# Oscillation results for second-order delay dynamic equations.

1 Introduction

In 1988 the theory of time scales was introduced by Stefan Hilger in his Ph.D. Thesis in order to unify continuous and discrete analysis (see [10]). Not only does this theory unify those of differential equations and difference equations, but it also extends these classical situations to cases "in between"--e.g., to the so-called q-difference equations. Moreover, the theory can be applied to other different types of time scales. Since its introduction, many authors have expounded on various aspects of this new theory, and we refer specifically to the paper by Agarwal et al. [1] and the references cited therein. A book on the subject of time scales by Bohner and Peterson [4] summarizes and organizes much of time scale calculus.

In recent years, there has been an increasing interest in studying the oscillation and nonoscillation of solutions of dynamic equations on a time scale (i.e., an arbitrary nonempty closed subset of the real numbers). This has lead to many attempts to harmonize the oscillation theory for the continuous and the discrete cases, to include them in one comprehensive theory, and to extend the results to more general time scales. We refer the reader to the papers [2,5,9,13,14], and the references cited therein.

Since we are interested in the oscillatory behavior of solutions near infinity, we assume throughout this paper that our time scale is unbounded above. We assume [t.sup.0] [member of] T and it is convenient to assume [t.sup.0] > 0. We define the time scale interval [[[t.sup.0], [infinity]).sub.T] by

[[t.sup.0], [infinity])t := [[t.sup.0], [infinity]) [intersection] T.

Our main interest is to consider the second-order linear dynamic equation

[y.sup.[DELTA][DELTA]](t) + q(t)y([tau](t)) = 0, t [member of] [[[t.sup.0], [infinity]).sub.T] (1.1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us recall that a solution of (1.1) is a nontrivial real-valued function y satisfying equation (1.1) for t [is greater than or equal to] [t.sub.0]. A solution y of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Our attention is restricted to those solutions y(t) of (1.1) which exist on some half-line [[[t.sub.y], [infinity]).sub.t] and satisfy sup{[absolute value of y(t)] : t > T} > 0 for any T [is greater than or equal to] [t.sub.y].

We note that (1.1) in its general form includes several types of differential and difference equations with delay arguments. In addition, different equations correspond to the choice of the time scale T. For example, when T = R, we have [y.sup.[DELTA]] = y', and so (1.1) becomes the delay differential equation

y"(t) + q(t)y([tau](t)) = 0.

In the case T = Z, [y.sup.[DELTA]] = [DELTA]y and (1.1) becomes the second-order delay difference equation

[DELTA][DELTA]y(t) + q(t)y([tau](t)) = 0

where [DELTA] denotes the forward difference operator. Finally, when T = {[q.sup.k.sub.o] : k [member of] [N.sub.0]} with [q.sub.0] > 1, (1.1) becomes the second-order delay [q.sub.0]-difference equation

y([q.sup.2.sub.o]t) - ([q.sub.0] + 1)y([q.sub.0]t) + [q.sub.0]y(t) + [q.sub.0]([q.sub.0] 1)[.sup.2]t[.sup.2]q(t)y([tau](t)) = 0.

In this paper we intend to use the method of upper and lower solutions to obtain oscillation criteria for (1.1) under certain conditions. We also use results about

[y.sup.[DELTA][DELTA]] (t) + q(t)[y.sup.[sigma]] (t) = 0

to obtain results for (1.1). Our results generalize those given in Erbe [6].

2 Preliminary Results

In this section, we state fundamental results needed to prove our main results. We begin with the following lemma.

Lemma 2.1 (See [7, Lemma 1.2]). Let y(t) be a solution of

[y.sup.[DELTA][DELTA]](t) + [n.summation over (i=1)] [q.sub.i](t)y([[tau].sub.i](t)) [is less than or equal to] 0

which satisfies

y(t) > 0, [y.sup.[DELTA]] (t) > 0, and [y.sup.[DELTA][DELTA]](t) < 0

for all [[tau].sub.i] (t) [is greater than or equal to] T > [t.sub.0]. Then for each 1 [is less than or equal to] i [is less than or equal to] n we have

y([[tau].sub.i](t)) [is greater than or equal to] ([[tau].sub.i](t) - T)/[sigma](t) - T) [y.sup.[sigma]](t), [[tau].sub.i](t) > T.

In order to prove our main results, we need a method for studying boundary value problems (BVP). Namely we will define functions called upper and lower solutions that, not only imply the existence of a solution of a certain BVP, but also provide bounds on that solution. Consider the second-order equation

[y.sup.[DELTA][DELTA]] = f (t,[y.sup.[sigma]]) (2.1)

where f is continuous on [[a, b].sub.T] x R.

Definition 2.2 (See [4, Definition 6.53]). We say that [alpha] [member of] [C.sup.2.sub.rd] is a lower solution of (2.1) on [[a], [[sigma].sup.2](b)].sub.T] provided

[[alpha].sup.[DELTA][DELTA]](t) [greater than or equal to] f(t,[[alpha].sub.[sigma]](t)) for all t [member of] [[[a, b].sub.T].

Similarly, [beta] [member of] [C.sup.2.sub.rd] is called an upper solution of (2.1) on [a, [[[sigma].sup.2](b)].sub.T] provided

([[beta].sup.[DELTA][DELTA]](t) [is less than or equal to] f (t,[[beta].sup.[sigma]](t)) for all t [member of] [[a, b].sub.T].

Theorem 2.3 (See [4, Theorem 6.54]). Let f be continuous on [[a, b].sub.T] x R. Assume that there exist a lower solution a and an upper solution [beta] of (2.1) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the BVP

[y.sup.[DELTA][DELTA]] = f(t,[y.sup.[sigma]]) on [[a,b].sub.T], y([alpha]) = A, y([[sigma].sub.2](b)) = B

has a solution y with

[alpha](t) [is less than or equal to] y(t) [is less than or equal to] [beta](t) for all t [member of] [a, [[[sigma].sup.2](b)].sub.T].

We conclude this section with the following generalization of [11, Theorem 7.4].

Theorem 2.4. Let f be continuous on [[a, b].sub.T] x R. Assume that there exist a lower solution [alpha] and an upper solution [beta] of (2.1) with [alpha](t) [is less than or equal to] [beta](t) for all t [member pf] [[a,[infinity]).sub.T]. Then for any [alpha](a) [is less than or equal to] c [is less than or equal to] [beta](a) the BVP

[y.sup.[DELTA][DELTA]] = f (t,[y.sup.[sigma]]), y(a) = c (2.2)

has a solution y with

[alpha](t) [is less than or equal to] y(t) [is less than or equal to] [beta] (t) for all t [member of] [[a, [infinity]).sub.T].

Proof. It follows from Theorem 2.3 that for each n [is greater than or equal to] 1 there is a solution [y.sub.n] (t) of [y.sup.[DELTA][DELTA]] = f (t,[y.sup.[sigma]]) on [[a,[t.sub.n]].sub.T] with [y.sub.n] ([alpha]) = c, [y.sub.n]([t.sub.n]) = [beta]([t.sub.n]) and [alpha](t) [is less than or equal to] [y.sub.n](t) [is less than or equal to] [beta] (t) on [[a,[t.sub.n]].sub.T,] where [t.sub.n] [right arrow] [infinity] as n [right arrow] [infinity]. Thus, for any fixed n [is greater than or equal to] 1, [y.sub.m] (t) is a solution on [[a,[t.sub.n]].sub.T] satisfying [alpha](t) [is less than or equal to] [y.sub.m](t) [is less than or equal to] [beta](t) for all m [is greater than or equal to] n. Hence, for m [is greater than or equal to] n, the sequence [y.sub.m] (t) is pointwise bounded on [[a,[t.sub.n]].sub.T.]

