Stevi´c Advances in Difference Equations (2017) 2017:283 DOI 10.1186/s13662-017-1350-8
RESEARCH
Open Access
Bounded and periodic solutions to the linear first-order difference equation on the integer domain Stevo Stevi´c* * Correspondence:
[email protected] Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Abstract The existence of bounded solutions to the linear first-order difference equation on the set of all integers is studied. Some sufficient conditions for the existence of solutions converging to zero when n → –∞, as well as when n → +∞, are also given. For the case when the coefficients of the equation are periodic, the long-term behavior of non-periodic solutions is studied. MSC: Primary 39A06; secondary 39A22; 39A23; 39A45 Keywords: linear first-order difference equation; bounded solution; periodic solution; difference equation on integer domain
1 Introduction Many classes of difference equations have been studied for a long time (see, for example, [–] and the references therein). The following difference equation: xn+ = qn xn + fn ,
()
where the coefficients (qn )n∈N and (fn )n∈N and the initial value x are given numbers, is called the linear first-order difference equation, which is one of the most important and useful solvable equations. The general solution to equation () is given by
xn = x
n– j=
qj +
n– n– fi qj , i=
n ∈ N.
()
j=i+
k– If, as usual, the conventions k– j=k aj = and j=k aj = , k ∈ Z are used, then () also holds for n = , that is, the formula holds for every n ∈ N . How formula () is obtained can be found, for example, in [, , ] (the case when sequences qn and fn are constant can be also found in [] and many other books dealing completely or partly with difference equations). Many nonlinear difference equations of interest are closely related to equation (). For example, some of the nonlinear equations in [, –] and systems in [, ] have been solved by transforming them to some special equations of the form in (), which shows its © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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importance (some of the equations and systems, such as the one in [], are transformed into special cases of the delayed version of equation (), which is obviously also solvable; see also [] and the comments on scaling indices of difference equations and systems). A deeper analysis can show that even the solvability of some product-type equations and systems is essentially influenced by the solvability of equation () (see, e.g., [] and the references therein). Usefulness of () has also been recently shown in [], where, among others, a small but nice result on convergence of its solutions was proved by using formula (), partially extending the result in the following problem in [] (in the eight edition of the Russian version of the problem book from it is Problem .). Problem Let a sequence (yn )n∈N be defined by a sequence (xn )n∈N as follows: y = x ,
yn = xn – αxn– ,
n ∈ N,
where |α| < . If limn→∞ yn = , find limn→∞ xn . For some other solvable equations, their applications, as well as invariants for some classes of equations, see, for example, [–, , –]. Note that the domain of the above defined sequences xn is N , and it is difficult to find papers which consider equation () on Z-the set of all integers, in the literature on difference equations. One of our aims is to fulfill the possible gap in the study of the equation. Motivated also by our recent paper [], here, among others, we study bounded solutions to equation (), but also on the whole Z. We also present some sufficient conditions for the existence of solutions to () converging to zero when n → –∞ as well as when n → +∞. Some of the results presented in the next section could be folklore, but we could not locate them in the literature. A solution xn to () is said to be (eventually) periodic with period T ∈ N if there is n ∈ N such that xn = xn+T
for n ≥ n .
If T = , then such a solution is called eventually constant []. Periodic solutions to () on N were studied in [], which was another motivation for this paper. Our main result on periodicity is a nice complement to that in []. Namely, for the case when the coefficients of equation () are periodic, we describe the long-term behavior of its non-periodic solutions when n → –∞ as well as when n → +∞. Assume that S ⊂ Z is an unbounded set and that f := (fn )n∈S is a sequence defined on S. Then, if f ∞,S := sup |fn | < +∞,
()
n∈S
we say that the sequence f is bounded on S. It is easy to see that the quantity defined in () is a norm on the space of all bounded sequences on S. From now on the set S will not be of special importance, so we will simply use the notation f ∞ instead of f ∞,S for any unbounded set S which appears as a domain.
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2 Bounded solutions to equation (1) on Z In this section we study the existence of bounded solutions to equation (). The cases when the domains are N and Z \ N are treated separately, while the results in the case when the domain is Z are obtained as some consequences of the considerations on the domains N and Z \ N. Our first result is, among others, an extension of the result in Problem , so it could be folklore. Theorem Assume that lim sup |qn | := q <
()
n→+∞
and that (fn )n∈N is a bounded sequence. Then the following statements are true. (a) Every solution to equation () is bounded. (b) If limn→+∞ fn = , then every solution to equation () converges to zero. Proof (a) Since () holds, we have that there is n ∈ N such that |qn | ≤
+q
for n ≥ n .
