PRINCIPAL AND NONPRINCIPAL SOLUTIONS OF IMPULSIVE DYNAMIC EQUATIONS WITH APPLICATIONS∗

In this paper, we introduce the concept of principal and nonprincipal solutions for second order impulsive dynamic equations on time scales. Polya and Trench Factorizations play an important role in this article. Firstly we establish these factorizations. Using these factorizations, we establish some new oscillation criteria for second impulsive dynamic equations on time scales.


Introduction
The theory of time scales was introduced by Hilger [7] in his Ph.D thesis in 1988 in order to unify continuous and discrete analysis, where a time scale T is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations.And the new theory of the so-called "dynamic equations" extends these classical cases to cases "in between", as e.g., to the so-called q-difference equations.Of course many other interesting time scales exist, and they give rise to plenty of applications.The theory of dynamic equations on time scales has been developing rapidly and has received much attention.We refer the reader to the book by Bohner & Peterson [2] and the references cited therein.
The concept of the principal solution was introduced in 1936 by Leighton & Morse [8].Since then the principal and nonprincipal solutions have been used successfully in connection with oscillation, see Bohner & Peterson [2], Özbekler & Zafer [9] , Zafer [10] and the references cited therein.
In recent years, impulsive dynamic equations on time scales have been investigated by Belarbi et al. [1], Benchohra et al. [3], [4], [5], Huang & Feng [6] and so forth.Principal and nonprincipal solutions of impulsive differential equations with application have been investigated by few authors and they gained some results, see Özbekler & Zafer [9] and Zafer [10].In the present work, using the similar method we continue our investigation to extend the work in Özbekler & Zafer [9] and Zafer [10] to second order impulsive dynamic equations on scale times.

Denote by PLC
Throughout the remainder of the paper, we assume that for each i = 1, 2, . . ., the points of impulses θ i are right dense(rd for short).We let T be a time scale with sup T = ∞, fix t 0 ∈ T and define T t0 = T ∩ [t 0 , ∞).In this paper we are concerned with oscillation of solutions of second-order impulsive dynamic equations of the form where r(t) > 0 and r, q, f are rd-continuous.Here we introduce the space This paper is organized as follows.In Section 2, we give some preliminaries and lemmas.In section 3, the main result concerning the existence of principal and nonprincipal solution of (1.1) is given, the proof is based on Polya and Trench Factorizations.And the section also contains two important applications, namely Wong and Leighton-Wintner theorems.An example is given to illustrate the relevance of the results.

Preliminaries and lemmas
Consider the linear operators ) has a solution v(t) with no generalized zeros in T t0 , then for any η ∈ D we have ) Proof.Assume that v(t) is a solution of (2.1) with no generalized zeros in T t0 .Then and for all η ∈ D. We complete the proof.

Lemma 2.2. (Trench Factorization). If (2.1) has a positive solution in T t0 , then for any η ∈ D we have
where Taking the derivative of both sides we get It follows that t0 and η ∈ D, we obtain Since t = θ i , i ∈ N are right dense, it follows that The proof is complete.

Main results
Theorem 3.1.If (1.1) has a positive solution in T t0 , then there exist linearly independent solutions u(t) and v(t) of (1.1) such that for t sufficiently large.The solutions u(t) and v(t) are called principal and nonprincipal solutions of (1.1), respectively.

It follows that
Not that v 0 (t 0 ) = 0, u(t) and v 0 (t) are two linearly independent solutions of (1.1) and Taking µ(t) = u(t) and η(t) = v 0 (t) in (2.2) and (2.3), we get Integrating (3.1) from t 0 to t, we obtain where Integrating both sides of (3.2) from t 0 to ∞ we get Let v(t) be any solution of (1.1) such that v(t) and u(t) are linearly independent.Then where c 2 and c 1 are constants with c 2 ̸ = 0.It follows that where remains the same.So without loss of generality we assume v(t) > 0 for t ∈ T t1 .It is easy to see that for t ∈ T t1 , Since the right side is continuous, by taking limit as t → θ ± we can get and v(t) u(t) is continuous for t ∈ T t1 , we can get the desired result that c 3 < 0 if t is large enough.This proof is complete.

Theorem 3.2. (Leighton-Wintner Theorem). If r(t) > 0 and
Proof.Suppose that Eq. (1.1) is nonoscillatory.Then by Theorem 3.1, there is a solution v(t) of (1.1) and a number t 2 ∈ T t0 such that v(t) > 0 in T t2 and It is not difficult to see that Then and Integrating both sides of (3.4) from T to t, we obtain Using (3.5) in (3.6), we get Let t 3 ∈ T t2 where t 3 is sufficiently large such that Therefore v(t) is decreasing on each interval which contradicts.The proof is complete.
the function f (t) and the sequence {f i } are as in (1.2).