SOOCHOW JOURNAL OF MATHEMATICS
Volume 33, No. 4, pp. 861-873, October 2007
QUALITATIVE BEHAVIOR OF HIGHER ORDER
DIFFERENCE EQUATION
BY
E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED
Abstract. In this paper we investigate some qualitative behavior of the solutions
of the recursive sequence
xn+1 =
dxn−l xn−k
+ a, n = 0, 1, . . .
cxn−s − b
where the initial conditions x−r , x−r+1 , x−r+2 , . . ., x0 are arbitrary positive real
numbers with xi 6= bc for i = −r, −r + 1, . . . , 0, a > b/c, r = max{l, k, s} is
nonnegative integer and a, b, c, d are positive constants. Also, we study two special
cases of this equation.
1. Introduction
In this paper we deal with some properties of the solutions of the recursive
sequence
xn+1 =
dxn−l xn−k
+ a, n = 0, 1, . . .
cxn−s − b
(1)
where the initial conditions x−r , x−r+1 , x−r+2 , . . . ,x0 are arbitrary positive real
numbers with xi 6=
b
c
for i = −r, −r + 1, . . . , 0, a > b/c, r = max{l, k, s} is
nonnegative integer and a, b, c, d are positive constants.
Also, we study two special cases of Eq.(1).
Recently there has been a lot of interest in studying the global attractivity,
boundedness character and the periodic nature of nonlinear difference equations.
Received; revised.
AMS Subject Classification. 39A10.
Key words. difference equations, stability, periodic solutions.
861
862
E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED
For some results in this area, see for example [4-8]. See also [1-3]. Here, we recall
some notations and results which will be useful in our investigation.
Let I be some interval of real numbers and let
f : I k+1 → I
be a continuously differentiable function. Then for every set of initial conditions
x−k , x−k+1 , . . . , x0 ∈ I, the difference equation
xn+1 = f (xn , xn−1 , . . . , xn−k ), n = 0, 1, . . . ,
(2)
has a unique solution {xn }∞
n=−k ([8]).
A point x ∈ I is called an equilibrium point of Eq.(2) if
x = f (x, x, . . . , x).
That is, xn = x for n ≥ 0, is a solution of Eq.(2), or equivalently, x is a fixed
point of f .
Definition 1. The difference equation (2) is said to be persistence if there
exist numbers m and M with 0 < m ≤ M < ∞ such that for any initial conditions
x−k , x−k+1 , . . . , x−1 , x0 ∈ (0, ∞) there exists a positive integer N which depends
on the initial conditions such that
m ≤ xn ≤ M
for all
n ≥ N.
Definition 2. (Stability)
(i) The equilibrium point x of Eq.(2) is locally stable if for every ǫ > 0, there
exists δ > 0 such that for all x−k , x−k+1 , . . . , x−1 , x0 ∈ I with
|x−k − x| + |x−k+1 − x| + · · · + |x0 − x| < δ,
we have
|xn − x| < ǫ for all n ≥ −k.
(ii) The equilibrium point x of Eq.(2) is locally asymptotically stable if x is
locally stable solution of Eq.(2) and there exists γ > 0, such that for all
x−k , x−k+1 , . . . , x−1 , x0 ∈ I with
|x−k − x| + |x−k+1 − x| + · · · + |x0 − x| < γ,
QUALITATIVE BEHAVIOR OF HIGHER ORDER DIFFERENCE EQUATION
863
we have
lim xn = x.
n→∞
(iii) The equilibrium point x of Eq.(2) is global attractor if for all x−k , x−k+1 , . . .,
x−1 , x0 ∈ I, we have
lim xn = x.
n→∞
(iv) The equilibrium point x of Eq.(2) is globally asymptotically stable if x is
locally stable, and x is also a global attractor of Eq.(2).
(v) The equilibrium point x of Eq.(2) is unstable if x is not locally stable.
