781
Internat. J. Math. & Math. Sci.
Vol. 9 No. 4 (1986) 781-784
QUASI-ADJOINT THIRD ORDER DIFFERENCE EQUATIONS:
OSCILLATORY AND ASYMPTOTIC BEHAVIOR
B. SMITH
Department of Mathematics
Texas Southern University
U.S.A.
Houston, Texas 77004
(Received August 6, 1985)
In this paper, asymptotic properties of solutions of
ABSTRACT.
A3Vn +
(E+)
0
Pn-iVn+l
are investigated via the quasi-adjoint equation
A3Un
(E-)
0
PnUn+2
A necessary and sufficient condition for the existence of oscillatory solutions of (E+)
An example showing that it is possible for (E+)
is given.
to have only nonoscillatory
solutions is also given.
KEY WORDS AND PHRASES. Difference equations, third order, oscillatory and nonoscillatory
solutions, quasi-adjoint eqtion.
1980 AMS SUBJECT CLASSIFICATION CODES. 39A10, 39A12.
i.
INTRODUCTION.
This study is concerned with the solutions of (E+), where
AVn Vn+l Vn
differencing operation
P
satisfying
n-i
> 0
for each
n >
V
denotes the forward
The coefficient function is a real sequence
By the graph of a solution
the polygonal path connecting the points
graph of
A
with the real axis is a node.
(n,V)
n
n > I.
A solution
V
{V
n
we mean
A point of contact of the
V
of (E+)
is said to be oscil-
latory if it has arbitrarily large nodes; otherwise it is said to be nonoscillatory.
Hereafter the term "solution" shall mean "nontrivial solution".
The oscillation criterion established by Lazer [I, Theorem 1.2], for the differential equation
y"’ + p(x)y’ + q(x)y
difference equation
(E+).
0, where
p(x) ! O,
q(x) > 0
is proved for the
We also include an example which demonstrates that is pos-
sible for (E+) to have only nonoscillatory solutions.
In studying (E+) we will make use of its quasi-adjoint equation (E-).
properties and definitions concerning (E+) and (E-) we refer to
For general
Fort’s book [2], and
to
the paper by this author.
2.
PRELIMINARY RESULTS.
Equation (E-) always has a nonoscillatory solution.
which satisfies
In fact, (E-) has a solution
B. SMITH
782
U AU
n
A2U n #
n
0, U
AU
> 0,
n
> 0,
n
A2Un
> 0,
sufficiently large (see [3, Theorem 2.2]).
for all n
The posltivlty of the coefficient function
places severe restrictions on the
p
behavior of the nonoscillatory solutions of (E+).
We have the following results concerning (E+).
LEMMA 2. I.
is a solution of (E+) satisfying
V
If.
Vm -> 0,
V
k
> 0,
k
AV k _< 0,
We show the lemma true for
PROOF.
_< 0.
Thus
dVm_l
implies
> 0
A2V k
> 0
< k < m.
for each
-Pm_2Vm
A2Vm
_> I, then
m > k
for some integer
AV _< 0,
m
m-l.
A2m--< A2Vm-I
< 0,
and we find
which in turn implies
V
If
d2Vm_l
Vm_
_< k
Repeating this process for each
THEOREM 2.2.
Note that
m-l.
k
> 0.
<
> 0.
Similarly,
A2Vm_l
> 0
Therefore the result holds for
m-I
proves the lemma.
is a nonoscillatory solution of
Vn AVn A2Vn
A3Vm-I A(A2Vm-I)
(E+) then either
sgnA2Vn # sgnAVn,
# 0, sgnVn
(2.1)
n > I, of
for each
V
n
AV
n
A2Vn #
sgnAVn
sgnVn
0
sgnA2Vn
for all sufficiently large n.
Assume that
PROOF.
Vn
generality
n > N
V
n
AVn
A2Vn
hence
M > N
exists,
> 0
A2Vn #
AV
for which
n
and
n
n > M.
0, for every
and
Note that
A3Vn A(A2Vn)
is decreasing and is eventually one sign.
V
(c)
n _> N.
for each
Vn
Vn
Vn
The cases
is a nonoscillatory solution of (E+), where without loss of
V
> O,
AV
n
> 0, AV
n
> 0,
AV
> 0,
AV
(d)
n
n
A2Vn
M
It follows that
are sign definite, for all
> 0,
< 0,
> 0,
A2Vn
d2Vn
d2Vn
n > M.
Hence
2Vn
> 0, n
_> M
(a)
> 0, n
_> M
(b)
< 0, n
_> M
(c)
< 0, n > M.
are clearly impossible since
apply Lemma 2.1 to the case
< 0 for al.!
The following cases must be considered:
< 0,
sufficiently large implies that
-Pn-IVn+l
sgnAJ-lvn
sgnAJVn
(d)
JVn AJ+Ivn
eventually.
