On the Difference Equation

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 409237, 20 pages doi:10.1155/2012/409237

Research Article On the Difference Equation xn an xn−k /bn cn xn−1 · · · xn−k ´ 1 Josef Dibl´ık,2, 3 Stevo Stevic, 4 3 ˇ ˇ ˇ Smarda Bratislav Iricanin, and Zdenek 1

Mathematical Institute of The Serbian Academy of Sciences and Arts, Knez, Mihailova 36/III, 11000 Beograd, Serbia 2 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic 3 Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic 4 Faculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia Correspondence should be addressed to Stevo Stević, [email protected] Received 27 May 2012; Accepted 2 July 2012 Academic Editor: Jean Pierre Gossez Copyright q 2012 Stevo Stević et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The behavior of well-defined solutions of the difference equation xn an xn−k /bn cn xn−1 · · · xn−k , n ∈ N0 , where k ∈ N is fixed, the sequences an , bn and cn are real, bn , cn / 0, 0, n ∈ N0 , and the initial values x−k , . . . , x−1 are real numbers, is described.

1. Introduction Recently there has been a huge interest in studying nonlinear difference equations and systems see, e.g., 1–33 and the references therein. Here we study the difference equation xn

xn−k , bn cn xn−1 · · · xn−k

n ∈ N0 ,

1.1

where k ∈ N is fixed, the sequences bn and cn are real, bn , cn / 0, 0, n ∈ N0 , and the initial values x−k , . . . , x−1 are real numbers. Equation 1.1 is a particular case of the equation xn

an xn−k bn cn xn−1 · · · xn−k

,

n ∈ N0 ,

1.2

2

Abstract and Applied Analysis

0, n ∈ with real sequences an , bn and cn . For an 0, n ∈ N0 , the equation is trivial, and, for an / N0 , it is reduced to equation 1.1 with bn bn / an and cn cn / an . Equation

xn

xn−k , b cxn−1 · · · xn−k

n ∈ N0 ,

1.3

where b, c ∈ R, which was treated in 32, is a particular case of equation 1.1. As in 32, here, we employ our idea of using a change of variables in equation 1.1 which extends the one in our paper 21 and is later also used, for example, in 4. For similar methods see also 22, 25. Equation 1.3 in the case k 2 was also studied in 1, 2, in a different way. The case when the sequences bn and cn are two-periodic was studied in 31 some related results are also announced in talk 3. For related symmetric systems of difference equations, see 27, 29. For some other recent results on difference equations and systems which can be solved, see, for example, 6, 7, 20–22, 30, 31, 33. Some classical results can be found, for example, in 11. Equation 1.1 is a particular case of the equation yn f yn−1 , . . . , yn−k , n yn−k ,

n ∈ N0 ,

1.4

where f : Rk1 → R is a continuous function. Numerous particular cases of 1.4 have been investigated, for example, in 9, 21, 23. In this paper we adopt the customary notation k k ik1 gi 1 and ik1 gi 0.

2. Case cn 0, n ∈ N0 Here we consider the case cn 0, n ∈ N0 . In this case equation 1.1 becomes

xn

xn−k , bn

n ∈ N0 ,

2.1

bn / 0, n ∈ N0 , from which it follows that for each i ∈ {1, . . . , k}

x−i xkm−i m , j1 bkj−i

m ∈ N0 .

Using formula 2.2 the following theorem can be easily proved.

2.2


3

0, n ∈ N0 . Then the following statements Theorem 2.1. Consider equation 1.1 with cn 0, bn / are true: a if lim inf |bkm−i | pi > 1,

2.3

m→∞

for some i ∈ {1, . . . , k}, then xkm−i → 0 as m → ∞; b if, for each i ∈ {1, . . . , k}, the limits pi in 2.3 are greater than 1, then xn → 0 as n → ∞; c if bkm−i 1, for every m ∈ N and for some i ∈ {1, . . . , k}, then xkm−i x−i , m ∈ N0 ; d if bkm−i −1, for every m ∈ N and for some i ∈ {1, . . . , k}, then xkm−i −1m x−i , m ∈ N0 ; e if lim sup |bkm−i | qi ∈ 0, 1,

2.4

m→∞

and x−i / 0, for some i ∈ {1, . . . , k}, then |xkm−i | → ∞, as m → ∞; f if, for each i ∈ {1, . . . , k}, the limits qi in 2.4 belong to the interval 0, 1 and x−i / 0, then |xn | → ∞ as n → ∞.

