Available at: http://www.pmf.ni.ac.rs/filomat
Existence and Global Attractivity of Periodic Solutions to Some Classes of Di fference Equations
Stevo Stevi´ca,b,c,d, Bratislav Iriˇcanine, Witold Kosmalaf, Zdenˇek ˇSmardag
aMathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
bDepartment of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China
cDepartment of Computer Science and Information Engineering, Asia University, 500 Lioufeng Rd., Wufeng, Taichung 41354, Taiwan, Republic of China
dBrno University of Technology, Faculty of Electrical Engineering and Communication, Department of Mathematics, Technicka 3058/10, CZ - 616 00 Brno, Czech Republic
eFaculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia
fDeptartment of Mathematical Sciences, Appalachian State University Boone, NC 28608, USA
gBrno University of Technology, CEITEC BUT, Purky ˇnova 656/123, 61600 Brno, Czech Republic
Abstract.Existence and global attractivity of periodic solutions to some subclasses of the following class of difference equations
xn+1= qnxn+ f (n, xn, xn−1, . . . , xn−k), n ∈ N0,
where k ∈ N0, (qn)n∈N0is a T-periodic sequence (T ∈ N), and f : N0× Rk+1→ R is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is continuous in other variables, are studied.
1. Introduction
Throughout the paper, by Z is denoted the set of all integers, while by Nl, where l ∈ Z, is denoted the set of all n ∈ Z such that n ≥ l. If k, l ∈ Z, then the notation n = k, l stands for the set of all n ∈ Z such that k ≤ n ≤ l.
Throughout the paper we use the conventions Xt
i=s
ci= 0 and Yt
i=s
ci= 1,
when t< s, t, s ∈ N0.
2010 Mathematics Subject Classification. Primary 39A10; Secondary 39A23
Keywords. Difference equation, existence of periodic solutions, global attractivity of periodic solutions.
Received: 20 November 2018; Accepted: 01 February 2019 Communicated by Jelena Manojlovi´c
The work of Bratislav Iriˇcanin was supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007, of Stevo Stevi´c by projects III 41025 and 44006. The work of Zdenˇek ˇSmarda was supported by the project CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme II.
Email addresses: sstevic@ptt.rs (Stevo Stevi´c), iricanin@etf.rs (Bratislav Iriˇcanin), kosmalawa@appstate.edu (Witold Kosmala), zdenek.smarda@ceitec.vutbr.cz (Zdenˇek ˇSmarda)
The linear first-order difference equation
xn+1 = qnxn+ fn, n ∈ N0, (1)
where (qn)n∈N0 and ( fn)n∈N0 are sequences of numbers is a very important and useful solvable difference equation (many classical solvable difference equations and systems, or their invariants, can be found, for example, in [1, 2, 7, 8, 11, 12, 14–16, 18, 19, 22–29, 34, 36, 61]). For some methods for solving equation (1), see, for example, [8, 12, 14, 25] (book [25] contains a nice presentation of three methods for solving it, essentially corresponding to the three methods for solving the linear differential equation of first order).
The general solution to equation (1) is xn= x0
Yn−1 j=0
qj+ Xn−1
i=0
fi
Yn−1 j=i+1
qj, n ∈ N0. (2)
To describe the usefulness of equation (1), we mention that many nonlinear difference equations and systems of difference equations are solved by transforming them into some of special cases of the equation, by using one or several changes of variables along with some algebra (see, for example, the difference equations in [30, 38, 41, 47, 55, 56], systems in [5, 42–45, 51, 52, 54, 55, 57–59], as well as the equations and systems in references therein).
An interesting case is when the sequences qn and fn are periodic. Recall that a sequence (an)n∈N0 is eventually periodic if there are T ∈ N and n0∈ N0, such that
an+T = an, for n ≥ n0.
When T= 1 for the sequence is said that is eventually constant (see, for example, [13]). If n0= 0 then it is said that the sequence is T-periodic, although many authors understand that every eventually T-periodic sequence is T-periodic. Some basic facts on periodic solutions to equation (1), can be found in [12], while some more complex ones can be found in [3]. For some other results on periodicity and related topics, e.g., [6, 37, 39, 40, 53] and the references therein.
