About Positivity of Green's Functions for Nonlocal Boundary Value Problems with Impulsive Delay Equations

The impulsive delay differential equation is considered (Lx)(t) = x′(t) + ∑i=1 m p i(t)x(t − τ i(t)) = f(t),  t ∈ [a, b], x(t j) = β j x(t j − 0),  j = 1,…, k,  a = t 0 < t 1 < t 2 < ⋯<t k < t k+1 = b,  x(ζ) = 0,  ζ ∉ [a, b], with nonlocal boundary condition lx = ∫a b φ(s)x′(s)ds + θx(a) = c, φ ∈ L ∞[a, b]; θ,  c ∈ R. Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained.


Introduction
Mathematical models with impulsive differential equations attract the topic attention of many researchers (see [1][2][3][4][5][6][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][9][10][11][12][13][14][15]. In this paper we use the approach proposed in the monograph [1] and developed in [16][17][18][19][20].
Various comparison theorems for solutions of the Cauchy and periodic problems for ordinary differential equations with impulses have been obtained in [17,[21][22][23][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].
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: The Scientific World Journal 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).

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:

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 ( , ) = 0 for > . Using the fact that the Cauchy function ( , ) for every fixed as a function of is a solution of satisfying the condition ( , ) = 1, we obtain the following theorem.   (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 0 ( , ) of the problem (29), (2), and (4) in the case = 3.
In the case of impulses we obtain the following theorem.

Estimation of Solutions
for every essentially bounded function and real .
Proof. By Theorem 1, problem (1)-(5) is equivalent to the integral equation: It is clear that where ‖ ‖ = max ≤ ≤ | ( )|. Thus we have got the following condition for existence of unique solution of problem (1)- (5): Denote where 0 ( , ) 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 The Scientific World Journal 9 Estimating the integrals in formula (11), we get the following inequalities: Let us write 0 = 1 + 2 , where 1 is upper triangular and 2 lower triangular parts of 0 and estimate ‖ 0 ‖. 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 write 0 = 1 + 2 , where 1 is upper triangular and 2 , lower triangular parts, and estimate ‖ 0 ‖: We obtain It is clear that We obtain the following solution estimation: Theorem 8 has been proven.

Sign Constancy of Green's Functions
The proofs of the following two assertions follow from the construction of Green's functions. Estimation of Green's function (11) leads us to the assertion. Proof. Without losing generality we assume that = 0.
Solution of problem (1)-(5) can be represented in the following form: Using Theorem 9, we obtain that the operator is positive.
Theorem 11 has been proven. 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. The proof of the following two theorems can be obtained analogously from the proof of Theorem 11.