with regularly varying coefficients

Asymptotic representation of intermediate solutions to a cyclic systems of second-order difference equations

with regularly varying coefficients

Aleksandra B. Kapeši´c

University of Niš, Faculty of Science and Mathematics, Department of Mathematics, Višegradska 33, 18000 Niš, Serbia

Received 25 February 2018, appeared 18 July 2018 Communicated by Stevo Stevi´c

Abstract. The cyclic system of second-order difference equations

∆(pi(n)|∆xi(n)|^αi−1∆xi(n)) =qi(n)|xi+1(n+1)|^βi−1xi+1(n+1),

for i = 1,N where x_N+1 = x₁, is analysed in the framework of discrete regu- lar variation. Under the assumption that α_i and β_i are positive constants such that α1α2· · ·αN>β1β2· · ·βNandp_iandq_iare regularly varying sequences it is shown that the situation in which this system possesses regularly varying intermediate solutions can be completely characterized. Besides, precise information can be acquired about the asymptotic behavior at infinity of these solutions.

Keywords: system of difference equations, Emden–Fowler type difference equation, nonlinear difference equations, intermediate solutions, asymptotic behavior, regularly varying sequence, discrete regular variation.

2010 Mathematics Subject Classification: 39A22, 39A12, 26A12.

1 Introduction

There has been some recent interest in studying of various systems of difference equations.

Since the mid of nineties there has been a considerable interest in symmetric systems (see, e.g., [3,38–40,44] and the references therein). If some parameters in symmetric systems are modified then are obtained more general systems which are now frequently called close-to- symmetric systems (see [49] and [50]); for some other systems of the type see, e.g. [37,41,47,48]

and the references therein. Multidimensional extensions of symmetric and close-to-symmetric systems are called cyclic systems of difference equations. Systems of the type were studied, for example, in [13,45,46].

The system of nonlinear difference equations which will be studied in this paper is the following cyclic one:

∆(p_i(n)|_∆xi(n)|^αi−1_∆xi(n)) +q_i(n)|x_i+1(n+1)|^βi−1x_i+1(n+1) =0, (E)

BEmail: aleksandra.trajkovic@pmf.edu.rs

wherei=1,N, x_N+1 =x₁, n∈N, and following conditions hold:

(a) α_i andβ_i,i=_1,N are positive constants such thatα₁α₂·_{. . .}·α_N > β₁β₂·_{. . .}·β_N; (b) p_i ={p_i(n)},q_i ={q_i(n)}are positive real sequences;

(c) All p_i,i=1,N simultaneously satisfy either

(I) S_i =

∑

1

p_i(n)^1/αi =_∞, or

(II) S_i =

∑

1

p_i(n)^1/αi <_∞.

In our case, whenα₁α₂· · ·α_N > β₁β₂· · ·β_N, the system (E) is said to be sub-half-linear. If opposite inequality hold then the system is super-half-linear and if equality hold then system is called half-linear.

We say that a solutionx= {x(n)}= {(x₁(n)_,x₂(n)_{, . . . ,}x_N(n))} ∈^N_R× · · · ×^N_R,^N_R= {f|f :N→_R}, of(E)oscillatesif for somei∈ {1, 2, . . . ,N}, and for every integern₀>0 there existsn>n₀such thatx_i(n)x_i(n+1)<0. A solutionx(n)of(E)is said to benonoscillatoryif there exists an integern₀>0 such that for eachi=_1,N,x_i(n)6=0 forn ≥n₀.

It can easily be seen that all components of positive solutionxof system (E) satisfy

c_i ≤x_i(n)≤C_i·P_i(n), i=1,N if (I) holds (1.1) or

k_iπ_i(n)≤x_i(n)≤K_i, i=1,N if (II) holds (1.2) where P_i(n) = _∑ⁿ_k=⁻₁^{1 1}

pi(_k)^1/αi and π_i(n) = _∑^∞_k=n ¹

pi(_k)^1/αi for i = 1,N, and c_i,C_i,k_i and K_i are positive real constants.