We claim that {[y.sub.m](t)} is equicontinuous on [[a,[t.sub.N]].sub.T] for any fixed n [is greater than or equal to] 1. Since f is continuous and [y.sub.m](t) [is less than or equal to] [beta](t) for all t [member of] [a, [[t.sub.n]].sub.T] , there is constant [K.sub.N] > 0 such that [absolute value of [y.sup.[DELTA][DELTA].sub.m](t)] = [absolute value of f (t,[y.sub.sup.[sigma].m](t))] [is less than or equal to] [K.sub.n] for all t [member of] [[a,[t.sub.N]].sub.T.] It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which gives that

[absolute value of [y.sub.[DELTA].sub.m](t)] [is less than or equal to] [absolute value of [y.sub.[DELTA].sub.m]([alpha])] + [absolute value of [K.sub.n]([t.sub.N] - a)].

Since {[y.sub.m](t)} is uniformly bounded on [[a,[t.sub.N]].sub.T] for all m [is greater than or equal to] N, it follows that [absolute value of [y.sub.[DELTA].sub.m]([alpha])] [less greater than or equal to] [L.sub.N] for some [L.sub.N] > 0 and all m [is greater than or equal to] N. Consequently,

[absolute value of [y.sub.[DELTA].sub.m](t)] [is less than or equal to] [L.sub.N] + [absolute value of [K.sub.n]([t.sub.N]-[alpha])]=: [M.sub.N],

and so,

[absolute value of [y.sub.m](t) - [y.sub.m](s)] = [absolute value of [[integral].sub.t.sub.s] [y.sup.[DELTA].sub.m][[DELTA].sub.s]] [less than or equal to] [M.sub.N] [absolute value of t - s] < [epsilon]

for all t, s [member of] [[a, [t.sub.N]].sub.T] provided [absolute value of t - s] < [delta] = [epsilon]/[M.sub.N]. Hence the claim holds.

So by Ascoli-Arzela and a standard diagonalization argument, {[y.sub.m](t)} contains a subsequence which converges uniformly on all compact subintervals [[a,[t.sub.N]].sub.T] of [[a, [infinity]).sub.T] to a solution y(t), which is the desired solution of (2.2) that satisfies [alpha](t) [is less than or equal to] y(t) [is less than or equal to] [beta] (t) for all t [[member of] [a, [infinity]).sub.T.]

3 Main Results

In this section we give four results concerning the oscillatory behavior of

[y.sup.[DELTA][DELTA]](t) + q (t) y ([tau](t)) = [y.sup.[sigma]] (t) (1.1)

on the time scale [[[t.sub.0], [infinity]).sub.T] where sup T = [infinity] and q [member of] [C.sub.rd] ([0, [infinity])T, [0, oo)). These are Theorems 3.1 and 3.9 and Corollaries 3.4 and 3.7.

Theorem 3.1. Assume that the equation

[y.sup.[DELTA][DELTA]](t) + [lambda] [[tau](t)/[sigma](t)] q(t)[y.sup.[sigma]] = (t) (3.1)

is oscillatory on [[[t.sub.0],[infinity]).sub.T] for some 0 < [lambda] < 1. Then all solutions of (1.1) are oscillatory.

Proof. Suppose, to the contrary, that (1.1) has an eventually positive solution u. That is, since [tau](t) [right arrow] [infinity] as t [right arrow] [infinity], there exists T [member of] [[t.sub.0], [infinity])T such that u(t) > 0 and u([tau](t)) > 0 for t [is greater than or equal to] T. As q(t) [is greater than or equal to] 0 on [[[t.sub.0], [infinity]).sub.T], we have

[u.sup.[DELTA][DELTA]] (t) = -q(t)u([tau](t)) [is less than or equal to] 0 for all t [is greater than or equal to] T, (3.2)

and so [u.sup.[DELTA]] decreases to a limit which must be nonnegative. In fact, we must have [u.sup.[DELTA]](t) > 0 on [[T,[infinity]).sub.T]. Indeed, if [u.sup.[DELTA]]([t.sub.1]) = 0 for some [t.sub.1] > T, then [u.sup.[DELTA]](t) [equivalent to] 0 on [[[t.sub.1], [infinity]).sub.T]. Consequently, from (1.1) we would have q(t) [equivalent to] 0 on [[[t.sub.1], [infinity]).sub.T], since u([tau](t)) > 0 on [[T,[infinity]).sub.T], contradicting the fact that (3.1) is oscillatory. So we have

u(t) > 0, [u.sup.[DELTA]](t) > 0, [u.sup.[DELTA][DELTA]] (t) [is less than or equal to] 0 on [[T,[infinity]).sub.T].

For any 0 < k < 1 there is a [T.sub.k] [is greater than or equal to] T such that

by Lemma 2.1. It follows that

[u.sup.[DELTA][DELTA]](t) + k [[tau](t)/[sigma](t)] q(t)[u.sub.[sigma]] [is less than or equal to] 0, t [is greater than or equal to] [T.sub.k]. (3.3)

Let z(t) = and [u.sup.[DELTA]](t)/ u(t)and Q(t) = k [[tau](t)/[sigma](t)]. Also, let

S[z] = [z.sup.2]/[1 + [mu](t)z].

Then

1 + [mu](t)z(t) = 1 + [mu](t) [[u.sup.[DELTA]](t)/u(t)] > 0

for t [is greater than or equal to] T and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by (3.2). Hence, by [7, Lemma 1.1], [u.sup.[DELTA][DELTA]] + [Qu.sup.[sigma]] = 0 is nonoscillatory. Choosing 0 < k < 1 such that k > [lambda], we have Q(t) > [lambda] [[tau](t)/[sigma](t)] =: R(t). By the Sturm-Picone comparison theorem [8, Lemma 6], we therefore have [u.sup.[DELTA][DELTA]] + R(t)[u.sup.[sigma]](t) = 0 is nonoscillatory. This contradiction proves the theorem.

Before we give the first corollary of Theorem 3.1 we prove the following.

Theorem 3.2. Assume there is a [t.sub.*] [is greater than or equal to] [alpha] [member of] T and [alpha] u [member of] [C.sup.1.sub.rd] [[t.sub.*],[infinity]) such that u(t) > 0 on [[[t.sub.*],[infinity]).sub.T] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the second-order dynamic equation

[u.sup.[DELTA][DELTA]](t) + q(t)[y.sup.[sigma]](t) = 0 (3.4)

is oscillatory on [[a,[infinity]).sub.T].