()
Let √ M := max , max |qj | .
()
j=,n –
Using () and () in (), as well as some standard estimates and sums, we have
|xn | ≤ |x |
n–
|qj | +
n–
j=
≤ |x |
n–
|fi |
i=
|qj | + f ∞
j=
n–
|qj |
()
j=i+
n – n– i= j=i+
|qj | +
n– n–
|qj |
i=n – j=i+
n – + q n–n n + q n–n M + f ∞ Mn –i– ≤ |x | i= + f ∞
n–
+ q n–i– i=n –
M n n Mn – – f ∞ ≤ |x | + M f ∞ + , there is n ∈ N such that |fn | < ε
for n ≥ n .
()
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Let n = max{n , n }, where n is from (), and √ M := max , max |qj | .
()
j=,n –
Using (), () and () in (), as well as some standard estimates and sums, we have
n – n– n– n– + q n–n n |xn | ≤ |x | M + f ∞ |qj | + ε |qj | i= j=i+ i=n j=i+
n – n–
+ q n–n n + q n–n + q n–i– n –i– ≤ |x | M + f ∞ M +ε i= i=n
n n M – + q n M n + q n ε ≤ |x | + f ∞ + +q +q M – –q
()
for n ≥ n . Letting n → +∞ in (), we get lim sup |xn | ≤ n→+∞
ε . –q
From this, and since ε is an arbitrary positive number, the result follows.
Remark Theorem is optimal in the sense that condition () cannot be replaced by the following one: there is n ∈ N such that |qn | <
for n ≥ n .
()
Indeed, assume that qn ∈ (, ), n ∈ N , is an increasing sequence such that the sequence
Qn :=
n–
qj ,
n ∈ N ,
j=
converges to Q ∈ (, ) as n → +∞, and that there are some numbers l and L such that < l ≤ fn ≤ L < ∞,
n ∈ N .
()
Then, by using () and the fact Q ≤ Qn ≤ , n ∈ N , we have n– fj |xn | = Qn x + ≥ Q –|x | + nl → +∞ Qj+
()
j=
as n → +∞, from which it follows that not only there are unbounded solutions to equation () in this case, but that even none of the solutions to the equation in the case is bounded. For example, let qn = –
, (n + )
n ∈ N
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and n ∈ N .
fn = + sin n, Then n–
– Qn = j=
(j + )
=
n– (j + )(j + ) j=
(j + )
=
n!(n + )! n+ = (n + )! (n + )
for n ∈ N , from which it follows that limn→+∞ Qn = /, ≤ Qn ≤ ,
n ∈ N ,
()
and along with () that n– n+ (j + ) ( + sin j) xn = x + . (n + ) j+ j=
()
Using (), the following obvious estimate ≤ fn ≤ ,
n ∈ N ,
()
and the triangle inequality in (), we get |xn | ≥
n – |x | ,
n ∈ N ,
from which it follows that each solution to equation () in this case is unbounded indeed. If we assume that fn is a positive sequence such that limn→+∞ fn = and ∞ j= fj = +∞ (for example, fn = /(n + ), n ∈ N ), and that qn is chosen as above, then from () and the fact Q ≤ Qn ≤ , n ∈ N , we have n– n– fj fj → +∞ |xn | = Qn x + ≥ Q –|x | + Qj+ j=
j=
as n → +∞, from which it follows that none of the solutions to equation () in this case converges to zero. Specially, every solution to the difference equation xn+ =
(n + )(n + ) , xn + (n + ) n+
n ∈ N ,
is unbounded. Now we consider the case lim infn→+∞ |qn | > . In this case, we may assume that qn = , n ∈ N , otherwise we can consider () for sufficiently large n for which, due to the condition lim infn→+∞ |qn | > , will hold qn = .
