Boundary value problem for the second order impulsive delay differential equations

We present some existence and uniqueness result for a boundary value problem for functional differential equations of second order with impulses at fixed points.


Introduction
Impulsive differential equations describe processes that are subjected to abrupt changes in their state at fixed or variable times and present a natural framework for mathematical modelling of several real-world problems (see [6,10]). In consequence, the study of impulsive differential equations is of great interest both for the theoretical and practical point of view.
The theory of impulsive differential equations and impulsive functional differential equations has been an important area of investigations in recent years. Among others the existence of solutions of the first and the second order impulsive functional differential equations by using the fixed point argument such as the Banach contraction principle, fixed point index theory and monotone iterative technique were discussed. We mention here the papers [1,2,3,4,5,7,8,9,12] and the references therein.
In the present paper we shall investigate the existence of the solutions of the boundary value problem for the second order delay differential systems with impulses at fixed points. The existence results for the boundary value problem for the second order delay differential equations of the above type without impulsive conditions have been studied in [11]. AMS (2010) Subject Classification: 34K10, 34A45.

Preliminaries
In this paper we consider the following second order boundary value problem with impulses at fixed points exists for all t ∈ [−τ, 0), and x(t + ) = x(t) for all but at most a finite number of points t ∈ [−τ, 0)}.
For any function x : [−τ, T ] → R n and any t ∈ J, we let x t denote the function Here represent the right and left limits of x(t), x (t) at t = t k , respectively, I 0k , I 1k : R n × R n → R n .
Boundary value problem for the second order impulsive delay differential equations

[29]
In order to define the concept of solution for (1) we introduce the following sets of functions We define are Banach spaces with the norms We shall prove an existence result for (1) by using the Banach contraction principle.

Auxiliary result
Let us start by defining what we mean by a solution of problem (1). Denote

Definition 3.1
A function x ∈ C is said to be a solution of (1) if x satisfies (1).
We need the following auxiliary lemma. (1) if and only if x is a solution of the following integral equation Proof. First we prove that the integrals We have For This implies that for any x ∈ P C 1 (J, R n ), f (t, x t ) is continuous on J except on a set of countable points. Then f (t, x t ) is Lebesque integrable on any bounded interval.
Assume that the function x ∈ C is a solution of (1). The function x can be written of the form Differentiating (2), we get Hence Using the boundary condition we obtain Thus Equation (2), together with (1) and (3) implies Boundary value problem for the second order impulsive delay differential equations

Main result
We introduce the following assumptions on the functions appearing in the problem (1): (H1) There exists a function m ∈ C(J, R + ) such that for any t ∈ J and x, y ∈ P C 1 (J, R n ).
If the assumptions (H1), (H2) hold and Proof. We transform the problem (1) into a fixed point problem. For x ∈ P C 1 (J, R n ), let where t ∈ J, x s (r) = x(s + r) = φ(s + r) for s + r ≤ 0. Differentiation of (6) gives For x, y ∈ P C 1 (J, R n ) we have for t ∈ J.
Boundary value problem for the second order impulsive delay differential equations

Lidia Skóra
This implies that In consequence Then Hence we have the following estimate Ax − Ay P C 1 ≤ α x − y P C 1 with α = β β−1 (τ M + C 1 ) + C 0 . Thus A is a contractive operator and by Banach fixed point theorem, A has a unique fixed point x ∈ P C 1 (J, R n ). The proof is complete.
As a consequence of the previous theorem, we have the following result. When µ ik = 0, i = 0, 1, k = 1, . . . , p we obtain existence result for the boundary problem for second order delay differential equation without impulses under different assumptions than in [11].