Proof. We prove this theorem by contradiction. So assume (3.4) is nonoscillatory on [[a,[infinity]).sub.T]. By [4, Theorem 4.61], y is a solution of (3.4) which is dominant at [infinity] such that for [t.sup.*] [is greater than or equal to] [alpha], sufficiently large,

[[infinity].integral over (t*) [[DELTA]t/y(t)[y.sup.[sigma]](t)] < [infinity],

and we may assume y(t) > 0 on [[[t.sup.*], [infinity]).sub.T]. Let [t.sub.*] and u be as in the statement of this theorem. Let T = max{[t.sub.*],[t.sup.*]}; then let

z(t):= y[DELTA](t)/y(t), t[is greater than or equal to] T

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

1 + [mu](t)z(t) > 0 for all t [is greater than or equal to] T.

Then by [4, Theorem 4.55], we have for t [is greater than or equal to] T

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Integrating from T to t, we obtain

z(t)[u.sup.2](t) [is less than or equal to] z(T)[u.sup.2](T) - [[integral].sup.t.sub.T]{(q(t)[u.sup.2]([sigma](t)) [[[u.sup.[DELTA]](t)].sup.2]}[DELTA]t

which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, then there is a [T.sub.1] > T such that for t [greater than or equal to] [T.sub.1]

z(t) = [y.sup.[DELTA]](t)/y(t) < 0.

This implies that [y.sup.[DELTA]] (t) < 0 for t [is greater than or equal to] [T.sub.1], and hence y is decreasing on [[[T.sub.1], [infinity]).sub.T]. However,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a contradiction.

The following example is illustrative.

Example 3.3. If [alpha] > 0 and

[[integral].sup.[infinity].sub.a][[sigma].sub.[alpha]](t)q(t)[DELTA]t = [infinity]

where 0 < [alpha] < 1, then [y.sup.[DELTA][DELTA]] + q(t)[y.sup.[sigma]] = 0 is oscillatory on [[a,[infinity]).sup.T]. We will show that this follows from Theorem 3.2. In the Potzsche chain rule [4, Theorem 1.90], let g(t) = t and f(t) = [t.sup.[alpha]/2], for 0 < [alpha] < 1. Then with u(t) = (f*g)(t) = [t.sup.[alpha]/2], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since [alpha] - 2 < 0. Therefore, it follows that [(u[DELTA](t)).sup.2] [is less than or equal to] [[[alpha].sup.2]/4] [t.sup.[alpha-2] for all t. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since 0 < [alpha] < 1 implies J

[infinity].integral over ([alpha])] [t.sup.[alpha]-2] [DELTA]t < [infinity].

Thus [y.sup.[DELTA][DELTA]] + q(t)[y.sup.[sigma]] = 0 is oscillatory on [a,[infinity])T by Theorem 3.2.

As a corollary to Theorem 3.1, we have the following.

Corollary 3.4. All solutions of

[y.sup.[DELTA][DELTA]] + q(t)y([tau](t)) = 0 (1.1)

are oscillatory in case either of the following holds:

(i) [[integral].sup.[infinity]][([sigma](t)).sup.[alpha]-1][tau](t)q(t) [DELTA]t = [infinity] for some [alpha] [member of] (0,1)

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. If (i) holds, then for any [lambda] > 0, [[sigma].sup.[alpha]](t)[lambda][[tau](t)/[sigma](t)]q(t)[DELTA]t = [infinity]. By Example 3.3, [y.sup.[DELTA][DELTA]] + [lambda] [[tau](t)/[sigma](t)]q(t)[y.sup.[sigma]](t) = 0 is oscillatory since 0 < [alpha] < 1. Hence, by Theorem 3.1, all solutions of [y.sup.[DELTA][DELTA]] + q(t)y([tau](t)) = 0, equation (1.1), are oscillatory.

Next assume (ii) holds. Then by [12, Theorem 4],

[y.sup.[DELTA][DELTA]] + [[tau](t)/[sigma](t)] q(t) [y.sup.[sigma]](t) = 0

is oscillatory. Since [mu](t) is bounded, we have that [y.sup.[DELTA][DELTA]] + [[tau](t)/[sigma](t)]q(t)[y.sup.[sigma]](t) = 0 is oscillatory by [14, Theorem 2.1] with f(t) = t. By the Sturm-Picone comparison theorem [8, Lemma 6], we have (1.1) is oscillatory since 0 < [tau](t) [less than or equal to] [tau] [less than or equal to] [sigma](t).

To prove our second corollary of Theorem 3.1, we will use the method of upper and lower solutions and the following lemma.

Lemma 3.5. If

[[integral].sup.[infinity]][tau](t)q(t)[DELTA]t = [infinity], (3.5)

then every bounded solution of equation (3.1) is oscillatory on [[[t.sub.0],[infinity]).sub.T].

Proof. Suppose that there exists an eventually positive and bounded solution y of (3.1). Then there exists T [member of] T such that

y(t) > 0, [y.sup.[DELTA]](t) > 0, [y.sup.[DELTA][DELTA]](t) [less than or equal to] 0 for all t [greater than or equal to] T [greater than or equal to] [t.sub.0],

and without loss of generality, there exist [alpha], [beta] [member of] R such that

0 < [alpha] < y(t) < [beta] for all t [greater than or equal to] T.

Let Y (t) = t[y.sup.[DELTA]](t). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e., there is a constant M > 0 such that

[y.sup.[DELTA]](t) [less than or equal to] -[M/t] for t [greater than or equal to] [??]

for some [??] [greater than or equal to] T, and this implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [3, Example 5.15], contradicting y(t) > 0 for all t [greater than or equal to] T. Thus every bounded solution of

[y.sup.[DELTA][DELTA]] + [lambda] [[tau](t)]/[sigma](t)]q(t)[y.sup.[sigma]](t) = 0

is oscillatory.

Example 3.6. Consider the delay dynamic equation

[y.sup.[DELTA][DELTA]](t) + [1/t[tau](t)]y([tau](t)) = 0 for t [greater than or equal to] [t.sup.0]. (3.6)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, by Lemma 3.5, every bounded solution of (3.6) oscillates on [[t.sub.0], [infinity]).

We can now state and prove our second corollary of Theorem 3.1.

Corollary 3.7. All bounded solutions of the linear second-order dynamic equation

[y.sup.[DELTA][DELTA]] + q(t)y([tau](t)) = 0 (1.1)

are oscillatory in case (3.5) holds.

Proof. Let u be a bounded nonoscillatory solution of (1.1) with u(t) > 0 and u([tau](t)) > 0 for t [greater than or equal to] T. Since [u.sup.[DELTA][DELTA]](t) [less than or equal to] 0 for all t, we have [u.sup.[DELTA]](t) > 0 on [[T,[infinity]).sup.T]. As in the proof of Theorem 3.1, for any 0 < k < 1 there exists a [T.sub.k] [greater than or equal to] T such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So [alpha], [beta] are lower and upper solutions, respectively, of

[y.sup.[DELTA][DELTA]] + [lambda] [[tau](t)/[sigma](t)] q(t)[y.sup.[sigma]](t) = 0. (3.1)

As u is increasing, [alpha](t) [less than or equal to] [beta](t) on [[[T.sub.k],[infinity]).sub.T]. Then by Theorem 2.4, there is a solution y(t) of (3.1) satisfying u(T) [less than or equal to] y(t) [less than or equal to] u(t) on [[[T.sub.k],[infinity]).sub.T]. As u is bounded, y is a bounded nonoscillatory solution of (3.1). This is a contradiction to Lemma 3.5 and proves the theorem.

