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Asymptotic Behavior of a Poincaré Difference EquationMihály Pituk aa Department of Mathematics and Computing , University of Veszprém , Hungary, P.O. Box 158, 8201 VeszprémPublished online: 29 Mar 2007.

To cite this article: Mihály Pituk (1997) Asymptotic Behavior of a Poincaré Difference Equation, Journal of Difference Equations and Applications, 3:1,33-53, DOI: 10.1080/10236199708808083

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Asymptotic Behavior of a Poincare Difference Equation MIHALY PiTUK Department of Mathematics and Computing, University of Veszprem, P. 0. Box 158, 8201 Veszprem, Hungary

In this paper. we describe the asymptotic behavior of the solutions of the difference equation

P

x(n + 1 ) = (a, + p, (n) )s ( t z - i). ,=n

provided that the characteristic equation of the corresponding equation with constant coetticients has a dominant root and the sequences (F, (n) ) (0 5 i 5 k ) are in I , and of bounded variation. The main tools in the proof are perrurhation theorems on asymptotic constancy and stability of the solutions. decomposition in the variation-of-constants formula and estimates on the complementary spaces.

AMS No. 39A10 KEYWORDS: Poincare difference equation. asymptotic behavior. variation of constants formula. uniform stability, asymptotic constancy (Rccc~v~.LI ML71 IS. 199.5; ir! fir?al fort?? Mo~wnber 20. 1995)

1. INTRODUCTION

Consider the difference equation

k

x (n + 1) = 2 (a , + p , (n ) ) x(n - i) i=o

where a, f 0 and

lim p,(n) = 0 (0 r i r k). n-Px

A fundamental result in the analysis of the asymptotic behavior of the solutions of (1.1) is a theorem due to H. Poincark (see, e.g., [I. Theorem 2.13.11) which states that if the roots A,,. X ,,..., X, of the polynomial

This work wa\ started while the author was visiting University of Pau (France). The author is grateful to Professor Ovide Arino for his hospitality and especially for his helpful idea concerning the proof of Theorem 4.2,

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M. PITIJK

have distinct moduli then every nontrivial solution .u of (1 . l ) satisfies

x(n + 1 ) lim = A, ,,+% x(n)

for some i E (0. 1: . .. k ] . Formula (1.4) does not yield an asymptotic approximation of the solutions. In

order to get more precise information about the asymptotic behavior of the solutions we need to make stronger assumptions on p,'s. Let us briefly mention some known results along this line. Evgrafov [5] has proved that if the zeros A , . A , . . . . . A, of (1.3) are distinct and

then (1.1) has solutions so. 1,: . ., x-, such that

.u,(n) = A: (1 + o(1)) as n 3 x (0 5 i 5 k ) . (1.6)

Further results were obtained by Gelfond and Kubenskaya [6] and Coffman [2] bv giving estimates of the o(1) term in (1.6). In a recent paper [16] Trench has shown that the conclusion of Evgrafov's theorem remains valid if (1.5) is replaced by weaker assumption allowing conditional convergence of the series Xzl,p,(n) (0 5 i 5 k ) (see [16, Theorem 31). Some of the previous results were extended to

systems of recurrence equations (see Mat6 and Nevai [14], Li [13] and Trench 1171). The aim of the present paper is to establish new asymptotic formulae for the

solutions of (1.1). In our investigations we will assume that the polynomial (1.3) has a dominant root Ao, i.e.. A, is a simple root of (1.3) and all other roots satisfy A < h,l. The perturbations are assumed to satisfy

and

which are substantially weaker than (1.5). Our main theorem shows that under the above assumptions the solutions of (1.1) satisfy the asymptotic formula

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A POINCARE DIFFERENCE FQ1 JATTON 35

I 1 k

x(n) = n (A, + c 2 Ao1pi(v)) ( F ; + o(1)) as n + x (1.9) 1'=11, i=0

where c is a nonzero constant (described in Theorem 4.1) independent of s and F; is a constant depending on solution I. From (1.9) one can observe the following interesting phenomenon. If (1.5) is fulfilled then (1.9) reduces to

where 5 is a constant. That is, in that case the terms p,(n) have no influence on the asymptotic behavior of the solutions. However. if (1.5) is not satisfied (e.g., when pi(n) = lln). then the asymptotic behavior of the solutions depends not only on the dominant root A,. but also on the perturbation terms p,(rz) (0 5 i 5 k ) .

