 49 (1992) 233 –239. [20] J. Popenda, The oscillation of solutions of dierence equations, Comput. Math. Appl. 28 (1994) 271–279. [21] A. N. Sharkovsky, Y.L. Maistrenko, E.Y. Romanenko, Dierence Equations and their Applications, Kluwer Academic Publishers, Dordrecht, 1993. [22] P.J.Y. Wong, R.P. Agarwal, Oscillation theorems and existence of positive monotone solutions for second order nonlinear dierence equations, Math. Comput. Modelling 21 (1995) 63 – 84. [23] P.J.Y. Wong, R.P. Agarwal, Oscillation theorems for certain second order nonlinear dierence equations, J. Math. Anal. Appl. 204 (1996) 813 – 829. [24] P.J.Y. Wong, R.P. Agarwal, Summation averages and the oscillation of second order nonlinear dierence equations, Math. Comput. Modelling 24 (9) (1996) 21–35. [25] P.J.Y. Wong, R.P. Agarwal, Oscillation criteria for nonlinear partial dierence equations with delays, Comput. Math. Appl. 32 (6) (1996) 57 – 86.