Abstract
We investigate the global behavior of the second-order difference equation 
, where initial conditions and all coefficients are positive. We find conditions on 
under which the even and odd subsequences of a positive solution converge, one to zero and the other to a nonnegative number; as well as conditions where one of the subsequences diverges to infinity and the other either converges to a positive number or diverges to infinity. We also find initial conditions where the solution monotonically converges to zero and where it diverges to infinity.
. Under various hypotheses on the parameters, we establish the existence of different behaviors of even and odd subsequences of solutions of (1.1). Our results are summarized below.
and 
, then we have the following.
, such that for each, one of its subsequences, 
, 
, converges to zero and the other diverges to infinity.
, which
;
;
.
and 
. Then for each positive solution 
, one of the subsequences, 
, 
, diverges to infinity and the other to a positive number that can be arbitrarily large depending on initial values. Further there, are positive initial values for which the corresponding solution, 
, increases monotonically to infinity.
and 
. Then for each positive solution 
, one of the subsequences, 
, 
, converges to zero and the other to a nonnegative number. Further, there are positive initial values for which the corresponding solution, 
, decreases monotonically to zero.
This transforms (1.1) to
and 
in (1.1). Then one has the following.
, such that for each, one of its subsequences, 
, 
, converges to zero and the other to infinity.
, which
;
;
.
be defined as 
. Note that for 
, 
is a decreasing function since 
. Also note that 
and 
. Hence 
has a unique positive fixed point 
.
and simplify, it including canceling the common factor 
from the numerator and denominator, thereby obtaining the following:
, 
and 
. Thus the numerator of 
has one and only one sign change. Therefore, by Descartes' rule of signs, the numerator of 
has exactly one positive root, 
.
and so, given that 
is the only positive root of the numerator of 
, we have 
for 
. Thus, since 
and 
is continuous, we must have 
for 
. Therefore,
for (2.1).
).
is a decreasing function so that 
is an increasing function, we have
Since 
is the only positive fixed point of 
, then we must have 
and 
).
and 
In both cases, the solution, 
, of (2.1) is divided into even and odd subsequences, 
and 
, where one subsequence converges monotonically to zero and the other to infinity.
from 
. To do this we first consider 
. Without loss of generality, we will assume that 
and so 
and 
.
, we have
, there exists 
such that
. We then have
,
since
. Then 
for all 
, and so 
for all 
. Induction then gives us 
for all 
. We thus have one of the following:
(
), then 
(
), then 
(
), then 
is a constant solution 
and 
, or 
and 
, then 
satisfies the following properties:
, with 
undefined when 
.
if 
.
and 
, we also obtain
, then 
, or 
.
be a positive solution of (1.1) with 
and 
. Then there exist 
and 
such that the following statements are true:
as 
,
as 
,
, and 
and 
are undefined; or if either 
or 
is not zero, then 
is a solution of (1.1).
.
, and so 
and 
are undefined by property (P1); or 
and
(Property (P1) hold. Finally, Statement 4 follows immediately from Statement 3 and Property (P4).
and 
in (1.1). Then for each positive solution, 
, one of the subsequences, 
, 
, converges to zero and the other to a nonnegative number.
, 
, and 
. Then it follows from Lemma 2.2 that for each positive solution of (1.1), 
, one of the subsequences, 
, 
, converges to zero and the other to a nonnegative number.
and 
in (1.1). Then for each positive solution 
, one of the subsequences, 
, 
, diverges to infinity and the other to a positive number or diverges to infinity.
and 
. Using the transformation 
convert (1.1) to the equation
, and so it follows from Lemma 2.2 that for each positive solution of (2.16), 
, one of the subsequences, 
, 
, converges to zero and the other to a nonnegative number. Hence, for each positive solution of (1.1), 
, one of the subsequences, 
, 
, diverges to infinity and the other to a positive number or diverges to infinity.
and 
in (1.1). Then there are positive initial values for which the corresponding solutions, 
, decrease monotonically to zero.
Given Descartes' rule of signs, we have that there exists a unique positive equilibrium, 
, where 
and 
Recall that 
and let 
for all 
. Then 
for all 
. It follows from induction that 
for all 
. Since 
, 
, with 
, decreases monotonically to zero.
and 
in (1.1). Then there are positive initial values for which the corresponding solution, 
, increases monotonically to infinity.
we obtain from Descartes' rule of signs, a unique positive equilibrium, 
, where 
and 
Recall that 
and let 
for all 
. Then 
for all 
. It follows from induction that 
for all 
. Since 
, 
, with 
, increases monotonically to infinity.
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. Journal of Difference Equations and Applications 2000,6(5):563-576. 10.1080/10236190008808246
. Mathematical Sciences Research Hot-Line 1998,2(5):1-16.
for 
. Journal of Difference Equations and Applications 2006,12(1):101-108. 10.1080/10236190500410109
. Journal of Difference Equations and Applications 1995,1(3):317-321. 10.1080/10236199508808030