3. Case bn 0, n ∈ N0 In this section we consider the case bn 0, n ∈ N0 . Note that in this case equation 1.1 becomes xn

xn−k , cn xn−1 · · · xn−k1 xn−k

n ∈ N0 ,

3.1

where cn / 0, n ∈ N0 . If xn is a well-defined solution of equation 3.1 i.e., a solution with 0, i 1, . . . , k, which implies xn / 0, n ∈ N0 , then initial values x−i / xn

cn−1 xn−2 · · · xn−k cn−1 1 xn−k , cn xn−1 xn−2 · · · xn−k1 cn xn−2 · · · xn−k1 cn

n ∈ N.

3.2

Hence for each i ∈ {0, 1, . . . , k − 1} xkm−i x−i

m c kj−i−1 j1

ckj−i

,

m ∈ N0 .

Using formula 3.3 we easily prove the next theorem.

3.3

4


0, n ∈ N0 . Then the following statements Theorem 3.1. Consider equation 1.1 with bn 0, cn / are true: a if ckm−i pi > 1, lim inf m → ∞ ckm−i−1

3.4

for some i ∈ {0, 1, . . . , k − 1}, then xkm−i → 0 as m → ∞; b if, for each i ∈ {0, 1, . . . , k − 1}, the limits pi in 3.4 are greater than 1, then xn → 0 as n → ∞; c if ckm−i−1 ckm−i , for every m ∈ N and for some i ∈ {0, 1, . . . , k − 1}, then xkm−i x−i , m ∈ N0 ; d if ckm−i−1 −ckm−i , for every m ∈ N and for some i ∈ {0, 1, . . . , k − 1}, then xkm−i −1m x−i , m ∈ N0 ; e if ckm−i lim sup c m→∞

km−i−1

qi ∈ 0, 1,

3.5

and x−i / 0 for some i ∈ {0, 1, . . . , k − 1}, then |xkm−i | → ∞, as m → ∞; f if, for each i ∈ {0, 1, . . . , k − 1}, x−i / 0 and the limits qi in 3.5 belong to the interval 0, 1, then |xn | → ∞ as n → ∞.

0 and cn / 0 4. Case bn / The case when bn / 0 and cn / 0 for every n ∈ N0 is considered in this section. If x−i0 0 for some i0 ∈ {1, . . . , k}, then from 1.1 we have that xkm−i0 0,

for m ∈ N0 .

4.1

From 4.1 and 1.1 we have that for each i ∈ {1, . . . , k} \ {i0 } xkm−i

xkm−1−i x−i m , bkm−i j1 bkj−i

m ∈ N0 .

4.2

From 4.1 we see that, for i i0 , 4.2 also holds. Hence Theorem 2.1 can be applied in this case. Note that if xn 0 for some n ∈ N0 , then 1.1 implies that there is an i0 ∈ {1, . . . , k} such that x−i0 0, and by the previous consideration we have that 4.2 also holds. 0, for each i ∈ {1, . . . , k}, then for every well-defined solution we have xn / 0 If x−i / for n ≥ −k note that there are solutions which are not well defined, that is, those for which xn−1 · · · xn−k −bn /cn , for some n ∈ N0 .


5

Multiplying equation 1.1 by xn−1 · · · xn−k1 and using the transformation yn

1 , xn xn−1 · · · xn−k1

n ≥ −1,

4.3

we obtain equation yn bn yn−1 cn ,

n ∈ N0 .

