J. Appl. Math. & Computing Vol. 18(2005), No. 1 - 2, pp. 229 - 234
ON THE RECURSIVE SEQUENCE xn+1 = α +
xp
n−1
xp
n
STEVO STEVIĆ
Abstract. The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the difference equation of the
form
xpn−1
xn+1 = α +
,
n = 0, 1, ...
xpn
is investigated, where all the coefficients are nonnegative real numbers.
AMS Mathematics Subject Classification : Primary 39A10
Key words and phrases : Equilibrium, positive solution, difference equation, boundedness, global attractivity.
1. Introduction
Recently there has been a great interest in investigating nonlinear difference
equations, see, for example, [1-9] and the references therein. The following
conjecture was formulated in [5]:
Conjecture 5.2.4 Every positive solution of the difference equation
xn−1
,
n = 1, 2, ...
xn+1 = β +
xn
is bounded if and only if β ≥ 1. Furthermore, if β = 1, then every positive
solution converges to a period two solution and, if β > 1, then every positive
solution converges to the equilibrium β + 1.
A first discussion of the problem is given in [2]. Motivated by [2], in [3] the
authors have considered the equation
xn+1 = α +
xpn−1
,
xpn
n = 0, 1, ...,
Received November 5, 2003. Revised January 27, 2004.
c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.
229
(1)
230
Stevo Stević
with positive initial conditions x−1 and x0 , under the assumptions α ∈ [0, ∞) and
p ∈ [1, ∞). They have investigated local stability, oscillation and boundedness
character of the positive solutions of Eq.(1).
The following four results, which we incorporate into a theorem, were proved
in [3].
Theorem A. Consider Eq.(1). Then the following statements are true:
(1) The equilibrium point x̄ = α + 1 of Eq.(1) is locally asymptotically stable
if α > 2p − 1 and unstable if α ∈ [0, 2p − 1).
(2) Let (xn ) be a positive solution of Eq.(1) which consists of a single semicycle. Then (xn ) converges monotonically to x̄.
(3) Let (xn ) be a positive nonequilibrium solution of Eq.(1) which consists
of at least two semi-cycles. Then (xn ) is oscillatory. Moreover, every
semi-cycle has length one except possibly the first semi-cycle.
(4) Let α ∈ [0, 1) and let (xn ) be a solution of Eq.(1) such that 0 < x−1 ≤ 1
and x0 ≥ 1/(1 − α)1/p . Then lim x2n = ∞ and lim x2n+1 = α.
n→∞
n→∞
The first result is a simple consequence of the linearized stability theorem [4].
The second one is also simple and it can be generalized as follows:
Theorem 1. Consider the difference equation
xn−k
xn+1 = α + f
,
n = 0, 1, ...,
xn
(2)
with positive initial conditions x−k , ..., x0 , where α ∈ [0, ∞), k ∈ N and the
function f is nonnegative, strictly increasing on the interval [0, ∞) and f (1) = 1.
Let (xn ) be a nonoscillatory solution of Eq.(2). Then (xn ) converges to x̄ = α+1.
Corollary 1. Consider Eq.(2) with k = 1, positive initial conditions x−1 and
x0 , where α ∈ [0, ∞) and the function f is nonnegative, strictly increasing on
the interval [0, ∞) and f (1) = 1. Let (xn ) be a solution of the equation which
consists of a single semi-cycle. Then (xn ) converges monotonically to x̄ = α + 1.
It was not shown in [3] the existence of solutions of Eq.(1) which consist of a
single semi-cycle. So we leave to the reader who are interested in this area the
following open problem.
Open problem 1. Investigate whether or not there is a nonequilibrium solution
of Eq.(1) which consists of a single semi-cycle.
The third result in Theorem A is a special case of Theorem 3.2 in [4].
The proof of the last result in Theorem A follows the lines of the proof of
Theorem 3.1 in [2]. This caused that they repeated the same mistake which
Recursive sequence
231
appears in that paper, that is, the proofs in both papers do not hold for α = 0.