We may assume that sequences qnand fnare periodic with the same period, since if qnis periodic with period T1and fnis periodic with period T2, then both sequences are periodic with period T= lcm(T1, T2) (the least common multiplier of natural numbers T1and T2).
Since in this paper qnwill always denote a T-periodic sequence, from now on we will use the following notation
λ :=
YT−1 j=0
qj. Note that the T-periodicity of qn, implies
n+T−1
Y
j=n
qj= λ, (3)
for every n ∈ N0.
In [3] is quoted the following result which essentially appears in [12].
Theorem 1. Assume that(qn)n∈N0 and ( fn)n∈N0 are two T-periodic sequences. Then the following statements are true.
(a) If
0 , λ , 1, (4)
then (1) has a unique T-periodic solution given by the initial condition
x0= PT−1
i=0 fiQT−1
j=i+1qj
1 −λ .
(b) If
λ = 1 and XT−1
i=0
fi
YT−1 j=i+1
qj= 0, (5)
then all solutions to equation (1) are T-periodic.
(c) If
λ = 1 and XT−1
i=0
fi
YT−1 j=i+1
qj, 0,
then equation (1) has no T-periodic solutions.
Remark 1.Difference equation (1) is a special case of the general first-order difference equation
xn+1 = f (n, xn), n ∈ N0, (6)
where f : N0× R → R. Note simply that for the case of equation (1) we have f (n, t) = qnt+ fn,
for n ∈ N0and t ∈ R.
The following condition
f (n+ T, x) = f (n, x), (7)
where n ∈ N0and x ∈ R, essentially corresponds to the T-periodicity of sequences qnand fn. By a direct calculation from (6) we have
xn= f (n − 1, f (n − 2, . . . f (1, f (0, x0)). . .)), n ∈ N,
from which it follows that if (bxn)n∈N0is a T-periodic solution to equation (6), then the following condition must hold
bxT= f (T − 1, f (T − 2, . . . f (1, f (0,bx0)). . .)) =bx0, (8) that is, the following nonlinear algebraic equation
f (T − 1, f (T − 2, . . . f (1, f (0, x)) . . .)) = x, (9)
must have a solution.
On the other hand, if equation (9) has a solution, saybx0, then (8) holds. From this, (6), (7) and (8), it follows that
bxT+1= f (T,bxT)= f (0,bx0)=bx1. Similarly, by an inductive argument, it follows that
bxn+T=bxn, for every n ∈ N0.
Hence, the following proposition holds.
Proposition 1. Consider equation(6), where f : N0× R → R, is a function satisfying condition (7). Then, the equation has a T-periodic solution if and only if the nonlinear equation (9) has a solution.
Since for the case of difference equation (1), by using formula (2) and (3) with n = 0, we have
f (T − 1, f (T − 2, . . . f (1, f (0, x0)). . .)) = λx0+ XT−1
i=0
fi
YT−1 j=i+1
qj, (10)
the first and third statements in Theorem 1 follow from Proposition 1, whereas if (5) holds, then from (10) we easily see that for every x0∈ R we have
x0= xT,
from which along with the periodicity of the sequences qnand fnthe second statement in Theorem 1 follows.
By using (2) and (3), for every solution (xn)n∈N0to difference equation (1), we have
xn+T = x0 n+T−1
Y
j=0
qj+
n+T−1
X
i=0
fi n+T−1
Y
j=i+1
qj
= λ x0
Yn−1 j=0
qj+ Xn−1
i=0
fi
Yn−1 j=i+1
qj
+
n+T−1
X
i=n
fi n+T−1
Y
j=i+1
qj
= λxn+ cn, for every n ∈ N0, where
cn:=
n+T−1
X
i=n
fi n+T−1
Y
j=i+1
qj. (11)
Hence, ifλ = 0, we have
xn+T = cn, (12)
for n ∈ N0.