4.4

Note that from 4.3, for every well-defined solution xn n≥−k of equation 1.1 such that 0, n ≥ −1. x−i / 0, for each i ∈ {1, . . . , k}, it follows that yn / 0, n ∈ N0 , we have that Since bn / ⎞

n ⎛ n

cj bi ⎝y−1 yn j

i0 bi

j0

i0

⎠,

n ∈ N0 .

4.5

From 4.3 and 4.5 we have that j y−1 n−1 j0 cj / yn−1 i0 bi 1 xn xn−k xn−k , j yn xn−1 · · · xn−k1 yn bn y−1 nj0 cj / i0 bi

4.6

for every n ∈ N0 . Hence, from 4.6, we obtain that

xmk−i x−i

m l1

j cj / i0 bi 1/α kl−i−1 j0 , j c bkl−i 1/α kl−i / b j i j0 i0

4.7

for every m ∈ N0 and each i 1, 2, . . . , k, where α

k

4.8

x−l .

l1

5. Case bn 1, n ∈ N0 Here we consider the case bn 1, n ∈ N0 . In this case, from 4.7 we have that for each i ∈ {1, . . . , k} xmk−i x−i

m 1α l1

kl−i−1

1α

j0

kl−i j0

cj

cj

,

m ∈ N0 .

Note that this formula includes also the case when x−i0 0 for some i0 ∈ {1, . . . , k}.

5.1

6

Abstract and Applied Analysis Now we formulate and prove a result in this case by using formula 5.1.

0, and Theorem 5.1. Consider equation 1.1 with bn 1, n ∈ N0 , sign cn sign c0 , n ∈ N, α / n

α cj / − 1,

n ∈ N0 .

5.2

αckl−i kl−i ∞, l1 1 α j0 cj

5.3

αckl−i kl−i 0, l→∞1 α j0 cj

5.4

j0

Then the following statements hold: a if for some i ∈ {1, . . . , k} ∞

lim

then xmk−i → 0 as m → ∞; b if 5.3 and 5.4 hold for every i ∈ {1, . . . , k}, then xn → 0 as n → ∞; c if for some i ∈ {1, . . . , k} the sum ∞

αckl−i kl−i l1 1 α j0 cj

5.5

converges, then the sequence xmk−i is also convergent; d if the sum in 5.5 is finite for every i ∈ {1, . . . , k}, then the sequences xkm−i are convergent. Proof. Let xn n≥−k be a solution of equation 1.1. Using condition sign cn sign c0 , n ∈ N, it is easy to see that if 5.4 holds for some i ∈ {1, . . . , k}, there is an m0 ∈ N such that for j ≥ m0 1 the terms in the product in 5.1 are positive and that the following asymptotic formula ln1 x x O x2

5.6


7

can be used with x being the fraction in the limit 5.4. From 5.1 and 5.6 we have that

kl−i−1 m 1 α j0 cj |xkm−i | |x−i | kl−i 1 α c l1 j0 j ⎛ |x−i |cm0 exp⎝

m

ln

lm0 1

⎛ |x−i |cm0 exp⎝

m

1α

kl−i−1 j0

1α

kl−i j0

cj

⎞ ⎠

cj ⎞⎞

⎛ ln⎝1 −

lm0 1

5.7

αckl−i ⎠⎠ 1 α kl−i j0 cj

⎛

⎞ m

αc o1 1 kl−i ⎠, |x−i |cm0 exp⎝− kl−i 1 α c lm0 1 j0 j

where

kl−i−1 m0 1 α j0 cj . cm0 kl−i 1 α c l1 j0 j

5.8

kl−i Using formula 5.7, the assumptions regarding the sum ∞ jm0 1 αckl−i /1α j0 cj and the comparison test for the series whose terms are of eventually the same sign, the results in the theorem easily follow.

6. Case bn −1, n ∈ N0 Here we consider the case bn −1, n ∈ N0 . In this case from 4.7 we have

m

xmk−i −1 x−i

m 1α l1

kl−i−1

1α

j0

kl−i j0

−1j1 cj

−1j1 cj

,

for every m ∈ N0 and each i 1, 2, . . . , k, where α is defined by 4.8.