In this note we also give a proof of this result, i.e. we prove the following result.
Theorem 2. Consider Eq.(1) with p ≥ 1 and α = 0. Let (xn ) be a solution
of Eq.(1) such that 0 < x−1 ≤ 1 and x0 ≥ 1/1 − ε for some ε ∈ (0, 1). Then
lim x2n = ∞ and lim x2n+1 = 0.
n→∞
n→∞
In [3] the authors also do not consider the case p ∈ (0, 1). Here we show that
unlike to the case p ≥ 1, in this case solutions of Eq.(1) are bounded. The fact
is contained in the next theorem.
Theorem 3. Consider Eq.(1). If p ∈ (0, 1), then every positive solution of
Eq.(1) is bounded.
2. Proofs of the main results
In this section we prove the main results of this paper. Before that we need
an auxiliary result which is contained in the next lemma.
Lemma 1. Consider Eq.(2) with positive initial conditions x−k , ..., x0 , where
α ∈ [0, ∞), k ∈ N and the function f is nonnegative, strictly increasing on the
interval [0, ∞) and f (1) = 1. Under these conditions Eq.(2) has no nontrivial
periodic solutions of period k (not necessarily prime).
Proof. Assume that p1 , ..., pk , p1 , ..., pk , ... is a periodic solution of Eq.(2). Since
in this case is xn−k = xn for n ∈ N, we have xn+1 = α + f (1) = α + 1 for all
n ∈ N, as desired.
2
Proof of Theorem 1. We may assume that xn ≥ α + 1 for every n ∈ N. The case
xn ≤ α + 1, n ∈ N is similar and will be omitted. We have
xn−k
≥ α + 1.
xn+1 = α + f
xn
Since f is increasing, it follows that
α + 1 ≤ xn ≤ xn−k ,
for
n = 1, 2, ... .
(3)
Hence there are finite lim xmk+i = pi , i = 1, ..., k, and p1 , ..., pk is a periodic
m→∞
solution of Eq.(2) of period k. By Lemma 1 we obtain that p1 = · · · = pk . The
proof is complete.
2
Proof of Corollary 1. It is a direct consequence of Theorem 1 and inequality (3)
with k = 1.
2
232
Stevo Stević
Proof of Theorem 2. We have
0 < x1 =
xp−1
1
p ≤ p ≤ 1,
x0
x0
i.e., x1 ∈ (0, 1].
Further we have
x2 =
and
0 < x3 =
Thus x3 ∈ (0, 1].
Also
x4 =
xp0
≥ xp0
xp1
xp1
1
1
1
p ≤ p ≤
p p ≤ p ≤ 1.
x2
x2
(x0 )
x0
xp2
p
p p
p2
p ≥ x2 ≥ (x0 ) = x0 .
x3
By induction, we obtain
x2n ≥ xp0
n
and
0 < x2n+1 ≤ 1.
Since x0 ≥ 1/1 − ε > 1 for some ε ∈ (0, 1), we obtain that lim x2n = ∞ and
n→∞
from this and by (1) we have that lim x2n+1 = 0, as desired.
2
n→∞
Remark 1. Note that when α = 0, Eq.(1) with positive initial conditions can
be solved by the change yn = ln xn . Indeed, in this case Eq.(1) is transformed
into the following linear second-order difference equation yn+1 +pyn −pyn−1 = 0,
whose general solution is
λ2 y0 − y1 n y1 − λ1 y0 n
yn =
λ +
λ ,
λ2 − λ1 1
λ2 − λ1 2
√ 2
−p± p +4p
where λ1,2 =
. From this we see that a solution of Eq.(1) is unbounded
2
p
−λ2
if and only if x−1 6= x0 .
Proof of Theorem 3. In view of Theorem 1 we may assume that a solution (xn ) of
Eq.(1) is oscillatory, since every nonoscillatory solution of Eq.(1) is convergent.
