MATHEMATICAL COMPUTER MODELLING PERGAMON
Mathematical
and Computer
Modelling
32 (2000) 653-660 www.elsevier.nl/locate/mcm
Asymptotic Properties of Solutions of a 2nth-Order Differential Equation on a Time Scale D. ANDERSON Department of Mathematics and Computer Science, Concordia College Moorhead, MN 56562, U.S.A. andersodQcord.edu A. PETERSON Department of Mathematics and Statistics University of Nebraska-Lincoln, Lincoln, NE 68588-0323, U.S.A.
[email protected] Abstract-In this paper, we are concernedwith a 2nth -order linear self-adjoint differential equation on a time scale. The results generalize known results for the corresponding ordinary differential equations and for difference equations. We define Type I and Type II solutions, prove the existence of these solutions, and verify asymptotic properties of these solutions. A quadratic functional corresponding to the differential equation on a time scale is defined and is used to prove several of the results in this paper. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-
Measurechains, Asymptotic behavior,Type I and Type II solutions. l.TYPE
INTRODUCTION
In this paper, we will be concerned with the 2nth-order differential equation Lz(t)
= [p(t)m”(t)]A*
+ q(t)z(a”(t))
= 0,
(1)
for t E T. We assume T is a closed subset of the real numbers Iwthat is unbounded above. We call a a time scale. DEFINITION.
For t E T define the forward jump operator 0 by f~(t) := inf{r E T : 7 > t}.
We define the backwards jump operator p for t E ‘I’, t > inf T, by p(t) := SUP{T E T : 7 4 t}. If a(t) = t we say t is right-dense, and if o(t) > t we say t is right-scattered. Similarly, if p(t) = t, then t is left-dense, and if p(t) < t, then t is left-scattered. We assume go(t) = t, and for any integer n > 0, we define ~9 : T -+ R by l?(t)
:= d (o”-‘(t))
.
0895-7177/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(00)00162-X
Typeset
by .&S-QJ-C
D. ANDERSON AND A. PETERSON
654
We assume throughout that the time scale T has the topology it inherits from the standard topology on W. We also assume p, q : T ---fW are continuous and p(t) > 03 on T. DEFINITION. Assume x : ‘II’ + R and fix t f T; define x*(t)
to be the number (provided it exists) with the property that given any e > 0, there is a neighborhood U oft such that
I[x (4)) for a11 s E U. We call x*(t)
- x(s)1 - x*(t) [4> - SIII f 14) - 4 ,
the delta derivative of x at t. Note, if ‘II’= Z, then x*(t)
= Ax(t)
:= x(t -I-1) - x(t).
Moreover, if ‘ll’= R, then x*(t)
= x’(t).
Hence, our results contain different&iequations and differenceequations as special cases. Finally, ifh>OandT=hZ={hn:nEZ), then xA(t) = X(t+ h) -X(t) h ’ DEFINITION. For n > 1, define x*“(t)
:= {cc*(t)}A”-1
;
assume x *O(t) = x(t). DEFINITION. We say x : T + IR is right-dense continuous provided for any t E T such that p(t) = t, the left-hand limit of x at t exists (and is finite). Also, if t E T and c(t) = t, then x is continuous at t. Note, if x : ‘ll’-S IIPis continuous, then x(a(t)) is right-dense continuous on ‘ll’. DEFINITION. We define
D := {x : ‘II’ -_) IR : xAk(oi(t)) is delta differentiable on T for 0 5 k 5 n - 1 and 0 < i _< n; [p (oi(t)) xAn (o*(t))]*’ is delta differentiable on T for 0 5 i, k 5 n - 1; and [p(t)x*“(t)]** is right-dense continuous on T.} We say x is a solution of Lx = 0 on T provided x E D and Lx(t) = 0 for ail t E T. Our results will hold only for those time scales ‘ll’ for which D is nonempty. Moreover, we need to assume below that the time scale T is such that [
x*“-1
(o”(t))]*
= xA’&(o”(t)).
This is true for time scales T = lR and If = hZ for any h > 0, but of course is not true in general. DEFINITION. Forx E D,set Fx(t)
:= x*+’
(o”-‘(t))
n-2 -(-l)"C(-l)ixqY"(t)) $4
p (C+(t)) [p(oi(t))xAR
xA” (u”-“(t)) (c7i(t))]Aa-1-i
;
(2)
this quadratic functionat F will be important in the study of the asymptotic properties of solutions of Lx(t) = 0. Note that F[kx(t)] = k2F(x(t)).
In the proof of the following lemma, we will make rapeated use of the product formula [f(t)9(tN” = f (O(t)) g*(t) + f*(t)g(t)* where f and g are interchangeable.
Asymptotic Properties
655
of (1) on T, tflen
LEMMA 1. If z is a sdution
[Fx(t)]* = p (u-1 (Q)[x””(on-v))]2+(-l)“q(t)xZ
(an(t)) ,
(3)
for t E T. In particular, if (--Vq(t) on T, then F is nondecreasing PROOF.
L 0,
(4)
along solutions x of (I) for t E T.
Assume x is a solution of (1). Then [Fz(t)]”
= x*“-I
(o”(t))
-I- xA” (C’(t)) - (-1)”
[p (C’(t))
xA” (cW(t,)]
p (u”-‘(t))
*
(5)
zAn (f?-‘(t))
nj(-l)ix*i (an(t))
[p (d(t))
(6)
xA” (uyt))]*‘+
(7)
i=o - (-ly
n~(_q”x*‘+’ (g”(t)) [p (,+1(t))
24” (#+1(t))]
*“-‘-’)
(8)
i=O
where (5) and (6) result from the product rule of the first advanced times the delta derivative of the second, plus the second times the derivative of the first, while (7) and (8) come from the product rule of the first times the delta derivative of the second, plus the derivative of the first times the second advanced. Next, evaluating the sum in (8) at n - 2 and reindexing, we obtain [Fx(t)lA
= x*“-’
(a”(t))
[p (o”-‘(t))
_ (-1>“(_1)“-2x*“-’
xA” (Cl@,)] (C?(t))
* fp
[p (C?-‘(t))
(o”-‘(t))
[P”
(o”-‘(t))]
2
xA’” (C-l(t))]*
n-2 -
(-r)n
C(-1)ixa’
(o”(t))
[p (o”(t)) cPn (cqt,)]
Arr--r
i=O
- (-1)n
n-2 C(-1)“-lx*’ i=l
= p (u”-‘(t))
(o”(t))
[p (o”(t)) ZA” (bi(t))]A’i-a
[x”” (u+1(t))]2 - (-1)%x
= p (an-1 w) [xA” (u”-W)]
2 + (-lYwx2
since x is a solution of (1).
(a”(t))
[p(t)xA” (t)] *‘l
(an(t))
> I
DEFINITIONS. If x(t) is a solution of (1) such that Fx(t) 2 0 in a neighborhood of W, we say x(t) is a Type I solution of (1). If x(t) is a solution of (1) such that Fx(t) < 0 for t E T, then we say x(t) is a strict Type I solution. If x solves (1) and Fx(t) > 0 near W, then x is a Type II solution. In view of Lemma 1, all solutions of (1) are Type I or Type II solutions if (4) holds. Hartman [l] defined generalized zeros of solutions of nth -order linear difference equations. We now would like to define what we mean by a generalized zero of order m, 0 < m < 2n - 1, for a nontrivial solution x(t) of (1) at to E ‘II’. If to = min’il’, then we say x(t) has a generalized zero of order m at to provided x*” (to) = 0, 0 2 k 5 m - 1. If to > min ‘II’, then x has a generalized zero of order m at to provided if to is left-dense then 8 (to) = 0,
D. ANDERSON AND A. PETERSON
656
0 5 k 5 m - 1; if to is left-scattered,
then
x(/dto))# 0
x*“(to)= 0,
and
for 0 5 k 5 m - 1, or to is right-scattered
with
xAk(to)
= 0,
forO c?-(a).
Then Qt,
657
s) is a solution of (1) satisfying
2n-1 c [uf i=O
(c?(a),
s)12 = 1.