The linearized equation of Eq.(2) about the equilibrium x is the linear difference equation
yn+1 =
k
X
∂f (x, x, . . . , x)
i=0
∂xn−i
yn−i .
Theorem A.([7]) Assume that pi ∈ R, i = 1, 2, . . . and k ∈ {0, 1, 2, . . .}.
Then
k
X
i=1
|pi | < 1
is a sufficient condition for the asymptotic stability of the difference equation
xn+k + p1 xn+k−1 + · · · + pk xn = 0, n = 0, 1, . . . .
Consider the following equation
xn+1 = g(xn , xn−1 , xn−2 ).
(3)
The following two theorems will be useful for the proof of our main results in this
paper.
Theorem B.([8]) Let [a, b] be an interval of real numbers and assume that
g : [a, b]3 → [a, b]
is a continuous function satisfying the following properties :
(a) g(x, y, z) is non-decreasing in x and y ∈ [a, b] for each z ∈ [a, b], and is
non-increasing in z ∈ [a, b] for each x and y ∈ [a, b].
864
E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED
(b) If (m, M ) ∈ [a, b] × [a, b] is a solution of the system
m = g(m, m, M ) and M = g(M, M, m)
then
m = M.
Then Eq.(3) has a unique equilibrium x ∈ [a, b] and every solution of Eq.(3)
converges to x.
Theorem C.([8]) Let [a, b] be an interval of real numbers and assume that
g : [a, b]3 → [a, b]
is a continuous function satisfying the following properties:
(a) g(x, y, z) is non-increasing in all three variables x, y, z ∈ [a, b].
(b) If (m, M ) ∈ [a, b] × [a, b] is a solution of the system
M = g(m, m, m) and m = g(M, M, M )
then
m = M.
Then Eq.(3) has a unique equilibrium x ∈ [a, b] and every solution of Eq.(3)
converges to x.
2. Periodic Solutions
In this section we study the existence of periodic solutions of Eq.(1). The
following theorem states the necessary and sufficient conditions that this equation
has periodic solutions.
Theorem 1. Eq.(1) has positive prime period two solutions if and only if
one of the following conditions is satisfied:
2
(i) If l, k, s are even and (b − ac)2 > 4d(b −abc−abd)
, b > a(c + d).
(c+d)
(ii) If l, k, s are odd and (ac + b)2 > 4ab(c − d), c > d.
2
2 −a2 c2 d−b2 d)
(iii) If l, k−even, s−odd and (ac + b)2 > 4(abc +abd(c−d)
, ab(c2 + d2 ) >
d(a2 c2 − b2 ), c 6= d.
QUALITATIVE BEHAVIOR OF HIGHER ORDER DIFFERENCE EQUATION
(iv) If l−even, k, s−odd and (ac + b)2 >
(v) If k−even, l, s−odd and (ac + b)2 >
(vi) If s−even, l, k−odd and (ac − b)2 >
4abc2
(c+d) , c > d.
4abc2
(c+d) .
4d(a2 c2 −abc−abd)
,
(c+d)
865
ac2 > b(c + d).
Proof. We will prove the theorem when Case (i) is true. The proof of other
cases is similar.
First suppose that there exists a prime period two solution
. . . , p, q, p, q, . . .
of Eq.(1). We will prove that condition (i) holds.
When l, k, s−even, we see from Eq.(1) that
p=
dq 2
+a
cq − b
q=
dp2
+ a.
cp − b
and
Then
cpq − bp = dq 2 + acq − ab,
(4)
cpq − bq = dp2 + acp − ab.
(5)
and
Subtracting (4) from (5) gives
b(q − p) = d(q 2 − p2 ) + ac(q − p).
Since p 6= q, it follows that
(b − ac)
.
d
Also, since p and q are positive, (b − ac) should be positive.
Again, adding (4) and (5) yields
p+q =
2cpq − b(p + q) = d(p2 + q 2 ) + ac(p + q) − 2ab.