By Theorem 3.2 in [4] the solution has semi-cycles of length one except possibly
for the first semi-cycle. Hence we may assume that x2n ∈ (α, α + 1] and x2n+1 ≥
α + 1 for every n ∈ N. The case x2n+1 ∈ (α, α + 1] and x2n ≥ α + 1 is similar
and will be omitted. We have
xp
xp
x2n+1 = α + 2n−1
≤ α + 2n−1
,
n ∈ N.
(4)
p
x2n
αp
Let (yn ) be the solution of the difference equation
yn+1 = α +
ynp
,
αp
y 1 = x1 .
Recursive sequence
233
By (4) and induction we see that x2n−1 ≤ yn , n ∈ N. Hence it is enough to
prove that the sequence (yn ) is bounded. Since the function
xp
f (x) = α + p , x ∈ (0, ∞),
α
is increasing and concave for p ∈ (0, 1) it follows that there is a unique fixed
point x∗ of the equation f (x) = x and that the function f satisfies the condition
(f (x) − x)(x − x∗ ) < 0,
x ∈ (0, ∞).
Using this fact it is easy to see that if y1 ∈ (0, x∗ ] the sequence is nondecreasing
and bounded above by x∗ and if y1 ≥ x∗ , it is nonincreasing and bounded below
by x∗ . Hence for every y1 ∈ (0, ∞) the sequence is bounded, from which the
result follows.
2
Corollary 2. Consider Eq.(1). Let p ∈ (0, 1) and α ≥ 1. Then every positive
solution of Eq.(1) converges to x̄.
Proof. Since by Theorem 3 in this case every solution (xn ) of Eq.(1) is bounded
it follows that there are finite lim inf xn = l and lim sup xn = L. Let us assume
that l < L. Letting lim inf and lim sup in (1) we obtain
α+
Lp
lp
≤
l
<
L
≤
α
+
Lp
lp
which implies that
αLp + lp ≤ Lp l < Llp ≤ αlp + Lp
i.e.,
(α − 1)Lp < (α − 1)lp ,
which is impossible for α ≥ 1. Hence the result follows.
References
1. R.P.Agarwal, Difference equations and inequalities, 2nd Edition, Pure Appl. Math. 228,
Marcel Dekker, New York, 2000.
2. A.M.Amleh, E.A.Grove, G.Ladas and D.A.Georgiou, On the recursive sequence yn+1 =
y
α + n−1
, J. Math. Anal. Appl. 233 (1999), 790-798.
yn
3. H.M.El-Owaidy, A.M.Ahmed and M.S.Mousa, On asymptotic behaviour of the difference
p
equation xn+1 = α +
xn−1
p
xn
, J. Appl. Math. & Computing 12 (1-2) (2003), 31-37.
4. M.R.S.Kulenović and G.Ladas, Dynamics of Second Order Rational Difference Equations,
Chapman & Hall/CRC, (2001).
5. G.Ladas, Open problems and conjectures, J. Differ. Equations Appl. 5 (1999), 211-215.
6. S.Stević, On the recursive sequence xn+1 = g(xn , xn−1 )/(A + xn ), Appl. Math. Lett. 15
(2002), 305-308.
7. S.Stević, On the recursive sequence xn+1 = xn−1 /g(xn ), Taiwanese J. Math. 6 (3) (2002),
405-414.
234
Stevo Stević
8. Z.Zhang, B.Ping and W.Dong, Oscillatory of unstable type second-order neutral difference
equations, J. Appl. Math. & Computing 9 No. 1 (2002), 87-100.
9. Z.Zhou, J.Yu and G.Lei, Oscillations for even-order neutral difference equations, J. Appl.
Math. & Computing 7 No. 3 (2000), 601-610.
Stevo Stević received his Ph.D at Belgrade University in 2001. He has written more than 80
original scientific papers and his research interests are mostly in analytic functions of one
and several variables, potential theory, difference equations, convergence and divergence
of infinite limiting, nonlinear analysis, fixed point theory, operators on function spaces,
inequalities and qualitative analysis of differential equations.
Mathematical Institute of Serbian Academy of Science, Knez Mihailova 35/I, 11000 Beograd
e-mail: [email protected]; [email protected]