Now we quote an interesting lemma, whose special cases have been essentially used in the literature (for example, in [33, 50]).
Lemma 1. Let(qn)n∈N0, ( fn( j))n∈N0, j = 1, p, be T-periodic sequences, and a sequence (bn)n≥kis defined by
bn:=
Xn+l j=n−k
p
X
t=1
fj(t)
n+mY
i=j+s
qi, n ≥ k, (13)
where k, l, m, p, s ∈ N0.
Then, the sequence (bn)n≥kis also T-periodic.
Proof. We have
bn+T=
n+T+l
X
j=n+T−k p
X
t=1
fj(t)
n+T+m
Y
i=j+s
qi=
n+l
X
j0=n−k p
X
t=1
fj(t)0+T
n+T+m
Y
i=j0+T+s
qi
=
n+l
X
j0=n−k p
X
t=1
fj(t)0
n+m
Y
i0=j0+s
qi0+T =
n+l
X
j0=n−k p
X
t=1
fj(t)0
n+m
Y
i0=j0+s
qi0= bn,
for all n ≥ k, from which the lemma follows.
If in (13) is chosen k= 0, l = T − 1, m = T − 1, p = 1, s = 1 and fn(1)= fn, n ∈ N0, then from Lemma 1 we obtain the following corollary.
Corollary 1. Let(qn)n∈N0and ( fn)n∈N0be two T-periodic sequences, and a sequence (cn)n∈N0is defined by (11). Then, the sequence (cn)n∈N0is also T-periodic.
From Corollary 1 and (12) the following result follows.
Corollary 2. Let(qn)n∈N0 and ( fn)n∈N0 be two T-periodic sequences, andλ = 0. Then, every solution (xn)n∈N0 to equation (1) is eventually T-periodic.
The relationship between the periodic and other solutions to equation (1) was not considered in [3]. The problem, among other ones, has been recently tackled in our paper [50], where we considered the equation not only on the domain N0, but on the set of all integers Z, which is possible if qn, 0, n ≤ −1 (for the case of equation (6) such a condition is in general case impossible to find). For a class of linear second-order difference equations some results of this type, among other ones, have been recently proved in [49].
Definition.For a sequence (an)n≥l, l ∈ Z, is said that it converges geometrically to a sequence ( ˜an)n≥lif there are L ≥ 0 and q ∈ (0, 1), such that
|an−˜an| ≤Lqn, for n ≥ l.
The following result, among other ones, was proved in [48].
Theorem 2. Assume(qn)n∈Zand ( fn)n∈Zare two T-periodic sequences, and that (4) holds. Then equation (1) has a unique T-periodic solution, and the following statements are true.
(a) If 0< |λ| < 1, then all the solutions to equation (1) converge geometrically to the periodic one as n → +∞, while they are getting away geometrically from the periodic one as n → −∞.
(b) If |λ| > 1, then all the solutions to equation (1) converge geometrically to the periodic one as n → −∞, while they are getting away geometrically from the periodic one as n →+∞.
Remark 2.Note that the conditionλ , 0 in Theorem 2 implies qn, 0, for n ∈ Z,
from which it follows that every solution to equation (1) is defined on the whole Z, in this case.
On the other hand, in [31] were studied positive solutions to the following difference equation
xn+1 = qnxn+ f (n, xn−k), n ∈ N0, (14)
where k ∈ N0, (qn)n∈N0is a positive T-periodic sequence, and f : N0× [0, ∞) → (0, ∞) is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is continuous in the second variable.
The main reason for studying only positive solutions to equation (14) in [31], is the fact that some special cases of the equation are discrete analogues of some models in biology. For example, the equation
xn+1= axn+ b
1+ xγn−k, n ∈ N0, where
min{a, b, γ} > 0
and k ∈ N0, is a discrete analogue of a model that has been used in studying of the blood cell production ([17]).
The following result was proved in [31].