Indeed, if (I) holds, then is easy to see that ∆x_i(n) > 0, for i = 1,N. This implies left inequality in (1.1). Also,p_i(n)_∆xi(n)^αi,i=1,Nare decreasing, so there exist positive constants b_i such that p_i(n)_∆xi(n)^αi ≤b_i. From previous, follows that

x_i(n)≤x₁(n) +b

1 αi

i P_i(n), i=1,N.

How allP_i(n)are increasing and divergent we get right inequality in (1.1).

Similarly, if (II) holds, then ∆xi(n) < 0, for i = 1,N, so right inequality in (1.2) holds.

On the other side, how −p_i(n) (−_∆x_i(n))^αi ≤ 0, and decrease for i = _1,N it follows that

−p_i(n) (−_∆xi(n))^αi ≤ h_i, h_i ∈_R⁺. Since, x_i,i= 1,Nare decreasing and positive, all x_i(_∞) = lim_n→_∞x_i(n)are finite and

x_i(n)≥ x_i(_∞) +h

1 αi

i π_i(n).

Using thatπ_i(n),i=1,N are decreasing and tends to zero we get left inequality in (1.2).

In the case (I) for each component x_i of solution x only one of next three possibilities holds:

(S1) lim_n→_∞ ^xi(n)

Pi(_n) =const>0,

(I M1) lim_n→_∞x_i(n) =_∞, lim_n→_∞ ^xi(n) Pi(n) =0,

(AC) lim_n→_∞x_i(n) =const>0.

In the case (II) for each component x_i of solution x only one of next three possibilities holds:

(AC) lim_n→_∞x_i(n) =const>0, (I M2) _limn→∞x_i(n) =_{0, limn→∞} ^xi(n)

πi(n) = _∞, (S2) limn→_∞ ^xi(n)

π_i(n) =const>0.

We consider a solutions whose all components are the same type. Solutions where the components are different types have not yet been considered in the existing literature. How we see, in the case (I) we have only increasing solutions. Solution is increasing if all its components are increasing sequences. Solutions of type (S1) are asymptotically equivalent to constant times P_i(n)and solutions of type(AC)are asymptotically equivalent to constant.

Solutions of type (I M1) are called intermediate solutions. In the case (II) we have only decreasing solutions. In this case, we again have solutions which are asymptotically equivalent to constant and solutions which are asymptotically equivalent to constant timesπ_i(n),n→_∞, marked as (S2). Here, also, we have intermediate solutions(I M2)but different kind then in the first case.

Our aim in this paper is to observe intermediate solutions(I M1)and(I M2), and to answer to following two questions:

1. Is it possible to establish sufficient and necessary conditions for the existence of inter- mediate solution of system (E)?

2. Is it possible to establish the unique explicit asymptotic formula for this solutions?

The results obtained in this paper can be extended to a system of Nequations of the first order only when Nis even.

The asymptotic behavior of nonoscillatory solutions for second-order difference equation has been studied in many papers, see, e.g. [2,6–10,19,34–43], the monograph [1] and refer- ences therein. Oscillation and existence criteria for positive solutions of discrete systems were considered in [28–30], however, in the existing literature, there are no results for cyclic system of difference equations of second order.

The recent development of asymptotic analysis of ordinary differential equations and cyclic system of differential equations by means of regular varying functions (see [24–27,32–42]

and monograph [31] for results up to 2000.), suggests investigating the discrete problem in the framework of regularly varying sequences. Thus, we limit ourselves to the system (E) with coefficientsp_i ={p_i(n)}_,q_i ={q_i(n)}which are regularly varying sequences and we es- tablish necessary and sufficient conditions for the existence of intermediate regularly varying solutions of (E) and obtain precise asymptotic representation of such solutions.

The theory of regularly varying sequences, sometimes called Karamata sequences, was initiated in 1930 by Karamata [20] and further developed in seventies by Galambos, Seneta and Bojani´c in [5,12] and recently in [11]. However, until the paper of Matucci and ˇRehák [35], relation between regularly varying sequences and difference equations has never been under consideration. In [35], as well as in succeeding papers [34,36,43], theory of regu- larly varying sequences has been further developed and applied in asymptotic analysis of second-order linear and half-linear difference equations. However, to our knowledge, theory

of regularly varying sequences has not been used for asymptotic analysis of any other type of second-order nonlinear difference equation, except by Agarwal and Manojlovi´c in [2], Kapeši´c and Manojlovi´c in [19] and Kharkov in [21] and [22]. Also, Kharkov in [23] give asymptotic representation of solutions of k-th order difference equations of Emden–Fowler type. Assum- ing that coefficients are normalized regularly varying sequences (introduced by Matucci and Rehak in [34]), asymptotic forms of positive intermediate solutions of Emden–Fowler second- order difference equation has been established in [2]. Thus, the purpose of this paper is to proceed further in this direction and to establish results which can be considered as a discrete analogue of results in the continuous case (see e.g. [17,18]).

