#### Abstract

The impulsive delay differential equation is considered with nonlocal boundary condition Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained.

#### 1. Introduction

Mathematical models with impulsive differential equations attract the topic attention of many researchers (see [1–7]); many important results on boundary value problems and stability of these equations have been obtained. One of possible approaches to study impulsive equations is the theory of generalized differential equations allowing researchers to consider systems with continuous solutions as well as systems with discontinuous solutions and discrete systems in the frame of the same theory [8–15]. In this paper we use the approach proposed in the monograph [1] and developed in [16–20].

Various comparison theorems for solutions of the Cauchy and periodic problems for ordinary differential equations with impulses have been obtained in [17, 21–24]. On the basis of the comparison theorems, tests of stability are proposed in [25, 26]. Theory of impulsive differential equations and inclusions was presented in the book [27].

Equations with nonlocal boundary conditions are applied in modelling different processes (see, e.g., the recent work of Skubachevskii [28]). Nonlocal problems for nonimpulsive functional differential equations were considered in ([25], Chapter 15). Existence results for nonlocal boundary value problems with impulsive equations are presented in [29–35]. There are almost no results on sign constancy of Green's function for impulsive boundary value problems. Concerning nonlocal impulsive boundary value problems, we know, there are no results about positivity/negativity of Green's function. In this paper we propose results of this sort.

In this paper we consider the following impulsive equation: with different types of boundary conditions:

We develop the ideas presented in [17] and we have obtained various results on the existence and uniqueness of solutions for impulsive boundary value problems. The main contribution of the presented paper is the formulation and proof of positivity/negativity conditions for Green's functions for the following impulsive functional differential boundary value problems: (1), (2), (4), and (5); (1), (2), (4), and (6); (1), (2), (4), (7).

#### 2. Solution's Representation Formulas

Define the space of piecewise continuous functions , isomorphic to the topological product , where is the space of measurable essential bounded functions , by the following equality: where .

It is clear that is absolutely continuous in , , satisfying the equality . It is clear that has discontinuity of the first kind and is continuous from the right at points , .

By [17], the general solution of (1)–(4) has the following representation:

Theorem 1. *If the boundary value problem (1)–(5) is uniquely solvable in the space for every essential bounded and , then its solution can be represented in the form
**
where Green's function of this problem is
**
and for . *

*Proof. *From the boundary condition (5), we obtain

Substituting (12) into (9), we obtain

The Cauchy function can be represented in the following form:
where in the rectangle , .

Let us take derivative in (9):

Since, according to the definition of the Cauchy function, , we obtain

Let us substitute (16) into (13) as follows:

Since , , it follows that .

Changing the order of integration in the third double integral, we get

Substituting now (16) into (12), we obtain
Substituting (19) into (18), we obtainIt is clear that Green's function of this problem is of the following form:
and , for .

One has
Theorem 1 has been proven.

Theorem 1 demonstrates the importance to know exactly or approximately the Cauchy function of impulsive equations (1)–(4).

Substituting , , we obtain the Corollary.

Corollary 2. *If generalized periodic problem (1)–(4), (7) is uniquely solvable, then its solution can be represented in the following form:
**
where the Green's function is as follows:
*

#### 3. Auxiliary Results

Let us construct the Cauchy function for several simple equations.

Consider now the following auxiliary equation:

Let us denote the Cauchy function of the nonimpulsive equation (25) by . It is known that . For every this function is absolutely continuous function with respect to , and for .

Using the fact that the Cauchy function for every fixed as a function of is a solution of satisfying the condition , we obtain the following theorem.

Theorem 3. *The Cauchy function of (25) and (2) can be represented in the following form:
**
where , Heaviside function, is defined by the following equality:
*

Now let us consider the following auxiliary equation:

Figure 1 describes the Cauchy function of the problem (29), (2), and (4) in the case .

In the case of impulses we obtain the following theorem.

Theorem 4. *The Cauchy function of the problem (29), (2) and its derivatives has the following representations:
*

Denote by Green's function of problem (29), (5), and by Green's function of generalized periodic problem (29), (7).

Let us denote .

Let . The computation of integrals in formula (11) leads us to the following equality:

Denoting , we obtain for

Denoting , we obtain for

Denoting , we obtain for

Let us denote .

The computation of integrals in formula (24) leads us to the following equality:

Denoting , we obtain for

Denoting , we obtain for

Denoting , we obtain for

Corollary 5. *Let us assume that ; then Green's function of problem (29), (5) in the case of three impulsive points (i.e., is described by Figure 2.*

Corollary 6. *Let us assume that ; then Green's function of generalized periodic problem (29), (7) in the case of three impulsive points (i.e. ) is described by Figure 3.*

#### 4. Estimation of Solutions

Let us denote , where ,

Theorem 7. *Let , , and
**
then there exists a unique solution of problem (1)–(5) and this solution can be estimated as follows:
**
for every essentially bounded function and real .*

*Proof. *By Theorem 1, problem (1)–(5) is equivalent to the integral equation:

Let us denote .

Define the operator by the equality .

We can write equation which is equivalent to problem (1)–(5):

If , then there exists the unique solution of (1)–(5) which can be represented in the form: .

It is clear that
where . Thus we have got the following condition for existence of unique solution of problem (1)–(5):

Denote
where can be written as

It is clear that

Estimation of the solution of the problem (1)–(5) can be done as follows:

Let us denote

Then the problem (1)–(5) has unique solution if

It is clear that

Estimating the integrals in formula (11), we get the following inequalities:

Let us write , where is upper triangular and lower triangular parts of and estimate . One has

From estimation of the integrals, we obtain

By (50)

Substituting (56) into (51) and (52) we obtain

Theorem 7 has been proven.

Let us denote ,

Theorem 8. *Let , , and
**
then there exists a unique solution of the generalized periodic problem (1)–(4), (7) and this solution can be estimated as follows:
**
for every essentially bounded function .*

*Proof. *By Corollary 2, problem (1)–(4), (7) is equivalent to the following integral equation:

Let us denote .

Define the operator by the equality .

We can write equation which is equivalent to problem (1)–(4), (7) as follows:

If , then there exists the unique solution of (1)–(4), (7) which can be represented in the form: .

It is clear that
where . Thus we have got the following condition for existence of unique solution of problem (1)–(4)*, *(7):

Estimation of the solution of the problem (1)–(4)*, *(7) can be done as follows: