Stability Analysis for Nonlinear Second Order Differential Equations with Impulses ∗

In this paper we investigate the impulsive equation � (r(t)x 0 ) 0 + a(t)x + f (t, x, x 0 ) = p(t), tt 0, t 6 tk, x(tk) = ckx(tk − 0), x 0 (tk) = dkx 0 (tk − 0), k = 1, 2, 3, . . . , and establish a couple of criteria to guarantee the equations of this type to possess the stability, including boundedness and asymptotic properties. Some examples are given to illustrate our results and the last one shows that, to some extent, our criteria have more comprehen- sive suitability than those given by G. Morosanu and C. Vladimirescu.

Then the null solution of ( 1) is stable.The questions posed here to answer are whether we can weaken the conditions in Theorem A, such as weakening the restrictions that h(t) > 0 and α > 1, and the conclusion is also true.To these ends, in this paper we consider a more general form than (1) and study the impulsive second order nonlinear differential equation (r(t)x ′ ) ′ + a(t)x + f (t, x, x ′ ) = p(t), t ≥ t 0 , t = t k , where Let N be the set of positive integers and R be the real axis.Before proceeding our discussions, we give the blanket assumptions for (2) as follows: where X = (x 1 , x 2 ) T and for all t ∈ [t 0 , ζ) and t = t k , and X(t k + 0) as well as X(t k − 0) exist and satisfy Let X(t) = X(t; t 0 , X 0 ) be a solution with X(t 0 ) = X 0 .It is clear that (3) has null solution when p(t) ≡ 0. The null solution of (3) is said to be stable if for any ε > 0, there exists a δ = δ(ε, t 0 ) such that ||X 0 || < δ implies that X(t) exists on [t 0 , ∞) and ||X(t)|| < ε for all t ≥ t 0 .

Preliminaries
For the convenience, we will view C(t), D(t), E(t) and F (t, u, v) as and whenever these notations are defined.Let U(t) = (u ij (t)) be any matrix.In this paper the norm of U(t) is defined by the maximum of the row sums of (|u ij (t)|).
First of all, we consider the relation between the solutions of (3) and the solutions of the following equation where and F is defined as in (5).
Let t 0 < ζ ≤ ∞.By a solution of (6) we mean a continuous function Now let Y (t) be a solution of (6).Then, by straightforward verifications, we learn that X(t) = E(t)Y (t) satisfies and it renders (3) into an identity when t = t k .Consequently, Conversely, suppose that X is a solution of (3).Then, for Y (t) = E −1 (t)X(t), we have In addition, it is easy to verify that Y (t) = E −1 (t)X(t) satisfies (6) when t = t k .So far the following result is obvious.
We next consider the solutions of (6).It is clear that the solutions of (6) exist by the theory of ordinary differential equations [9].Specially, if where then, the fundamental matrix of the linear system corresponding to (6): ds sin ds cos Let Y (t) be a solution of (6) with Y (t 0 ) = Y 0 , then it satisfies that As a special case, we consider For example, we consider r(t) = 2 + t + sin t, t ≥ 0 and t k = (2k − 1)π.
Then it holds that r ′ (t k ) = 0 for all k ∈ N. At this stage we set Then, similarly to [1,8], from (2) it follows that when t ≥ t 0 and EJQTDE, 2012 No. 29, p. 5 where X and I k are defined as in (3), and Analogously to (6), we consider the following equation where Y = (y 1 , y 2 ) T , and .
Since the relation between the solutions of ( 10) and the solutions of ( 11) is similar to Lemma 1, we wish to refrain from the repeating statements.Let us set as well as Then it follows that Now we take into account the following system corresponding to (11): It is easy to verify that the fundamental matrix of ( 12) is given by ds sin Subsequently, the solution Y of (11) with Y (t 0 ) = Y 0 satisfies that EJQTDE, 2012 No. 29, p. 6

Main results
In the sequel we give the stability criteria for (2).Recall that the definitions of C(t), D(t) and E(t) have been defined in (4).For simplicity, we introduce another notations as follows.Let λ 1 (c) and λ 2 (c) be denoted, respectively, by The notations λ 1 (d) and λ 2 (d) can be defined similarly.
Theorem 1 Suppose that the following conditions hold: Then the null solution of ( 3) is stable.
We observe that when c k ≡ d k on N, C(t) −1 D(t) = C(t)D(t) −1 = 1 for all t ≥ t 0 .Hence the following result is clear.
Corollary 1 Suppose that the following conditions hold: Then the null solution of ( 3) is stable.
We notice that, by similar arguments, we may show that the solution X(t; t 0 , X 0 ) of (3) exists on [t, 0 , ∞) for any X 0 ∈ R 2 under the provisions in Theorem 1.Now we consider the case that p(t) is of constant sign and is EJQTDE, 2012 No. 29, p. 9 not identically zero.In this case we impose the assumption (in (H2)) that f (t, u, v) is monotone decreasing in u and learn that the vector function Recall the Comparison Theorem [10].Briefly speaking, if F (t, x) : R 2+1 → R 2 and F is quasi-monotone increasing in x, and if ψ is the maximal solution of x ′ = F (t, x) with ψ(t 0 ) = ψ 0 for t ≥ t 0 , then ϕ ′ ≤ F (t, ϕ) with ϕ(t 0 ) ≤ ψ 0 for t ≥ t 0 implies that ϕ(t) ≤ ψ(t) for t ≥ t 0 .Hence, with the aid of comparison theorem we may also show that the solution X(t; t 0 , X 0 ) of (3) exists on [t 0 , ∞) for any X 0 ∈ R 2 .For simplicity we ignore the details of proof.
Next we consider the boundedness for (3).
Theorem 2 Suppose that the following conditions hold: (i) Then every solution of (3) is bounded.
Proof.We first assume that Y (t) is the solution of (6) with Y (t 0 ) = Y 0 .Then X(t) = E(t)Y (t) is a solution of (3).Let M be defined as in (14) and Furthermore, let w(t) be defined as in (16).Note that the function B in (8) satisfies that for the time being.Case 1. Suppose that α = 1.Then, similarly to (15) we have which shows that every solution of ( 3) is bounded when α = 1.Case 2. Suppose that α > 1.For any given ε > 0, we take T > t 0 so that Analogously to (8) we have which leads to By the same manner as (19), we have from ( 22) that where where Since ε is arbitrary, we can ensure that ( 24) is valid for Y (T ).Further, from ( 22) and ( 24) we learn that R(t) is bounded on [T, ∞), which implies that the solution X(t) of ( 3) is bounded on [t 0 , ∞) when α > 1.The proof is complete.
The following result is concerned with the asymptotic behavior of (10) (or (2)) under the assumptions (9).It is based on the fact that the solution X(t; t 0 , x 0 ) of ( 10) exists on [t 0 , ∞) for any X 0 ∈ R 2 .The reasons are similar to the proof of Theorem 1 and the statements before entering Theorem 2, and therefore we skip them.