Our main result concerning Eq: (1:1) is fnrmulated in Thenrorn 4.1 in Section 4. Its proof is based on a new result (cf. Theorem 4.2) on asvnzptutic constancy (convergence to a constant at infinity) of the solutions of certain nonautonomous linear difference equations. To prove the above theorems. we use a technique similar to the perturbation technique known from the theory of delay differential equations (see [9], [lo]. [ll]).

The paper is organized as follows. In Section 2, after introducing some notations we formulate some auxiliary results which will be useful in the proof of our main theorems. In Section 3, we show that a linear perturbation with I,-coefficients of a uniformly stable linear equation has the same convergence and stability properties than the original equation. Finally, in Section 4, we give a proof of our main theorem about the asymptotic behavior of the solutions of (1.1). At the end of the section a conjecture is formulated about the existence of the solutions of (1.1) with certain asymptotic behavior.

2. PRELIMINARIES

Let N and @ denote the set of nonnegative integers and the set of complex numbers, respectively. Consider the nonhomogeneous linear difference equation

where (1,: N + C (0 5 i 5 k ) and g: N -+ C. Denote by 9 the set of initial sequences, i.e.

The norm of an initial sequence cp E 9. is defined by

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36 M. PITUK

For any given initial point no E N and initial sequence cp E 3 . there is a unique solution v of (2.1) through (no, rp), denoted by y = y(.; 4,. c p ) . such that it satisfies (2.1) for i 7 r no and

!(no + I : rlo. 9 ) = rpu) for j E ( - k . - k t 1.. .. 0).

The homogeneous equation corresponding to Eq. (2.1) has the form

k

x(n + 1) = 2 a,(n)x(n - i ) . (2.2) i-0

For n, E N fixed, let u( . . no) denote the solution of (2.2) through (no. qo). where

0 for j E {-k. - k + I; . ., -1) 1 for j = 0.

The function u = u(n, no), defined for n , n,, E N, n 1 n, - k, is called the jifimdamentd sohtion of (2.2 ).

The variiltiorz-of-co~~statzts fornrda for Eq. (2.1) can be stated as follows (cf. 14, Theorem 71).

Lemma 2.1. For any (nu, q) E N X 9. we have

where y(.; no. c p ) and x(.: n,,, c p ) are rhe corresponding solutions of (2.1) and (2.2). ~ o c n ~ ~ t i i ~ o l w . ,,., ,---. . -. I'

As it will be illustrated later (see the proof of Theorem 4.2), in some cases it is useful to consider the solutions in the "state-space" 4. For this purpose, we present a vector form of formula (2.3). Before we formulate it, we need the following notation. Given n E N and a sequence z: (n - k, n - k + 1; . .. n ) + @, denote by z,, an element of 9. defined by

Formula (2.3) now can be written in the following equivalent form:

If the coefficients in (2.2) are constants (independent of 11) . then x(n: n,,, r p ) = .u(n - n,,: 0. c p ) for 17 r 17, - k and cp E 9. Specially. u(n. n,,) = zi(n - n,,) for n r no - k , where 17 is defined by G(n) = u ( n , 0 ) for n e - k . In that case (2.3) and (2 .3 ' ) reduce to

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A POINCARE DIFFERENCE EQUATION 37

11-1

y(t7: no. cp) = x ( n : nu. cp) + ): ii ( n - j - l )g( j ) . n 2 n,, (2.4) I = " , ,

and

respectively. In the proof of our main theorems, we will use some properties of the

autonomous equation

where b, E @ (1 5 i 5 k) . 'I'he behavior of the solutions of (2.5) is determined by the roots of its characteristic equation

The following result is well-known.

