Canonical, Noncanonical, and Semicanonical Third Order Dynamic Equations on Time Scales

The notion of third order semicanonical dynamic equations on time scales is introduced so that any third order equation is either in canonical, noncanonical, or semicanonical form. Then a technique for transforming each of the two types of semicanonical equations to an equation in canonical form is given. The end result is that oscillation and other asymptotic results for canonical equations can then be applied to obtain analogous results for semicanonical equations


Introduction
The study of qualitative properties of solutions of second order dierential equations of the form (r(t)x ′ ) ′ + q(t)x γ (t) = 0, t ≥ t 0 , with r, q : [t 0 , ∞) → R + and γ the ratio of odd positive integers, is often divided into two parts depending on whether equation ( 1) is in canonical form, that is, or is in nonconanical form This is also the case for second order dierence equations and dynamic equations on time scales.In the case of third order equations, (b(t)(a(t)x ′ ) ′ ) ′ + q(t)x γ (t) = 0, with a, b : [t 0 , ∞) → R + and q and γ as above, the situation is more complicated due to the presence of two coecients in the lead term.
Here we will consider the more general setting of the third order dynamic equation where T is a time scale with The basic notation and terminology for time scales can be found in the well-known monograph by Bohner and Peterson [2] and will be used without further mention.We will say that equation and it is in noncanonical form if then we will say that equation (E) is in semicanonical form.
In this paper we wish to show that under certain conditions, a semicanonical equation, i.e., equation (E) with either (S 1 ) or (S 2 ) holding, can be written as an equivalent equation in canonical form.One advantage of dealing with equations in canonical form is that we can apply the famous Kiguradze lemma to classify the behavior of nonoscillatory solutions, and the number of possible types is less for canonical equations than it is for noncanonical ones.
Interest in the relationship between canonical, noncanonical and semicanonical equations can be traced back to the now classic 1974 paper of Trench [10].This has attracted the attention of other authors as can be seen, for example, from the recent papers [1,3,4,5,6,7,8].The motivation for examining this classication scheme for dynamic equations on time scales stems partially from some recent results for dierence equations in [9].

Semicanonical Equations of Type (S 1 ).
In this case, we let .
Here is our theorem in this case.
can be written as the canonical operator Proof.Recalling that σ(t) is the forward jump operator and dierentiating, we obtain so this together with condition (C 1 ) proves that the operator (3) is in canonical form.

Now we let
Our result in this case is the following.
can be written as the canonical operator Proof.Once again by a straightforward dierentiation, this together with (C 2 ) shows that the operator B σ (t)Qx(t) is in canonical form.

Discussion
In order to develop some insight and intuition about the results obtained in Sections 2 and 3 above, let us begin by considering a dierential equation (T = R) in which the coecients are powers of t, namely, That is, for equation (E) to be in canonical form, we must have α ≤ 1 and β ≤ 1.On the other hand, (E) is in noncanonical form provided α > 1 and β > 1.Finally, for the semicanonical cases, (S 1 ) holds provided α > 1 and β ≤ 1, and for (S 2 ) to hold, we need α ≤ 1 and β > 1.
By Theorem 2.1, if α+β ≤ 2, then equation (E) is semicanonical, but the equation involving the operator P given in (3), namely, This means that the equation becomes ) is in canonical form.As a consequence of Theorem 2.1, we have the following result.
is also semicanonical ((S 2 ) holds) and the transformed equation is which is a canonical equation.Here, in view of Theorem 4.2, x(n) is a solution of (D 2 ) if and only if it is a solution of (9).
In conclusion, to demonstrate how the results in this paper can be utilized, consider the simple case of the semicanonical dierential equation ((S 2 ) holds) Here, b(t) = t 2 and a(t) = 1, so B(t) = 1 t and Qx(t) = (tx ′ (t)) ′′ which is in canonical form.Then any conditions that ensure that a solution x(t) of the canonical equation oscillates or possesses some other asymptotic property, implies that x(t) is a solution of (10) with that same behavior.For example, by [1, Theorem 3.1], if then any positive nonoscillatory solution of (11) belongs to the class {x > 0 : x ′ < 0, (tx ′ ) ′ > 0} and the same is true of any positive nonoscillatory solution of (10).Applications to dierence equations can be found in [9].As a nal remark, let us point out that in the case of the time scale being the real numbers so that we are talking about dierential equations, our result in Theorem 2.1 in which condition (S 1 ) holds, agrees exactly with what can be obtained from Trench [10, Lemma 1].As pointed out in [6], the coecients obtained from [10, Lemma 2] are too complicated to make easy comparisons to results such as our Theorem 3.1 above or others.

Theorem 4 . 1 .
Under conditions (S 1 ) and (C 1 ), the semicanonical equation (E) has a solution x(t) if and only if the canonical equation