- Email: [email protected]
Global attractivity in a nonlinear difference equation
Global At&activity in a Nonlinear Difference Ch. G. Philos, I. K. Pumaras,
Equation
and Y. G. Sficas
Department of Mathematics University of Ioannina P.O. Box 1186 45110 Zoannina, Greece
ABSTRACT We consider the (nonlinear) difference equation
x, =
a+
m h c-_, k-1
where a and b, (k = 1,2,. . . , m) and we are interested in
n = 0,1,2
,...)
xrt-k
are
nonnegative numbers with B = CrElbk > 0, solutions are attracted by the positive
equilibrium L = (a/2) +
In this note we consider a nonlinear difference equation and we deal with the question of whether the (unique) positive equilibrium of the equation is a global attractor of all positive solutions. Recently, there has been a lot of interest in studying the global attractivity of nonlinear difference equations. For some recent results about the global attractivity of scalar nonlinear difference equations, see Jaroma, Kocid and Ladas [I], Jaroma, Kuruklis and Ladas [2], Karakostas, Philos and S&as [3], Koci6 and Ladas [4-71, Kocid, Ladas and Rodrigues [8], and Kurukhs and Ladas [9]. See also the paper of Franke and Yakubu [IO] for some results about global attractivity in systems of difference equations. Difference equations appear in the formulation and analysis of several discrete-time systems, as well as in the study of discretization methods for differential equations. Such equations also appear naturally as discrete analogous of differential and delay differential equations which model various APPLIED MATHEMATICS
AND COMPUTATZON
0 Elsevier Science Inc., 1994 655 Avenueof the Americas, New York, NY 10010
62:249-258 (1994)
249
0096-3003/94/$7.00
CH. PHILOS, I. PURNARAS, AND Y. SFICAS
250
diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. Moreover, the application of the theory of difference equations is rapidly broadening to various fields such as numerical analysis, control theory, finite mathematics, and computer science. So, the study of difference equations has acquired a new significance within the past two decades. Consider
the difference
equation
x, =
*
a+ c-7 kc1
where
bk
w
xn-k
a and b, (k = 1,2,. . . , m> are nonnegative
numbers
with
2 b, >0.
B =
k=l
By a solution of (E), we mean a sequence (x n>n>-m of nonzero numbers which satisfies (E) for all integers n >= 0. It is clear that a solution (X n>nz-m of the difference equation (E) is positive (in the sense that all its terms are positive numbers) positive.
if and only if the initial values
x _m, x _-m+1, . . . , x_ 1 are
The number
is the unique
ositive equilibrium
of the difference
equation
(E). (For a = 0,
L equals to $ B .> Our aim in this note is to establish that all positive solutions of(E) with a > 0 are attracted by the positive equilibrium L of the equation as well as to give sufficient conditions under which all positive solutions of the difference equation (E) with a = 0 are also attracted by its positive equilibrium a. It must be noted that in the case where a = 0 it is possible to have positive solutions of(E) which are not attracted by the equilibrium 6. As an example, let us consider the difference equation
P *tl =-++ x
where p, q are positive number
n-
4 1
nonnegative numbers w with w # Jp+q,
x,-3
with p + q > 0. For this equation admits
an arbitrary the positive
Global Attractivity
2-periodic
XZn-l
of a Daference
251
Equation
solution (x,,),, z _s with
=
0
and
-l,O,l,...
(n=
xzn=(p+q)/bJ
).
Our main result is the following theorem.
THEOREM. (i) Assume that a > 0. Then L is a global attractor of all positive solutions of the difference equation (El. (ii) Assume that a = 0. Let u, 1 5 Y 5 m, be an integer such that b, > 0, and suppose that there exists a positive integer p with 2~” s m such that > 0. Then L = 6 is a global attractor of all positive solutions of the 2&k-ence equation (E). Before we can establish our theorem we need result, which is interesting in its own right.
to prove
PROPOSITION. Every positive solution of the difference bounded from above and from below by positive constants.