It follows by (6), (7) and the relation
p2 + q 2 = (p + q)2 − 2pq for all p, q ∈ R,
(6)
(7)
866
E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED
that
pq =
b2 − abc − abd
.
d(c + d)
(8)
It is clear now from Eq.(6) and Eq.(8) that p and q are the two positive distinct
roots of the quadratic equation
dt2 − (b − ac)t +
and so
(b − ac)2 >
b2 − abc − abd
= 0,
(c + d)
(9)
4d(b2 − abc − abd)
.
(c + d)
Therefore Inequality (i) holds.
Second suppose that (i) is true. We will show that Eq.(1) has a prime period
two solution.
Assume that
p=
(b − ac) −
q
(b − ac)2 −
4d(b2 −abc−abd)
(c+d)
(b − ac) +
q
(b − ac)2 −
4d(b2 −abc−abd)
(c+d)
and
q=
2d
2d
,
.
We see from (i) that
(b − ac)2 >
4d(b2 − abc − abd)
.
(c + d)
Therefore p and q are distinct real numbers.
Set
x−r = p, x−r+1 = q, . . . , and x−1 = q, x0 = p.
We wish to show that
x1 = x−1 = q and x2 = x0 = p.
It follows from Eq.(1) that
x1 =
dx−l x−k
dp2
+a=
+a
cx−s − b
cp − b
QUALITATIVE BEHAVIOR OF HIGHER ORDER DIFFERENCE EQUATION
"
d
= "
c
(b−ac)−
r
(b−ac)2 −
4d(b2 −abc−abd)
(c+d)
2d
(b−ac)−
r
(b−ac)2 −
4d(b2 −abc−abd)
(c+d)
2d
#
#2
867
+ a,
−b
or,
q
i
2
4b2 d
2b2 − 2abc − (c+d)
− 2b (b − ac)2 − 4d(b −abc−abd)
(c+d)
q
x1 =
i .
h
2
−abc−abd)
2 bc − ac2 − 2bd − c (b − ac)2 − 4d(b (c+d)
h
Multiplying the denominator and numerator by
q
h
i
2 −abc−abd)
we get
(c + d) bc − ac2 − 2bd + c (b − ac)2 − 4d(b (c+d)
4b3 d2
x1 =
h
x1 =
h
b − ac +
−
4ab2 cd2
+
4b2 d2
q
(b − ac)2 −
4d(b2 −abc−abd)
(c+d)
8b2 d3
i
,
or,
q
(b − ac)2 −
4d(b2 −abc−abd)
(c+d)
2d
i
= q.
Similarly as before one can easily show that
x2 = p.
Then it follows by induction that
x2n = p and x2n+1 = q for all n ≥ −r.
Thus Eq.(1) has the positive prime period two solution
. . . , p, q, p, q, . . .
where p and q are the distinct roots of the quadratic equation (9) and then the
proof is complete.
3. Local Stability of Eq.(1)
In this section we study the local stability character of the equilibrium point
of Eq.(1) in the case c = d.
868
E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED
The equilibrium points of Eq.(1) are given by the relation
x=
dx2
+ a.
cx − b
If c = d, then the only equilibrium point of Eq.(1) is given by
x=
ab
.
ac + b
Let g : (0, ∞)3 −→ (0, ∞) be a function defined by
g(u, v, w) =
Therefore
cuv
+ a.
cw − b
∂g(u, v, w)
cv
=
,
∂u
cw − b
∂g(u, v, w)
cu
=
,
∂v
cw − b
∂g(u, v, w)
c2 uv
.
=−
∂w
(cw − b)2
Then we see that
∂g(x, x, x) −ac
=
= −c0 ,
∂u
b
∂g(x, x, x) −ac
=
= −c1 ,
∂v
b
∂g(x, x, x) −a2 c2
= −c2 .
=
∂w
b2
Then the linearized equation of Eq.(1) about x is
yn+1 + c0 yn−l + c1 yn−k + c2 yn−s = 0.
(10)
Theorem 2. Assume that
√
2b > (ac + b).