Throughout this paper extensive use is made of the symbol ∼ to denote the asymptotic equivalence of two positive sequences, i.e.

x_n ∼y_n, n→_∞ ⇔ lim

n→_∞

y_n x_n =1.

The main results are given in Section 4. Intermediate solution of system (E) are solution of system of equations

x_i(n) =c_i+

n−1 k

∑

1 p_i(k)

∑

q_i(s)x_i+₁(s+1)^βi

!_α¹

i

, i=1,N if (I)holds, (1.3)

x_i(n) =

∑

1 p_i(k)

k−1 s

∑

q_i(s)x_i+1(s+1)^βi

!¹

αi

, i=1,N if (II)holds, (1.4) for some constants n0 ≥ 1 and c_i > 0. It follows therefore that intermediate solution of (E) satisfies asymptotic relations

x_i(n)∼

n−1 k

∑

1 p_i(k)

∑

q_i(s)x_i+1(s+₁)^βi

!_α¹

i

, n→_∞, i=_1,N, (1.5) or

x_i(n)∼

∑

1 p_i(k)

k−1 s

∑

q_i(s)x_i+1(s+1)^βi

!_α¹

i

, n→_∞, i=1,N (1.6)

in case(I)or(II)respectively.

The proof of our main results are essentially based on the fact that a through knowledge of the existence and asymptotic behavior of regularly varying solution of (1.5) and (1.6) can be acquired. In fact, one direction of proof of main theorems is an immediate consequence of manipulation of (1.5) and (1.6) by means of regular variation. The other direction is proved in two steps – first showing the existence of solution of the system of relations (1.5) and (1.6) with the help of fixed point techniques and in the next step using Stolz–Cesarò theorem showing that such solution is regularly varying.

Our main tools are, besides theory of regularly varying sequences presented in Section 2, the fixed point technique and Stolz–Cesarò theorem. Thus, we recall two variants of Stolz–

Cesarò theorem as well as Knaster’s fixed point theorem [1, Theorem 5.2.1].

Lemma 1.1. If f = {fn} is a strictly increasing sequence of positive real numbers, such that lim_n→_∞ f_n=∞, then for any sequence g={g_n}of positive real numbers one has the inequalities:

lim inf

n→_∞

∆f_n

∆gn

≤lim inf

n→_∞

f_n gn

≤lim sup

n→_∞

f_n gn

≤lim sup

n→_∞

∆f_n

∆gn.

In particular, if the sequence{_∆f_n/∆gn}has a limit then

nlim→_∞

f_n

g_n = lim

n→_∞

∆f_n

∆gn

. (1.7)

Lemma 1.2. Let f ={f_n}, g= {g_n}be sequences of positive real numbers, such that (i) lim_n→_∞ f_n =lim_n→_∞g_n =0;

(ii) the sequence g is strictly monotone;

(iii) the sequence{_∆fn/∆gn}has a limit.

Then, a sequence{f_n/g_n}is convergent and(1.7)hold.

Lemma 1.3(Knaster’s fixed point theorem). Let X be a partially ordered Banach space with ordering

≤. Let M be a subset of X with the following properties: the infimum of M belongs to M and every nonempty subset of M has a supremum which belongs to M. Let F : M → M be an increasing mapping, i.e. x ≤y impliesFx ≤ Fy.ThenF has a fixed point in M.

2 Regularly varying sequences

We state here definitions and some basic properties of regularly varying sequences which will be essential in establishing our main results on the asymptotic behavior of nonoscillatory solutions stated and proved in the next section. For a comprehensive treatise on regular variation the reader is referred to Bingham et al. [4].