PROOF. Let
(x,),
z _*
be
a positive
solution
of (E).
the following
equation
Clearly,
(E)
is
if the
solution is bounded from above by a positive constant C, then it is also bounded from below by the positive number c = a + B/C. Moreover, if the solution is bounded from below by a constant c > 0, then it is also bounded from above by C = a + B/c > 0. Now, assume for the sake of contradiction that lim sup x, = a n-+m For any integer
n > -m,
and
lim inf xn = 0. n-m
we define
L, = min A E (n,n (
+ l,...}:x,
We observe that (L,), , _-m is an increasing l,-m-t2,...}withlim~,,L,=~andsuchthat
>
sequence
lim xL = 03. n-+m n
max
-m~r6n-l
x,
1
of integers
,
in { -m
+
252
CH. PHILOS, I. PURNARAS, AND Y. SFICAS
We also observe that for any integer
for -m
XT < XLn Furthermore,
en= max
n > -m,
for each integer
n > -m,
A E {-m,
+ l,...,
1
-m
5 r 5 L,, - 1. we set
L, - I}:
xA =
min
x,
--m~r~L,-l - -
Clearly, CL,>,> _ is also an increasing sequence of integers 1, -m + 2, . . . ] with lim n ,J,, = 00. Moreover, we have
)
in I-m
m
. +
lim xp = 0. n-cc n Next, we consider an integer (E) we obtain for all integers
XL,, =
a+
xe, = a+
g
k=l
no > -m n >= n,
bk
B
---MU+XL,-k
or
rL _y/ _I ax, n
n
n
+ B
“4
bk cm ->>a+-
k=l
such that I,,,” > /no 2 0. Then from
B
*t’“,-k
or
r!nrr,” ’
UXL,
+
B.
xL,
Thus. we have UXL
which is a contradiction. proposition.
n < ax, n This
for every n >= n,,
contradiction
completes
the
proof
of our
The method used in proving our theorem relies on the theory of the limiting (and full limiting) se q uences and their relation with a recursive sequence of real numbers. Karakostas, Philos, and Sficas [3] presented this theory (see Section 2 in [3]) and used it for the study of the dynamics of some discrete population models; see also Karakostas [ll] for a detailed analysis of this theory. The theory of the limiting (and full limiting) sequences is obtained by considering the discrete version of the so-called Bebutov semi-
Global Attractivity of a Diference Equation
253
dynamical system, being known for continuous functions (see, for example, Karakostas [12] and Sib&y [13]). On the basis of this theory, we have the following facts, which will be used to prove our theorem: Let (x,,),, t _m be a positive solution of the difference equation (El. By our proposition, there exist two positive constants C and c such that for all integers
ClX,ZC -
n 2
- m.
(1)
Then there exist full limiting sequences (Y,>, E z and (.z,Jn E E of positive numbers (Z is the set of all integers) such that c=<.zOsYnsYYo=
limsupx, h+m
for all n E Z
(2)
for all n E Z.
(3)
and liminfx, A-m
= z0 5 z, s Y0 5 C
Each one of the sequences ( ynjnE z and (x,), E z is a full solution of the difference equation (E) in the sense that it satisfies (E) for a11 integers n. So, we have
yn=a+
m bk c--
k=l
for every n E Z
Yn-k
and
z, = a+
m
C-
k=l
bk
for every n E ‘Z.
%-k
(5)
Now, we are in a position to prove our theorem.
PROOF OF THE THEOREM. Let (x,1,, t _m be a positive solution of the difference equation (El. From our proposition, it follows that (1) holds for some positive constants C and c. Thus, there exist full limiting sequences and (.z,),,~B of positive numbers such that (2) and (3) are satisfied. (YAEZ These sequences are full solutions of (El, which means that (4) and (5) are true. Since L is the unique positive equilibrium of (El, in order to show that
254
CH. PHILOS,
I. PURNARAS,
AND Y. SFICAS
lim.,,r, = L, it suffices to verify that y0 = za. We will examine the two cases where a > 0 or a = 0. (i) Assume that
a > 0.By using (2>, from (4) we obtain
m
Y0=a+
B
b,
C-ra+-. k=l
Similarly,
separately
Y-k
20
in view of (3), from (5) it follows that
20
=
a+
m bk
B
C-za+-. k=l
z-k
Yo
Thus, we have
ay, I yozo - B
5
az,.
This ensures that y. = zo, since a > 0 and z. 5 yo. (ii) Assume that a = 0. First of all, we show that
yozo = B.
(6)
By (4) and (21, we have
k=l Y-k
%I
Also, (5) and (3) give
So, (6) follows immediately. Set now Z = { 1,2,. . . , m] - {v, 2 Z_LV)(I may be the empty set). By using (2), from (4) we derive
+-
b 2PV Y-2pLy
b
b2wv
A+
-+
Y-Y
Y--2pv
+
Global Attractivity
of a Diflerence
255
Equation
where we use the convention
that
On the other hand, (6) gives 1
b
2?+_
20
ZO
Cbk.
ksl
So, we have
b L+
bzgv
bzl*v bv ---~--+---. 20
ZO
Y-2p
Y-Y
But, our hypothesis guarantees that b, > 0 and bzpLY> 0. Moreover, of (3, y-Y 2 z. and y_2yV 2 zo. Hence, we always have
Y-Y = y-zpv
=
(7)
x0.