Then the positive equilibrium point of Eq.(1) is locally asymptotically stable.
Proof. It is follows by Theorem A that, Eq.(10) is asymptotically stable if
|c2 | + |c1 | + |c0 | < 1.
2 2
−ac −ac −a2 c2 +
+
< 1, or 2ac + a c < 1,
b b b2 b
b2
QUALITATIVE BEHAVIOR OF HIGHER ORDER DIFFERENCE EQUATION
and so
a2 c2 + 2abc + b2 < 2b2 ⇒
869
√
2b > (ac + b).
The proof is complete.
4. Global Attractor of the Equilibrium Point of Eq.(1)
In this section we investigate the global attractivity character of solutions of
Eq.(1).
Theorem 3. If b = ac, c = d, then the equilibrium point x of Eq.(1) is global
attractor.
Proof. Let p, q be real numbers and assume that g : [p, q]3 −→ [p, q] is a
function defined by
g(u, v, w) =
Therefore
cuv
+ a.
cw − b
∂g(u, v, w)
cv
=
,
∂u
cw − b
∂g(u, v, w)
cu
=
,
∂v
cw − b
c2 uv
∂g(u, v, w)
=−
.
∂w
(cw − b)2
Case (1) If cw − b > 0, then we can easily see that the function g(u, v, w)
increasing in u, v and decreasing in w.
Suppose that (m, M ) is a solution of the system
m = g(m, m, M ) and M = g(M, M, m).
Then from Eq.(1), we see that
cm2
cM 2
+ a, M =
+a
cM − b
cm − b
cM m − bm = cm2 + acM − ab, cM m − bM = cM 2 + acm − ab,
m=
then
b(M − m) = c(m2 − M 2 ) + ac(M − m), b = ac.
870
E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED
Thus M = m.
It follows by the Theorem B that x is a global attractor of Eq.(1).
Case (2) If cw − b < 0, then we can easily see that the function g(u, v, w)
decreasing in u, v, w.
Suppose that (m, M ) is a solution of the system
M = g(m, m, m) and m = g(M, M, M ).
Then from Eq.(1), we see that
cM 2
cm2
+ a, m =
+a
cm − b
cM − b
cM m − bM = cm2 + acm − ab, cM m − bm = cM 2 + acM − ab,
M=
then
b(m − M ) = c(m2 − M 2 ) + ac(m − M ), b = ac.
Thus M = m.
It follows by the Theorem C that x is a global attractor of Eq.(1) and then the
proof is complete.
5. Special Cases of Eq.(1)
5.1 Case (1)
In this section we study the following special case of Eq.(1)
xn+1 = 1 −
xn xn−1
,
1 − xn
(11)
where the initial conditions x−1 , x0 are arbitrary real numbers with x−1 , x0 ∈
R/{0, 1}, and x−1 + x0 6= 1.
5.1.1 The solution form of Eq.(11)
Theorem 4. Let {xn }∞
n=−1 be a solution of Eq.(11). Then for n = 0, 1, . . .
(n − 1)(h + k − 1) + hk
,
1−h
n(1 − h) − (n − 1)k − hk
x2n = 1 −
,
k
x2n−1 = 1 −
QUALITATIVE BEHAVIOR OF HIGHER ORDER DIFFERENCE EQUATION
871
where x−1 = k, x0 = h, (n−1)(1−h) 6= (n−2)k+hk, and (n−1)(h+k−1) 6= −hk.
Proof. For n = 0 the result holds. Now suppose that n > 0 and that our
assumption holds for n − 1. That is;
(n − 2)(h + k − 1) + hk
,
1−h
(n − 1)(1 − h) − (n − 2)k − hk
x2n−2 = 1 −
.
k
x2n−3 = 1 −
Now, it follows from Eq.(11) that
x2n−1 = 1 −
=1−
x2n−3 x2n−2
1 − x2n−2
1 − (n−2)(h+k−1)+hk
1−
1−h
(n−1)(1−h)−(n−2)k−hk
k
(n−1)(1−h)−(n−2)k−hk
k
.