> 0
for all
n
To complete the proof
(a).
The exlstnece of a solution
V
of (E+), satisfying the conditions (2.1) is estab-
lished in [3].
Denote by (S-)
the solution space of
For (U,V) e (S-) x (S+)
F
n
F[Un,Vn]
(E-)
and
(S+)
the solution space of
(E+).
define
Un+IA2Vn+I -AUnAVn+ + Vn+IA2Un.
(2.3)
783
QUASI-ADJOINT THIRD ORDER DIFFERENCE EQUATIONS
The function defined by (2.3) is a constant determined by the
It is easy to verify.
U
initial values of
V, hence,
and
concomitant for solutions of
F
(E-)
represents the discrete LaGrange billnear
n
Using (2.3), proofs of the following
(E+).
and
two results can be modeled after the analogous results appearing in
U
Suppose that
THEOREM 2.3.
is a nonoscillatory solution of
[3].
(E-).
(E+)
If
has an oscillatory solution, then there are two independent oscillatory solutions of
(E+)
that satisfy
A2U
AV
A()n + (Un n-i
Un+l Vn+l
COROLLARY 2.4.
solutions for
(E+)
If
solution space (S+)
(2.4)
0.
has an oscillatory solution, there exists a basis for the
nonoscillatory solutions and
k
consisting of
oscillatory
3-k
k=0, i.
Since the nodes of linearly independent solutions of (2.4) separate each
REMARK.
other, and those of linearly dependent solutions coincide, it follows that solutions of
(2.4) are oscillatory.
See for example, Fort [3, p.221].
We conclude this section with the following easily verifiable result.
LEMMA 2.5.
Y
Let
G(r, Xn
(r
(AX)
n
be a sequence.
Then,
(1)
n
I, where
n+l, n
r
(ll)
AX
If
(ill) If
a
> 0
m
Xm <
3.
Xn + b Yn ))
G(r, (a
> 0
n < r < n+l, n > I.
n)
a G(r, X
b
is defined by
n
+
b G(r,
<
Xm+
0)
Yn
are constants.
for some integral m,
G(r, Xm) <
Xn
n) + X
n
and
{X
X
The graph of a sequence
(X
G(r, Xm) >
for each
n, then
m
then
Xm+l).
G(r, Xn)
<
> 0
0)
for each
r.
MAIN RESULT.
(E+)
In case
has oscillatory solutions, our main result shows that more strin-
gent requirements are imposed on the nonoscillatory solutions of (E+), than those
posed by Theorem 2.2.
’ntroduced"
We will show that solutions satisfying relations (2.2) cannot be
into the solution space
(S+) without "forcing" out all the oscillatory
solutions.
THEOREM.
A necessary and sufficient condition for (E+) to have oscillatory solu-
PROOF.
The sufficiency is clear.
the relations (2.1)
each
n
I.
by Lemma 2 5
the relations (2.1) are satisfied.
V
tions is that for any nonoscillatory solution
If every nonoscillatory solution
G(r, Vn
V
satisfies
for such a solution is of one sign for
It follows that any solution with a node is oscillatory; thus we see that
To prove the
initial values can be used to construct oscillatory solutions of (E+).
necessity, suppose that (E+) has an oscillatory solution, and that (E+) has a solution
Z
satisfying the relations (2.2).
a basis for
By Corollary 2.3 and the above remark, there exists
(S+) consisting of one nonoscillatory solution R satisfying conditions
(2.1) and two oscillatory solutions
T oscillatory. Now
and
S
+
T
and
+
with every linear combination of
Zn CIRn C2Sn C3Tn
where
CI
C
2
C
3
S
are scalars,
B. SMITH
184
C + C + C # 0.
Conditions (2.2) imply
.
lira
n+
sequence of nodes of
{C2S n + C3Tn}
G(ri,Zn)
IZnl
Then at each
CIG(ri,R
The left member of (2.5) is bounded as
i
Let
r
{r i}
be an increasing
i
(2.5)
This contradiction completes the proof
of the theorem.
In conclusion we present an example showing that it is possible for every solution
of (E+) to be nonoscillatory. The relations (2.2) are satisfied by the funciton V
dfined by
V
n
n
+ 2 -n+l
It is easily seen that
A3 vn +
vn+l
V
is a solution of
0
4[ (n+l) 2n+l)
Consequently every solution of (E) is nonoscillatory.
ACKNOWLEDGEMENT.
This research was partially supported by Texas Southern University
Faculty Research #16512.
REFERENCES
y"’ + p + qy
14
LAZER, A. C., The behavior of solutions of the differential equation
0, Pacific J. Math., 17(1966),435-466.
2.
FORT, T., Finite Differences and Difference Equations in the real Domain, Oxford
University Press, London, 1948.
3.
SMITH, B., Oscillatory and asymptotic behavior in certain third order difference
equations, to appear, Rocky Mt. J. Math.