Lemma 2.2. The soliiiioiis of (2.5) are asytnptoticalli; constan: if and only if p, = 1 13 LI durriiri~iiii i-uui 07" (2.6) (if., p,, = 1 i f i ~ i i i i i p l ~ i.oiii aiid all oilier roots sdsJCL. lpl < I).

If yo = 1 is a dominant root of (2.6), then every solution x of (2.5) can be written in the form

where c E C, qi's are polynomials and pi (1 5 i 5 m) are roots of (2.6) different from y,, = 1. Evidently. c = lim,,,, x(n) . The value of the last limit can be expressed in terms of the initial conditions.

Lemma 2.3. [3. Theorem 11 If p , = 1 is a dominant root of (2.6), then for every g E 9 the h i t

exists rind its value is given by

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Let % and Y denote the set of constant initial sequences and the stable subspace of 9, respectively, i.e.

% = { c p E $Iq(-k) = cp(-k + 1 ) = . . . = cp(O)]

and

From representation (2.7). it follows that ever cp F $can he written uniquely in x - - - - the form cp = cp' + cp'. where cp* E 'e and rp E 9. This means that Scan be decomposed into a direct sum, .a = % @ Y. According to Lemma 2.3, the projections 6' and cp' of rp can be given explicitely by

where I(cp) is given by (2.8) and cp, E $is defined by cp,G) = 1 for j E { - k : . ., 01. Clearly, the projections of the solutions are of the same form:

for n r nu r 0 and cp E 9. The following lemma is a consequence of the Stable Subspace Theerem (see [!2, Theerem 4.71 and its p e f . see a!se [!5] fer a different proof). It provides estimates of the projections of the solutions on the above complementary spaces.

Lemma 2.4. Asszime that y, = 1 is a dominant root of (2.6). If p > 0 satisfies

where pi ( 1 s i s k ) are all (not necessarily distinct) roots of (2.6) different from yo = 1. then there exists a positive constant K such that the estimates

and

hold for all n 2 no r 0 and q E 9..

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3. L,-PERTURBATION OF A UNIFORMLY STABLE EQUATION

Consider the equation

where a,: M + C(0 I i I k ) . Throughout this section. we will assume that the zero solution of (3.1) is riniformly stable. i.e., for every E > 0 there exists 6 > 0 such that for every no E N and cp E 4with I l cp l l < 6, we have Ix(n; no, c p ) l < E for all n r no.

With Eq. (3.1). consider the perturbed equation

k

y(n + 1 ) = C (ui(n) + b,(n))y(n - i). i=n

Our aim in this section is to show that if the sequences bi: N -+ @ ( O 5 i 5 k ) are in I,, i.e.,

then Eq. (3.2) has the samc convergence and stability properties than the unperturbed equation (3.1).

Theorem 3.1. Suppose (3.3) holds. I f the zero solution of Eq. (3.1) is uniformly stable then so is the zero solution of Eq. (3.2).

Proof. By Lemma 2.1, we have

for all n E N and cp E 4.. Since the zero solution of (3.1) is uniformly stable. there exists a constant K SO that

for all n. j E M, n r j. Consequently, for n 1 n,,,

11-1 k

ly(n; no. c p ) l I sup Ix(n: n,,, c p ) J + K Z C IbiO')ll)l(i - i; I 2 o . ( F ) I

which by Gronwall's inequality implies

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40 M. PITUK

ly(n: no, c p ) l 5 sup Ix(n; no, c p ) l exp ( K n-l C C Ibi(j)l I 5 K * sup Ix(n; no.

where

Taking into account that the zero solution of (3.1) is uniformly stable, the last estimate implies the uniform stability of the zero solution of (3.2).

The next theorem deals with asymptotic constancy of a perturbed equation.

Theorem 3.2. Assume that the zero solution of (3.1) is uniformly stable and (3.3) holds. I f the solutions o,f Eq. (3.1) are asymptotically consfant, then so are the solutions of Eq. (3.2).

In the proof of Theorem 3.2 we need the following lemma.

Lemma 3.1. Assuine that a:: M x N i C is bounded ad, for every fixed j E N, rlze limit

C(j) = lim v(n, j) n+=

then. for each n,, E N,

the last series being absolutely convergent.