6.1

8


Theorem 6.1. Consider equation 1.1 with α / 0, bn −1, n ∈ N0 , and n

α −1j1 cj / − 1,

n ∈ N0 .

6.2

α−1kl−i1 ckl−i ∞, kl−i j1 cj l1 1 α j0 −1

6.3

α−1kl−i1 ckl−i 0, kl−i j1 l→∞1 α cj j0 −1

6.4

j0

Then the following statements hold: a if for some i ∈ {1, . . . , k} ∞

lim

∞

l1

2 ckl−i

1α

kl−i j0

−1j1 cj

2 < ∞,

6.5

then xmk−i → 0 as m → ∞; b if for every i ∈ {1, . . . , k}, 6.3, 6.4, and 6.5 hold, then xn → 0 as n → ∞; c if for some i ∈ {1, . . . , k} ∞

α−1kl−i1 ckl−i −∞, kl−i j1 cj l1 1 α j0 −1

6.6

conditions 6.4 and 6.5 hold, and x−i / 0, then |xmk−i | → ∞ as m → ∞; d if for every i ∈ {1, . . . , k}, conditions 6.4, 6.5, and 6.6 hold, and x−i / 0, i ∈ {1, . . . , k}, then |xn | → ∞ as n → ∞; e if for some i ∈ {1, . . . , k} the sum ∞

α−1kl−i1 ckl−i kl−i j1 cj l1 1 α j0 −1

6.7

converges and condition 6.5 holds, then the sequences x2mk−i and x2m1k−i are also convergent; f if for every i ∈ {1, . . . , k} the sum in 6.7 converges and condition 6.5 holds, then the sequences x2km−j , j 1, . . . , 2k are convergent.


9

Proof. Let xn n≥−k be a solution of equation 1.1. By 6.4 we see that irrespectively on i ∈ {1, . . . , k}, there is an m1 ∈ N such that for j ≥ m1 1 the terms in the product in 6.1 belong to the interval 1/2, 3/2 and that asymptotic formulae ln1 x x −

x2 O x3 2

6.8

can be used with −x being the fraction in 6.4. From this and 6.1 we have that kl−i−1 j1 m 1 α j0 −1 cj |xkm−i | |x−i | kl−i j1 cj l1 1 α j0 −1 ⎛ |x−i |c1 m1 exp⎝

m

1α

ln

kl−i−1

1α

lm1 1

j0

kl−i j0

−1j1 cj

−1j1 cj

⎞ ⎠

⎞⎞ kl−i1 c α−1 kl−i ⎠⎠ |x−i |c1 m1 exp⎝ ln⎝1 − kl−i j1 1 α c −1 j lm1 1 j0 ⎛

m

⎛

⎛ m

⎜ ⎜ |x−i |c1 m1 exp⎝− ⎝ ⎛

lm1 1

⎞⎞ 2 2 α c o1 1 α−1 ckl−i ⎟⎟ kl−i kl−i 2 ⎠⎠, j1 1 α j0 −1 cj 2 1 α kl−i −1j1 c j j0 kl−i1

6.9 where kl−i−1 j1 m1 1 α j0 −1 cj . c1 m1 kl−i j1 c −1 j l1 1 α j0

6.10

Using formula 6.9, the assumptions of the theorem and some well-known convergence tests for series, the results in a–f easily follow.

7. Case bn bnk , cn cnk , n ∈ N0 In this section we consider equation 1.1 for the case bn bnk , cn cnk , n ∈ N0 , that is, when the sequences bn and cn are k-periodic. First we show the existence of k-periodic solutions of equation 4.4. If y 0 , y1 , . . . , y k−1

7.1

y 1 b1 y 0 c1 , y 2 b2 y 1 c2 , . . . , y 0 bk yk−1 ck .