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Theorem Assume that (qn )n∈N ⊂ C \{} is a sequence satisfying the following condition: lim inf |qn | := qˆ > ,
()
n→+∞
and that (fn )n∈N is a bounded sequence of complex numbers. Then the following statements are true. (a) There is a unique bounded solution to equation (). (b) If fn → as n → +∞, then the bounded solution also converges to zero as n → +∞. Proof (a) From (), we have n– lim qj = +∞. n→+∞
()
j=
Thus, from () and () we see that for a bounded solution to () it must be
x = –
∞
i
fi
j= qj
i=
,
()
and that the sum on the right-hand side of () is finite (see []). Using () in (), it follows that
xn = –
n– j=
qj
∞ i=n
i
fi
j= qj
=–
∞
i
fi
j=n qj
i=n
,
n ∈ N .
()
Condition () implies that there is n ∈ N such that |qn | ≥
+ qˆ >
for n ≥ n .
()
Let √ M := min / , min |qj | .
()
j=,n –
Note that M > due to the assumption qn = , n ∈ N . Hence, by using () and (), we have ∞ ∞ f i ≤ f i i ∞ q j= j j= |qj | i=
i=
n –
∞
i–n + + n ≤ f ∞ i M i=n + qˆ j= |qj | i=
≤ f ∞ + n < ∞, n M ( – M ) M (ˆq – )
from which it follows that x defined in () is finite.
()
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From () and (), we have ∞
|xn | ≤
i=n
∞
i+–n f ∞ , = ≤ f ∞ i qˆ – + qˆ j=n |qj | i=n |fi |
for n ≥ n , from which the boundedness of the sequence in () follows. It is directly verified that the sequence satisfies equation (), from which along with the unique choice of x in () it follows that it is a unique bounded solution to the equation. (b) Since fn tends to zero as n → +∞, it follows that () holds for, say, n ≥ n . Using () and () in (), we have ∞
|xn | ≤
i
|fi |
j=n |qj |
i=n
for n ≥ n .
()
Indeed, assume that qn > , n ∈ N , is an decreasing sequence such that the sequence Qn :=
n–
qj ,
n ∈ N ,
j=
converges to Q > as n → +∞, and that there are some numbers l and L such that () holds. Then, by using () and the fact Q ≥ Qn ≥ , n ∈ N , we have n– fj |xn | = Qn x + ≥ –|x | + nl/Q → +∞ Qj+
()
j=
as n → +∞, from which it follows that all the solutions to equation () are unbounded in the case, so, it does not have bounded solutions. For example, let qn = +
, n + n +
n ∈ N ,
and fn = + cos n,
n ∈ N .
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Then Qn =
n–
+ j=
j + j +
=
n– j=
(j + ) (n + ) = (j + )(j + ) n+
for n ∈ N , from which it follows that limn→+∞ Qn = , ≤ Qn ≤ ,
n ∈ N ,
()
and along with () that n– (n + ) j+ xn = x + . ( + cos j) n+ (j + ) j=
()
Using (), the following obvious estimate ≤ fn ≤ ,
n ∈ N ,
()
and the triangle inequality in (), we get |xn | ≥ n – |x |,
n ∈ N ,
from which it follows that each solution to equation () in this case is unbounded. If we assume that fn is a positive sequence such that limn→+∞ fn = and ∞ j= fj = +∞ (for example, fn = / ln(n + ), n ∈ N ), and that qn is chosen as above, then from () and the fact ≤ Qn ≤ Q, n ∈ N , we have n– n– fj |xn | = Qn x + fj → +∞, ≥ –|x | + Qj+ Q j=
j=
as n → +∞, from which it follows that none of the solutions to equation () in this case converges to zero. Specially, every solution to the difference equation xn+ =
(n + ) xn + , (n + )(n + ) ln(n + )
n ∈ N ,
is unbounded. Now we consider the case when n ∈ Z \ N. If in () is qn = for every n ∈ Z \ N , then the sequence xn is not only well-defined on the set N , but also for every n ∈ Z. Indeed, if n ≤ , then from () we have x–n =
x–(n–) f–n – , q–n q–n
n ∈ N.
()
Using one of the methods for solving equation (), from () the following is obtained: x–n = for n ∈ N.
x –
n
j– j= f–j l= q–l n j= q–j
()
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Closed form formulas () and () together present the general solution to equation () on Z, when qn = , for n ∈ Z \ N . Now we formulate and prove the corresponding results to Theorems and concerning bounded solutions to equation (). The results are dual to Theorems and and are essentially obtained from them by using the change of variables yn = x–n . However, there are some different details which are used later in the text. Because of this and for the completeness, we will sketch their proofs. First, we consider equation () for the case lim inf |q–n | =: q˜ > .
()
n→∞
Theorem Assume that (q–n )n∈N ⊂ C \ {} is a sequence satisfying condition (), and that (f–n )n∈N is a bounded sequence of complex numbers. Then the following statements are true. (a) Every solution to () is bounded. (b) If limn→+∞ f–n = , then every solution to () converges to zero as n → +∞. Proof (a) From () it follows that there is n ∈ N such that ≤ |q–n | + q˜
for n ≥ n .
()
Let √ M := min / , min |q–j | .
()
j=,n –
Using () and () in (), as well as some standard estimates and sums, we have |f–j | |x | n + |x–n | ≤ n j= |q–j | l=j |q–l | j= n
≤
|x | n – M
+
f ∞ n – ( – M )M
+
() f ∞ , there is n ∈ N such that |f–n | < ε
for n ≥ n .
Let n = max{n , n }, where n is from (), and √ M := min / , min |q–j | .
()
()
j=,n –
Using (), () and () in (), as well as some standard estimates and sums, we have
+ q˜ n – ε n + q˜ n – n |xn | ≤ |x | + f ∞ + M + q˜ M + q˜ – M q˜ – for n ≥ n .
()
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Letting n → +∞ in () and using the fact that ε is an arbitrary positive number, the result follows. Now we consider the case lim supn→+∞ |q–n | < . If so, then a bounded solution (x–n )n∈N to () is obtained only if
x =
∞
f–j
j–
j=
q–l ,
()
l=
and for such chosen x , it is obtained j– j=n+ f–j l= q–l n l= q–l
∞ x–n =
=
∞
f–j
j=n+
j–
()
q–l
l=n+
for n ∈ N. Theorem Assume that (q–n )n∈N ⊂ C \ {} is a sequence such that lim sup |q–n | := q < ,
()
n→∞
and that (f–n )n∈N is a bounded sequence of complex numbers. Then the following statements are true. (a) There is a unique bounded solution to (). (b) If limn→+∞ f–n = , then the bounded solution x–n also converges to zero as n → +∞. Proof (a) From () we have that there is n ∈ N such that |q–n |
, there is n ∈ N such that () holds for n ≥ n . From this, () and (), we have that
|x–n | ≤
∞ j=n+
|f–j |
j– l=n+
|q–l | < ε
∞
+ q j–n– j=n+
=
ε –q
()
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for n ≥ max{n , n }. Letting n → +∞ in () and since ε is an arbitrary positive number, we obtain limn→+∞ x–n = , as desired. From Theorems - the following four interesting corollaries are obtained. From Theorems and we obtain the following corollary. Corollary Consider equation () for n ∈ Z. Assume that (qn )n∈Z and (fn )n∈Z are sequences of complex numbers such that q–n = , n ∈ N, lim supn→+∞ |qn | < and lim infn→+∞ |q–n | > , and sup |fn | < ∞.
()
n∈Z
Then the following statements are true. (a) Every solution to () is bounded on Z. (b) If lim fn = ,
()
n→±∞
then, for every solution (xn )n∈Z to (), we have limn→±∞ xn = . From Theorems and we obtain the following corollary. Corollary Consider equation () for n ∈ Z. Assume that (qn )n∈Z and (fn )n∈Z are sequences of complex numbers such that q–n = , n ∈ N, lim supn→+∞ |qn | < and lim supn→+∞ |q–n | < , and that () holds. Then the following statements are true. (a) There is a unique bounded solution to () on Z. (b) If () holds, then, for the bounded solution (xn )n∈Z , we have limn→±∞ xn = . From Theorems and we obtain the following corollary. Corollary Consider equation () for n ∈ Z. Assume that (qn )n∈Z and (fn )n∈Z are sequences of complex numbers such that q–n = , n ∈ N, lim infn→+∞ |qn | > and lim infn→+∞ |q–n | > , and that () holds. Then the following statements are true. (a) There is a unique bounded solution to () on Z. (b) If () holds, then, for the bounded solution (xn )n∈Z , we have limn→±∞ xn = . From Theorems and we obtain the following corollary. Corollary Consider equation () for n ∈ Z. Assume that (qn )n∈Z and (fn )n∈Z are sequences of complex numbers such that q–n = , n ∈ N, lim infn→+∞ |qn | > and lim supn→+∞ |q–n | < , and that () holds. Then the following statements are true. (a) There is a unique bounded solution to () on Z if and only if
x =
∞ j=
f–j
j– l=
q–l = –
∞ i=
i
fi
j= qj
.