Proof. By the assumption on v. there exists a constant K, > 0 such that

and obviously the same estimate holds for F. From (3.4), it follows that g is bounded, i.e., there exists K, > 0 so that

Ig(n)l 5 K?. n E N .

Let N E N. Then, for n r N, we have

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A POINCARE DIFFERENCE EQUATION

Let E > 0 be given. Find N so large that

Since

N- l

lim C I tl(n, j ) - $j) I = 0, n+z j = r q j .

there exists n , ( r N) so that

Then for n 2 n , ,

which completes the proof of the lemma.

Proof of Theorem 3.2. Let (n,, c p ) E N x 4. By the variation-of-constants formula

for n 2 no. where

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42 M. PITUK

By Theorem 3.1, the solution )I(.; n,,, q ) is bounded which, together with (3.3), implies that the sequence {g(n)] is in I , . The asymptotic constancy of y ( - ; no. cp) now follows from (3.5) by Lemma 3.1.

The last result of this section is about asymptotic stability. Recall that the zero solution of (3.1) is usynzptotically stable if for every (tz,,. q ) E W x 3 .

lim x(n; n,, cp) = 0. "-7.

Theorem 3.3. Suppose (3.3) holds. If the zero solrrtion of Eq. (3.1) is iirziformly and asymptotically stable, then so is the zero solution of Eq. (3.2).

Proof. The statement of the theorem is an immediate consequence of (3.5) and Lemma 3.1.

4. MAIN RESULTS

We are in a position ro state and prove our main result concerning the Poincare difference equation

where ai E C(0 5 i 5 k ) , a, f 0, and pi: N --+ C(0 5 i r k ) .

Theorem 4.1. Asszme that the characteristic function

of the unperturbed equation

has a dominant root A,, E C. Fzirthermore. asstme that

and

Then every solution x of (4.1) can be written in the form

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A POINCARE DIFFERENCE EOUATION 43

where n,, E N is sllficiently large (see the Remark below,) and ( = t[x] is a constant depending on solution x. Moreover. there exists a sollition x of (4.1) for which 5[x] # 0.

Rernark. no in the statement of Theorem 4.1 is to be chosen such that each term of the product in (4.6) is different from zero. This condition is certainly satisfied for n,, large enough, since p,(n) -+ 0 as n -+ x (0 5 i 5 k ) and A, # 0.

Theorem 4.1 will be deduced from the following result on asymptotic constancy which is of independent interest.

Theorem 4.2. Consider thc cquation

k k

y(n + 1) - y(n) = C (hi + r , ( n ) ) ( y ( n ) - y(n - i)) + s,(i?)y(n - i ) (4.7) i= 1 r = C I

where bi E C. r; N -+ C(1 5 i 5 k ) and si: N + C(O 5 i 5 k ) . Assume that the sol~rrions of the autonomous equation

k

x (n + 1) - x ( n ) = 1 bi ( x (n ) - x(n - i ) ) (4.8) i= l

are asymptotic all^ constant, or, eqrrivalently, po = 1 is a doniinant root of the characteristic eq~iation

of Eq. (4.8). I f

lim ri(n) = 0 ( 1 5 i 5 k ) n+z

and

are satisfied, then the following statements are valid.

( i ) The zero solution o f Eq. (4.7) is ~in~formly stable. (ii) Tl7e solutions of Eq. (4.7) are asyinptotically constant.

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(iii) Eq. (4.7) hirs a solz~tinn \$.irk a non;en> limit at injinit?..

Proof. First we give a proof in the case when

s , (n)=O ( O s i s k ) .

That is. we consider the equation

k

z(n + 1) - z(n) = (b, + ri(n)) (z(n) - z(n - i)). (4.13) i- 1

Introduce linear mappings Y, 2,,: 4+

Y(g) = p(0j +

and

for cp E $and n E N. The norm of 2 is defined by

l l 3 = sup{lY(q)l Iq E 4 , Ilql l = 1).