Theorem 3. Assume that k ∈ N0, (qn)n∈N0 is a positive T-periodic sequence such that qn ∈ (0, 1], n ∈ N0, and f : N0 × [0, ∞) → (0, ∞) is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is nonincreasing and continuous in the second variable and for each n ∈ {0, 1, . . . , T −1} there are nonnegative constants Lnsuch that
|f (n, x) − f (n, y)| ≤ Ln|x − y|, (15)
for every x, y ∈ [0, ∞), and
n+k
X
j=n
Lj n+k
Y
i=j+1
qi< 1, (16)
for each n ∈ {0, 1, . . . , T − 1}.
Then equation (14) has a unique positive T-periodic solution ( ˜xn)n≥−k, and every positive solution (xn)n≥−kto the equation satisfies
n→∞lim( ˜xn−xn)= 0.
Note that the main difference between Theorems 2 and 3, is that the later one considers only positive solutions to equation (14), unlike the former one which considers arbitrary solutions to equation (1). For some related results see [32]. It is a natural problem to get some related results to Theorem 3 for some more general equations. In the recent paper [33] a generalization of Theorem 3 has been given.
Motivated by the problem, as well as some results in [31, 33, 46, 48–50], here we consider the existence and global attractivity of periodic solutions to some subclasses of the following class of difference equations
xn+1 = qnxn+ f (n, xn, xn−1, . . . , xn−k), n ∈ N0, (17) where k ∈ N0, f : N0× Rk+1 → R is a function such that for each n ∈ N0 it is continuous in other k+ 1 variables and
f (n+ T, t1, . . . , tk+1)= f (n, t1, . . . , tk+1), for some T ∈ N and every tj∈ R, j = 1, k + 1.
Let l∞T(Nm), where m ∈ Z \ N, be the space of all T-periodic sequences x = (xn)n≥m, with the following norm
kxkps= max
m≤n≤m+T−1|xn|. (18)
It is known that l∞T(Nm) with norm (18) is a Banach space.
One of the standard methods for showing the existence of a specific type of solutions to difference equations is application of fixed-point theorems. Beside the contraction mapping theorem, which was formulated and proved in [4] (some interesting applications of the theorem have been recently presented in [46], [50] and [60]), one of the fixed-point theorems which is frequently applied is the Schauder fixed point theorem ([35]). For some applications of the theorem, see, for example [9, 10, 20, 21, 33] and the related references therein.
2. Main results
This section states and proves the main results in this paper. The first result is about the existence of periodic solutions to equation (17) and is motivated by Theorem 1 in [33].
Theorem 4. Assume(qn)n∈N0is a T-periodic sequence such thatλ , 0, 1, function f : N0× [a, b]k+1→ R, a, b ∈ R, is continuous on [a, b]k+1for each n ∈ N0, and is T-periodic in the first variable, and the following conditions
a ≤ min
n=0,T−1 min
a≤tj≤b,j=1,k+1 n+T−1
X
i=n
f (i, t1, . . . , tk+1) 1 −λ
n+T−1
Y
j=i+1
qj
≤ max
n=0,T−1 max
a≤tj≤b,j=1,k+1 n+T−1
X
i=n
f (i, t1, . . . , tk+1) 1 −λ
n+T−1
Y
j=i+1
qj≤b,
(19)
hold.
Then, equation (17) has a T-periodic solution.
Proof. Let
Aa,b:= {(xn)n≥−k∈l∞T(N−k) : a ≤ xn≤b, n ≥ −k}.
It is easy to see that Aa,bis a convex and compact subset of the linear space l∞T(N−k).
Using (2) and (3), we have that for every solution (xn)n≥−kto equation (17) holds
xn+T =x0 n+T−1
Y
j=0
qj+
n+T−1
X
i=0
f (i, xi, . . . , xi−k)
n+T−1
Y
j=i+1
qj
=λ x0
Yn−1 j=0
qj+ Xn−1
i=0
f (i, xi, . . . , xi−k) Yn−1 j=i+1
qj
+
n+T−1
X
i=n
f (i, xi, . . . , xi−k)
n+T−1
Y
j=i+1
qj
=λxn+ dn, (20)
where
dn:=
n+T−1
X
i=n
f (i, xi, . . . , xi−k)
n+T−1
Y
j=i+1
qj, (21)
for n ∈ N0.
