Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 281-286
281
OSCILLATION AND NONOSCILLATION IN NONLINEAR
THIRD ORDER DIFFERENCE EQUATIONS
B. SMITH and W.E. TAYLOR, JR.
Department of Mathematlcs
Texas Southern University
Houston, TX 77004
(Received August 18, 1988 and in revised form June 20, 1989)
KEY
AND
WORDS
PHRASES.
Difference
equations,
third
oscillatory
order,
and
nonoscillatory solutlons.
1980 AMS SUBJECT CLASSIFICATION CODES. 39AI0, 39A12.
I.
INTRODUCTION.
This paper is concerned with the oscillatory behavior of solutions of the third
order nonllnear difference equation
a(Pna2Vn) + Qnf(Vn+l AVn+ I, A2vn+ 1) -0,
V+I
where & is the forward difference operator i.e., &V
n
throughout that the conditions below are satisfied:
(I)
(II)
(III)
n
P
n
>
O, AP n > 0 and
Qn >
0 for n
n-
n
I, 2,
-V
n
...,
(.i)
It will be assumed
O, 1, 2,
[
3
f: R
R is continuous and xf(x, y, z)
>
0 for x
*
O.
By a solution of (I.I) we mean a real sequence V satisfying equation (1.1) for
A solutlon V of (1.1) is called nonoscillator if it is eventually
1, 2,
positive or eventually negative.
determining
equations
results of
oscillatlon
has
criteria
been investigated
Otherwise, it is called oscillatory.
for
certain
by Hooker and Patula
[2] were generallzed by Li [3].
general equation than those studied in [2] and
slmilar equations in [4].
second
order
[1],
The problem of
nonllnear
and Szmanda
difference
[2].
The
This paper examines a sllghtly more
[3].
The authors began a study of
282
2.
B. SMITH AND W.E. TAYLOR, JR.
MAIN RESULTS.
LEMMA 2.1.
Suppose V is a nonoscillatory solution of (1.1).
sgnVn
sgn&Vn
Then, either
sgnA2Vn
(2.1)
sgn&Vn
(2.2)
for all n sufficiently large, or
sgna2Vn
sgnVn
#
AV
for all sufficiently large n, and llm
&2Vn
llm
n
0.
PROOF. Assume V is a nonoscillatory solution of (1.1), where Vn > 0 for all
N.
N, where N is a positive integer. The proof is similar if V. < 0 for all n
) N.
is
Thus
n
Note that
for
each
0,
<
-Qnf(Vn+l,
I)
I,
)
N
M
for
which
and
is eventually sign definite. A positive integer
exists
decreasing
are of one sign when n ) M. The following cases must be considered:
AVn and
n
(a)
> 0, &V > 0,
> 0, n M,
n
(Pn2Vn
Vn+
Pn &2Vn
2Vn+
AZV
Vn
>
O, &Vn
<
O,
A2Vn >
O, n )M,
>
0
AVn
<
0,
A2V
<
0, n
)M
0, &V
n
>
0,
A2Vn <
0, n
)M.
(b)
V
(c)
V
(d)
Vn >
n
2Vn
n
A2V
Case (c) is impossible because if AV
0 for all sufficiently large n, then
n
n >
sgnV -sgnAVn eventually. We show that (d) is also impossible. If (d) holds, then
n
from above P
n
2Vn is
be such that P
n
negative and decreasing for all n sufficiently large.
A-Vn < k for
we obtain
.
R-I
&VR AVM < k n
Letting R +
implies
(d) cannot hold.
AVR
-
all n
M. Then
A2Vn
<
--,
n
M.
Let k
<
0
Summing from M to R-
n
n
is eventually negative, but this contradicts
(d), therefore
This completes the proof of the lemma.
We continue our study of (1.1) by considering a functional which plays a vital
role in our investigation.
Similar functionals have been used to study solutions of
differential equations (Taylor [5]).
LEMMA 2.2.
Let V be a solution of (I. I).
F[Vn]
Then
Fn 2VnPn A2Vn Pn-I(AVn)
OSCILLATION AND NON-OSCILLATION IN NONLINEAR DIFFERENCE EQUATIONS
283
is nonlncreaslng, in fact
AFn
&2Vn+l) Pn (&2vn)2
-2QnVn+If(Vn+l’AVn+l’
APn_I AVn )2
Since F n is monotone for any nontrivial solution of (I.I) we have that F n is of
one sign for all n sufficiently large. Using this we will examine solutions of (I.I)
0 for each n and those for which Fm < 0 for some positive integer m.