7.2

is such a solution, then we have that

10


By successive elimination, or by Kronecker theorem note that system 7.2 is linear, we get k−1 yi

if

k

j1 bj

j0

j−1 cσ j i s0 bσ s i , 1 − kj1 bj

7.3

i 1, k,

1, where σ is the permutation defined by / σi i − 1,

7.4

i 2, k, σ1 k,

and σ i σ ◦ σ i−1 , σ 0 Id, where Id denotes the identity. It is easy to see that 4.4 along with k periodicity of sequences bn and cn implies

ykmi

⎞ ⎛ j−1 k k−1

⎝ bj ⎠ykm−1i cσ j i bσ s i ,

7.5

s0

j0

j1

for every m ∈ N0 and i ∈ {1, 2, . . . , k}, such that km − 1 i ≥ −1. 1, its Since 7.5 is a linear first-order difference equation, we have that when kj1 bj / general solution is ⎛ ykmi ⎝

k

⎞m bj ⎠ yi

1−

j1

1−

k j1 bj

m

k

j1 bj

j−1 k−1

cσ j i bσ s i .

7.6

s0

j0

By letting m → ∞ in 7.6 we obtain the following corollary. Corollary 7.1. Consider equation 4.4 with bn bnk , cn cnk , n ∈ N0 . Assume that k bj < 1. j1

7.7

Then for every solution yn of the equation we have that lim ykmi yi ,

7.8

m→∞

for every i ∈ {1, 2, . . . , k}, that is, yn converges to the k-periodic solution in formula 7.3. Let j−1 k−1

j cσ i bσ s i , Li : j0

s0

i 1, k,

q :

k bj . j1

7.9


11

From now on we will use the following convention: if i, j ∈ N0 , then we regard that Lj Li , if i ≡ j mod k. Also if a sequence mj j∈N0 is defined by the relation mj fLj , where f is a real function, then we will assume that mj mi , if i ≡ j mod k. Using 7.6 and notation 7.9 in the relation xn yn−1 /yn xn−k see 4.6, for the case q / 1, we have that xkmi xi−k

m yi−1 − Li−1 / 1 − q qj Li−1 / 1 − q j yi − Li / 1 − q q Li / 1 − q j0

m Li−1 1 1 − q yi−1 /Li−1 − 1 qj xi−k · , Li 1 1 − q yi /Li − 1 qj j0

7.10

for every m ∈ N0 and each i ∈ {2, . . . , k}, and xkm1

m yk − Lk / 1 − q qj−1 Lk / 1 − q x1−k y1 − L1 / 1 − q qj L1 / 1 − q j0 x1−k

j0

7.11

1 − q yk /Lk − 1 qj−1 · . L1 1 1 − q y1 /L1 − 1 qj

m Lk

1

Now we present some results, which are applications of formulae 7.10 and 7.11.

7.1. Case q −1 If q −1, then by 7.5 we get ykmi −ykm−1i Li Li − Li − ykm−2i ykm−2i ,

m ∈ N,

7.12

for km − 2 i ≥ −1; that is, ykmi is two-periodic for each i ∈ {1, . . . , k}. Hence yn is a 2k-periodic solution of equation 4.4, in this case. Hence from the relation xn yn−1 /yn xn−k see 4.6, for each i ∈ {1, 2, . . . , k}, we have xkm−i

ykm−i−1 ykm−i−1 ykm−1−i−1 xkm−i−k xkm−i−2k , ykm−i ykm−i ykm−1−i

7.13

for km − 1 ≥ i. From 7.13 and by 2k periodicity of yn , we get x2klj

for each j ∈ {−k 1, . . . , −1, 0, 1, . . . , k}.

yj−1 yjk−1 yj yjk

l xj ,

l ∈ N0 ,

7.14

12


From 7.14, the behavior of solutions of equation 1.1, in this case, easily follows. For example, if pj :

yj−1 yjk−1 1, yj yjk

7.15

for each j ∈ {−k 1, . . . , −1, 0, 1, . . . , k}, then the solution xn n≥k of 1.1 is 2k-periodic.

7.2. Case q 1 If q 1 and α / 0, then from 7.5 we obtain ykmi ykm−1i Li ,

m ∈ N0 , i 1, k,

7.16

when km − 1 i ≥ −1, from which along with 4.6, it follows that xkmi xi

m yi−1 jLi−1 j1

xkm1

yi jLi

,

m ∈ N, i 2, k,

m yk j − 1 Lk x1 , y1 jL1 j1

7.17 m ∈ N.