Two main approaches are known in the basic theory of regularly varying sequences: the approach due to Karamata [20], based on a definition that can be understood as a direct discrete counterpart of simple and elegant continuous definition (see Definition 2.3), and the approach due to Galambos and Seneta, based on purely sequential definition.

Definition 2.1 (Karamata [20]). A positive sequencey = {y(k)},k ∈ _Nis said to beregularly varying of indexρ ∈_Rif

klim→_∞

y([λk])

y(k) =λ^ρ for∀λ>0, where[u]denotes the integer part ofu.

Definition 2.2 (Galambos and Seneta [12]). A positive sequencey = {y(k)},k ∈ _Nis said to beregularly varying of indexρ∈_Rif there exists a positive sequence{α(k)}satisfying

klim→_∞

y(k)

α(k) =C, 0<C<_∞, lim

k→_∞k∆α(k−1) α(k) =ρ.

Definition 2.3. A measurable function f : (a,∞) → (0,∞) for some a > 0 is said to be regularly varying at infinity of indexρ∈_Rif

tlim→_∞

f(λt)

f(t) =λ^ρ for all λ>0.

Ifρ = 0, thenyis said to beslowly varying. The totality of regularly varying sequences of indexρand slowly varying sequences denoted, respectively, byRV(ρ)andS V.

Bojani´c and Seneta have shown in [5] that Definition2.1and Definition2.2are equivalent.

The concept of normalized regularly varying sequences were introduced by Matucci and Rehak in [35], where they also offered a modification of Definition2.2, i.e. they proved that second limit in Definition2.2can be replaced with

klim→_∞k∆α(k) α(k) = ρ.

Definition 2.4. A positive sequencey= {y(k)},k∈_Nis said to benormalized regularly varying of indexρ∈ _Rif it satisfies

klim→_∞

k∆y(k) y(k) =ρ.

Ifρ=0, thenyis called anormalized slowly varying sequence.

In what follows, N RV(ρ) and N S V will be used to denote the set of all normalized regularly varying sequences of indexρand the set of all normalized slowly varying sequences.

Typical examples are:

{logk} ∈ N S V, {k^ρlogk} ∈ N RV(ρ), {1+ (−1)^k/k} ∈ S V \ N S V.

There exist various necessary and sufficient conditions for a sequence of positive numbers to be regularly varying (see [5,12,34,35]) and consequently each one of them could be used to define regularly varying sequence. The one that is the most important is the following Representation theorem (see [5, Theorem 3]), while some other representation formula for regularly varying sequences were established in [35, Lemma 1].

Theorem 2.5(Representation theorem). A positive sequence{y(k)},k ∈ _Nis said to be regularly varying of indexρ ∈_Rif and only if there exists sequences{c(k)}and{δ(k)}such that

klim→_∞c(k) =c0 ∈(0,∞) and lim

k→_∞δ(k) =0, and

y(k) =c(k)k^ρ exp

∑

δ(i) i

! .

In [5] very useful imbedding theorem was proved, which gives possibility of using the continuous theory in developing a theory of regularly varying sequences. However, as noticed in [5], such development in not generally close and sometimes far from a simple imitation of arguments for regularly varying functions.

Theorem 2.6(Imbedding theorem). If y= {y(n)}is regularly varying sequence of indexρ ∈ _R, then function Y(t) defined on [0,∞) by Y(t) = y([t]) is a regularly varying function of index ρ.

Conversely, if Y(t)is a regularly varying function on[0,∞)of indexρ, then a sequence{y(k)}, y(k) = Y(k), k ∈_Nis regularly varying of indexρ.

Next, we state some important properties ofRV sequences useful for the development of asymptotic behavior of solutions of (E) in the subsequent section (for more properties and proofs see [5,34]).

Theorem 2.7. Following properties hold:

(i) y∈ RV(ρ)if and only if y(k) =k^ρl_k,where l={l(k)} ∈ S V.

(ii) Let x ∈ RV(ρ₁) and y ∈ RV(ρ₂). Then, xy ∈ RV(ρ₁+ρ₂), x+y ∈ RV(ρ), ρ = max{ρ₁,ρ₂}and1/x ∈ RV(−ρ₁).

(iii) If y∈ RV(ρ), thenlim_k→∞ ^y(_y^k₍⁺_k)¹⁾ =1.

