R E S E A R C H
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 whenn→–∞, as well as whenn→+∞, 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+=qnxn+fn, ()
where the coefficients (qn)n∈Nand (fn)n∈Nand the initial valuexare given numbers, is called thelinear first-order difference equation, which is one of the most important and usefulsolvableequations. The general solution to equation () is given by
xn=x n–
j= qj+
n–
i= fi
n–
j=i+
qj, n∈N. ()
If, as usual, the conventionskj=–kaj= andjk=–kaj= ,k∈Zare used, then () also holds
forn= , that is, the formula holds for everyn∈N. How formula () is obtained can
be found, for example, in [, , ] (the case when sequencesqnandfnare 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
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∈Nbe defined by a sequence(xn)n∈Nas follows:
y=x, yn=xn–αxn–, n∈N,
where|α|< .Iflimn→∞yn= ,findlimn→∞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 sequencesxnisN, and it is difficult to find
papers which consider equation () onZ-the set of all integers, in the literature on differ-ence 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 wholeZ. We also present some sufficient conditions for the existence of solutions to () converging to zero whenn→–∞as well as whenn→+∞. Some of the results presented in the next section could be folklore, but we could not locate them in the literature.
A solutionxnto () is said to be (eventually) periodic with periodT∈Nif there isn∈N
such that
xn=xn+T forn≥n.
IfT= , then such a solution is called eventually constant [].
Periodic solutions to () onNwere 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 whenn→–∞as well as whenn→+∞.
Assume thatS⊂Zis an unbounded set and thatf := (fn)n∈Sis a sequence defined onS.
Then, if
f∞,S:=sup n∈S|
fn|< +∞, ()
we say that the sequencef is bounded onS. It is easy to see that the quantity defined in () is a norm on the space of all bounded sequences onS. From now on the setSwill not be of special importance, so we will simply use the notationf∞ instead off∞,Sfor any
2 Bounded solutions to equation (1) onZ
In this section we study the existence of bounded solutions to equation (). The cases when the domains areNandZ\Nare treated separately, while the results in the case when the
domain isZare obtained as some consequences of the considerations on the domainsN
andZ\N.
Our first result is, among others, an extension of the result in Problem , so it could be folklore.
Theorem Assume that
lim sup
n→+∞ |qn|:=q< ()
and that(fn)n∈Nis a bounded sequence.Then the following statements are true.
(a) Every solution to equation()is bounded.
(b) Iflimn→+∞fn= ,then every solution to equation()converges to zero.
Using () and () in (), as well as some standard estimates and sums, we have
Letn=max{n,n}, wherenis from (), and
Using (), () and () in (), as well as some standard estimates and sums, we have
|xn| ≤ |x|
From this, and sinceεis an arbitrary positive number, the result follows.
Remark Theorem is optimal in the sense that condition () cannot be replaced by the following one: there isn∈Nsuch that
asn→+∞, 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= –
and
fn= +sinn, n∈N.
Then
Qn= n–
j=
–
(j+ )
=
n–
j=
(j+ )(j+ ) (j+ ) =
n!(n+ )! (n+ )! =
n+ (n+ )
forn∈N, from which it follows thatlimn→+∞Qn= /,
≤Qn≤, n∈N, ()
and along with () that
xn= n+ (n+ )
x+ n–
j=
( +sinj)(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 thatfnis a positive sequence such thatlimn→+∞fn= and
∞
j=fj= +∞
(for example,fn= /(n+ ),n∈N), and thatqnis chosen as above, then from () and the
factQ≤Qn≤,n∈N, we have
|xn|=Qn
x+
n–
j= fj Qj+
≥Q
–|x|+
n–
j= fj
→+∞
asn→+∞, 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+ ) (n+ ) xn+
n+ , n∈N,
is unbounded.
Theorem Assume that(qn)n∈N⊂C\{}is a sequence satisfying the following condition:
lim inf
n→+∞|qn|:=qˆ> , ()
and that(fn)n∈Nis a bounded sequence of complex numbers.Then the following statements
are true.
(a) There is a unique bounded solution to equation().
(b) Iffn→asn→+∞,then the bounded solution also converges to zero asn→+∞.
and that the sum on the right-hand side of () is finite (see []). Using () in (), it follows that
Condition () implies that there isn∈Nsuch that
|qn| ≥
From () and (), we have
|xn| ≤
∞
i=n |fi|
i j=n|qj|
≤ f∞ ∞
i=n
+qˆ
i+–n
=f∞
ˆ q– ,
forn≥n, from which the boundedness of the sequence in () follows. It is directly ver-ified that the sequence satisfies equation (), from which along with the unique choice of
xin () it follows that it is a unique bounded solution to the equation.
(b) Sincefntends to zero asn→+∞, it follows that () holds for, say,n≥n. Using ()
and () in (), we have
|xn| ≤
∞
i=n |fi|
i j=n|qj|
<ε
∞
i=n
+qˆ
i+–n
= ε
ˆ
q– ()
forn≥max{n,n}.
Lettingn→+∞in () the following is obtained:
lim sup n→+∞ |xn| ≤
ε
ˆ q– .