In order to prove our last result, which in an extension of [2, Theorem 4.4], we need the following lemma.

Lemma 3.8. Let y(t) be a positive solution of (1.1) defined on [[T,[infinity]).sub.T] for some T > 0 that satisfies [y.sup.[DELTA]](t) [greater than or equal to] 0 and [y.sup.[DELTA][DELTA]](t) [less than or equal to] 0 on [[T,[infinity]).sub.T]. If (3.5) holds, then there exists a [T.sub.1] [greater than or equal to] T such that

[y(t)/[y.sup.[DELTA]](t)] [greater than or equal to] t and y(t)/t is decreasing (3.7)

on [[[T.sub.1],[infinity]).sub.T].

Proof. Let y(t) be as in the statement of the lemma and assume (3.5) holds. Also let Y(t) := y(t) - t[y.sup.[DELTA]](t). Then [Y.sup.[DELTA]](t) = -[sigma](t)[y.sup.[DELTA][DELTA]](t) [greater than or equal to] 0 for t [member of] [[T,[infinity]).sub.T]. This implies that Y(t) is increasing on [[T,[Infinity]).sub.T].

We claim there is a [T.sub.1] [member of] [T,[infinity]) such that Y(t) [greater than or equal to] 0 on [[[T.sub.1],[infintiy]).sub.T]. If not, then Y(t) < 0 on [[[T.sub.1],[infintiy]).sub.T]. Therefore

[(y(t)/t).sup.[DELTA]] = t[y.sup.[DELTA]](t) - y(t)/t[sigma](t) = - Y(t)/t[sigma](t) > 0, t [member of] [[[T.sub.1], [infintiy]).sub.T]

which implies that y(t)/t is increasing on [[[T.sub.1],[infinity]).sub.T]. Choose [T.sub.2] [member of] [[[T.sub.1],[infinity]).sub.T] such that [tau](t) [greater than or equal to] [tau]([T.sub.2]) for t [greater than or equal to] [T.sub.2]. Then

[y([tau](t))/[tau](t)] [greater than or equal to] y([tau]([T.sub.2]))/[tau]([T.sub.2]) =: D > 0,

which gives y([tau](t)) [greater than or equal to] D[tau](t) for t [greater than or equal to] [T.sub.2]. Now by integrating both sides of (1.1) from [T.sub.2] to t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (3.5). Hence there is a [T.sub.1] [member of] [[T,[infinity]).sub.T] such that Y(t) [greater than or equal to] 0 on [[[T.sub.1],[infinity]).sub.T], and the first part of (3.7) holds. Moreover,

[(y(t)/t).sup.[DELTA]] = [t[y.sup.[DELTA]](t) - y(t)]/t[sigma](t) = -[Y(t)/t[sigma](t)] [less than or equal to] 0, t [member of] [[[T.sub.1],[infinity]).sub.T],

and we have that y(t)/t is decreasing on [[[T.sub.1],[infinity]).sub.T]. This completes the proof of the lemma.

Theorem 3.9. Assume (3.5) holds. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.8)

then every solution of (1.1) is oscillatory on [[[t.sub.0],[infinity]).sub.T].

Proof. To the contrary, suppose y is a nonoscillatory solution of (1.1). Then there exists T [member of] T such that

y(t) > 0, [y.sup.[DELTA]](t) > 0, and [y.sup.[DELTA][DELTA]](t) [less than or equal to] 0 for all t [greater than or equal to] T [greater than or equal to] [t.sub.0].

It follows that for s [greater than or equal to] t [greater than or equal to] T we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From the above inequality and Lemma 3.8, it follows that for sufficiently large t [member of] T

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction to (3.8). This completes the proof.

Example 3.10. Let h > 0 and T = hZ = {hk : k [member of] Z}. In this case (1.1) becomes

y(t + 2h) - 2y(t + h) + y(t) + [h.sup.2]q(t)y([tau](t)) = 0. (3.9)

Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then every bounded solution of (3.9) oscillates on [[[t.sub.0],[infinity]).sub.T]. Additionally, if,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then every solution of (3.9) is oscillatory on [[[t.sub.0],[infinity]).sub.T].

4 Conclusion and Future Directions

In this paper we have obtained sufficient conditions for the oscillatory behavior of

[y.sup.[DELTA][DELTA]](t) + q(t)y([tau](t)) = 0.

This was done by comparing nonoscillatory solutions of the delay dynamic equation with the solutions of a corresponding linear dynamic equation and then using known properties of the linear equation to obtain a desired contradiction.

Possibilities for further exploration include changing the leading term to [(p(t)[y.sup.[DELTA]]).sup.[DELTA]] where p(t) > 0 on the time scale interval [[[t.sub.0],[infinity]).sub.T] and [[integral].sup.[infinity]][[DELTA]t/p(t)] = [infinity], and replacing the delay [tau](t) with the advance [xi]: T [right arrow] T where [sigma](t) [less than or equal to] [xi](t) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

References

[1] Ravi Agarwal, Martin Bohner, Donal O'Regan, and Allan Peterson. Dynamic equations on time scales: a survey. J. Comput. Appl. Math., 141(1-2):1-26, 2002. Dynamic equations on time scales.

[2] Ravi Agarwal, Martin Bohner, and Samir H. Saker. Oscillation of second order delay dynamic equations. Can. Appl. Math. Q., 13(1):1-17, 2005.

[3] Martin Bohner and Gusein Sh. Guseinov. Improper integrals on time scales. Dynam. Systems Appl., 12(1-2):45-65, 2003. Special issue: dynamic equations on time scales.

[4] Martin Bohner and Allan Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser Boston Inc., Boston, MA, 2001.

[5] Martin Bohner and Samir H. Saker. Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mountain J. Math., 34(4):1239-1254, 2004.

[6] Lynn Erbe. Oscillation criteria for second order nonlinear delay equations. Canad. Math. Bull., 16:49-56, 1973.

[7] Lynn Erbe and Allan Peterson. Some oscillation results for second order linear delay dynamic equations. volume 53 of Adv. Stud. Pure Math., pages 237-245. Math. Soc. Japan, Tokyo, 2009.

[8] Lynn Erbe, Allan Peterson, and Pavel Rehak. Comparison theorems for linear dynamic equations on time scales. J. Math. Anal. Appl., 275(1):418-438, 2002.

[9] Lynn Erbe, Allan Peterson, and Samir H. Saker. Oscillation criteria for second-order nonlinear dynamic equations on time scales. J. London Math. Soc. (2), 67(3):701-714, 2003.

[10] Stefan Hilger. Analysis on measure chains--a unified approach to continuous and discrete calculus. Results Math., 18(1-2):18-56, 1990.

[11] Lloyd K. Jackson. Subfunctions and second-order ordinary differential inequalities. Advances in Math., 2:307-363 (1968), 1968.

[12] Pavel Rehak. How the constants in Hille-Nehari theorems depend on time scales. Adv. Difference Equ., pages Art. ID 64534, 15, 2006.

[13] Samir H. Saker. Oscillation of nonlinear dynamic equations on time scales. Appl. Math. Comput., 148(1):81-91, 2004.