If (xn)n≥−kis a T-periodic sequence, then the following sequence fen:= f (n, xn, . . . , xn−k), n ∈ N0,
is also T-periodic. Indeed, by using the T-periodicity of function f in the first variable and T-periodicity of (xn)n≥−k, we have
efn+T = f (n + T, xn+T, . . . , xn+T−k)= f (n, xn, . . . , xn−k)= efn, for every n ∈ N0.
Since sequences qn and efn are T-periodic, by Lemma 1 it follows that for every T-periodic sequence (xn)n≥−kthe corresponding sequence dndefined in (21) is also T-periodic.
Let bT be an operator on the set Aa,bdefined by
bTxn= Pn+T−1
i=n f (i, xi, . . . , xi−k)Qn+T−1
j=i+1 qj
1 −λ , (22)
for n ∈ N0.
Then, from (19) and (22), we have
a ≤ bTxn≤b, (23)
for every n ∈ N0and (xn)n≥−k∈Aa,b. The periodicity of the sequence (bTxn)n∈N0follows from the T-periodicity of the sequence defined in (21). Hence, bT(Aa,b) ⊆ Aa,b.
Since for each n ∈ {0, 1, . . . , T − 1}, f (n, t1, . . . , tk+1) is continuous on the set [a, b]k+1, it is not difficult to see, by using the standardε-δ technique, that the operator bT is continuous on the set Aa,b.
By using the Schauder fixed point theorem, operator bT has a fixed point in the set Aa,b, that is, there is a sequence (bxn)n≥−k∈Aa,bsuch that
bTbxn=bxn, for n ∈ N0, which can be written as
bxn=
Pn+T−1
i=n f (i,bxi, . . . ,bxi−k)Qn+T−1
j=i+1 qj
1 −λ ,
for n ∈ N0.
Since sequences qnandbxnare T-periodic, as well as the function f in the first variable, we have
bxn+1 = Pn+T
i=n+1 f (i,bxi, . . . ,bxi−k)Qn+T j=i+1qj
1 −λ
=qn+TPn+T−1
i=n f (i,bxi, . . . ,bxi−k)Qn+T−1
j=i+1 qj
1 −λ
+ f (n+ T,bxn+T, . . . ,bxn+T−k) − f (n,bxn, . . . ,bxn−k)Qn+T j=n+1qj
1 −λ
= qnbxn+ f (n,bxn, . . . ,bxn−k), (24)
which shows that the sequence (bxn)n≥−kis a T-periodic solution to difference equation (17).
Remark 3.The introduction of the operator defined in (22) is quite natural and is a folklore thing. Namely, if a solution (xn)n≥−kto equation (17) is T-periodic, then from (20) it follows that, in the caseλ , 1, it must be
xn= dn
1 −λ, n ∈ N0, which strikingly suggests the introduction of the operator.
Theorem 5. Consider the following difference equation
xn+1 = qnxn+ f (n, xn), n ∈ N0, (25)
where (qn)n∈N0is a positive T-periodic sequence such thatλ < 1, f : N0× R → R is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is nonincreasing in the second variable, and that the functions
1l(x)= qlx+ f (l, x), (26)
are nondecreasing for each l ∈ {0, 1, . . . , T − 1}.
If equation (25) has a T-periodic solution, then every solution to the equation converges to the periodic one geometrically as n →+∞.
Proof. First note that if x0≤
bx0, then from (25) and the monotonicity of function 11(x), it follows that x1= q1x0+ f (1, x0) ≤ q1bx0+ f (1,bx0)=bx1. (27)
Assume that we have proved xn≤
bxn, (28)
for some n ∈ N, and let n = mT + l, where m ∈ N0and l ∈ {0, 1, . . . , T − 1}. Then from the assumption and the monotonicity of function 1l(x), T-periodicity of the sequence qnand function f in the first variable, it follows that
xn+1 = qnxn+ f (n, xn)= qlxn+ f (l, xn)
≤qlbxn+ f (l,bxn)= qnbxn+ f (n,bxn)=bxn+1. (29) From (27), (29) and by the induction, it follows that (28) holds for every n ∈ N0.
