PICONE TYPE FORMULA FOR NON-SELFADJOINT IMPULSIVE DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS SOLUTIONS

As the impulsive differential equations are useful in modelling many real processes observed in physics, chemistry, biology, engineering, etc., see [1, 11, 13, 20, 21, 22, 25, 26, 27], there has been an increasing interest in studying such equations from the point of view of stability, asymptotic behavior, existence of periodic solutions, and oscillation of solutions. The classical theory can be found in the monographs [9, 18]. Recently, the oscillation theory of impulsive differential equations has also received considerable attention, see [2, 14] for the Sturmian theory of impulsive differential equations, and [15] for a Picone type formula and its applications. Due to difficulties caused by the impulsive perturbations the solutions are usually assumed to be continuous in most works in the literature. In this paper, we consider second order non-selfadjoint linear impulsive differential equations with discontinuous solutions. Our aim is to derive a Picone type identity for such impulsive differential equations, and hence extend and generalize several results in the literature. We consider second order linear impulsive differential equations of the form


Introduction
As the impulsive differential equations are useful in modelling many real processes observed in physics, chemistry, biology, engineering, etc., see [1,11,13,20,21,22,25,26,27], there has been an increasing interest in studying such equations from the point of view of stability, asymptotic behavior, existence of periodic solutions, and oscillation of solutions.The classical theory can be found in the monographs [9,18].Recently, the oscillation theory of impulsive differential equations has also received considerable attention, see [2,14] for the Sturmian theory of impulsive differential equations, and [15] for a Picone type formula and its applications.Due to difficulties caused by the impulsive perturbations the solutions are usually assumed to be continuous in most works in the literature.In this paper, we consider second order non-selfadjoint linear impulsive differential equations with discontinuous solutions.Our aim is to derive a Picone type identity for such impulsive differential equations, and hence extend and generalize several results in the literature.
We consider second order linear impulsive differential equations of the form where ∆z(t) = z(t + )−z(t − ) and z(t ± ) = lim τ →t ± z(τ ).For our purpose, we fix t 0 ∈ R and let I 0 be an interval contained in [t 0 , ∞).We assume without further mention that EJQTDE, 2010 No. 35, p. 1 (i) {p i }, {p i }, {q i } and {q i } are real sequences and {θ i } is a strictly increasing unbounded sequence of real numbers; (ii) k, r, p, m, s, q ∈ PLC(I 0 ) := h : Note that if z ∈ PLC(I 0 ) and ∆z(θ i ) = 0 for all i ∈ N, then z becomes continuous and conversely.If τ ∈ R is a jump point of the function z(t) i.e. ∆z(τ ) = 0, then there exists a j ∈ N such that θ j = τ .Throughout this work, we denote by j τ , the index j satisfying θ j = τ .
By a solution of the impulsive system (1.1) on an interval I 0 ⊂ [t 0 , ∞), we mean a nontrivial function x which is defined on I 0 such that x, x ′ , (kx ′ ) ′ ∈ PLC(I 0 ) and that x satisfies (1.1) for all t ∈ I 0 .

The Main Results
Let I be a nondegenerate subinterval of I 0 .In what follows we shall make use of the following condition: k(t) = m(t) whenever r(t) = s(t) for all t ∈ I. (C) We will see that condition (C) is quite crucial in obtaining a Picone type formula as in the case of nonimpulsive differential equations.If (C) fails to hold then a device of Picard is helpful.The Picone type formula is obtained by making use of the following Picone type identity, consisting of a pair of identities.
Proof.Let t ∈ I.If t = θ i , then we have Rearranging we get EJQTDE, 2010 No. 35, p. 3 If t = θ i , then the computation becomes more involved.We see that Proof.Using (2.1) and (2.2), and employing Lemma 2.1 with where we easily see that (2.3) holds.
The following corollary is an extension of the classical comparison theorem of Leighton [ (2.6) As ǫ → 0 + the left-hand side of (2.6) tends to Using (2.5) and (2.7) in (2.6) we get and similarly, if x(t) is continuous at t = a (i.e.x(a ± ) = 0) but not at t = b, then As a consequence of Theorem 2.2 and Corollary 2.1, we have the following oscillation criterion.
Corollary 2.3.Suppose for any given T ≥ t 0 there exists an interval (a, b) ⊂ [T, ∞) for which either the conditions of Theorem 2.2 or Corollary 2.1 are satisfied, then every solution y of (1.2) is oscillatory.