From this and by using the fact thatεis an arbitrary positive number, we obtain thatxn→
asn→+∞, as desired.
Remark Theorem is optimal in the sense that condition () cannot be replaced by the following one: there isn∈Nsuch that
|qn|> forn≥n. ()
Indeed, assume thatqn> ,n∈N, is an decreasing sequence such that the sequence
Qn:= n–
j=
qj, n∈N,
converges toQ> asn→+∞, and that there are some numberslandLsuch that () holds. Then, by using () and the factQ≥Qn≥,n∈N, we have
|xn|=Qn
x+
n–
j= fj Qj+
≥–|x|+nl/Q→+∞ ()
asn→+∞, 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
Then
and along with () that
xn=
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 thatfnis a positive sequence such thatlimn→+∞fn= and
asn→+∞, 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+=
Using one of the methods for solving equation (), from () the following is obtained:
Closed form formulas () and () together present the general solution to equation () onZ, whenqn= , forn∈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 variablesyn=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
n→∞ |q–n|=:q˜> . ()
Theorem Assume that(q–n)n∈N⊂C\ {}is a sequence satisfying condition(),and that(f–n)n∈Nis a bounded sequence of complex numbers.Then the following statements are true.
(a) Every solution to()is bounded.
(b) Iflimn→+∞f–n= ,then every solution to()converges to zero asn→+∞. Proof (a) From () it follows that there isn∈Nsuch that
|q–n|
≤
+q˜ forn≥n. ()
Let
M:=min
/√, min j=,n–
|q–j|
. ()
Using () and () in (), as well as some standard estimates and sums, we have
|x–n| ≤n|x| j=|q–j|
+
n
j= |f–j|
n l=j|q–l|
()
≤ |x| Mn–
+ f∞ ( –M)Mn–
+f∞
˜
q– <∞
forn≥n, from which the boundedness of (xn)n∈Nfollows.
(b) Sincef–nconverges to zero asn→+∞, we have that for everyε> , there isn∈N
such that
|f–n|<ε forn≥n. ()
Letn=max{n,n}, wherenis from (), and
M:=min
/√, min j=,n–
|q–j|
. ()
Using (), () and () in (), as well as some standard estimates and sums, we have
|xn| ≤ |x|
+q˜
M
n–
+q˜
n
+f∞
+q˜
M
n–
+q˜
n
–M+
ε
˜
q– ()
Lettingn→+∞in () and using the fact thatεis an arbitrary positive number, the
result follows.
Now we consider the caselim supn→+∞|q–n|< . If so, then a bounded solution (x–n)n∈N
to () is obtained only if
x=
∞
j= f–j
j–
l=
q–l, ()
and for such chosenx, it is obtained
x–n=
∞
j=n+f–j
j– l=q–l
n l=q–l
=
∞
j=n+ f–j
j–
l=n+
q–l ()
forn∈N.
Theorem Assume that(q–n)n∈N⊂C\ {}is a sequence such that
lim sup
n→∞ |q–n|:=q< , ()
and that(f–n)n∈Nis a bounded sequence of complex numbers.Then the following statements are true.
(a) There is a unique bounded solution to().
(b) Iflimn→+∞f–n= ,then the bounded solutionx–nalso converges to zero asn→+∞. Proof (a) From () we have that there isn∈Nsuch that
|q–n|<
+q
forn≥n. ()
By using () and some simple estimates in (), we have
|x–n| ≤
∞
j=n+ |f–j|
j–
l=n+
|q–l|<f∞
∞
j=n+
+q
j–n–
=f∞
–q ()
forn≥n, from which the boundedness of sequence () easily follows. A simple calcu-lation shows that the sequence satisfies equation (). Sincexis uniquely determined by the convergent series in (), it follows that the sequence is a unique bounded solution to ().
(b) Sincelimn→+∞f–n= , we have that for everyε> , there isn∈Nsuch that ()
holds forn≥n.
From this, () and (), we have that
|x–n| ≤
∞
j=n+ |f–j|
j–
l=n+ |q–l|<ε
∞
j=n+
+q
j–n–
= ε
forn≥max{n,n}. Lettingn→+∞in () and sinceεis an arbitrary positive number, we obtainlimn→+∞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∈Zand(fn)n∈Zare sequences of complex numbers such that q–n= ,n∈N,lim supn→+∞|qn|< andlim infn→+∞|q–n|>
,and
sup
n∈Z|fn|<∞. ()
Then the following statements are true.
(a) Every solution to()is bounded onZ. (b) If
lim
n→±∞fn= , ()
then,for every solution(xn)n∈Zto(),we havelimn→±∞xn= .
From Theorems and we obtain the following corollary.
Corollary Consider equation()for n∈Z.Assume that(qn)n∈Zand(fn)n∈Zare sequences of complex numbers such that q–n= ,n∈N,lim supn→+∞|qn|< andlim supn→+∞|q–n|<
,and that()holds.Then the following statements are true.
(a) There is a unique bounded solution to()onZ.
(b) If ()holds,then,for the bounded solution(xn)n∈Z,we havelimn→±∞xn= .
From Theorems and we obtain the following corollary.
Corollary Consider equation()for n∈Z.Assume that(qn)n∈Zand(fn)n∈Zare sequences of complex numbers such that q–n= ,n∈N,lim infn→+∞|qn|> andlim infn→+∞|q–n|> , and that()holds.Then the following statements are true.
(a) There is a unique bounded solution to()onZ.
(b) If ()holds,then,for the bounded solution(xn)n∈Z,we havelimn→±∞xn= .
From Theorems and we obtain the following corollary.
Corollary Consider equation()for n∈Z.Assume that(qn)n∈Zand(fn)n∈Zare sequences of complex numbers such that q–n= ,n∈N,lim infn→+∞|qn|> andlim supn→+∞|q–n|<
,and that()holds.Then the following statements are true.
(a) There is a unique bounded solution to()onZif and only if
x=
∞
j= f–j
j–
l= q–l= –
∞
i= fi
i j=qj
.
3 Periodic solutions to equation (1)
In this section we study equation () in the case when the sequences qnandfnare
pe-riodic with the same periodT. Note that if sequenceqnis periodic with periodT and
sequence fnis periodic with periodT, whereT = T, thenT := lcm(T,T) (the least
common multiple of integersTandT) is a common period of these two sequences, since
T=kT=kTfor some integerskandk. 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∈Nand(fn)n∈Nare two periodic sequences with period T.
Then the following statements are true.
(a) If
λ:=
T–
j=
qj= , ()
then()has a uniqueT-periodic solution given by the initial condition
x=
T– i= fi
T– j=i+qj
–λ . ()
(b) If
λ= and
T–
i= fi
T–
j=i+
qj= , ()
then()has a one-parameter family ofT-periodic solutions. (c) If
λ= and
T–
i= fi
T–
j=i+
qj= , ()
then()has noT-periodic solutions.
The case in (a) is interesting for guaranteeing the uniqueness of aT-periodic solution to equation (). Note that, due to theT-periodicity of sequencesqnandfn, from () we see
that a solution to equation () will be periodic with periodTif and only if
x=xT=x T–
j= qj+
T–
i= fi
T–
j=i+ qj,
Theorem Assume that(q–n)n∈Nand(f–n)n∈Nare two periodic sequences with period T such that q–n= ,n= ,T.Then the following statements are true.
(a) If
ˆ
λ:=
T
j=
q–j= , ()
then()has a uniqueT-periodic solution given by the initial condition
ˆ x=
T
j=f–j
j– l=q–l
–λˆ . ()
(b) If
ˆ
λ= and
T
j= f–j
j–
l=
q–l= , ()
then all the solutions to()areT-periodic. (c) If
ˆ
λ= and
T
j= f–j
j–
l=
q–l= , ()
then()has noT-periodic solutions.
Proof (a)-(c) Sinceq–nandf–nare periodic, by using () we see that a solution to () will
beT-periodic if and only if
ˆ
x=x–ˆ T= ˆ
x–Tj=f–jjl–=q–l
T
j=q–j
, ()
from which, ifλˆ= , () follows. Ifλˆ = , then from the second equality in (), we see that if () holds we havex–ˆ T=xˆ , so all the solutions areT-periodic, while if () holds,
suchxˆdoes not exist, so there are noT-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 fromand.Then equation()has a unique T -periodic solution,say, (xˆn)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
(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→+∞.
Using again the periodicity ofpnandqn, we have
dn+T=
which means that the sequencednisT-periodic.
Let Then from () it follows that
|xn–xˆn| ≤Cx
T|
λ|n forn∈N. ()
If|λ|< , then from () it follows that all the solutions to equation () converge geo-metrically to the periodic solution to the equation asn→+∞. Now note that ifxn=xˆn for somen∈N, then it will bexn=xˆnforn≥n. Moreover, since due to the condition
λ= , we haveqn= ,n∈N, it follows thatxn=xˆnfor everyn∈N. So, for an arbitrary
solutionxndifferent fromxˆn, it must bexn=xˆnfor everyn∈N. Hence, if|λ|> , then
from () it follows that
|xn–xˆn| ≥
that is, every solutionxndifferent fromxˆngets away geometrically from the periodic
so-lution asn→+∞.
Now we are going to consider the casen≤. From Theorem (a) we see that equation () has a uniqueT-periodic solution if () holds withxˆ given in (). T-periodic solution to () onZis obtained.
Using (), we obtain
forn∈N, where
From () we see that ifx–˜ nis the periodic solution to equation (), then it must be
˜
Then from () it follows that
|x–n–x–˜ n| ≤
Cx
(√T|λ|
)n. ()
we havex–n=x–˜ nfor everyn∈N. So, for an arbitrary solutionx–ndifferent fromx–˜ n, it
must bex–n=x–˜ nfor everyn∈N. Hence, if <|λ|< , then from () it follows that
|x–n–x–˜ n| ≥
T|
λ|–n min l=,T–
T|
λ|lx–l– cl
λ––
> , n∈N,
and consequently, every solutionx–ndifferent fromx˜–ngets away geometrically from the
periodic solution asn→+∞, 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.
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Received: 14 July 2017 Accepted: 4 September 2017 References
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