Multiple the denominator and numerator by k(1 − h) gives
(1 − h − (n−2)(h+k−1) − hk) (k − (n−1)(1−h) + (n−2)k + hk)
((n − 1)(1 − h) − (n − 2)k − hk) (1 − h)
((1 − h)(n − 1) − (n − 2)k − hk) ((n − 1)(h + k − 1) + hk)
=1−
,
((n − 1)(1 − h) − (n − 2)k − hk) (1 − h)
x2n−1 = 1 −
then we have
x2n−1 = 1 −
Also, we get from Eq.(11)
x2n = 1 −
=1−
(n − 1)(h + k − 1) + hk
.
(1 − h)
x2n−2 x2n−1
1 − x2n−1
1−
1 − (n−1)(1−h)−(n−2)k−hk
k
(n−1)(h+k−1)+hk
(1−h)
(n−1)(h+k−1)+hk
(1−h)
,
or,
(k − (n−1)(1−h) + (n−2)k + hk) (1 − h − (n−1)(h+k−1) − hk)
((n − 1)(h + k − 1) + hk) k
((n − 1)(h + k − 1) + hk) ((1 − h)(1 + n − 1) − (n − 1)k − hk)
=1−
.
((n − 1)(h + k − 1) + hk) k
x2n = 1 −
Thus we obtain
x2n = 1 −
n(1 − h) − (n − 1)k − hk
.
k
872
E. M. ELABBASY, H. EL-METWALLY AND E. M. ELSAYED
Hence, the proof is completed.
Remark 1. If x−1 + x0 = 1 then every solution of Eq.(11) is periodic with
period two otherwise the solution is unbounded.
5.2 Case (2)
In this section we study the following special case of Eq.(1)
xn+1 = 1 −
xn−1 xn−2
,
1 − xn
(12)
where the initial conditions x−2 , x−1 , x0 are arbitrary real numbers with x−2 ,
x−1 , x0 ∈ R/{0}, and x0 ∈ R/{1}.
5.2.1 The solution form of Eq.(12)
Theorem 5. Let {xn }∞
n=−2 be a solution of Eq.(12). Then for n = 0, 1, . . .
x2n−1 =
n−1
X
p
(−1)
p=0
x2n =
n−1
X
p
(−1)
p=0
where x−2 = r, x−1 = k, x0 = h,
r
1−h
p
1−h
r
p
n
+ (−1)
+ (−1)
p
p=0 (−1)
P−1
n
r
1−h
n
k,
1−h
r
n
h,
(A)p = 0.
Remark 2. If x0 + x−2 = 1 then the solution is bounded and periodic with
period four otherwise every solution of Eq.(12) is unbounded.
References
x
[1] C. Cinar, On the positive solutions of the difference equation xn+1 = 1+xn−1
, Appl.
n xn−1
Math. and Comput., 150(2004), 21-24.
x
[2] C. Cinar, On the positive solutions of the difference equation xn+1 = 1+axn−1
, Appl.
n xn−1
Math. and Comput., 158:3(2004), 809-812.
x
[3] C. Cinar, On the difference equation xn+1 = −1+xn−1
, Appl. Math. and Comput.,
n xn−1
158(2004), 813-816.
[4] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equation xn+1 =
αxn−k
Q
, Journal of Concrete and Applicable Mathematics, 5:2(2007), 101-113.
β+γ k
x
i=0
n−i
QUALITATIVE BEHAVIOR OF HIGHER ORDER DIFFERENCE EQUATION
873
[5] H. El-Metwally, E. A. Grove and G. Ladas, A global convergence result with applications to
periodic solutions, J. Math. Anal. Appl., 245(2000), 161-170.
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periodic character of some difference equations, J. Differ. Equations Appl., 7(2001), 1-14.
[7] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher
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with Open Problems and Conjectures, Chapman & Hall/CRC Press, 2001.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]