(b) If () holds, then, for the bounded solution (xn )n∈Z , we have limn→±∞ xn = .
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3 Periodic solutions to equation (1) In this section we study equation () in the case when the sequences qn and fn are periodic with the same period T. Note that if sequence qn is periodic with period T and sequence fn is periodic with period T , where T = T , then T := lcm(T , T ) (the least common multiple of integers T and T ) is a common period of these two sequences, since T = k T = k T for some integers k and k . Hence, the case leads to the investigation of equation () whose coefficients are periodic with the same period. Periodic solutions to equation () have been studied in []. Among others, the authors of [] quote, in an equivalent form, the following basic result, which is certainly folklore. Theorem Assume that (qn )n∈N and (fn )n∈N are two periodic sequences with period T. Then the following statements are true. (a) If
λ :=
T–
qj = ,
()
j=
then () has a unique T-periodic solution given by the initial condition
x =
T– T– i= fi j=i+ qj
.
()
T– T– fi qj = ,
()
–λ
(b) If
λ = and
i=
j=i+
then () has a one-parameter family of T-periodic solutions. (c) If
λ = and
T– T– fi qj = , i=
()
j=i+
then () has no T-periodic solutions. The case in (a) is interesting for guaranteeing the uniqueness of a T-periodic solution to equation (). Note that, due to the T-periodicity of sequences qn and fn , from () we see that a solution to equation () will be periodic with period T if and only if
x = xT = x
T– j=
qj +
T– T– fi qj , i=
j=i+
from which, if () holds, it follows that x must be given by (). However, the authors of [] did not consider there the relationship between the periodic and other solutions to equation (). We prove a nice result which gives an answer to the natural problem. Before this, we formulate and prove a closely related result to Theorem .
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Theorem Assume that (q–n )n∈N and (f–n )n∈N are two periodic sequences with period T such that q–n = , n = , T . Then the following statements are true. (a) If λˆ :=
T
q–j = ,
()
j=
then () has a unique T-periodic solution given by the initial condition T xˆ =
j= f–j
j–
l= q–l
– λˆ
.
()
(b) If T
λˆ = and
f–j
j=
j–
q–l = ,
()
l=
then all the solutions to () are T-periodic. (c) If T
λˆ = and
j=
f–j
j–
q–l = ,
()
l=
then () has no T-periodic solutions. Proof (a)-(c) Since q–n and f–n are periodic, by using () we see that a solution to () will be T-periodic if and only if xˆ = xˆ –T =
xˆ –
j– j= f–j l= q–l , T j= q–j
T
()
from which, if λˆ = , () follows. If λˆ = , then from the second equality in (), we see that if () holds we have xˆ –T = xˆ , so all the solutions are T-periodic, while if () holds, such xˆ does not exist, so there are no T-periodic solutions in this case, which completes the proof. Now we formulate and prove the main result in this section. The result deals with the asymptotic behavior of solutions to equation () in the case when the quantity in () is different from and . Theorem Assume that (qn )n∈Z and (fn )n∈Z are two periodic sequences with period T and that the quantity in () is different from and . Then equation () has a unique T-periodic solution, say, (ˆxn )n∈Z , and the following statements are true. (a) If |λ| < , then all the solutions to () converge geometrically to the periodic one as n → +∞, while they are getting away geometrically from the periodic one as n → –∞.
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(b) If |λ| > , then all the solutions to () converge geometrically to the periodic one as n → –∞, while they are getting away geometrically from the periodic one as n → +∞. Proof Unique existence of a T-periodic solution (ˆxn )n∈Z to () follows from Theorem (a). Assume that (xn )n∈Z is an arbitrary solution to (). Then, due to the T-periodicity of qn , we have λ=
n+T–
()
qj
j=n
for every n ∈ Z (note that due to an assumption we have λ = ), from which along with (), we obtain xn+T = x
n+T–
qj +
n+T–
j=
n+T–
i=
n–
= λ x
fi
qj
j=i+
n+T– n+T– n– n– qj + fi qj + fi qj = λxn + dn ,
j=
i=
j=i+
i=n
()
j=i+
where dn :=
n+T–
fi
n+T–
i=n
n ∈ N .