(iv) If l ∈ S V and l(k)∼ L(k), k→_{∞, then L}∈ S V.

(v) If y ∈ N RV(ρ), then{n^−σy(n)}is eventually increasing for each σ < ρ and {n^−µy(n)} is eventually decreasing for eachµ>ρ.

In view of the statement(i)of the previous theorem, if fory ∈ RV(ρ)

klim→_∞

y(k) k^ρ = lim

k→_∞l(k) =const>0,

then y = {y(n)} is said to be a trivial regularly varying sequence of index ρ and is denoted by y ∈ tr− RV(ρ). Otherwise y is said to be a nontrivial regularly varying sequence of index ρ, denoted byy∈ ntr− N RV(ρ).

Next Theorem can be found in [2] for normalized regularly varying sequences, but it clearly hold for all regularly varying sequences because its proof is based on the Mean Value Theorem and property(iii)from Theorem2.7which holds for all RV sequences (not only for N RV).

Theorem 2.8. If f = {f(n)} ∈ RV is a strictly decreasing sequence, such thatlim_n→_∞ f(n) = 0, then for eachγ∈_R

nlim→_∞f(n)^−γ

∑

f(k)^γ−1 −_∆f(k) = ¹

γ. (2.1)

If g ={g(_n)} ∈ RV is a strictly increasing sequence such thatlimn→_∞g(_n) =_{∞, then}

nlim→_∞g(n)^−γ

n−1 k

∑

g(k)^γ−1_∆g(k) = ¹

γ. (2.2)

The following theorem can be concerned asthe discrete analog of the Karamata’s integration theorem and plays a central role in the proof of our main results in the Section 3. Proof of this Theorem can be found in [5] and [19] and partially in [43].

Theorem 2.9. Let l ={l(n)} ∈ S V.

(i) If α>−1, then limn→_{∞ n}α+1¹l(n)∑ⁿ_k=1k^αl(k) = ₁₊¹

α; (ii) If α<−1, then limn→_{∞ n}α+1¹l(n)∑^∞k=nk^αl(k) =−₁₊¹_α; (iii) If ∑^∞k=1

l(k)

k < _∞, then ∑^∞k=n l(k)

k ∈ S V and lim_n→_{∞ l}(¹n)∑^∞k=n l(k)

k =_∞;

(iv) If ∑^∞k=1 l(k)

k = _∞, then ∑ⁿk=1

l(k)

k ∈ S V and lim_n→_∞l(¹n)∑ⁿk=1 l(k)

k =_∞.

Remark 2.10. It is easy to see, in view od Theorem2.7(iii) and Theorem2.9(i), that forl∈ S V, ifα> −1, we have

n−1 k

∑

k^αl(k)∼ (n−1)^α+1l(n−1) α+1 ∼ ⁿ

α+1l(n) α+1 ∼

∑

k^αl(k), n→_∞, and since limn→_∞∑ⁿ_k=⁻1¹k^αl(_k) =∞, we also get

∑

k^αl(k)∼

∑

k^αl(k), n→_∞. If lim_n→_∞∑ⁿk=1k⁻¹l(k) =_{∞, we have}

∑

k⁻¹l(k)∼

∑

k⁻¹l(k), n→_∞.

Definition 2.11. A vector x ∈ ^N_R×. . .×^N_R is said to be regularly varying of index (ρ₁,ρ₂, . . . ,ρ_N) if x_i ∈ RV(ρ_i)for i = 1,N. If allρ_i are positive (or negative), thenx is called regularly varying of positive (or negative) index(ρ₁,ρ₂, . . . ,ρ_N). The set of all regularly vary- ing vectors of index(ρ₁,ρ₂, . . . ,ρ_N)is denoted byRV(ρ₁,ρ₂, . . . ,ρ_N).

3 Preliminaries and preparation results

In this section assume thatp_i ∈ RV(λ_i)andq_i ∈ RV(µ_i)and represent them with

p_i(n) =n^λil_i(n), q_i(n) =n^µim_i(n), l_i,m_i ∈ S V, i=1,N. (3.1) We also assume that all sequences p_i, i = 1,N satisfy either (I) or (II). Condition (I) (resp. (II)) holds if and only if

λ_i < α_i or λ_i = α_i and

∑

n⁻¹l_i(n)^−α1i =_∞, (3.2) resp.

