Lyapunov-type inequalities for nonlinear impulsive systems with applications

We obtain new Lyapunov-type inequalities for systems of nonlinear impulsive differential equations, special cases of which include the impulsive Emden–Fowler equations and half-linear equations. By applying these inequalities, sufficient conditions are derived for the disconjugacy of solutions and the boundedness of weakly oscillatory solutions.


Introduction
Impulsive differential equations have played an important role in recent years because they provide better mathematical models in both physical and social sciences.Although there is an extensive literature on the Lyapunov-type inequalities for linear and nonlinear ordinary differential equations [1,2,7,[9][10][11][12] and systems [3,8,14,16] as well as linear impulsive differential equations [5] and impulsive systems [4,6], there is not much done for nonlinear systems with or without impulse [15].The present work stems from the corresponding ones in [11,15].
Note that many equations can be written as a special case of (1.1).For instance, impulsive Emden-Fowler type differential equation If we set α = β, then we obtain the impulsive half-linear equation (1.3) Therefore, our results will be applied to important equations as special cases.
Let us recall the definition of a (generalized) zero.
Definition 1.1 ([4,5]).A real number c is called a generalized zero of a function f if f (c − ) = 0 or f (c + ) = 0.If f is continuous at c, then c becomes a real zero.If no such zero exists then we will write f (t) = 0.

Main results
Let k be a piece-wise constant function defined by and {k j } be sequence given by We will also use the usual notations Our first result is the following.
Theorem 2.1.Let (x(t), u(t)) be a real solution of the impulsive system (1.1) and α be the conjugate number of γ, that is, If x(t) has two consecutive generalized zeros at t 1 and t 2 , then we have the following Lyapunov-type inequality: where It is easy to see that (2.4) Integrating the first equation in (2.4) from t 1 to t 2 and using β + 2 (t) = max {β 2 (t), 0} and (η i /ξ i ) + = max {η i /ξ i , 0} yields, From the first equation in (2.3), we have and Integrating (2.6) from t 1 to τ, we get Similarly, by integrating (2.7) from τ to t 2 , we have Then we observe that Applying Hölder's inequality to the right-hand side of (2.11) with indices α and γ and then using inequality (2.5) lead to 2z(τ) exp 1 2 Finally, we use (2.2) in the last inequality to see that (2.1) holds.
If we take β = α, then the M dependence drops.
Theorem 2.2.Let (x(t), u(t)) be a real solution of the impulsive system (1.1) and β be the conjugate number of γ, that is, If x(t) has two consecutive generalized zeros at t 1 and t 2 , then we have the following Lyapunov-type inequality: Corollaries below are immediate.
If the impulsive Emden-Fowler equation (1.2) has a real solution x(t) having two consecutive generalized zeros at t 1 and t 2 , then we have the following Lyapunov-type inequality: x(t) k(t) . (2.12) Corollary 2.4.Let β be the conjugate number of γ.
If the impulsive half-linear equation (1.3) has a real solution x(t) having two consecutive generalized zeros at t 1 and t 2 , then we have the following Lyapunov-type inequality: Theorem 2.5.Let α be the conjugate number of γ and M be given by (2.12).
If the impulsive Emden-Fowler equation (1.2) has a real solution x(t) having two consecutive generalized zeros at t 1 and t 2 , then there exists τ ∈ (t 1 , t 2 ) such that the following inequalities hold: Proof.(i) The proof is obtained by applying the proof of the Theorem 2.1 step by step for the intervals (t 1 , τ) and (τ, t 2 ) separately and using z (τ) = 0.
(ii) Let τ = τ n and τ n < s < τ n+1 .Set If we repeat the procedure in the proof of Theorem 2.1, for the interval (t 1 , s), we get On the other hand, one can show that (2.14) Substituting (2.14) into (2.13),we have , we obtain the desired inequality Now, let τ = τ n and s < τ n < τ n+1 .By the same procedure worked on (s, t 2 ), we get , which in a similar manner above leads to , and so as s → τ − , we obtain Theorem 2.6.Let β be the conjugate number of γ.
If the impulsive half-linear equation (1.3) has a real solution x(t) having two consecutive zeros at t 1 and t 2 , then there exists τ ∈ (t 1 , t 2 ) such that the following inequalities hold: Remark 2.7.It may not be plausible at first to have a constant M in the inequalities, however they are still very useful in several applications, see the next section.Moreover, if β and γ are conjugate, then M disappears.

Disconjugacy
In this section, by using the inequalities obtained in Section 2, we establish some disconjugacy results.
We first recall the disconjugacy definition.