Next, by taking into account (61, (7), (41, and (21, we obtain
b 2+Yo
c - bk m
B
bzpv+ --,;;bk= 1
-
=z,=y_,=
Yo
Yo
=---+
k=l
bv
Y-2”
b,
>-+ = Y-2Y
b 2w Y-(Zp+W
kel
b 2PV Y -_(zp+
b
bzpv
4
-+ =
Y-ZV
y--Y-k
+c--
YO kc1
b 2w Y -(z/.L+l)v
.
bk
Y-v-k
+‘cb,
lb
and consequently
-II.+-> Yo Yo
because
CH. PHILOS,
256 Since b, > 0, must have
I. PURNARAS, AND Y. SFICAS
bSpLY> 0, and (because of (2)) y_2y 6 yo, y_(211+l)v 5 yo, we
Y-2v
Y-(ZflL+l)Y = Yo.
=
(8)
If p = I, then (7) and (8) give z. = y-pV = y. and the proof is complete. So, we suppose that p > 1. By using (6)
b -II+_ 20
b2pv 20
+ ;
k;Ih =
$
=-+
(8)
= y. =
y_2y
4 <---+ Y-3”
=
b 2PU
6, Y-3v
=
(4), and (2)
Y -(2pL+2)v
bk E ___ k=l Y-52-k +
b2pv Y -_(Z*LtZ)v
we get
+
c--
ker
bk
Y-Zv-k
tkFIbk
and so
b b+w z+-<-+ =
20
20
bv Y-3”
> 0,and (in view of (2)) But b, > 0,btpLY the last inequality implies that Y-3”
=
Y -(2p1_2)v
b 2cLv Y -(2pLc2)v
*
y-3V 2 .zo, y_(sfi+z)”
=
zo.
(9)
Next, by taking into account (6) (9), (4) and (2) and using arguments with those in obtaining (8) we can conclude that
Y-4v
=
Y-(2w+3)v
=
2 ~0. Thus,
the
same
(10)
Yo.
In the case where p = 2, from (7) and (10) it follows that z. = y_4V = y. and the proof is complete. If /..L> 2, then, repeating the above procedure, we finally obtain Y -2pv
=
Y -[zp+(2p-l)]Y
=
Yo*
Global
Attractiuity of a Difierence Equation
257
This and (7) give 2.0
and so the proof of the theorem
Y -2/Lv
=
Yo
is complete.
We have seen that the equation
REMARK.
P
9
xn=-+_ has 2-periodic
(PLo>qLo,p+q>o)
x 11- 3
X,-l
difference
=
solutions.
This is also true for the more general
case of the
equation
xn
PI
=-+-
P3
X,-l
+
***
Pzr+
+
xn-(zr+
x,&-3
1 1) ’
where r is a positive integer and p,, p,, . . . , pzr+ I are nonnegative numbers > 0. Indeed, for an arbitrary number w with P = p, + 3 + a+. +P~,+~ with 0 < w + P I’, the 2-periodic
XZn-,
=
w
and
X2n
sequence =
P/w
(x,),
> _ _(2r+ 1) defined by
(n = -?-,-r+
l,...)
is a solution. Note that the difference equation under consideration is not subject to the hypothesis posed in our theorem for the case where a = 0.
The authors thank Professor of this paper.
G. Ladas for helpful conversations
on the subject
REFERENCES 1
2
J. H. Jaroma, V. Lj. Kocid, and G. Ladas, Global asymptotic stability of a second-order difference equation, in Proceedings of the International Conference on Theory and Applications of Dafirential Equations, The Univ. of Texas-Pan American, 1991. J. H. Jaroma, S. A. Kuruklis, and G. Ladas, Oscillations and stability of a discrete delay logistic model, Ukrainian Math. J. 43:734-744 (1991).
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258 3
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Math.
dynamics
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(1991). in a discrete
model
(1990).
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in nonlinear
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The
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V. Lj. Kocid and G. Ladas, ence equations,
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A&.
Amer.
Proc.
Anal.