[14] B. G. Zhang and Zhu Shanliang. Oscillation of second-order nonlinear delay dynamic equations on time scales. Comput. Math. Appl., 49(4):599-609, 2005.

Raegan Higgins

Texas Tech University

Department of Mathematics & Statistics

Lubbock, TX 79409-1042, U.S.A.

raegan.higgins@ttu.edu

In 1988 the theory of time scales was introduced by Stefan Hilger in his Ph.D. Thesis in order to unify continuous and discrete analysis (see [10]). Not only does this theory unify those of differential equations and difference equations, but it also extends these classical situations to cases "in between"--e.g., to the so-called q-difference equations. Moreover, the theory can be applied to other different types of time scales. Since its introduction, many authors have expounded on various aspects of this new theory, and we refer specifically to the paper by Agarwal et al. [1] and the references cited therein. A book on the subject of time scales by Bohner and Peterson [4] summarizes and organizes much of time scale calculus.

In recent years, there has been an increasing interest in studying the oscillation and nonoscillation of solutions of dynamic equations on a time scale (i.e., an arbitrary nonempty closed subset of the real numbers). This has lead to many attempts to harmonize the oscillation theory for the continuous and the discrete cases, to include them in one comprehensive theory, and to extend the results to more general time scales. We refer the reader to the papers [2,5,9,13,14], and the references cited therein.

Since we are interested in the oscillatory behavior of solutions near infinity, we assume throughout this paper that our time scale is unbounded above. We assume [t.sup.0] [member of] T and it is convenient to assume [t.sup.0] > 0. We define the time scale interval [[[t.sup.0], [infinity]).sub.T] by

[[t.sup.0], [infinity])t := [[t.sup.0], [infinity]) [intersection] T.

Our main interest is to consider the second-order linear dynamic equation

[y.sup.[DELTA][DELTA]](t) + q(t)y([tau](t)) = 0, t [member of] [[[t.sup.0], [infinity]).sub.T] (1.1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us recall that a solution of (1.1) is a nontrivial real-valued function y satisfying equation (1.1) for t [is greater than or equal to] [t.sub.0]. A solution y of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Our attention is restricted to those solutions y(t) of (1.1) which exist on some half-line [[[t.sub.y], [infinity]).sub.t] and satisfy sup{[absolute value of y(t)] : t > T} > 0 for any T [is greater than or equal to] [t.sub.y].

We note that (1.1) in its general form includes several types of differential and difference equations with delay arguments. In addition, different equations correspond to the choice of the time scale T. For example, when T = R, we have [y.sup.[DELTA]] = y', and so (1.1) becomes the delay differential equation

y"(t) + q(t)y([tau](t)) = 0.

In the case T = Z, [y.sup.[DELTA]] = [DELTA]y and (1.1) becomes the second-order delay difference equation

[DELTA][DELTA]y(t) + q(t)y([tau](t)) = 0

where [DELTA] denotes the forward difference operator. Finally, when T = {[q.sup.k.sub.o] : k [member of] [N.sub.0]} with [q.sub.0] > 1, (1.1) becomes the second-order delay [q.sub.0]-difference equation

y([q.sup.2.sub.o]t) - ([q.sub.0] + 1)y([q.sub.0]t) + [q.sub.0]y(t) + [q.sub.0]([q.sub.0] 1)[.sup.2]t[.sup.2]q(t)y([tau](t)) = 0.

In this paper we intend to use the method of upper and lower solutions to obtain oscillation criteria for (1.1) under certain conditions. We also use results about

[y.sup.[DELTA][DELTA]] (t) + q(t)[y.sup.[sigma]] (t) = 0

to obtain results for (1.1). Our results generalize those given in Erbe [6].

2 Preliminary Results

In this section, we state fundamental results needed to prove our main results. We begin with the following lemma.

Lemma 2.1 (See [7, Lemma 1.2]). Let y(t) be a solution of

[y.sup.[DELTA][DELTA]](t) + [n.summation over (i=1)] [q.sub.i](t)y([[tau].sub.i](t)) [is less than or equal to] 0

which satisfies

y(t) > 0, [y.sup.[DELTA]] (t) > 0, and [y.sup.[DELTA][DELTA]](t) < 0

for all [[tau].sub.i] (t) [is greater than or equal to] T > [t.sub.0]. Then for each 1 [is less than or equal to] i [is less than or equal to] n we have

y([[tau].sub.i](t)) [is greater than or equal to] ([[tau].sub.i](t) - T)/[sigma](t) - T) [y.sup.[sigma]](t), [[tau].sub.i](t) > T.

In order to prove our main results, we need a method for studying boundary value problems (BVP). Namely we will define functions called upper and lower solutions that, not only imply the existence of a solution of a certain BVP, but also provide bounds on that solution. Consider the second-order equation

[y.sup.[DELTA][DELTA]] = f (t,[y.sup.[sigma]]) (2.1)

where f is continuous on [[a, b].sub.T] x R.

Definition 2.2 (See [4, Definition 6.53]). We say that [alpha] [member of] [C.sup.2.sub.rd] is a lower solution of (2.1) on [[a], [[sigma].sup.2](b)].sub.T] provided

[[alpha].sup.[DELTA][DELTA]](t) [greater than or equal to] f(t,[[alpha].sub.[sigma]](t)) for all t [member of] [[[a, b].sub.T].

Similarly, [beta] [member of] [C.sup.2.sub.rd] is called an upper solution of (2.1) on [a, [[[sigma].sup.2](b)].sub.T] provided

([[beta].sup.[DELTA][DELTA]](t) [is less than or equal to] f (t,[[beta].sup.[sigma]](t)) for all t [member of] [[a, b].sub.T].

Theorem 2.3 (See [4, Theorem 6.54]). Let f be continuous on [[a, b].sub.T] x R. Assume that there exist a lower solution a and an upper solution [beta] of (2.1) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the BVP

[y.sup.[DELTA][DELTA]] = f(t,[y.sup.[sigma]]) on [[a,b].sub.T], y([alpha]) = A, y([[sigma].sub.2](b)) = B

has a solution y with

[alpha](t) [is less than or equal to] y(t) [is less than or equal to] [beta](t) for all t [member of] [a, [[[sigma].sup.2](b)].sub.T].

We conclude this section with the following generalization of [11, Theorem 7.4].

Theorem 2.4. Let f be continuous on [[a, b].sub.T] x R. Assume that there exist a lower solution [alpha] and an upper solution [beta] of (2.1) with [alpha](t) [is less than or equal to] [beta](t) for all t [member pf] [[a,[infinity]).sub.T]. Then for any [alpha](a) [is less than or equal to] c [is less than or equal to] [beta](a) the BVP

[y.sup.[DELTA][DELTA]] = f (t,[y.sup.[sigma]]), y(a) = c (2.2)

has a solution y with

[alpha](t) [is less than or equal to] y(t) [is less than or equal to] [beta] (t) for all t [member of] [[a, [infinity]).sub.T].