Corollary 7.2. Consider equation 1.1. Let q 1, α / 0, and pi : Li−1 /Li , i ∈ {1, . . . , k}. Then the following statements hold true. a If | pi | < 1, for some i ∈ {1, . . . , k}, then xkmi → 0 as m → ∞. b If | pi | > 1, or Li 0 and Li−1 / 0, for some i ∈ {1, . . . , k}, then |xkmi | → ∞ as m → ∞, 0. if xi / c If pi 1, for some i ∈ {2, . . . , k}, and yi−1 − yi /Li > 0, then |xkmi | → ∞ as m → ∞, 0. if xi / d If pi 1, for some i ∈ {2, . . . , k}, and yi−1 − yi /Li < 0, then xkmi → 0 as m → ∞. e If pi 1, for some i ∈ {2, . . . , k}, and yi−1 yi , then the sequence xkmi m∈N0 is convergent. f If p1 1, and yk − L1 − y1 / L1 > 0, then |xkm1 | → ∞ as m → ∞, if x1 / 0. g If p1 1, and yk − L1 − y1 /L1 < 0, then xkm1 → 0 as m → ∞. h If p1 1, and yk L1 y1 , then the sequence xkm1 m∈N0 is convergent. Proof. The statements in a and b follow from the facts that yi−1 jLi−1 pi , lim j → ∞ yi jLi

i ∈ {2, . . . , k}

7.18


13

0, if Li / yi−1 jLi−1 ∞, lim j →∞ yi jLi

i ∈ {2, . . . , k}

7.19

if Li 0 and Li−1 / 0, yk j − 1Lk lim p1 , j → ∞ y1 jL1

7.20

yk j − 1Lk lim ∞, j → ∞ y1 jL1

7.21

if L1 / 0, and

if L1 0 and Lk / 0. Now assume that pi 1 and let xn n≥−k be a solution of equation 1.1. It is easy to see that there is an m2 ∈ N such that for j ≥ m2 1 the terms in the products in 7.17 are positive and that the following asymptotic formulae 1 x−1 1 − x O x2 ,

ln1 x x O x2

7.22

can be applied with x yi−1 − yi /jLi , when i ∈ {2, . . . , k} or with x yk − L1 − y1 /jL1 . Using these formulae, for the case i ∈ {2, . . . , k}, we have that xkmi xi

m yi−1 jLi−1 j1

yi jLi ⎛

⎞ jL y i−1 i−1 ⎠ xi cm2 exp⎝ ln y jL i i jm2 1 m

⎞ − y y 1 i−1 i ⎠ xi cm2 exp⎝ ln 1 O 2 jLi j jm2 1 ⎛

m

7.23

⎞ m

− y y 1 i−1 i ⎠, xi cm2 exp⎝ O 2 jLi j jm2 1 ⎛

where cm2

m2 yi−1 jLi−1 j1

yi jLi

.

7.24

14

Abstract and Applied Analysis Letting m → ∞ in 7.23, using the facts that m

1 −→ ∞ j jm2 1

as m −→ ∞

7.25

2 and that the series ∞ jm2 1 O1/j converges, we get statements c–e. If p1 1, that is L1 Lk / 0, then by using 7.22 we get xkm1

m yk j − 1 Lk x1 y1 jL1 j1 ⎞ yk j − 1 Lk ⎠ x1 dm2 exp⎝ ln y1 jL1 jm2 1 ⎛

m

⎞ − L − y y 1 k 1 1 ⎠ x1 dm2 exp⎝ ln 1 O 2 jL j 1 jm2 1 ⎛

m

7.26

⎞ m

− L − y y 1 k 1 1 ⎠, x1 dm2 exp⎝ O 2 jL j 1 jm2 1 ⎛

where m2 yk j − 1 Lk dm2 . y1 jL1 j1 Letting m → ∞ in 7.26, using 7.25 and the fact that the series we get statements f–h, as desired.