qj ,
j=i+
Using again the periodicity of pn and qn , we have dn+T =
n+T–
fi
i=n+T
=
n+T–
n+T–
qj =
n+T–
j=i+
fl
l=n
n+T–
fl+T
l=n
qk+T =
k=l+
n+T– l=n
n+T–
qj
j=l+T+
fl
n+T–
qk = dn ,
()
k=l+
which means that the sequence dn is T-periodic. From () we see that if xˆ n is the periodic solution to (), it must be xˆ n =
dn , –λ
n ∈ N .
()
Also, we have xmT+l = λm xl + dl
m–
λj ,
m, l ∈ N.
()
j=
Hence, since λ = , we have m– dl dl m j λ – |xmT+l – xˆ mT+l | = λ xl + dl = |λ|m xl – – λ – λ j= for every m, l ∈ N .
()
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Let dl Cx := max √ x – . T l l – λ l=,T– ( |λ|) Note that since λ ∈/ {, }, the quantity is well defined. Then from () it follows that |xn – xˆ n | ≤ Cx
T
n |λ|
for n ∈ N .
()
If |λ| < , then from () it follows that all the solutions to equation () converge geometrically to the periodic solution to the equation as n → +∞. Now note that if xn = xˆ n for some n ∈ N, then it will be xn = xˆ n for n ≥ n . Moreover, since due to the condition λ = , we have qn = , n ∈ N , it follows that xn = xˆ n for every n ∈ N . So, for an arbitrary solution xn different from xˆ n , it must be xn = xˆ n for every n ∈ N . Hence, if |λ| > , then from () it follows that |xn – xˆ n | ≥
T
dl √ xl – – λ > , T l=,T– ( |λ|)l
n |λ| min
n ∈ N ,
that is, every solution xn different from xˆ n gets away geometrically from the periodic solution as n → +∞. Now we are going to consider the case n ≤ . From Theorem (a) we see that equation () has a unique T-periodic solution if () holds with xˆ given in (). From () we see that λˆ = λ. Using this fact along with the periodicity of q–n and f–n , we obtain T xˆ =
j= f–j
T
j–
l= q–l
=
j= fT–j
j–
l= q–l
–λ – λˆ T– T–i– T– T–i– fi l= q–l fi l= qT–l = i= = i= –λ –λ T– T– fi s=i+ qs . = i= –λ
()
From () and () we see that xˆ = x , so, there is only one ‘initial value’ for which a T-periodic solution to () on Z is obtained. Using (), we obtain
x–(n+T) =
x –
n+T
j– j= f–j l= q–l n+T l= q–l
n j–
n+T f–j x – j= f–j l= q–l – = n+T n λ q l= –l l=j q–l j=n+ =
x–n – cn λ
()
Stevi´c Advances in Difference Equations (2017) 2017:283
Page 16 of 17
for n ∈ N , where cn =
n+T
f–j n+T l=j
j=n+
.
q–l
()
Since cn+T =
n+T
f–j
n+T
j=n+T+
=
n+T
q–l
f–k
n+T
l=k+T q–l
k=n+
=
l=j
n+T
f–k n+T
k=n+
s=k
q–s
n+T
=
fT–j n+T
j=n+T+ n+T
=
k=n+
l=j
f–k n+T l=k+T
q–l
qT–l
= cn ,
we see that cn is a T-periodic sequence for n ∈ N. From () we see that if x˜ –n is the periodic solution to equation (), then it must be x˜ –n =
cn , –
λ–
n ∈ N .