λ_i > α_i or λ_i = α_i and

∑

n⁻¹l_i(n)^−α1i <_∞. (3.3) In this paper we do not consider cases when λ_i = α_i for one or all i(these cases lead to ρ_i = 0) because of computational difficulty. Therefore, we have requirements of positivity or negativity for the regularity indices of solutions.

Therefore, if the case(I)is satisfied, thenλ_i <α_iand for the sequencesP_i ={P_i(n)}given byP_i(n) =_∑ⁿ_k=⁻₁¹p_i(k)^−α1i_,i=_1,Nwe have

P_i(n)∼ ^αi α_i−λ_in

αi−λi

αi l_i(n)^−α1i. (3.4) In the case(II), when λ_i >α_i for the sequences π_i ={π_i(n)}given byπ_i(n) =_∑^∞_k=np_i(k)^−α1i, i=1,Nwe have

π_i(n)∼ ^αi λ_i−α_in

αi−λi

αi l_i(n)^−α1i_{. (3.5)}

In what follows to simplfy notation we denote A_N = α₁α₂·. . .·α_N, B_N = β₁β₂·. . .·β_N and use matrix

M=

























 1 ^β_α¹

1

β1β2

α1α2 . . . ^β_α^{1β2···βN−1}

1α2···αN−1

1 ^β_α²

2

β2β3

α2α3 . . . ^β_α^{2β3···βN−1}

2α3···αN−1

1 ^β_α³

3 . . . ^β_α^{3···βN−1}

3···α_N−1

. .. ... ... 1 ^β_α^N−1

N−1

1



























, (3.6)

whose elements will be denoted by M = (M_ij), where the lower triangular elements are omitted for economy of notation. In fact, the i−th row of (M_ij) is obtained by shifting the vector

1, β_i

α_i, β_iβ_i+1

α_iα_i+1

, . . . ,β_iβ_i+1· · ·β_i+(N−2) α_iα_i+1· · ·α_i+(N−2)

!

, α_N+j =α_j, β_N+j = β_j, j=1,N−2

(i−₁)-times to the right cyclically, so that the lower triangular elementsM_ij,i> j, satisfy the relation

M_ijM_ji = ^β1β2· · ·β_N

α₁α₂· · ·α_N, i> j, i=2,N.

The following theorem gives us necessary and sufficient condition for the existence of regularly varying solution x of positive index (ρ₁,ρ₂, . . . ,ρ_N) of the system of asymptotic relations (1.5).

Theorem 3.1. Let p_i ∈ RV(λ_i),q_i ∈ RV(µ_i) and suppose that λ_i < α_i, i = 1,N. The sys- tem of asymptotic relations (1.5) has regularly varying solution x ∈ RV(ρ₁,ρ₂, . . . ,ρ_N) with ρ_i ∈

0,^αi−_α^λi

i

,i=1,N if and only if 0<

∑

M_ijα_j−λ_j+µ_j+1

α_j < ^αi−λ_i α_i

1− ^BN A_N

, i=1,N (3.7)

holds, in which caseρ_i are uniquely determined by

ρ_i = ^AN A_N−B_N

∑

M_ijα_j−λ_j+µ_j+1

α_j , i=1,N (3.8)

and the asymptotic behavior of any such solution is governed by the unique formula

x_i(n)∼n^ρi







∏



 l_j(n)⁻

1 αjm_j(n)

1 αj

D_j





Mij





ANAN−_BN

, n→_∞, i=1,N. (3.9)

Proof. Letx∈ RV(ρ₁,ρ₂, . . . ,ρ_N)with allρ_i >0 be a solution of (1.5) expressed in the form x_i(n) =n^ρiξ_i(n), ξ_i ∈ S V, i=1,N. (3.10) From (3.4) and (1.1) eachρ_i must satisfyρ_i < ^αi−λi

αi ,i=1,N. Using (3.1) and (3.10), we have

∑

q_i(k)x_i+1(k+1)^βi ∼

∑

k=n

k^µi+βiρi+1m_i(k)ξ_i+1(k)^βi, n≥ n₀, i=1,N. (3.11)

