Open Access

On the Solvability of Second-Order Impulsive Differential Equations with Antiperiodic Boundary Value Conditions

Boundary Value Problems20082008:864297

https://doi.org/10.1155/2008/864297

Received: 3 July 2008

Accepted: 10 November 2008

Published: 26 November 2008

Abstract

We prove existence results for second-order impulsive differential equations with antiperiodic boundary value conditions in the presence of classical fixed point theorems. We also obtain the expression of Green's function of related linear operator in the space of piecewise continuous functions.

1. Introduction and Preliminaries

Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known that many biological phenomena involving threshold, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems do exhibit impulse effects. The branch of modern, applied analysis known as "impulsive" differential equations provides a natural framework to mathematically describe the aforementioned jumping processes. The reader is referred to monographs [14] and references therein for some nice examples and applications to the above areas.

In this paper, we mainly study the following second-order impulsive differential equations with antiperiodic boundary value conditions:
(1.1)

where and is continuous on , are continuous functions.

In [412], the authors studied the existence of antiperiodic solutions for first-order, second-order, or high-order differential equations without impulses, and in [3, 1316] the authors were concerned with the antiperiodic solutions of first-order impulsive differential equations. Also we should mention the work by Cabada et al. in [17] which is concerned with a certain th order linear differential equation with constant impulses at fixed times and nonhomogeneous periodic boundary conditions. So far, to the best of our knowledge, this is the first work to deal with the antiperiodic solutions to second-order differential equations with nonconstant impulses. Our method to prove the existence of antiperiodic solutions is based on the works in [13, 18, 19]. We should point out that it is Christopher C. Tisdell who started with this method.

The article is organized as follows. In Section 2, we present the expression of Green's functions of related linear operator in the space of piecewise continuous functions. Section 3 contains the main results of the paper and is devoted to the existence of solutions to (1.1). There, differential inequalities are developed and applied to prove the existence of at least one solution to (1.1). In Section 4, a couple of examples are given to illustrate how the main results work.

To understand the notation used above and the ideas in the remainder of the paper, we now briefly introduce some appropriate concepts connected with impulsive differential equations. Most of the following notation can be found in [1, 2, 4, 5]. We assume that , exist and . We introduce and denote the Banach space by
(1.2)

with the norm where is the usual Euclidean norm and will be the Euclidean inner product.

In a similar fashion to the above, define and denote the Banach space by
(1.3)

with the norm

The following fixed point theorem is our main tool to prove the existence of at least one solution to (1.1).

Schaefer's Fixed Point Theorem (19)

Let be a Banach space and let be a completely continuous operator. Then, either
  1. (i)

    the operator equation has a solution for , or

     
  2. (ii)

    the set is unbounded.

     

2. Expression of Green's Function

In this part, we present the expression of Green's functions for second order impulsive equations with antiperiodic conditions.

Lemma 2.1.

Assume and are two constants. Let , . Then for any , solves
(2.1)
if and only if is the solution of integral equation
(2.2)
where
(G)
(H)

Proof.

Assume is a solution of (2.1) and let for . We have
(2.3)
Then for ,
(2.4)
This implies . Consequently, from the impulsive condition in (2.1) we get that
(2.5)
where . Now we integrate (2.3) from to and use (2.5) to obtain
(2.6)
It follows that
(2.7)
Similarly, we have for that
(2.8)
To sum up, we have for that
(2.9)
Since , we can deduce in a similar way as to deal with to obtain
(2.10)

Now we are in position to show the expression of for To do that, we need to compute in (2.10). In what follows we present the expression of for step by step and then obtain the general form of for .

First of all, for , we have
(2.11)
See that
(2.12)
Consequently,
(2.13)
Integrate by parts to get
(2.14)
Thus,
(2.15)
Similarly, we have for that
(2.16)
Now we consider for . Clearly,
(2.17)
Noting that , we have
(2.18)
where is denoted by Similarly, for there holds
(2.19)
Thus, for ,
(2.20)
By the boundary condition of (2.1), we have
(2.21)
Substituting (2.21) into (2.20), and also noting that for
(2.22)

we see that is the solution of (2.2).

Now assume is a solution of (2.2). Then for
(2.23)
(2.24)
It is easy to verify
(2.25)
For we compute straightforwardly to get
(2.26)
which implies
(2.27)

Now, we prove is a solution of (2.1). Then the proof is completed.

For later use, we present the following estimations:
(2.28)

Corollary 2.2.

Assume in (2.1) that and . Then for any , is the solution of
(2.29)
if and only if is the solution of integral equation
(2.30)
where
(2.31)
Obviously, there hold
(2.32)

We now give Green's function of (2.1) for .

Lemma 2.3.

