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:
Let us denote
Then the problem (1)–(4), (7) has unique solution if
It is clear that
Estimating the integrals in formula (24), we get the following inequalities:
Let us write , where is upper triangular and , lower triangular parts, and estimate :
We obtain
It is clear that
We obtain the following solution estimation:
Theorem 8 has been proven.
5. Sign Constancy of Green's Functions
The proofs of the following two assertions follow from the construction of Green's functions.
Theorem 9. Let then Green's function of problem (29), (5) is positive for .
Theorem 10. Let then Green's function of problem (29), (5) is negative for .
Estimation of Green's function (11) leads us to the assertion.
Theorem 11. Let conditions (40) and (75) be fulfilled, for ; then problem (1)–(5) is unique solvable and its Green's function is positive for .
Proof. Without losing generality we assume that .
Solution of problem (1)–(5) can be represented in the following form:
where
Using Theorem 9, we obtain that the operator is positive.
The conditions of Theorem 7 imply that .
Now it is clear that
It follows from the positivity of the operator that for every nonnegative we get according to (78) and .
It is clear that
From nonnegativity of for every nonnegative , we obtain that .
Theorem 11 has been proven.
Theorem 12. Let conditions (40) and (76) be fulfilled, for then problem (1)–(5) is unique solvable and its Green's function is negative for .
We prove this assertion analogously to the proof of Theorem 11.
The proofs of the following two assertions follow from the construction of Green's functions.
Theorem 13. Let then Green's function of problem (29), (7) is negative for .
Theorem 14. Let then Green's function of problem (29), (7) is positive for .
The proof of the following two theorems can be obtained analogously from the proof of Theorem 11.
Theorem 15. Let conditions (60) and (82) be fulfilled, for ; then problem (1)–(4), (7) is unique solvable and its Green's function is positive for .
Theorem 16. Let conditions (60) and (81) be fulfilled, for ; then problem (1)–(4), (7) is unique solvable and its Green's function is negative for .
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.