The convergence of (3.11) as n → _∞ implies that µ_i +β_iρ_i+1 ≤ −1, i = 1,N. If for some i equality holds, then since

1 p_i(n)

∑

q_i(k)x_i+1(k+1)^βi

!¹

αi

∼n⁻

λi

αil_i(n)^−α1i

∑

k⁻¹m_i(k)ξ_i+1(k)^βi

!¹

αi

, (3.12) from (1.5) and Theorem2.9we find that

x_i(n)∼ ^αi α_i−λ_in

αi−λi

αi l_i(n)^−α1i

∑

k⁻¹m_i(k)ξ_i+₁(k+₁)^βi

!_α¹

i

∈ RV

α_i−λ_i α_i

as n → ∞. This implies that ρ_i = ^αi−λi

αi which is a contradiction with our assumption. It follows thatµ_i+β_iρ_i+1 <−1 fori=1,N. Application of Theorem2.9 to (3.11) gives

1 p_i(n)

∑

q_i(k)x_i+1(k+1)^βi

!_α¹

i

∼ ⁿ

−λi+µi+βiρi+1+1

αi l_i(n)^−α1im_i(n)^α1iξ_i+1(n)

βi αi

(−(µ_i+β_iρ_i+1+1))^α1i

, (3.13)

whenn → _∞,i = 1,N. Because x_i(n)→ _∞, n → ∞, we conclude from (3.13) that it must be (−λ_i+µ_i+β_iρ_i+1+1)/α_i ≥ −1,i=1,N. Here, also, the equality should be ruled out. If the equality holds for somei, then summing (3.13) fromn₀ ton−1 yields

x_i(n)∼ 1

α_i−λ_{i n}−1

k

∑

k⁻¹l_i(k)^−α1im_i(k)^α1iξ_i+1(k)

βi

αi ∈ S V, n→_∞,

which is impossible. Therefore, (−λ_i+µ_i +β_iρ_i+1+1)/α_i > −1, i = 1,N. Summing (3.13) fromn₀ ton−1 and applying Theorem2.9we conclude that

x_i(n)∼ ⁿ

−λi+µi+βiρi+1+1

αi +1

l_i(n)^−α1im_i(n)^α1iξ_i+₁(n)

βi αi

(−(µ_i+β_iρ_i+1+1))^{α1i −λi+µi+βiρi+1+1}

αi +1, n→_∞, i=_1,N. (3.14) From previous relation we see that

ρ_i = −λ_i+µ_i+β_iρ_i+1+1

α_i +1, i=1,N, ρ_N+1=ρ₁ which is equivalent with

ρ_i− ^βi

α_iρ_i+1= ^αi−λ_i+µ_i+₁

α_i , i=_1,N, ρ_N+1 =ρ₁. (3.15) We now have a linear cyclic system of equations in whichρ_i are unknown. To find ρ_i, let we denote with Athe coefficient matrix of system (3.15), i.e.

A= A β₁

α₁,β₂

α₂, . . . ,β_N α_N

=

























1 −^β_α¹

1 0 . . . 0 0

0 1 −^β2

α2 . . . 0 0 ... ... . .. ... ... ... ... . .. ... ... 0 0 0 . . . 1 −^βN−1

α_N−1

−^βN

αN 0 0 . . . 0 1

























. (3.16)

This matrix is nonsingular because, according to condition(a), det(A) =1− ^β1β2· · ·β_N

α₁α₂· · ·α_N >0, (3.17) so system (3.15) has an unique solution and matrix Ais invertible

A⁻¹ = ^AN

A_N−B_NM (3.18)

where matrix Mis given by (3.6).

Solving the system (3.15) we get the unique solutionρ_i, i= 1,N which is given explicitly by (3.8). From (3.8) we can see that all ρ_i satisfy 0 < ρ_i < ^αi−_α^λi

i , i = 1,N if and only if (3.7) holds.