For any , is the solution of
(2.33)
if and only if satisfies the integral equation
(2.34)
where
(2.35)
Since the proof is very similar to that of Lemma 2.1, we omit it here. We can check easily that satisfies (2.34) and hence is a solution of (2.33). Also we get by straightforward computation that
(2.36)

Recall that a mapping between Banach spaces is compact if it is continuous and carries bounded sets into relatively compact sets.

Lemma 2.4.

Suppose that and are continuous. Define an operator as
(2.37)

where and are as given in Lemma 2.1. Then is a compact map.

Proof.

Noting the continuity of and , this follows in a standard step-by-step process and so it is omitted.

3. Main Results

In this section, we prove the existence results for (1.1) in presence of Schaefer's fixed-point theorem.

Theorem 3.1.

Suppose that and are continuous. If for some and , there exist nonnegative constants , and such that
(3.1)
(3.2)
(3.3)

where is the Euclidean inner product, . Then (1.1) has at least one solution.

Proof.

Define an integral operator as
(3.4)
where and follow the forms of (G) and (H) in Lemma 2.1. By Lemma 2.4, is a compact mapping. Also, it follows from Lemma 2.1 that is a fixed point of if and only if satisfies
(3.5)
which is equivalent to (1.1). Consequently, all that we need to do is to verify that has at least one fixed point. With this in mind, we assume is a solution of
(3.6)
That is,
(3.7)

It is equivalent to say that satisfies

(3.8)
Firstly, we see that for ,
(3.9)
Further more, by the antiperiodic boundary condition we have
(3.10)
As a result,
(3.11)
Now we show that any potential solution of (3.6) is bounded a priori. By (3.2) and (3.11), we obtain
(3.12)
Taking the supremum and rearranging, we get by (3.3) that
(3.13)
Differentiating both sides of (3.7) and noting (2.23), we obtain
(3.14)
where
(3.15)
Thus,
(3.16)

Now we have shown that any possible solution of (3.6) is bounded by which is independent of . By Scheafer's fixed theorem we know that has at least one fixed point. Therefore, the proof is completed.

Suppose both and in Theorem 3.1. We obtain the following theorem.

Theorem 3.2.

Assume that and are continuous. If for some there exist nonnegative constants , and such that
(3.17)

where is the Euclidean inner product, , then (1.1) has at least one solution.

Proof.

Consider the mapping
(3.18)
(3.19)
where
(3.21)
By Lemma 2.4, is a compact mapping. Consider the equation
(3.22)
To show that has at least one fixed point, we apply Schaefer's theorem by showing that all potential solutions to
(3.23)
are bounded a priori, with the bound being independent of . With this in mind, let be a solution of (3.23). Note that is also a solution to
(3.24)
On one hand, we see that for
(3.25)
On the other hand, by the antiperiodic boundary condition we have
(3.26)
It therefore follows that
(3.27)
Consequently,
(3.28)

where .

We compute directly to get
(3.29)
Differentiating both sides of (3.19), we obtain
(3.30)
where
(3.31)
Thus,
(3.32)

Then the proof is completed.

Similarly, we can prove the following existence result for in Theorem 3.2.

Theorem 3.3.

Suppose that and are continuous. If there exist nonnegative constants and such that
(3.33)

where is the Euclidean inner product, , then (1.1) has at least one solution.

4. Examples

In this part, we show how our main theorems work by a couple of examples.

Example 4.1.

The scalar second-order impulsive equations with antiperiodic boundary value condition
(4.1)

where have at least one solution.

Proof.

Let and in Theorem 3.1. For , we have and
(4.2)
On the other hand, for
(4.3)
Noting , we have for and that
(4.4)

Moreover, , . Then the conclusion follows from Theorem 3.1.

Example 4.2.

Consider antiperiodic value problem
(4.5)

We claim that (4.5) has at least one solution.

Proof.

Let and in Theorem 3.2. Choosing , we have for that
(4.6)
Since , we have Thus, for and ,
(4.7)

Moreover, , . Then the conclusion follows from Theorem 3.2.

Declarations

Acknowledgments

This research is supported by Ad Futura Scientific and Educational Foundation of the Republic of Slovenia, the Ministry of Higher Education, Science and Technology of the Republic of Slovenia; the Nova Kreditna Banka Maribor; TELEKOM Slovenije; National Natural Science Foundation of China (10671127); National Natural Science Foundation of Shanghai (08ZR1416000); and Foundation of Science and Technology Commission of Shanghai Municipality (06XD14034).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Shanghai Normal University
(2)
Center for Applied Mathematics and Theoretical Physics, University of Maribor

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Copyright

© Y. Xing and V. Romanovski. 2008

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