Now assume that ( ˜xn)n∈N0is a T-periodic solution to equation (25), and that (xn)n∈N0is another solution to the equation.
First, assume that x0≤
ex0. Then, by (28) we have xn≤
exn, (30)
for every n ∈ N0.
If there is n0 ∈ N0such that xn0 =exn0, then from (25) is obtained that xn=exnfor n ≥ n0, from which the result follows in this case.
Otherwise, from (30), we have
yn:=exn−xn> 0, (31)
for every n ∈ N0.
Sinceexnis a solution to equation (25), we have
xn+1+ yn+1= qn(xn+ yn)+ f (n, xn+ yn), n ∈ N0. (32) Using (25) in (32), it follows that
yn+1= qnyn+ f (n, yn+ xn) − f (n, xn), (33)
for n ∈ N0.
From (33) along with the monotonicity of f (n, t) in the second variable and (31), it follows that
yn+1≤qnyn, (34)
for every n ∈ N0, from which it follows that
0< yn≤y0
Yn−1 j=0
qj, n ∈ N0. (35)
Hence, if n= mT + l for some m ∈ N0and l ∈ {0, 1, . . . , T − 1}, form (35) and by using (3), we get
0< ˜xn−xn= yn= ymT+l≤y0 mT+l−1
Y
j=0
qj
= y0
Yl−1
j=0
qj
λm= y0
Yl−1
j=0
qj
√T
λ
(T
√λ)n
≤ ( ˜x0−x0) max
l=1,T
Yl−1
j=0
qj
√T
λ
(T
√
λ)n, (36)
for every n ∈ N0.
Sinceλ ∈ (0, 1), by letting n → +∞ in (36) the result follows in this case.
Now assume thatex0< x0. Then, by (28) we have
exn≤xn, (37)
for every n ∈ N0.
If there is n1∈ N, such thatexn1 = xn1, then as above we have xn=exnfor n ≥ n1, from which the result follows in this case.
Otherwise, from (37), we have
yn:=exn−xn< 0, (38)
for every n ∈ N0, and that (33) holds, from which along with (38) and the monotonicity of f (n, t) in the second variable, it follows that
qnyn≤yn+1, (39)
for every n ∈ N0.
From (38) and (39), we have
0< (−yn) ≤ (−y0) Yn−1
j=0
qj, n ∈ N0. (40)
Hence, if n= mT + l for some m ∈ N0and l ∈ {0, 1, . . . , T − 1}, then form (40), as in (36), we get
0< xn−
exn= (−ymT+l) ≤ (−y0)
mT+l−1
Y
j=0
qj
≤ (x0−
ex0) max
l=1,T
Yl−1
j=0
qj
√T
λ
(T
√λ)n, (41)
for every n ∈ N0.
Sinceλ ∈ (0, 1), by letting n → +∞ in (41) the result follows in this case, finishing the proof of the theorem.
Remark 4.Since in Theorem 5 any solution to equation (25) converges to the chosen periodic one, it follows that the periodic solution is unique.
From Theorems 4 and 5 the following corollary follows.
Corollary 3. Consider equation(25), where (qn)n∈N0 is a positive T-periodic sequence such thatλ < 1, f : N0× [a, b] → R, a, b ∈ R, is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is a continuous and nonincreasing in the second variable, the functions in (26) are nondecreasing for each l ∈ {0, 1, . . . , T − 1}, and that the following conditions are satisfied
a ≤ min
n=0,T−1min
a≤t≤b n+T−1
X
i=n
f (i, t) 1 −λ
n+T−1
Y
j=i+1
qj≤ max
n=0,T−1max
a≤t≤b n+T−1
X
i=n
f (i, t) 1 −λ
n+T−1
Y
j=i+1
qj≤b. (42)
Then, equation (25) has a unique T-periodic solution (exn)n∈N0, such that
a ≤exn≤b, n ∈ N0, (43)
and every solution to the equation converges to the periodic one geometrically as n →+∞.