Obviously. (4.10) implies that

Using the notation from Section 2, Eqs. (4.8) and (4.13) can be written as

and

Define

Y = (q E 3 1 lim x(n:O.q) = 0). n+y

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The assumption about Eq. (4.8) implies (cf. Section 2) that $can be written as a direct sum, 9 = .% @ Y Similarly as in Section 2, the values of the projections n, and TT, of cp E $,will be denoted by 7it(cp) = cp" and n,(cp) = cF', respectively.

Let no E N and cp E 4. By the application of Lemma 2.4. we conclude that if pi (1 5 i 5 k ) are roots of (3.9) different from p,, = 1 and p > 0 satisfies

then there exists a constant K independent of no and cp, such that

'f . ILT, (.,nu,cp)ll 5 ~llqll, n 2 n,, (4.15)

The variation-of-constants formula (cf. (2.4'), for Eqs. (4.8') and (4.13') gives

for n r no, where L? denotes the fundamental solution of (4.8), i.e., L7= x(.;O,cpo), where cp,(j) = 0 for j E {-k, - k + 1,. . ., - 1) and cpo(0) = 1. Applying projection n, to Eq. (4.17) and denoting for brevity x = x(.;n,,cp), z = z(.;no,cp), we obtain

n - l

Using the linearity of 2, and the fact that Lti(cp) = 0 for cp E (e and j E N, we get

Finally, since x: is independent of n (cf. (2.10)), we have

Applying projection ;rr, to Eq. (4.17) we obtain by a similar argument

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From (4.16) and (4.19). we obtain

(Here we used the fact that II i:ll 5 ~llq,llp" = ~ p " for n E N.) Choose q E (1, Up). Multiplying the last inequality by qn-"". we obtain

for n r no. Define

Then the last estimate yields for n r n,,,

where K~ = ~ q ( 1 - pq)-'. Consequently

Let N be so large that

1 sup I lY)l < -. j>N K l

(In view of (4.14) such N certainly exists.) Then for all n, r N and cp E 9, we have

where

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A POINCARF DTFFFRFNCF FQI JATTON

Consequently.

whenever n,, e N and cp E 9. Hence

Y lim z,, (.; no, cp) = 0. n+7-

Let no r N. where N is chosen as before. From (4.18). in virtue of (4.15) and (4.20). we obtain for m > n 2 no,

where K~ is a constant. Since q > 1, {z:] is a Cauchy sequence (in $). Therefore the limit lirn,,, z: = iim,,, ;:(.; no, cp) exists in 4. But

f z(n; n,,. q ) = z,,(O; n,, q ) = zll(O; no, cp) + zT(0; no, cp)

and hence (cf(4.21))

which completes the proof of statement (ii). Note that statement (iii) is trivial in case (4.12), since each constant sequence is a solution of (4.13).

Now we prove that the zero solution of (4.13) is uniformly stable. From (4.18), according to estimates (4.15) and (4.20). it follows for all n e no r N and cp E 4.

where K~ is a constant independent of n , and cp. That is,

f . llz,, (., n,, cp)ll 5 ~, l lcp l l (4.22)

whenever n L n,, 2 N and cp E 9. Estimates (4.20) and (4.22). imply

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48 M. PITUK

for all rz r rz, r Nand cp E 4, where K~ = K~ + K ~ . On the other hand. it is an easy consequence of Gronwall's inequality that for every fixed N E N there exists a constant K~ = K ~ ( N ) such that

for all 0 5 no r n I N and cp E 4. Finally, from (4.23) and (4.24) we see that

for all n, no E N. n r nu, and cp E 4. where K, = max{K5. Kg, KSK6]. Since K, is independent of no and cp. this implies the uniform stability of the zero solution of (4.13). Thus, we have proved the theorem in case (4.12).