Proof. It follows from Theorem 2.3 that for each n [is greater than or equal to] 1 there is a solution [y.sub.n] (t) of [y.sup.[DELTA][DELTA]] = f (t,[y.sup.[sigma]]) on [[a,[t.sub.n]].sub.T] with [y.sub.n] ([alpha]) = c, [y.sub.n]([t.sub.n]) = [beta]([t.sub.n]) and [alpha](t) [is less than or equal to] [y.sub.n](t) [is less than or equal to] [beta] (t) on [[a,[t.sub.n]].sub.T,] where [t.sub.n] [right arrow] [infinity] as n [right arrow] [infinity]. Thus, for any fixed n [is greater than or equal to] 1, [y.sub.m] (t) is a solution on [[a,[t.sub.n]].sub.T] satisfying [alpha](t) [is less than or equal to] [y.sub.m](t) [is less than or equal to] [beta](t) for all m [is greater than or equal to] n. Hence, for m [is greater than or equal to] n, the sequence [y.sub.m] (t) is pointwise bounded on [[a,[t.sub.n]].sub.T.]

We claim that {[y.sub.m](t)} is equicontinuous on [[a,[t.sub.N]].sub.T] for any fixed n [is greater than or equal to] 1. Since f is continuous and [y.sub.m](t) [is less than or equal to] [beta](t) for all t [member of] [a, [[t.sub.n]].sub.T] , there is constant [K.sub.N] > 0 such that [absolute value of [y.sup.[DELTA][DELTA].sub.m](t)] = [absolute value of f (t,[y.sub.sup.[sigma].m](t))] [is less than or equal to] [K.sub.n] for all t [member of] [[a,[t.sub.N]].sub.T.] It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which gives that

[absolute value of [y.sub.[DELTA].sub.m](t)] [is less than or equal to] [absolute value of [y.sub.[DELTA].sub.m]([alpha])] + [absolute value of [K.sub.n]([t.sub.N] - a)].

Since {[y.sub.m](t)} is uniformly bounded on [[a,[t.sub.N]].sub.T] for all m [is greater than or equal to] N, it follows that [absolute value of [y.sub.[DELTA].sub.m]([alpha])] [less greater than or equal to] [L.sub.N] for some [L.sub.N] > 0 and all m [is greater than or equal to] N. Consequently,

[absolute value of [y.sub.[DELTA].sub.m](t)] [is less than or equal to] [L.sub.N] + [absolute value of [K.sub.n]([t.sub.N]-[alpha])]=: [M.sub.N],

and so,

[absolute value of [y.sub.m](t) - [y.sub.m](s)] = [absolute value of [[integral].sub.t.sub.s] [y.sup.[DELTA].sub.m][[DELTA].sub.s]] [less than or equal to] [M.sub.N] [absolute value of t - s] < [epsilon]

for all t, s [member of] [[a, [t.sub.N]].sub.T] provided [absolute value of t - s] < [delta] = [epsilon]/[M.sub.N]. Hence the claim holds.

So by Ascoli-Arzela and a standard diagonalization argument, {[y.sub.m](t)} contains a subsequence which converges uniformly on all compact subintervals [[a,[t.sub.N]].sub.T] of [[a, [infinity]).sub.T] to a solution y(t), which is the desired solution of (2.2) that satisfies [alpha](t) [is less than or equal to] y(t) [is less than or equal to] [beta] (t) for all t [[member of] [a, [infinity]).sub.T.]

3 Main Results

In this section we give four results concerning the oscillatory behavior of

[y.sup.[DELTA][DELTA]](t) + q (t) y ([tau](t)) = [y.sup.[sigma]] (t) (1.1)

on the time scale [[[t.sub.0], [infinity]).sub.T] where sup T = [infinity] and q [member of] [C.sub.rd] ([0, [infinity])T, [0, oo)). These are Theorems 3.1 and 3.9 and Corollaries 3.4 and 3.7.

Theorem 3.1. Assume that the equation

[y.sup.[DELTA][DELTA]](t) + [lambda] [[tau](t)/[sigma](t)] q(t)[y.sup.[sigma]] = (t) (3.1)

is oscillatory on [[[t.sub.0],[infinity]).sub.T] for some 0 < [lambda] < 1. Then all solutions of (1.1) are oscillatory.

Proof. Suppose, to the contrary, that (1.1) has an eventually positive solution u. That is, since [tau](t) [right arrow] [infinity] as t [right arrow] [infinity], there exists T [member of] [[t.sub.0], [infinity])T such that u(t) > 0 and u([tau](t)) > 0 for t [is greater than or equal to] T. As q(t) [is greater than or equal to] 0 on [[[t.sub.0], [infinity]).sub.T], we have

[u.sup.[DELTA][DELTA]] (t) = -q(t)u([tau](t)) [is less than or equal to] 0 for all t [is greater than or equal to] T, (3.2)

and so [u.sup.[DELTA]] decreases to a limit which must be nonnegative. In fact, we must have [u.sup.[DELTA]](t) > 0 on [[T,[infinity]).sub.T]. Indeed, if [u.sup.[DELTA]]([t.sub.1]) = 0 for some [t.sub.1] > T, then [u.sup.[DELTA]](t) [equivalent to] 0 on [[[t.sub.1], [infinity]).sub.T]. Consequently, from (1.1) we would have q(t) [equivalent to] 0 on [[[t.sub.1], [infinity]).sub.T], since u([tau](t)) > 0 on [[T,[infinity]).sub.T], contradicting the fact that (3.1) is oscillatory. So we have

u(t) > 0, [u.sup.[DELTA]](t) > 0, [u.sup.[DELTA][DELTA]] (t) [is less than or equal to] 0 on [[T,[infinity]).sub.T].

For any 0 < k < 1 there is a [T.sub.k] [is greater than or equal to] T such that

by Lemma 2.1. It follows that

[u.sup.[DELTA][DELTA]](t) + k [[tau](t)/[sigma](t)] q(t)[u.sub.[sigma]] [is less than or equal to] 0, t [is greater than or equal to] [T.sub.k]. (3.3)

Let z(t) = and [u.sup.[DELTA]](t)/ u(t)and Q(t) = k [[tau](t)/[sigma](t)]. Also, let

S[z] = [z.sup.2]/[1 + [mu](t)z].

Then

1 + [mu](t)z(t) = 1 + [mu](t) [[u.sup.[DELTA]](t)/u(t)] > 0

for t [is greater than or equal to] T and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by (3.2). Hence, by [7, Lemma 1.1], [u.sup.[DELTA][DELTA]] + [Qu.sup.[sigma]] = 0 is nonoscillatory. Choosing 0 < k < 1 such that k > [lambda], we have Q(t) > [lambda] [[tau](t)/[sigma](t)] =: R(t). By the Sturm-Picone comparison theorem [8, Lemma 6], we therefore have [u.sup.[DELTA][DELTA]] + R(t)[u.sup.[sigma]](t) = 0 is nonoscillatory. This contradiction proves the theorem.

Before we give the first corollary of Theorem 3.1 we prove the following.

Theorem 3.2. Assume there is a [t.sub.*] [is greater than or equal to] [alpha] [member of] T and [alpha] u [member of] [C.sup.1.sub.rd] [[t.sub.*],[infinity]) such that u(t) > 0 on [[[t.sub.*],[infinity]).sub.T] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the second-order dynamic equation

[u.sup.[DELTA][DELTA]](t) + q(t)[y.sup.[sigma]](t) = 0 (3.4)

is oscillatory on [[a,[infinity]).sub.T].