Proof. From Theorem 4 it follows that equation (25) has a T-periodic solution. By Theorem 5 it follows that every solution to the equation converges to the periodic one geometrically as n →+∞, from which it follows that it is unique (Remark 4).
Example 1. Here we present an example of equation (25) which satisfies the conditions in Theorem 5. Let (qn)n∈N0be a positive T-periodic sequence such thatλ < 1, and
f (n, x) = − eqlx
1+ |x|, (44)
where n= Tm + l, for some m ∈ N0, l ∈ {0, . . . , T − 1}, 0 <eql< ql, for every l ∈ {0, . . . , T − 1}. It is clear from (44) that f is decreasing for each l ∈ {0, . . . , T − 1}, and consequently for each n ∈ N0, and that by definition it is T-periodic in the first variable. On the other hand, since
1(n, x) = qnx+ f (n, x) =qnx|x|+ (qn− eqn)x 1+ |x| , it is easy to see that 1 is increasing in the second variable for each n ∈ N0.
Now note that the corresponding difference equation
xn+1 =qnxn|xn|+ (qn− eqn)xn
1+ |xn| , n ∈ N0, (45)
has the trivial solution
xn= 0, n ∈ N0, (46)
which is periodic of all periods, hence, it is also T-periodic. Since, all the conditions of Theorem 5 are satisfied, it follows that all the solutions to equation (45) converge geometrically to solution (46) as n →+∞.
Theorem 6. Consider equation(25), where (qn)n∈Z is a positive T-periodic sequence such that λ > 1, and f : N0× R → R is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is nondecreasing in the second variable.
If equation (25) has a T-periodic solution, then each other solution to the equation is getting away geometrically from the periodic one as n →+∞.
Proof. Since ql > 0 for every l ∈ {0, 1, . . . , T − 1}, this fact along with the monotonicity of the function f (l, x) in the second variable implies that the functions defined in (26) are strictly increasing on R. Hence, by an argument as in the proof of Theorem 5 we see that if x0 ≤
bx0, then xn≤
bxn (47)
for every n ∈ N0, and that in (47) strict inequality holds if and only if x0 <bx0.
Let (exn)n∈N0be the T-periodic solution to (25) and (xn)n∈N0be another solution to the equation.
First, assume that x0<ex0. Then, if yn:=exn−xn, we have yn> 0, for every n ∈ N0. From (25) and (32), we get (33), from which along with the monotonicity of f (n, x) in the second variable, it follows that
yn+1≥qnyn, (48)
for every n ∈ N0, and consequently
yn≥y0
Yn−1 j=0
qj, (49)
for n ∈ N0.
Hence, if n= mT + l for some m ∈ N0and l ∈ {0, 1, . . . , T − 1}, form inequality (49) we obtain
˜xn−xn= yn≥y0
Yl−1
j=0
qj
λm
= y0
Yl−1
j=0
qj
√T
λ
(T
√ λ)n
≥ (ex0−x0) min
l=1,T
Yl−1
j=0
qj
√T
λ
(T
√
λ)n, (50)
for every n ∈ N0. Sinceλ > 1, and
min
l=1,T
Yl−1
j=0
qj
√T
λ
> 0, (51)
by letting n →+∞ in (50) the result follows in this case.
Now, assume thatex0< x0. Then, we have yn< 0, for every n ∈ N0. From this along with (33), and the monotonicity of f (n, x) in the second variable, it follows that
yn+1≤qnyn, (52)
for every n ∈ N0, and consequently
(−yn) ≥ (−y0) Yn−1
j=0
qj, (53)
for n ∈ N0.
