The Scientific World Journal

Volume 2014, Article ID 978519, 13 pages

http://dx.doi.org/10.1155/2014/978519

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

Department of Mathematics and Computer Science, Ariel University, Ariel, Israel

Received 18 August 2013; Accepted 22 October 2013; Published 13 February 2014

Academic Editors: K. Ammari, F. J. Garcia-Pacheco, and K. Zhu

Copyright © 2014 Alexander Domoshnitsky and Irina Volinsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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*

*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*

*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*

*Conflict of Interests*

*The authors declare that they have no conflict of interests regarding the publication of this paper.*

*References*

*References*

- N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina,
*Introduction to the Theory of Functional Differential Equations*, Advanced Series in Mathematical Science and Engineering, World Federation, Atlanta, Ga, USA, 1995. - D. Baĭnov and P. Simeonov,
*Impulsive Differential Equations*, vol. 66 of*Pitman Monographs and Surveys in Pure and Applied Mathematics*, Longman Scientific & Technical, Harlow, UK, 1993. View at MathSciNet - V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, vol. 6 of*Series in Modern Applied Mathematics*, World Scientific, Singapore, 1989. View at MathSciNet - S. G. Pandit and S. G. Deo,
*Differential Systems Involving Impulses*, vol. 954 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1982. View at MathSciNet - A. M. Samoilenko and N. A. Perestyuk,
*Differential Equations with Impulse Effect*, Villa Skola, Kiev, Russia, 1987, (Russian). - A. M. Samoilenko and A. N. Perstyuk,
*Impulsive Differential Equations*, World Scientific, Singapore, 1992. - S. T. Zavalishchin and A. N. Sesekin,
*Dynamic Impulse Systems: Theory and Applications*, vol. 394 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at MathSciNet - M. Federson and S. Schwabik, “Generalized ODE approach to impulsive retarded functional differential equations,”
*Differential and Integral Equations*, vol. 19, no. 11, pp. 1201–1234, 2006. View at Google Scholar · View at MathSciNet - M. Federson and S. Schwabik, “A new approach to impulsive retarded differential equations: stability results,”
*Functional Differential Equations*, vol. 16, no. 4, pp. 583–607, 2009. View at Google Scholar · View at MathSciNet - Z. Halas and M. Tvrdý, “Continuous dependence of solutions of generalized linear differential equations on a parameter,”
*Functional Differential Equations*, vol. 16, no. 2, pp. 299–313, 2009. View at Google Scholar · View at MathSciNet - J. Kurzweil, “Generalized ordinary differential equations and continuous dependence on a parameter,”
*Czechoslovak Mathematical Journal*, vol. 7, no. 82, pp. 418–449, 1957. View at Google Scholar · View at MathSciNet - S. Schwabik,
*Generalized Ordinary Differential Equations*, Eorkd Scientific, Singapore, 1992. - M. Tvrdý, “Generalized differential equations in the space of regulated functions (boundary value problems and controllability),”
*Mathematica Bohemica*, vol. 116, no. 3, pp. 225–244, 1991. View at Google Scholar · View at MathSciNet - M. Tvrdý, “Differential and integral equations in the space of regulated functions,”
*Memoirs on Differential Equations and Mathematical Physics*, vol. 25, pp. 1–104, 2002. View at Google Scholar · View at MathSciNet - M. Ashordia, “Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations,”
*Czechoslovak Mathematical Journal*, vol. 46, no. 121, pp. 385–404, 1996. View at Google Scholar · View at MathSciNet - A. Domoshnitsky and M. Drakhlin, “On boundary value problems for first order impulse functional-differential equations,” in
*Boundary Value Problems for Functional Differential Equations*, J. Henderson, Ed., pp. 107–117, World Scientific, Singapore, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - A. Domoshnitsky and M. Drakhlin, “Nonoscillation of first order impulse differential equations with delay,”
*Journal of Mathematical Analysis and Applications*, vol. 206, no. 1, pp. 254–269, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - A. Domoshnitsky, M. Drakhlin, and E. Litsyn, “On N-th order functional-differential equations with impulses,” in
*Memoirs on Differential Equations and Mathematical Physics*, vol. 12, pp. 50–56, 1997. View at Google Scholar - A. Domoshnitsky, “Boundary value problem for first order impulsive functional-di
*¤*erential nonlinear equations,”*Functional Differential Equations*, vol. 4, pp. 37–45, 1997. View at Google Scholar - A. Domoshnitsky, M. Drakhlin, and E. Litsyn, “On boundary value problems for N-th order functional differential equations with impulses,”
*Advances in Mathematical Sciences and Applications*, vol. 8, no. 2, pp. 987–996, 1998. View at Google Scholar - D. D. Baĭnov, S. G. Hristova, S. C. Hu, and V. Lakshmikantham, “Periodic boundary value problems for systems of first order impulsive differential equations,”
*Differential and Integral Equations*, vol. 2, no. 1, pp. 37–43, 1989. View at Google Scholar · View at MathSciNet - J. Li, J. J. Nieto, and J. Shen, “Impulsive periodic boundary value problems of first-order differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 226–236, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - J. J. Nieto and R. Rodríguez-López, “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 318, no. 2, pp. 593–610, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - S. C. Hu and V. Lakshmikantham, “Periodic boundary value problems for second order impulsive differential systems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 13, no. 1, pp. 75–85, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Agarwal, L. Berezansky, E. Braverman, and A. Domoshnitsky,
*Nonoscillation Theory of Functional Differential Equations with Applications*, Springer, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - V. Lakshmikantham and X. Liu, “On quasistability for impulsive differential systems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 13, no. 7, pp. 819–828, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - M. Benchohra, J. Henderson, and S. K. Ntouyas,
*Impulsive Differential Equations and Inclusions*, vol. 2 of*Contemporary Mathematics and Its Applications*, Hindawi Publishing Corporation, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - A. L. Skubachevskii, “Nonclassical boundary value problems,”
*Journal of Mathematical Sciences*, vol. 155, no. 2, pp. 199–330, 2008. View at Google Scholar - M. Benchohra, J. Henderson, and S. K. Ntouyas, “An existence result for first-order impulsive functional differential equations in Banach spaces,”
*Computers & Mathematics with Applications*, vol. 42, no. 10-11, pp. 1303–1310, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Fan, “Impulsive problems for semilinear differential equations with nonlocal conditions,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 2, pp. 1104–1109, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Fan and G. Li, “Existence results for semilinear differential equations with nonlocal and impulsive conditions,”
*Journal of Functional Analysis*, vol. 258, no. 5, pp. 1709–1727, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - S. Ji and G. Li, “Existence results for impulsive differential inclusions with nonlocal conditions,”
*Computers & Mathematics with Applications*, vol. 62, no. 4, pp. 1908–1915, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. Ji and S. Wen, “Nonlocal Cauchy problem for impulsive differential equations in Banach spaces,”
*International Journal of Nonlinear Science*, vol. 10, no. 1, pp. 88–95, 2010. View at Google Scholar · View at MathSciNet - S. Ji and G. Li, “A unified approach to nonlocal impulsive differential equations with the measure of noncompactness,”
*Advances in Difference Equations*, vol. 2012, article 182, 2012. View at Publisher · View at Google Scholar - L. Zhu, Q. Dong, and G. Li, “Impulsive differential equations with nonlocal conditions in general Banach spaces,”
*Advances in Difference Equations*, vol. 2012, article 10, 2012. View at Google Scholar

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