Proof. We prove this theorem by contradiction. So assume (3.4) is nonoscillatory on [[a,[infinity]).sub.T]. By [4, Theorem 4.61], y is a solution of (3.4) which is dominant at [infinity] such that for [t.sup.*] [is greater than or equal to] [alpha], sufficiently large,

[[infinity].integral over (t*) [[DELTA]t/y(t)[y.sup.[sigma]](t)] < [infinity],

and we may assume y(t) > 0 on [[[t.sup.*], [infinity]).sub.T]. Let [t.sub.*] and u be as in the statement of this theorem. Let T = max{[t.sub.*],[t.sup.*]}; then let

z(t):= y[DELTA](t)/y(t), t[is greater than or equal to] T

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

1 + [mu](t)z(t) > 0 for all t [is greater than or equal to] T.

Then by [4, Theorem 4.55], we have for t [is greater than or equal to] T

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Integrating from T to t, we obtain

z(t)[u.sup.2](t) [is less than or equal to] z(T)[u.sup.2](T) - [[integral].sup.t.sub.T]{(q(t)[u.sup.2]([sigma](t)) [[[u.sup.[DELTA]](t)].sup.2]}[DELTA]t

which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, then there is a [T.sub.1] > T such that for t [greater than or equal to] [T.sub.1]

z(t) = [y.sup.[DELTA]](t)/y(t) < 0.

This implies that [y.sup.[DELTA]] (t) < 0 for t [is greater than or equal to] [T.sub.1], and hence y is decreasing on [[[T.sub.1], [infinity]).sub.T]. However,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a contradiction.

The following example is illustrative.

Example 3.3. If [alpha] > 0 and

[[integral].sup.[infinity].sub.a][[sigma].sub.[alpha]](t)q(t)[DELTA]t = [infinity]

where 0 < [alpha] < 1, then [y.sup.[DELTA][DELTA]] + q(t)[y.sup.[sigma]] = 0 is oscillatory on [[a,[infinity]).sup.T]. We will show that this follows from Theorem 3.2. In the Potzsche chain rule [4, Theorem 1.90], let g(t) = t and f(t) = [t.sup.[alpha]/2], for 0 < [alpha] < 1. Then with u(t) = (f*g)(t) = [t.sup.[alpha]/2], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since [alpha] - 2 < 0. Therefore, it follows that [(u[DELTA](t)).sup.2] [is less than or equal to] [[[alpha].sup.2]/4] [t.sup.[alpha-2] for all t. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since 0 < [alpha] < 1 implies J

[infinity].integral over ([alpha])] [t.sup.[alpha]-2] [DELTA]t < [infinity].

Thus [y.sup.[DELTA][DELTA]] + q(t)[y.sup.[sigma]] = 0 is oscillatory on [a,[infinity])T by Theorem 3.2.

As a corollary to Theorem 3.1, we have the following.

Corollary 3.4. All solutions of

[y.sup.[DELTA][DELTA]] + q(t)y([tau](t)) = 0 (1.1)

are oscillatory in case either of the following holds:

(i) [[integral].sup.[infinity]][([sigma](t)).sup.[alpha]-1][tau](t)q(t) [DELTA]t = [infinity] for some [alpha] [member of] (0,1)

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. If (i) holds, then for any [lambda] > 0, [[sigma].sup.[alpha]](t)[lambda][[tau](t)/[sigma](t)]q(t)[DELTA]t = [infinity]. By Example 3.3, [y.sup.[DELTA][DELTA]] + [lambda] [[tau](t)/[sigma](t)]q(t)[y.sup.[sigma]](t) = 0 is oscillatory since 0 < [alpha] < 1. Hence, by Theorem 3.1, all solutions of [y.sup.[DELTA][DELTA]] + q(t)y([tau](t)) = 0, equation (1.1), are oscillatory.

Next assume (ii) holds. Then by [12, Theorem 4],

[y.sup.[DELTA][DELTA]] + [[tau](t)/[sigma](t)] q(t) [y.sup.[sigma]](t) = 0

is oscillatory. Since [mu](t) is bounded, we have that [y.sup.[DELTA][DELTA]] + [[tau](t)/[sigma](t)]q(t)[y.sup.[sigma]](t) = 0 is oscillatory by [14, Theorem 2.1] with f(t) = t. By the Sturm-Picone comparison theorem [8, Lemma 6], we have (1.1) is oscillatory since 0 < [tau](t) [less than or equal to] [tau] [less than or equal to] [sigma](t).

To prove our second corollary of Theorem 3.1, we will use the method of upper and lower solutions and the following lemma.

Lemma 3.5. If

[[integral].sup.[infinity]][tau](t)q(t)[DELTA]t = [infinity], (3.5)

then every bounded solution of equation (3.1) is oscillatory on [[[t.sub.0],[infinity]).sub.T].

Proof. Suppose that there exists an eventually positive and bounded solution y of (3.1). Then there exists T [member of] T such that

y(t) > 0, [y.sup.[DELTA]](t) > 0, [y.sup.[DELTA][DELTA]](t) [less than or equal to] 0 for all t [greater than or equal to] T [greater than or equal to] [t.sub.0],

and without loss of generality, there exist [alpha], [beta] [member of] R such that

0 < [alpha] < y(t) < [beta] for all t [greater than or equal to] T.

Let Y (t) = t[y.sup.[DELTA]](t). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e., there is a constant M > 0 such that

[y.sup.[DELTA]](t) [less than or equal to] -[M/t] for t [greater than or equal to] [??]

for some [??] [greater than or equal to] T, and this implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [3, Example 5.15], contradicting y(t) > 0 for all t [greater than or equal to] T. Thus every bounded solution of

[y.sup.[DELTA][DELTA]] + [lambda] [[tau](t)]/[sigma](t)]q(t)[y.sup.[sigma]](t) = 0

is oscillatory.

Example 3.6. Consider the delay dynamic equation

[y.sup.[DELTA][DELTA]](t) + [1/t[tau](t)]y([tau](t)) = 0 for t [greater than or equal to] [t.sup.0]. (3.6)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, by Lemma 3.5, every bounded solution of (3.6) oscillates on [[t.sub.0], [infinity]).

We can now state and prove our second corollary of Theorem 3.1.

Corollary 3.7. All bounded solutions of the linear second-order dynamic equation

[y.sup.[DELTA][DELTA]] + q(t)y([tau](t)) = 0 (1.1)

are oscillatory in case (3.5) holds.

Proof. Let u be a bounded nonoscillatory solution of (1.1) with u(t) > 0 and u([tau](t)) > 0 for t [greater than or equal to] T. Since [u.sup.[DELTA][DELTA]](t) [less than or equal to] 0 for all t, we have [u.sup.[DELTA]](t) > 0 on [[T,[infinity]).sup.T]. As in the proof of Theorem 3.1, for any 0 < k < 1 there exists a [T.sub.k] [greater than or equal to] T such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So [alpha], [beta] are lower and upper solutions, respectively, of

[y.sup.[DELTA][DELTA]] + [lambda] [[tau](t)/[sigma](t)] q(t)[y.sup.[sigma]](t) = 0. (3.1)

As u is increasing, [alpha](t) [less than or equal to] [beta](t) on [[[T.sub.k],[infinity]).sub.T]. Then by Theorem 2.4, there is a solution y(t) of (3.1) satisfying u(T) [less than or equal to] y(t) [less than or equal to] u(t) on [[[T.sub.k],[infinity]).sub.T]. As u is bounded, y is a bounded nonoscillatory solution of (3.1). This is a contradiction to Lemma 3.5 and proves the theorem.