Hence, if n= mT + l for some m ∈ N0and l ∈ {0, 1, . . . , T − 1}, form (53), as above, we have
xn−
exn= (−yn) ≥ (−y0)
Yl−1
j=0
qj
λm= (−y0)
Yl−1
j=0
qj
√T
λ
(T
√λ)n
≥ (x0−
ex0) min
l=1,T
Yl−1
j=0
qj
√T
λ
(T
√
λ)n, (54)
for every n ∈ N0.
Sinceλ > 1 and (51) holds, by letting n → +∞ in (54) the result follows in this case, completing the proof of the theorem.
Remark 5.Since in Theorem 6 any other solution to difference equation (25) is getting away geometrically from the periodic one, it follows that the periodic solution is unique.
Combining Theorem 4 and Theorem 6 it is easy to see that the following corollary holds.
Corollary 4. Consider difference equation (25), where (qn)n∈N0is a positive T-periodic sequence such that λ > 1, f : N0× [a, b] → R, a, b ∈ R, is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is continuous and nondecreasing in the second variable, and that the conditions in (42) are satisfied. Then, the equation has a unique T-periodic solution (exn)n∈N0 satisfying (43), and each other solution to the equation is getting away geometrically from the periodic one as n →+∞.
The following theorem is proved similarly to Theorem 6, thus the proof is omitted.
Theorem 7. Consider equation(17), where (qn)n∈N0is a positive T-periodic sequence such thatλ > 1, f : N0×Rk+1→ R is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is nondecreasing in the other k + 1 variables.
If equation (17) has a T-periodic solution ( ˜xn)n∈N0, then each other solution to the equation which does not oscillate about ( ˜xn)n∈N0is getting away geometrically from the periodic one as n →+∞.
Remark 6. It is interesting to note that if we consider equation (17), where (qn)n∈N0 is a positive T-periodic sequence such thatλ < 1, f : N0× Rk+1 → R is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is nonincreasing in the other k + 1 variables, and if we assume that the functions
1l(t1, . . . , tk+1)= qlt1+ f (l, t1, . . . , tk+1), (55) are nondecreasing in variables tj, j = 1, k + 1, for each l ∈ {0, 1, . . . , T − 1}, then if equation (17) has a T- periodic solution ( ˜xn)n∈N0, each other solution to the equation which does not oscillate about ( ˜xn)n∈N0 will converge to the periodic one geometrically as n →+∞. The result looks like a generalization of Theorem 5 for such solutions. However, the posed conditions on the monotonicity of functions f and 1l, l = 1, k + 1, imply that the function f is constant in variables t2, . . . , tk+1, so that the result does not improve essentially Theorem 5 for such solution.
From the same reason, it is not possible to extend Theorem 3 in [31], in this way. For example, the following result, which is motivated also by Theorem 2 in [33] holds, but it is not an essential improvement of the theorems.
Theorem 8. Assume k ∈ N0, (qn)n∈N0 is a positive T-periodic sequence such thatλ < 1, f : N0× Rk+1 → R is a T-periodic function in the first variable, which for each n ∈ {0, 1, . . . , T − 1} is nonincreasing in the other k + 1 variables, the functions in (55) are nondecreasing in variables tj, j= 1, k + 1, for each l ∈ {0, 1, . . . , T − 1}, there are nonnegative T-periodic sequences (L(l)n)n∈N0, l = 1, k + 1, such that the following condition holds
|f (n, t1, . . . , tk+1) − f (n,et1, . . . ,etk+1)| ≤
k+1
X
l=1
L(l)n|tl−etl|, (56)
for every tj,etj∈ [0, ∞), j = 1, k + 1, and
j=0,T−1max qj≤ 1 and max
n=0,T−1 n+k
X
j=n k+1
X
l=1
L(l)j
n+k
Y
i=j+1
qi< 1, (57)
or
max
n=0,T−1 n+k+T−1
X
j=n k+1
X
l=1
L(l)j
n+k+T−1
Y
i=j+1
qi< 1. (58)
If equation (17) has a positive T-periodic solution (exn)n≥−k, then for every solution (xn)n≥−k to the equation the following relation holds
n→∞lim(exn−xn)= 0. (59)
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