7.27 ∞

jm2 1

O1/j 2 converges,

7.3. Case q / ±1 If q / ± 1, then from 7.6 we get ykmi qm si ti ,

m ∈ N0 ,

7.28

where si yi

Li , q−1

ti

Li , 1−q

i 1, k,

7.29

from 4.6 it follows that xkmi xi

m qj si−1 ti−1 j1

qj si ti

,

m ∈ N0 ,

7.30


15

for i ∈ {2, . . . , k}, and xkm1

m qj−1 sk tk x1 yk , qs1 t1 j2 qj s1 t1

m ∈ N0 .

7.31

Note that ti tj , if i ≡ j mod k. Corollary 7.3. If 0 < |q| < 1, α / 0, and qj si ti / 0, for every j ∈ N0 and i ∈ {1, . . . , k}, then the following statements hold true. a If |ti−1 | < |ti |, for some i ∈ {1, . . . , k}, we have that xkmi → 0 as m → ∞. b If |ti−1 | > |ti |, and si / 0 if ti 0 for some i ∈ {1, . . . , k}, we have that |xkmi | → ∞ as 0. m → ∞, if xi / c If ti−1 ti / 0, for some i ∈ {1, . . . , k}, then xkmi is convergent. d If ti−1 ti 0, and |si−1 | < |si | for some i ∈ {2, . . . , k}, then |xkmi | → 0 as m → ∞. e If t1 tk 0, and |sk | < |qs1 |, then xkm1 → 0 as m → ∞. f If ti−1 ti 0, and |si−1 | > |si | for some i ∈ {2, . . . , k}, then |xkmi | → ∞ as m → ∞, if 0. xi / g If t1 tk 0, and |sk | > |qs1 |, then |xkm1 | → ∞ as m → ∞, if x1 / 0. h If ti−1 ti 0, and si−1 si / 0 for some i ∈ {2, . . . , k}, then xkmi is constant. i If t1 tk 0, and sk qs1 / 0, then xkm1 is constant. j If ti−1 ti 0, and si−1 −si / 0 for some i ∈ {2, . . . , k}, then xkmi −1m xi . m−1 k If t1 tk 0, and sk −qs1 / 0, then xkm1 x1 yk −1 /qs1 .

l If ti−1 −ti / 0, for some i ∈ {1, . . . , k}, then the subsequences x2kmi and x2kmki are convergent. Proof. Since we have that qj si−1 ti−1 ti−1 , j → ∞ qj si ti ti lim

i ∈ {2, . . . , k}

7.32

when ti / 0, qj s t i−1 i−1 lim ∞, j → ∞ qj si ti

i ∈ {2, . . . , k}

7.33

when |ti−1 | > ti 0 and si / 0, qj−1 sk tk tk , j → ∞ qj s1 t1 t1 lim

7.34

16


0, and when t1 / qj−1 s t k k lim ∞, j → ∞ qj s1 t1

7.35

when |tk | > t1 0 and s1 / 0, the statements in a and b easily follow from 7.30–7.35. 0, then c If ti−1 ti / xkmi xi

m

1q

j

j1

si−1 − si ti

o qj ,

7.36

for i ∈ {2, . . . , k}, and if t1 tk / 0, then xkm1

m x1 yk sk − qs1 1 qj−1 o qj qs1 t1 j2 t1

7.37

from which the statement in c easily follows. d–k If ti−1 ti 0, then xkmi xi

m si−1 j1

si

,

7.38

for i ∈ {2, . . . , k}, and if t1 tk 0, then xkm1

m x1 yk sk qs1 j2 qs1

7.39

from which the statements in d–k easily follow. l If ti−1 −ti / 0, then we have that

xkmi

m j si−1 si xi − 1−q o qj ti j1

7.40

for i ∈ {2, . . . , k}, and if t1 −tk / 0, then xkm1

m x1 yk sk qs1 − 1 − qj−1 o qj qs1 t1 j2 t1

from which the statement in l easily follows.

