Trench's Perturbation Theorem for Dynamic Equations

We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.


Introduction
We consider the second-order linear dynamic equation together with its linear perturbation where we assume that T is a time scale, that is, a nonempty closed subset of the real numbers T that is unbounded above, p : T→R, r : T→R + , f : T→C, are rd-continuous, and (1.1) is nonoscillatory, that is, rxx σ > 0 eventually for all solutions of (1.1).For the theory of time scales, we refer the reader to [1,2] and we mention here only that the forward shift and the derivative of a function z : T→R are given by z σ (t) = z(t) and z Δ (t) = z (t) if T = R and z σ (t) = z(t + 1) and z Δ (t) = z(t + 1) − z(t) if T = Z, and that the product rule and the quotient rule for differentiable z 1 ,z 2 : T→R read [1, Theorem 1.20] where z σ = z • σ for a function z : T→R.By [1,Theorem 4.61], since (1.1) is nonoscillatory, there exists a solution x 1 of (1.1) such that lim t→∞ (u(t)/x 1 (t)) = ∞ for all of x 1 linearly independent solutions u of (1.1).
Let u be any solution of (1.1) that is linearly independent of x 1 , that is, and define x 2 = u/c so that which implies that Note that x 1 and x 2 are called recessive and dominant solutions of the nonoscillatory dynamic equation (1.1).
The main result of this paper gives conditions on f , x 1 , and x 2 that guarantee that the perturbed equation (1.2) is also nonoscillatory and has solutions y 1 and y 2 that behave asymptotically like x 1 and x 2 .Our results unify corresponding results by William F. Trench for differential equations [3,4] and for difference equations [5] and extend them to other dynamic equations, for example, to q-difference equations [6].Note that the unification process forces us to give many of the calculations from [5] in a "shifted form" for the discrete case.
In the next section, we will derive some preparatory results while the main theorem is stated and proved in Section 3. The paper concludes with some corollaries given in Section 4.

Some auxiliary results
Suppose now and in the remainder of this paper that (2.4) Proof.We use (1.6), the product rule, the definition of the integral, and [1, Theorems 1.75 and 1.16(iv)] to find and therefore where we have used (1.6), (2.2), and its consequence φ σ ≤ φ.
In the sequel, we use the Landau "O" symbol defined in the standard way for asymptotic behavior as t→∞ and consider the collection of differentiable functions which can easily be seen to be a Banach space when equipped with the norm (2.9) Proof.Note that the fourth equal sign of the first calculation in the proof of Lemma 2.1 implies and therefore due to (2.2) and (1.6).By the product rule and (1.6), and therefore due to (2.4) from Lemma 2.1.
For z ∈ Ꮾ, let us define the operator

The perturbation result
In this section, we use the auxiliary results from Section 2 to prove the following main theorem of this paper.

Applications
In this last section, we state and prove some simple consequences of Theorem Then For T ≥ t, we have by the product rule that Note now that ( Gϕ)(T)→0 as T→∞ since due to (4.3) and (4.1).Note also that the quantity on the right of the last equal sign in the above calculation must have a finite limit as T→∞ since the quantity on the left of the first equal sign converges as T→∞ due to (4.1).Therefore, Hence, using (2.2), -: Ꮾ→Ꮾ is a contraction.By Banach's fixed point theorem,has a unique fixed point ζ ∈ Ꮾ.Hence, ζ = -ζ = u + ᏸζ.Define Corollary 4.3.Assume that α is a positive function tending monotonically to ∞ and define Hence, we can choose φ as in the statement of the corollary in place of (2.2) and all results as presented in this paper still hold with this φ.Now and therefore, (2.15) is satisfied.