In order to prove our last result, which in an extension of [2, Theorem 4.4], we need the following lemma.

Lemma 3.8. Let y(t) be a positive solution of (1.1) defined on [[T,[infinity]).sub.T] for some T > 0 that satisfies [y.sup.[DELTA]](t) [greater than or equal to] 0 and [y.sup.[DELTA][DELTA]](t) [less than or equal to] 0 on [[T,[infinity]).sub.T]. If (3.5) holds, then there exists a [T.sub.1] [greater than or equal to] T such that

[y(t)/[y.sup.[DELTA]](t)] [greater than or equal to] t and y(t)/t is decreasing (3.7)

on [[[T.sub.1],[infinity]).sub.T].

Proof. Let y(t) be as in the statement of the lemma and assume (3.5) holds. Also let Y(t) := y(t) - t[y.sup.[DELTA]](t). Then [Y.sup.[DELTA]](t) = -[sigma](t)[y.sup.[DELTA][DELTA]](t) [greater than or equal to] 0 for t [member of] [[T,[infinity]).sub.T]. This implies that Y(t) is increasing on [[T,[Infinity]).sub.T].

We claim there is a [T.sub.1] [member of] [T,[infinity]) such that Y(t) [greater than or equal to] 0 on [[[T.sub.1],[infintiy]).sub.T]. If not, then Y(t) < 0 on [[[T.sub.1],[infintiy]).sub.T]. Therefore

[(y(t)/t).sup.[DELTA]] = t[y.sup.[DELTA]](t) - y(t)/t[sigma](t) = - Y(t)/t[sigma](t) > 0, t [member of] [[[T.sub.1], [infintiy]).sub.T]

which implies that y(t)/t is increasing on [[[T.sub.1],[infinity]).sub.T]. Choose [T.sub.2] [member of] [[[T.sub.1],[infinity]).sub.T] such that [tau](t) [greater than or equal to] [tau]([T.sub.2]) for t [greater than or equal to] [T.sub.2]. Then

[y([tau](t))/[tau](t)] [greater than or equal to] y([tau]([T.sub.2]))/[tau]([T.sub.2]) =: D > 0,

which gives y([tau](t)) [greater than or equal to] D[tau](t) for t [greater than or equal to] [T.sub.2]. Now by integrating both sides of (1.1) from [T.sub.2] to t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (3.5). Hence there is a [T.sub.1] [member of] [[T,[infinity]).sub.T] such that Y(t) [greater than or equal to] 0 on [[[T.sub.1],[infinity]).sub.T], and the first part of (3.7) holds. Moreover,

[(y(t)/t).sup.[DELTA]] = [t[y.sup.[DELTA]](t) - y(t)]/t[sigma](t) = -[Y(t)/t[sigma](t)] [less than or equal to] 0, t [member of] [[[T.sub.1],[infinity]).sub.T],

and we have that y(t)/t is decreasing on [[[T.sub.1],[infinity]).sub.T]. This completes the proof of the lemma.

Theorem 3.9. Assume (3.5) holds. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.8)

then every solution of (1.1) is oscillatory on [[[t.sub.0],[infinity]).sub.T].

Proof. To the contrary, suppose y is a nonoscillatory solution of (1.1). Then there exists T [member of] T such that

y(t) > 0, [y.sup.[DELTA]](t) > 0, and [y.sup.[DELTA][DELTA]](t) [less than or equal to] 0 for all t [greater than or equal to] T [greater than or equal to] [t.sub.0].

It follows that for s [greater than or equal to] t [greater than or equal to] T we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From the above inequality and Lemma 3.8, it follows that for sufficiently large t [member of] T

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction to (3.8). This completes the proof.

Example 3.10. Let h > 0 and T = hZ = {hk : k [member of] Z}. In this case (1.1) becomes

y(t + 2h) - 2y(t + h) + y(t) + [h.sup.2]q(t)y([tau](t)) = 0. (3.9)

Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then every bounded solution of (3.9) oscillates on [[[t.sub.0],[infinity]).sub.T]. Additionally, if,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then every solution of (3.9) is oscillatory on [[[t.sub.0],[infinity]).sub.T].

4 Conclusion and Future Directions

In this paper we have obtained sufficient conditions for the oscillatory behavior of

[y.sup.[DELTA][DELTA]](t) + q(t)y([tau](t)) = 0.

This was done by comparing nonoscillatory solutions of the delay dynamic equation with the solutions of a corresponding linear dynamic equation and then using known properties of the linear equation to obtain a desired contradiction.

Possibilities for further exploration include changing the leading term to [(p(t)[y.sup.[DELTA]]).sup.[DELTA]] where p(t) > 0 on the time scale interval [[[t.sub.0],[infinity]).sub.T] and [[integral].sup.[infinity]][[DELTA]t/p(t)] = [infinity], and replacing the delay [tau](t) with the advance [xi]: T [right arrow] T where [sigma](t) [less than or equal to] [xi](t) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

References

[1] Ravi Agarwal, Martin Bohner, Donal O'Regan, and Allan Peterson. Dynamic equations on time scales: a survey. J. Comput. Appl. Math., 141(1-2):1-26, 2002. Dynamic equations on time scales.

[2] Ravi Agarwal, Martin Bohner, and Samir H. Saker. Oscillation of second order delay dynamic equations. Can. Appl. Math. Q., 13(1):1-17, 2005.

[3] Martin Bohner and Gusein Sh. Guseinov. Improper integrals on time scales. Dynam. Systems Appl., 12(1-2):45-65, 2003. Special issue: dynamic equations on time scales.

[4] Martin Bohner and Allan Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser Boston Inc., Boston, MA, 2001.

[5] Martin Bohner and Samir H. Saker. Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mountain J. Math., 34(4):1239-1254, 2004.

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[8] Lynn Erbe, Allan Peterson, and Pavel Rehak. Comparison theorems for linear dynamic equations on time scales. J. Math. Anal. Appl., 275(1):418-438, 2002.

[9] Lynn Erbe, Allan Peterson, and Samir H. Saker. Oscillation criteria for second-order nonlinear dynamic equations on time scales. J. London Math. Soc. (2), 67(3):701-714, 2003.

[10] Stefan Hilger. Analysis on measure chains--a unified approach to continuous and discrete calculus. Results Math., 18(1-2):18-56, 1990.

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[12] Pavel Rehak. How the constants in Hille-Nehari theorems depend on time scales. Adv. Difference Equ., pages Art. ID 64534, 15, 2006.

[13] Samir H. Saker. Oscillation of nonlinear dynamic equations on time scales. Appl. Math. Comput., 148(1):81-91, 2004.

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Raegan Higgins

Texas Tech University

Department of Mathematics & Statistics

Lubbock, TX 79409-1042, U.S.A.

raegan.higgins@ttu.edu

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Author: | Higgins, Raegan |
---|---|

Publication: | International Journal of Difference Equations |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jun 1, 2010 |

Words: | 4854 |

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