()
Also, we have x–(mT+l) = λ–m x–l – cl
m–
λ–j
()
j=
for m ∈ N. Hence, since λ = , we have m– c l –m |x–(mT+l) – x˜ –(mT+l) | = λ x–l – cl λ–j – – λ – j= cl –m = |λ| x–l – – λ –
()
for every m, l ∈ N . Let Cx := max
l T |λ| x–l –
l=,T–
cl . λ– –
Then from () it follows that Cx . |x–n – x˜ –n | ≤ √ T ( |λ|)n
()
If |λ| > , then from () it follows that all the solutions to equation () converge geometrically to the periodic solution to the equation as n → +∞. Now note that if x–n = x˜ –n for some n ∈ N, then it will be x–n = x˜ –n for n ≥ n . Moreover, since q–n = , n ∈ N , then
Stevi´c Advances in Difference Equations (2017) 2017:283
Page 17 of 17
we have x–n = x˜ –n for every n ∈ N . So, for an arbitrary solution x–n different from x˜ –n , it must be x–n = x˜ –n for every n ∈ N . Hence, if < |λ| < , then from () it follows that |x–n – x˜ –n | ≥
T
l –n cl |λ| min T |λ| x–l – – > , λ – l=,T–
n ∈ N ,
and consequently, every solution x–n different from x˜ –n gets away geometrically from the periodic solution as n → +∞, which finishes the proof of the theorem. Competing interests The author declares that he has no competing interests. Authors’ contributions The author has contributed solely to the writing of this paper. He read and approved the manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 14 July 2017 Accepted: 4 September 2017 References 1. Agarwal, RP: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Dekker, New York (2000) 2. Agarwal, RP, Popenda, J: Periodic solutions of first order linear difference equations. Math. Comput. Model. 22(1), 11-19 (1995) 3. Andruch-Sobilo, A, Migda, M: Further properties of the rational recursive sequence xn+1 = axn–1 /(b + cxn xn–1 ). Opusc. Math. 26(3), 387-394 (2006) 4. Berezansky, L, Braverman, E: On impulsive Beverton-Holt difference equations and their applications. J. Differ. Equ. Appl. 10(9), 851-868 (2004) 5. Brand, L: A sequence defined by a difference equation. Am. Math. Mon. 62(7), 489-492 (1955) 6. Brand, L: Differential and Difference Equations. Wiley, New York (1966) 7. Demidovich, B: Problems in Mathematical Analysis. Mir, Moscow (1989) 8. Iriˇcanin, B, Stevi´c, S: Eventually constant solutions of a rational difference equation. Appl. Math. Comput. 215, 854-856 (2009) 9. Jordan, C: Calculus of Finite Differences. Chelsea, New York (1956) 10. Karakostas, G: Convergence of a difference equation via the full limiting sequences method. Differ. Equ. Dyn. Syst. 1(4), 289-294 (1993) 11. Krechmar, VA: A Problem Book in Algebra. Mir, Moscow (1974) 12. Levy, H, Lessman, F: Finite Difference Equations. Dover, New York (1992) 13. Mitrinovi´c, DS, Adamovi´c, DD: Sequences and Series. Nauˇcna Knjiga, Beograd (1980) (in Serbian) 14. Mitrinovi´c, DS, Keˇcki´c, JD: Methods for Calculating Finite Sums. Nauˇcna Knjiga, Beograd (1984) (in Serbian) 15. Papaschinopoulos, G, Schinas, CJ: Invariants for systems of two nonlinear difference equations. Differ. Equ. Dyn. Syst. 7(2), 181-196 (1999) 16. Papaschinopoulos, G, Schinas, CJ: Invariants and oscillation for systems of two nonlinear difference equations. Nonlinear Anal. TMA 46(7), 967-978 (2001) 17. Papaschinopoulos, G, Stefanidou, G: Asymptotic behavior of the solutions of a class of rational difference equations. Int. J. Difference Equ. 5(2), 233-249 (2010) 18. Stevi´c, S: Existence of a unique bounded solution to a linear second order difference equation and the linear first order difference equation. Adv. Differ. Equ. 2017, Article ID 169 (2017) 19. Stevi´c, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On some solvable difference equations and systems of difference equations. Abstr. Appl. Anal. 2012, Article ID 541761 (2012) 20. Stevi´c, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On the difference equation xn = an xn–k /(bn + cn xn–1 · · · xn–k ). Abstr. Appl. Anal. 2012, Article ID 409237 (2012) 21. Stevi´c, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On the difference equation xn+1 = xn xn–k /(xn–k+1 (a + bxn xn–k )). Abstr. Appl. Anal. 2012, Article ID 108047 (2012) 22. Stevi´c, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On a solvable system of rational difference equations. J. Differ. Equ. Appl. 20(5-6), 811-825 (2014) 23. Stevi´c, S, Iriˇcanin, B, Šmarda, Z: Two-dimensional product-type system of difference equations solvable in closed form. Adv. Differ. Equ. 2016, Article ID 253 (2016)