Oscillation
of Nicholson’s
I. PURNARAS,
and Y. G. S&as,
Nonlinear
V. Lj. Koci& and G. Ladas,
equations, 6
Ch. G. Philos,
models,
PHILOS,
attractivity
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difference
(1992).
in second-order
nonlinear
differ-
Appl., to appear.
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difference
equations,
to appear. 8
V. Lj. Kocih, G. Ladas, and I. W. Rodrigues, Math.
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S. A. Kurukhs logistic
10
J.
E.
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Nonlinear 11
12
Quart. and
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sequences
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delay
(1992). attractors
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G. Karakostas,
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36:66-74
in the by
(19891,
space
difference
of sequences equations,
and study
of
McYO~/.L(YTLK.~~
in Greek.
and topological
dynamics,
Ann. Mat.
Pura Appl.
(1982).
K. S. Sibirsky,
Introduction
to Topological
Dynamics,
Noordhoff,
Leyden,
1975.
Equation
and Y. G. Sficas
Department of Mathematics University of Ioannina P.O. Box 1186 45110 Zoannina, Greece
ABSTRACT We consider the (nonlinear) difference equation
x, =
a+
m h c-_, k-1
where a and b, (k = 1,2,. . . , m) and we are interested in
n = 0,1,2
,...)
xrt-k
are
nonnegative numbers with B = CrElbk > 0, solutions are attracted by the positive
equilibrium L = (a/2) +
In this note we consider a nonlinear difference equation and we deal with the question of whether the (unique) positive equilibrium of the equation is a global attractor of all positive solutions. Recently, there has been a lot of interest in studying the global attractivity of nonlinear difference equations. For some recent results about the global attractivity of scalar nonlinear difference equations, see Jaroma, Kocid and Ladas [I], Jaroma, Kuruklis and Ladas [2], Karakostas, Philos and S&as [3], Koci6 and Ladas [4-71, Kocid, Ladas and Rodrigues [8], and Kurukhs and Ladas [9]. See also the paper of Franke and Yakubu [IO] for some results about global attractivity in systems of difference equations. Difference equations appear in the formulation and analysis of several discrete-time systems, as well as in the study of discretization methods for differential equations. Such equations also appear naturally as discrete analogous of differential and delay differential equations which model various APPLIED MATHEMATICS
AND COMPUTATZON
0 Elsevier Science Inc., 1994 655 Avenueof the Americas, New York, NY 10010
62:249-258 (1994)
249
0096-3003/94/$7.00
CH. PHILOS, I. PURNARAS, AND Y. SFICAS
250
diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. Moreover, the application of the theory of difference equations is rapidly broadening to various fields such as numerical analysis, control theory, finite mathematics, and computer science. So, the study of difference equations has acquired a new significance within the past two decades. Consider
the difference
equation
x, =
*
a+ c-7 kc1
where
bk
w
xn-k
a and b, (k = 1,2,. . . , m> are nonnegative
numbers
with
2 b, >0.
B =
k=l
By a solution of (E), we mean a sequence (x n>n>-m of nonzero numbers which satisfies (E) for all integers n >= 0. It is clear that a solution (X n>nz-m of the difference equation (E) is positive (in the sense that all its terms are positive numbers) positive.
if and only if the initial values
x _m, x _-m+1, . . . , x_ 1 are
The number
is the unique
ositive equilibrium
of the difference
equation
(E). (For a = 0,
L equals to $ B .> Our aim in this note is to establish that all positive solutions of(E) with a > 0 are attracted by the positive equilibrium L of the equation as well as to give sufficient conditions under which all positive solutions of the difference equation (E) with a = 0 are also attracted by its positive equilibrium a. It must be noted that in the case where a = 0 it is possible to have positive solutions of(E) which are not attracted by the equilibrium 6. As an example, let us consider the difference equation
P *tl =-++ x
where p, q are positive number
n-
4 1
nonnegative numbers w with w # Jp+q,
x,-3
with p + q > 0. For this equation admits
an arbitrary the positive
Global Attractivity
2-periodic
XZn-l
of a Daference
251
Equation
solution (x,,),, z _s with
=
0
and
-l,O,l,...
(n=
xzn=(p+q)/bJ
).
Our main result is the following theorem.
THEOREM. (i) Assume that a > 0. Then L is a global attractor of all positive solutions of the difference equation (El. (ii) Assume that a = 0. Let u, 1 5 Y 5 m, be an integer such that b, > 0, and suppose that there exists a positive integer p with 2~” s m such that > 0. Then L = 6 is a global attractor of all positive solutions of the 2&k-ence equation (E). Before we can establish our theorem we need result, which is interesting in its own right.
to prove
PROPOSITION. Every positive solution of the difference bounded from above and from below by positive constants.