Consider now the general case. According to the previous part of the proof, the zero solution of (4.13) is uniformly stable and the solutions of (4.13) are asymptoti- cally constant. By assumption (4.11). Eq. (4.7) is an /,-perturbation of (4.13). Consequently, statements (i) and (ii) of the theorem are immediate consequences of Theorems 3.1 and 3.2. To prove statement (iii). suppose for contradiction that every solution of (4.7') tends to zero at infinity. This means that the zero solution of (4.7) is uniformly and asymptotically stable. By Theorem 3.3, perturbing Eq. (4.7) by a term - s,(n)x(n - i), where {s,(n)] (0 I i 5 k) are in I,, we obtain an equation with the same stability properties. Therefore the zero solution of the equation

k

x(n + 1) - x(n) = x (b, + r,(n)) (x(n) - x(n - i)) i= 1

k

= x (bi + ri(n)) (x(n) - x(n - i)) i= I

is asymptotically stable. Since each constant sequence is a solution of the last equation. asymptotic stability cannot hold and we have a contradiction.

The proof of Theorem 4.2 is complete.

Before we give a proof of Theorem 4.1. we state an easy lemma on I,-sequences.

Lemma 4.1. Let pi: fW -+ @, i = 1, 2. If

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Proof. The statement is an immediate consequence of the Schwarz inequality:

Proof of Theorem 4.1. We shall prove Theorem 4.1 by applying Theorem 4.2. To this aim. consider the transformation

where

An easy computation shows that y fulfils

where

and

We may write

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50 M. PITUK

where In denotes thc main value of the logarithmic function By the Taylor theorem

1 lnih,, + ;) = In A,, + - : + O(:-) a\ : 1 O

A,,

which, together with (4.1) and Lemma 4.1, ~mpllcs that

where {v(n)] is in I,. Using the fact that exp : = 1 + z + 0 ( z 2 ) as z + 0 and Lemma 4.1 again, we obtain

where { ~ ( r l ) ] is in I , . Consequently

where {:(TI)) is in I , . Since 2: ,, rr1h,'-' = I . Eq. (4.26) can be written in the form

where

Our aim is t o apply Theorem 1.2 to Fq. (4.26'). Observe that

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hr = lim c,(n) = -a,h,'-' (1 5 i 5 k ) . 11+7.

Therefore the limiting autonomous equation of (4.26') is the following

A-

x(n + 1) - x(n) = - x ark,'-' (.u(n) - x(t1 - i)) 1- 1

The characteristic equation of (4.27) has the form

It is easily seen that p i q a root of (4.28) if and only if A = uA,, is a root o f f (defined by (4.2)). Consequently, if A,, is a dominant root off. then p,, = I is a dominant root of (4.28). Now we show that {s(n)J is in 1,. Rewrite s(n) as follows:

From assumption (4.5), it follows that the first sum on the right-hand side of the last equation is in I,. Further, we have

Consequently, the sequence {s(n)J is in 1, and all the hypotheses of Theorem 4.2 are satisfied. By the application of Theorem 4.2 we conclude that the solutions of Eq. (4.26) are asymptotically constant. Moreover, there exists a solution of (4.26) with a nonzero limit at infinity. In view of relation (4.25) between the solutions of Eqs. (4.1) and (4.26). this completes the proof of Theorem 4.1.

Example. Consider the equation

The roots of the characteristic function f(A) = A - 4 + 5A-I - 2 K 2 of the corresponding equation with constant coefficients are A,, = 2, A , = A, = 1. Clearly,

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52 M. PITUK

A, = 2 is a dominant root and Theorem 4.1 can be applied. (Note that the previous results ([2], [5], [a], [16]) do not apply to Eq. (4.29).) From (4.6), it follows by simple computations that for every solution x of (4.29) the limit

x ( n ) ([XI = lim - n+l n2"

exists and is finite. Moreover. there exists a solution x of (4.29) such that ([x] Z 0, i.e., (4.30) gives a genuine asymptotic representation.

We close this section by a conjecture related to Theorem 4.1.

Conjecture. Suppose that A, is a simple root of the characteristic equation

0.f Eq. (4.3) atzd all other roots of (4.31) satisfy IAI f IX-I. If (4.4) and (4.5) are fil@lIed. then Eq. (4.1) has a solution x such that

where f is the characteristic filnctiorz defined by (4.2) and rz, is szificierztly large. The second statement of Theorem 4.1 confirmes the conjecture in the case when

A, is a dominant root of (4.31).

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