Device of Picard
If the condition (C) fails to hold, then we introduce the so called device of Picard [16] (see also [7, p. 12]), and thereby obtain different versions of Corollary 2.1.
Clearly, for any h ∈ PLC(I), It is not difficult to see that EJQTDE, 2010 No. 35, p. 7 Assuming r ′ , s ′ ∈ PLC(I), and taking h = (r − s)/2 we get Thus we obtain the following results in a similar manner as in the previous section.
Theorem 3.1.Let r ′ , s ′ ∈ PLC(I) and x be a solution of (1.1) having two consecutive generalized zeros a and b in I. Suppose that are satisfied for all t ∈ [a, b], and that and that the inequalities in (3.

Further results
The lemma below, cf.[2, Lemma 1.] and [14,Lemma 3.1.],is used for comparison purposes.The proof is a straightforward verification.Lemma 4.1.Let ψ be a positive function for t ≥ α with ψ ′ , ψ ′′ ∈ PLC[α, ∞), where α is a fixed real number.Then the function is a solution of equation where a j j ∈ PLC[α, ∞), j = 0, 1, 2, {e i } and {ẽ i } are real sequences, with In view of Lemma 4.1 and Corollary 2.2, we can state the next theorem.
If one of the inequalities in (4.9) and (4.10) is strict, then every solution x of (1.1) is oscillatory.
It is clear that an impulsive differential equation with a known solution can be used to obtain more concrete oscillation criteria.For instance, consider the impulsive differential equation It is easy to verify that x(t) = x i (t), where x i (t) = (e + i)(t − i) + i e t−i , t ∈ (i − 1, i], (i ∈ N), is an oscillatory solution with generalized zeros τ i = i and ξ i = i(e + i − 1)(e + i) −1 ∈ (i − 1, i).Indeed, x(τ i )x(τ + i ) < 0 and x(ξ i ) = 0, i ∈ N. Applying Corollary 2.2, we easily see that equation (1.1) with θ i = i is oscillatory if there exists an n 0 ∈ N such that, for each fixed i ≥ n 0 and for all t ∈ (i − We finally note that it is sometimes possible to exterminate the impulse effects from a differential equation.In our case, if the condition The oscillatory nature of x and z are the same.However, the restriction imposed is quite severe. 10, Corollary 1].Theorem 2.2 (Leighton type comparison).Let x be a solution of (1.1) having two generalized zeros a, b ∈ I. Suppose that (C) holds, and that b a

)Corollary 3 . 1 .
for all i for which θ i ∈ [a, b].If either (3.1) or (3.2) is strict in a subinterval of [a, b], or one of the inequalities in (3.3) is strict for some i, then every solution y of (1.2) must have at least one generalized zero in [a, b].Suppose that the conditions (3.1)-(3.2) are satisfied for all t ∈ [t * , ∞) for some integer t * ≥ t 0 , and that the conditions in (3.3) are satisfied for all i for which θ i ≥ t * .If r ′ , s ′ ∈ PLC[t * , ∞) and one of the inequalities (3.1)-(3.3) is strict, then (1.2) is oscillatory whenever a solution x of (1.1) is oscillatory.Theorem 3.2 (Leighton type comparison).Let r ′ , s ′ ∈ PLC[a, b] and x be a solution of (1.1) having two generalized zeros a, b ∈ I such that b a q

3 )Corollary 3 . 2 .
hold for all i for which θ i ∈ [a, b].Then every solution y of (1.2) must have at least one generalized zero on [a, b].From Theorem 3.1 and Theorem 3.2, we have the following oscillation criterion.Suppose for any given t 1 ≥ t 0 there exists an interval (a, b) ⊂ [t 1 , ∞) for which either the conditions of Theorem 3.1 or Theorem 3.2 are satisfied, then (1.2) is oscillatory.EJQTDE, 2010 No. 35, p. 8
Definition 1.1.A function z ∈ PLC(I 0 ) is said to have a generalized zero at t = t * if z(t + ∈ I 0 .A solution is called oscillatory if it has arbitrarily large generalized zeros, and a differential equation is oscillatory if every solution of the equation is oscillatory. * )z(t * ) ≤ 0 for t * ∈ [a, b], and that (2.5) holds for all i for which θ i ∈ [a, b].If either (2.10) or (2.11) is strict in a subinterval of [a, b], or one of the inequalities in (2.5) is strict for some i ∈ N, then every solution y of (1.2) must have at least one generalized zero on [a, b].Suppose that the conditions (2.10)-(2.11)are satisfied for all t ∈ [t * , ∞) for some integer t * ≥ t 0 , and that (2.5) is satisfied for all i for which θ i ≥ t * .If one of the inequalities in (2.5) or in (2.10)-(2.11) is strict, then every solution y of (1.2) is oscillatory whenever a solution x of (1.1) is oscillatory.