PROOF. Let
(x,),
z _*
be
a positive
solution
of (E).
the following
equation
Clearly,
(E)
is
if the
solution is bounded from above by a positive constant C, then it is also bounded from below by the positive number c = a + B/C. Moreover, if the solution is bounded from below by a constant c > 0, then it is also bounded from above by C = a + B/c > 0. Now, assume for the sake of contradiction that lim sup x, = a n-+m For any integer
n > -m,
and
lim inf xn = 0. n-m
we define
L, = min A E (n,n (
+ l,...}:x,
We observe that (L,), , _-m is an increasing l,-m-t2,...}withlim~,,L,=~andsuchthat
>
sequence
lim xL = 03. n-+m n
max
-m~r6n-l
x,
1
of integers
,
in { -m
+
252
CH. PHILOS, I. PURNARAS, AND Y. SFICAS
We also observe that for any integer
for -m
XT < XLn Furthermore,
en= max
n > -m,
for each integer
n > -m,
A E {-m,
+ l,...,
1
-m
5 r 5 L,, - 1. we set
L, - I}:
xA =
min
x,
--m~r~L,-l - -
Clearly, CL,>,> _ is also an increasing sequence of integers 1, -m + 2, . . . ] with lim n ,J,, = 00. Moreover, we have
)
in I-m
m
. +
lim xp = 0. n-cc n Next, we consider an integer (E) we obtain for all integers
XL,, =
a+
xe, = a+
g
k=l
no > -m n >= n,
bk
B
---MU+XL,-k
or
rL _y/ _I ax, n
n
n
+ B
“4
bk cm ->>a+-
k=l
such that I,,,” > /no 2 0. Then from
B
*t’“,-k
or
r!nrr,” ’
UXL,
+
B.
xL,
Thus. we have UXL
which is a contradiction. proposition.
n < ax, n This
for every n >= n,,
contradiction
completes
the
proof
of our
The method used in proving our theorem relies on the theory of the limiting (and full limiting) se q uences and their relation with a recursive sequence of real numbers. Karakostas, Philos, and Sficas [3] presented this theory (see Section 2 in [3]) and used it for the study of the dynamics of some discrete population models; see also Karakostas [ll] for a detailed analysis of this theory. The theory of the limiting (and full limiting) sequences is obtained by considering the discrete version of the so-called Bebutov semi-
Global Attractivity of a Diference Equation
253
dynamical system, being known for continuous functions (see, for example, Karakostas [12] and Sib&y [13]). On the basis of this theory, we have the following facts, which will be used to prove our theorem: Let (x,,),, t _m be a positive solution of the difference equation (El. By our proposition, there exist two positive constants C and c such that for all integers
ClX,ZC -
n 2
- m.
(1)
Then there exist full limiting sequences (Y,>, E z and (.z,Jn E E of positive numbers (Z is the set of all integers) such that c=<.zOsYnsYYo=
limsupx, h+m
for all n E Z
(2)
for all n E Z.
(3)
and liminfx, A-m
= z0 5 z, s Y0 5 C
Each one of the sequences ( ynjnE z and (x,), E z is a full solution of the difference equation (E) in the sense that it satisfies (E) for a11 integers n. So, we have
yn=a+
m bk c--
k=l
for every n E Z
Yn-k
and
z, = a+
m
C-
k=l
bk
for every n E ‘Z.
%-k
(5)
Now, we are in a position to prove our theorem.
PROOF OF THE THEOREM. Let (x,1,, t _m be a positive solution of the difference equation (El. From our proposition, it follows that (1) holds for some positive constants C and c. Thus, there exist full limiting sequences and (.z,),,~B of positive numbers such that (2) and (3) are satisfied. (YAEZ These sequences are full solutions of (El, which means that (4) and (5) are true. Since L is the unique positive equilibrium of (El, in order to show that
254
CH. PHILOS,
I. PURNARAS,
AND Y. SFICAS
lim.,,r, = L, it suffices to verify that y0 = za. We will examine the two cases where a > 0 or a = 0. (i) Assume that
a > 0.By using (2>, from (4) we obtain
m
Y0=a+
B
b,
C-ra+-. k=l
Similarly,
separately
Y-k
20
in view of (3), from (5) it follows that
20
=
a+
m bk
B
C-za+-. k=l
z-k
Yo
Thus, we have
ay, I yozo - B
5
az,.