7.41


17

0, for every j ∈ N0 and i ∈ {1, . . . , k}, then the Corollary 7.4. If |q| > 1 and α / 0, and qj si ti / following statements hold true. a If |si−1 | < |si |, for some i ∈ {2, . . . , k}, then xkmi → 0 as m → ∞. b If |sk | < |qs1 |, then xkm1 → 0 as m → ∞. 0 and ti / 0, for some i ∈ {2, . . . , k}, then |xkmi | → ∞ as c If |si−1 | > |si |, or si 0, si−1 / 0. m → ∞, if xi / 0 and t1 / 0, then |xkm1 | → ∞ as m → ∞, if x1 / 0. d If |sk | > |qs1 |, or if s1 0, sk / 0, for some i ∈ {2, . . . , k}, then the sequence xkmi m∈N0 is convergent. e If si−1 si / f If si−1 si 0 and |ti−1 | < |ti | for some i ∈ {2, . . . , k}, then xkmi → 0 as m → ∞. g If s1 sk 0 and |tk | < |t1 |, then xkm1 → 0 as m → ∞. h If si−1 si 0 and |ti−1 | > |ti | for some i ∈ {2, . . . , k}, then |xkmi | → ∞ as m → ∞, if 0. xi / 0. i If s1 sk 0 and |tk | > |t1 |, then |xkm1 | → ∞ as m → ∞, if x1 / j If si−1 si 0 and ti−1 ti for some i ∈ {2, . . . , k}, then the sequence xkmi is constant. k If s1 sk 0 and t1 tk , then the sequence xkm1 is constant. l If si−1 si 0 and ti−1 −ti for some i ∈ {2, . . . , k}, then the sequence xkmi is twoperiodic. m If s1 sk 0 and t1 −tk , then the sequence xkm1 is two periodic. 0, then the sequence xkm1 m∈N0 is convergent. n If sk qs1 / 0, for some i ∈ {2, . . . , k}, then the sequences x2kmi m∈N0 and o If si−1 −si / x2kmki m∈N0 , are convergent. 0, then the sequences x2km1 m∈N0 and x2kmk1 m∈N0 , are convergent. p If sk −qs1 / Proof. a–d These statements follow correspondingly from the next relations which are derived using formulae 7.30 and 7.31: qj si−1 ti−1 si−1 j → ∞ qj si ti si

7.42

qj s t i−1 i−1 lim ∞ j → ∞ qj si ti

7.43

lim

for i ∈ {2, . . . , k} if si / 0, and

for i ∈ {2, . . . , k} if si 0, si−1 / 0 and ti / 0; qj−1 sk tk sk , j → ∞ qj s1 t1 qs1 lim

7.44

18


0, and if s1 / qj−1 s t k k lim ∞ j → ∞ qj s1 t1

7.45

if s1 0, sk / 0 and t1 / 0. e If si−1 si / 0, then from 7.30 we get xkmi xi

m ti−1 − ti 1 q−j o q−j , si−1 j1

7.46

for i ∈ {2, . . . , k}, from which e follows. f–m If si−1 si 0 for some i ∈ {2, . . . , k}, then for i ∈ {2, . . . , k} we have qj si−1 ti−1 ti−1 ti qj si ti

7.47

qj−1 sk tk tk t1 qj s1 t1

7.48

while when s1 sk 0, we have

from which the statements f–i easily follow. n If sk qs1 / 0, then we have xkm1

m x1 yk tk − t 1 1 q−j o q−j , qs1 t1 j2 s1

7.49

from which along with the assumption |q| > 1 the statement follows. o and p If si−1 −si / 0, then m ti−1 ti − 1 − q−j o q−j si j1

7.50

m x1 yk tk t 1 . − 1 − q−j o q−j qs1 t1 j2 s1

7.51

xkmi xi

for i ∈ {2, . . . , k}, and xkm1

From 7.50 and 7.51 the statements in o and p correspondingly follow.


19

Acknowledgment The second author is supported by Grant P201/10/1032 of the Czech Grant Agency Prague and by the Council of Czech Government grant MSM 00 216 30519. The fourth author is supported by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.

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