Using (3.1) and (3.10) we can transform (3.12) in the following form x_i(n)∼ ⁿ

αi+1

αi p_i(n)^−α1iq_i(n)^α1ix_i+₁(n)

βi αi

D_i , n→_∞, (3.19)

where

D_i = (α_i−λ_i−α_iρ_i)^α1iρ_i, (3.20) for i= 1,N. Without difficulty we can obtain following explicit formula for eachx_i from the cyclic system of asymptotic relations (3.19)

x_i(n)∼









∏





 n

αj+1

αj p_j(n)⁻

1 αjq_j(n)

1 αj

D_j







Mij







ANAN−_BN

, n→_∞, i=1,N. (3.21)

Previous relation can be rewritten in the following form

x_i(n)∼n^ρi







∏



 l_j(n)⁻

1 αjm_j(n)

1 αj

D_j





Mij





ANAN−_BN

, n→_∞, i=1,N.

implying that regularity index of x_i is exactlyρ_i.

Suppose now that (3.7) holds and defineρ_i with (3.8) andD_i with (3.20). Denote

X_i(n) =









∏





 n

αj+1

αj p_j(n)⁻

1 αjq_j(n)

1 αj

D_j







Mij







ANAN−_BN

, i=1,N. (3.22)

Clearly,X_i ∈ RV(ρ_i),i=1,NandX_i’s satisfy the system of asymptotic relations (1.5), i.e.

n−1 k

∑

1 p_i(k)

∑

q_i(_s)_Xi+1(_s+₁)^βi

!_α¹

i

∼X_i(_n)_{, n}→_∞, i=_{1,N, (3.23)} for any n>n₁, whereX_N+1(n) =X₁(n)_{. Indeed,}X_i(n)can be expressed as

X_i(n) =n^ρiχ_i(n)_, χ_i(n) =







∏



 l_j(n)⁻

1 αjm_j(n)

1 αj

D_j





Mij





ANAN−_BN

, (3.24)

and using Theorem2.9, we obtain 1

p_i(n)

∑

q_i(k)X_i+1(k+1)^βi

!¹

αi

∼ ⁿ

ρ_i−1l_i(n)^−α1im_i(n)^α1iχ_i+1(n)

βi αi

(α_i−λ_i−α_iρ_i)^α1i

, and

n−1 k

∑

1 p_i(k)

∑

q_i(s)X_i+1(s+1)^βi

!¹

αi

∼ ⁿ

ρ_il_i(n)^−α1im_i(n)^α1iχ_i+1(n)

βi αi

D_i , (3.25)

asn→∞. Since for the elements of matrixM hold M_i+1,i

β_i α_i = ^BN

A_N, M_i+1,j

β_i

α_i = M_ij, forj6=i, (3.26) where M_N+1,j = M_1,j, j=1,N, relation (3.25) can be transformed as

l_i(n)^−α1im_i(n)^α1i

D_i χ_i+1(n)

βi

αi = ^li(n)^−α1im_i(n)^α1i D_i









∏



 l_j(n)⁻

1 αjm_j(n)

1 αj

D_j





M_i+1,j βi αi









ANAN−_BN

=







∏



 l_j(n)⁻

1 αjm_j(n)

1 αj

D_j





Mij





ANAN−_BN

=χ_i(n),

so from (3.25) we obtain thatX_i,i=_1,Nsatisfy (3.23).

In the same way, we can solve the second problem. Assuming that (II)holds, we are in position to find necessary and sufficient condition that system of asymptotic relations (1.6) possesses regularly varying solutionxof negative index(ρ₁,ρ₂, . . . ,ρ_N).

Theorem 3.2. Let p_i ∈ RV(λ_i),q_i ∈ RV(µ_i) and suppose that λ_i > α_i, i = 1,N. System of asymptotic relations(1.6)has regularly varying solutionx∈ RV(ρ₁,ρ2, . . . ,ρ_N)withρ_i ∈ ^αi−_α^λi

i , 0

, i=1,N if and only if

α_i−λ_i α_i

1− ^BN AN

<

∑

M_ijα_j−λ_j+µ_j+1

α_j <0 (3.27)

in which caseρ_i are given by(3.8)and the asymptotic behavior of any such solution is governed by the unique formula

x_i(n)∼









∏





 n

αj+1

αj p_j(n)⁻

1 αjq_j(n)

1 αj

W_j







M_ij







ANAN−_BN

, n→_∞, i=1,N, (3.28)

where

W_i = (λ_i−α_i+α_iρ_i)^α1i(−ρ_i)_, i=_1,N. (3.29)