This ensures that y. = zo, since a > 0 and z. 5 yo. (ii) Assume that a = 0. First of all, we show that
yozo = B.
(6)
By (4) and (21, we have
k=l Y-k
%I
Also, (5) and (3) give
So, (6) follows immediately. Set now Z = { 1,2,. . . , m] - {v, 2 Z_LV)(I may be the empty set). By using (2), from (4) we derive
+-
b 2PV Y-2pLy
b
b2wv
A+
-+
Y-Y
Y--2pv
+
Global Attractivity
of a Diflerence
255
Equation
where we use the convention
that
On the other hand, (6) gives 1
b
2?+_
20
ZO
Cbk.
ksl
So, we have
b L+
bzgv
bzl*v bv ---~--+---. 20
ZO
Y-2p
Y-Y
But, our hypothesis guarantees that b, > 0 and bzpLY> 0. Moreover, of (3, y-Y 2 z. and y_2yV 2 zo. Hence, we always have
Y-Y = y-zpv
=
(7)
x0.
Next, by taking into account (61, (7), (41, and (21, we obtain
b 2+Yo
c - bk m
B
bzpv+ --,;;bk= 1
-
=z,=y_,=
Yo
Yo
=---+
k=l
bv
Y-2”
b,
>-+ = Y-2Y
b 2w Y-(Zp+W
kel
b 2PV Y -_(zp+
b
bzpv
4
-+ =
Y-ZV
y--Y-k
+c--
YO kc1
b 2w Y -(z/.L+l)v
.
bk
Y-v-k
+‘cb,
lb
and consequently
-II.+-> Yo Yo
because
CH. PHILOS,
256 Since b, > 0, must have
I. PURNARAS, AND Y. SFICAS
bSpLY> 0, and (because of (2)) y_2y 6 yo, y_(211+l)v 5 yo, we
Y-2v
Y-(ZflL+l)Y = Yo.
=
(8)
If p = I, then (7) and (8) give z. = y-pV = y. and the proof is complete. So, we suppose that p > 1. By using (6)
b -II+_ 20
b2pv 20
+ ;
k;Ih =
$
=-+
(8)
= y. =
y_2y
4 <---+ Y-3”
=
b 2PU
6, Y-3v
=
(4), and (2)
Y -(2pL+2)v
bk E ___ k=l Y-52-k +
b2pv Y -_(Z*LtZ)v
we get
+
c--
ker
bk
Y-Zv-k
tkFIbk
and so
b b+w z+-<-+ =
20
20
bv Y-3”
> 0,and (in view of (2)) But b, > 0,btpLY the last inequality implies that Y-3”
=
Y -(2p1_2)v
b 2cLv Y -(2pLc2)v
*
y-3V 2 .zo, y_(sfi+z)”
=
zo.
(9)
Next, by taking into account (6) (9), (4) and (2) and using arguments with those in obtaining (8) we can conclude that
Y-4v
=
Y-(2w+3)v
=
2 ~0. Thus,
the
same
(10)
Yo.
In the case where p = 2, from (7) and (10) it follows that z. = y_4V = y. and the proof is complete. If /..L> 2, then, repeating the above procedure, we finally obtain Y -2pv
=
Y -[zp+(2p-l)]Y
=
Yo*
Global
Attractiuity of a Difierence Equation
257
This and (7) give 2.0
and so the proof of the theorem
Y -2/Lv
=
Yo
is complete.
We have seen that the equation
REMARK.
P
9
xn=-+_ has 2-periodic
(PLo>qLo,p+q>o)
x 11- 3
X,-l
difference
=
solutions.
This is also true for the more general
case of the
equation
xn
PI
=-+-
P3
X,-l
+
***
Pzr+
+
xn-(zr+
x,&-3
1 1) ’
where r is a positive integer and p,, p,, . . . , pzr+ I are nonnegative numbers > 0. Indeed, for an arbitrary number w with P = p, + 3 + a+. +P~,+~ with 0 < w + P I’, the 2-periodic
XZn-,
=
w
and
X2n
sequence =
P/w
(x,),
> _ _(2r+ 1) defined by
(n = -?-,-r+
l,...)
is a solution. Note that the difference equation under consideration is not subject to the hypothesis posed in our theorem for the case where a = 0.
The authors thank Professor of this paper.
G. Ladas for helpful conversations
on the subject
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