Thus, for each k, and s E T with s > (~~~(a) ( recall T is unbounded above), from the set of points
in IR2” we can pick a convergent sequence
Let (13) forOIi tr for all j, 2 jtl. Then
for all j, 2 j,, . Taking the limit as j ---f 00, we get that
for 1 I k 5 n. As tl > a*(u)
for all t > @(a), By (ll)-(13),(15)
for 1 5 k 6 n. It follows that the yk are Type I solutions of (1) for 1 5 k 5 n. we have that Yf
If yf’-‘(,n(u))
was arbitrary,
(64)
= 0,
forO to in T such that p(t)sqt,] [
A”-1 < 0,
for all t > tl in T, lizgf for t E T using (21)-(23), expression, we obtain
[p(t)rA”
(24)
and so on. By continuing this process of integrating each successive lirngf
fort E T, i = n-2,12-3,. that
(t)] Arr--l = -co,
[p(t)~~-(t)]~’
= -00,
. . , 1,O. Using (20) and similar reasoning, we see for i = n, n- 1, . . . , 1,O hrngf zAi (t) = -co,
as well. Consequently, limil&f z(t) = -00,
D. ANDERSON AND A. PETERSON
660
a contradiction of our initial assumption that z(t) > 0 for t 2 to in T. Thus, every unbounded solution of (1) is oscillatory under conditions (19) and (20). I EXAMPLE 7. The differential equation
xAAAA(t)
+
2
(u”(t)) =
0,
for t E T, satisfies the hypotheses of most of the theorems in this paper. Note that if T = R+, where lR+ is the set of nonnegative real numbers, then q(t)
=
fP~%os
x2(t) = f&JZlzJtsin
Q(t)
(1 () pt
,
Jz
,
qt
= e-(fi~%os
qt (
x4(t) = e-(J2/2)tsin
, )
$t (
, )
are solutions on R+. Note that zl(t) and zz(t) are unbounded Theorem 7 are oscillatory near 00. It can be shown that
and it follows that q(t)
and q(t)
near 00 and as guaranteed
by
are Type II solutions. Also, it can be shown that
so x3(t) and zq(t) are Type I solutions. Hence, the conjecture noted after Theorem 4 holds. Note that Peil and Peterson [2] studied the asymptotic behavior of solutions of (1) with T 3 Z. For fourth-order difference equations see [3]. For fourth-order ordinary differential equations, see [4,5]. For results of the type in this paper for third-order difference equations, see [6]. For results for third-order differential equations on a time scale, see [7].
REFERENCES 1. P. Hartman, Difference equations: Disconjugacy, principal solutions, Green’s functions, complete monotonicity, Zbnsoctions of the American Mathematical Society 246, l-30, (December 1978). 2. T. Peil and A. Peterson, Asymptotic behavior of solutions of a two-term difference equation, Rock Mountain Journal of Mathematics 24 (l), 233-251, (Winter 1994). 3. B. Smith and W. Taylor, Oscillation and asymptotic behavior of certain fourth order difference equations, Rocky Mountain Jounal Mathematics 16, 403-406, (1986). 4. G. Jones, Oscillatory solutions of a fourth order linear differential equation, Lecture Notes, In Pure Appl. Math., Vol. LW,, (Edited by S. Elaydi), pp. 261-266, Marcel Dekker. 5. M. Svec, Sur le comportement esymptotique des integrals de lequation differentielle yc4) + Q(r)v = 0, Czech. Math. J. 8, 230-244, (1958). 6. A. Peterson, A quadratic functional for a third order linear difference equation, Journal of Diflerence Equations 3, 463-472, (1998). 7. M. Morelli and A. Peterson, A third order differential equation on a time scale, Maths. Comput. Modelling, (this issue). in Applied AnaIysis 2 (4), 521-529, a. D. Anderson, A 2nth-order linear difference equation, Communications (1998). 9. CD. Ahlbrandt and A.C. Peterson, The (n, n)-disconjugacy of a 2nth -order linear difference equation, Computers Math. Appfic. 28 (l-3), l-9, (1994).