)0.
THEOREM 2.1. Let V be a nontrivial solution of (I.I) for which F[V
where F
n
)
n]__
Then
the following are true:
(i)
)" QnVn+l f (vn+l’
AVn+I’
(ii)
[
Pn(A2Vn )2 <
and
(ill)
). Apn_
PROOF.
(R),
2
(aVn) <
A2Vn+l
(R)’
-.
> 0 for each
Since F
<
n
differencing Fn and summing from 0 to k-I we find
k-I
F
0
FO
k
2
0
k-I
0
Thus,
QnVn+l f (vn+l’Avn+l’
k-I
Pn(A2Vn )2
0
k-1
2
0
QnVn+if(Vn+l,AVn+l,
A2Vn+I
APn_I(AVn )2.
A2Vn+l
+
k-I
k-I
Pn(A2Vn )2 + 0 APn-I(AVn)2 < F0"
0
Allowing k to tend to infinity establishes each of (1), (li) and (ill) since F 0 is
independent of k.
Suppose that f(x,y,z) ) r > 0 for x
x
0 for each n.
be a solution of (I.I) for which F[Vn]
THEOREM 2.2.
<iv)
[ vn2 <.,
(v)
llm
PROOF.
V
n
llm
AVn
A2Vn
lira
To prove (iv), observe that
Vn+if(Vn+
AVn+
A2Vn+l rV2n+l
)
Thus
where
lim inf
Qn’
so we have
O.
0 and lim inf
Then
>
O.
Let V
284
B. SMITH AND W.E. TAYLOR, JR.
2
ar0 Vn+l
A2Vn+I )"
QnVn+l f( Vn+l’ AV n+l’
Now apply (i), of Theorem 2.1, and the proof of (iv) is complete.
The relations (v) follow as a consequence of (iv).
EXAMPLE 2.1. As an illustration of Theorem 2.2 consider equation (1.1) with
22n(_.___n-l)
0
-n
2
2n+ 2
Pn
n,
x 3 + x.
f(x,y,z)
+
Then
3
A(nA2Vn) + 22n(n-1) (Vn+l
22n+2+1
+
Vn+
0
The sequence defined by 2 -n is a solution of this equation for which F
f(x,y,z.)
r > 0, and
x
solution of (1.1) approaches zero as n / -.
THEOREM
2.3.
If
Qn
then
every
-.
0 as n
nonoscillatory
Suppose V is an eventually positive solution of (1.1) that is bounded away
from zero, i.e. V >
> 0 for all n sufficiently large. Because of Lemma 2.1, an
n
integer M exists so that the relations (2.1) or (2.2) are satisfied by V for all
where r is a positive constant. From (1.1)
Now f(Vn+l, AVn+I,
n > M.
PROOF.
A2Vn+l
rVn+
we find
&(Pn&2Vn)
A(PnA2Vn)
ThUS,
-Qnf (Vn+l’ AVn+l’
A2Vn+l )"
(2.3)
-rQnVn+ 1.
Summing both sides of (2.3) from M to k-1 we find
PkA2Vk PMA2V
(
But as k
to tend to
-,
k-1
(2.4)
the right hand side of (2.4) tends to
-,
k-1
M
A2Vk <
and hence
-,
which in turn forces
PkA2Vk
0 eventually, a contradiction of relations (2.1) and
A similar argument treats the case of an eventually negative solution.
completes the proof of the theorem.
then every
COROLLARY 2.1. If f(x,y,z)_ ) r > 0, for x
O, and Qn
(2.2).
x
This
,
nonosclllatory solution of (I.I) satisfies the relations (2.2).
We are now in a position to show that oscillatory solutions exist under certain
conditions.
Suppose f.(.x,y,z)
and
is bounded.
r > 0, for x
0,
Qn
x
V
is
for
then
V
F[V
0
some
a
oscillatory.
(I.I)
of
for
If
is
which
solution
n,
n] <
PROOF. Suppose V is a nonoscillatory solutlon of (1.1). We may suppose without
THEOREM 2.4.
any
generality
loss
nontncreasing as n
"’
Vn > 0 and F[Vn] < 0 for all n
N, since F[Vn]
-.
0 and
0 as n
From Theorem 2.3, V / O, AV
n
n
n
that
-.
Pn
A2V
is
OSCILLATION AND NON-OSCILLATION IN NONLINEAR DIFFERENCE EQUATIONS
Pn
This together with the boundedness of
impossible
since
<
Fn
0
and AF
n
<
implies that F[V
0 for
large
n
n]
/
This is clearly
0.
follow
proof
the
and
285
by
contradiction.
Under certain conditions the bounded solutions of (I.I) behave rather nicely.
Similar results appeared in [2] and [3].
THEOREM
Suppose
2.5.
nQn
and
Pn
3,
constant.
Then
every
bounded
solution of (1.1) is either oscillatory or tends to zero notontcally.
PROOF.
By Lemma 2.1 a bounded nonoscillatory solution V satisfies
sgnV n
sgn Pn
A2Vn
Assume that Vn
for all n sufficiently large.
lim
AV
n
A where Ao
0
A2V
that P
n
n
Consider the sequence
The
fact
f
is
0.
0 as n
follows
n(Pn AVn )"
rn
0 eventually and suppose
from
the
boundedness
of
Vn
and
(II).
Note that
&2Vn+1
nQnf(Vn+l’ AVn+1’
f(Vn+l, AVn+I,__
continuous
>
Note also
Pn+lA2Vn+l
&rn
Since
>
sgn AVn
A-Vn+ I)
(2.5)
)"
0, 0)
>
0 as n
for all n
>
N.
f(A0,
-,
so
there
exists N such that
f(Vn+l, AVn+l,
A2Vn+ I) > 1/2f(Ao, O, 0)
Therefore from (2.5) we have
Ar
<
n
n
Summing, from N to n
rn+1 < rN
-,
rn
the theorem.
As n
-
A2Vn+1 Qnf(Ao O,
A2Vn+ -. 0f(Ao o, o).
< Pn+l
-,
+
AVn+2
O)
VN+I 1/2f(A0’
a contradiction.
n
0, 0)
Therefore A
0
0.
JQj.
This completes the proof of
Finally we have
2m+I
THEOREM 2.6. Suppose .(R) Qn
a
f(x,y,z), a 0 and
(R), f(ax, ay az)
Then every solution of (1.) is either
f(x, y + h, z) > f(x,y,z) for h Y 0.
bounded or oscillatory.
B. SMITH AND W.E. TAYLOR, JR.
286
PROOF.
Suppose V n is an unbounded nonoscillatory solution of (I.I).
Without loss
of generality we have
V
n
>
>
0, AV n
0, Pn
p
Differencing
qn
0,
Consider the functional
for all n sufficiently large.
qn
A2V n >
2Vn
n
V
n
we find
-Qn f(vn+l’
al n
2Vn+ )_
AVn+I’
Vn+
P V
n
n
2V n
VnVn+
+)
Vn+
v+
n+ I_
AZVn+I
2m
Vn+ f(l 0, Vn+
2m
V Qnf(l, 0, 0)
N
2
Summing we obtain
V
qmK0
But this implies
qm
2m
m-I
N f(l,0,0)
2
N
IQn"
as m
(R),
a contradiction since
Pn’
A2vn
and V n are positive
for all n sufficiently large.
EXAMPLE 2.2.
It is possible for equations of the form of (I. I) to have unbounded
(-2) n is a solution of
The sequence Vn
oscillatory solutions.
A3V n +
II
18(4n+I) (AVn+1)
3
+
3Vn+1
0.
Note that this example does not violate the conclusion of Theorem 2.6.
Note also that
(III) is not satisfied.
REFERENCES
J.W. and PATULA, W.T., A Second-order Nonlinear Difference Equatlon:
Oscillation and Asymptotic Behavior, J. Math. Anal. Appl. 91 (1983), 9-29.
I.
HOOKER,
2.
SZMANDA, B., Oscillation Theorems for Nonlinear Second-order Difference
Equations, J. Math. Anal. Appl. 79 (1981), 90-95.
3.
Li, Z.H., A Note on the Oscillatory Property for Nonlinear Difference Equations
and Differentlal Equations, J. Math. Anal. App1. 103 (1984), 344-352.
4.
SMITH, B. and TAYLOR, W.E., Jr., Aymptottc Behavior of Solutions of a Third
Order Difference Equation, Port. Math. 44 (1987), 113-117.
5.
TAYLOR, W.E., Jr., Asymptotic Behavior of Solutions of a Fourth Order Nonlinear