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Boundary Value Problems

, 2008:197205 | Cite as

Multiplicity Results via Topological Degree for Impulsive Boundary Value Problems under Non-Well-Ordered Upper and Lower Solution Conditions

  • Xu Xian
  • Donal O'Regan
  • RP Agarwal
Open Access
Research Article

Abstract

Some multiplicity results for solutions of an impulsive boundary value problem are obtained under the condition of non-well-ordered upper and lower solutions. The main ideas of this paper are to associate a Leray-Schauder degree with the lower or upper solution.

Keywords

Existence Result Lower Solution Integrodifferential Equation Impulsive Differential Equation Multiplicity Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In this paper, we study multiplicity of solutions of the impulsive boundary value problem

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window .

Impulsive differential equations arise naturally in a wide variety of applications, such as spacecraft control, inspection processes in operations research, drug administration, and threshold theory in biology. In the past twenty years, a significant development in the theory of impulsive differential equations was seen. Many authors have studied impulsive differential equations using a variety of methods (see [1, 2, 3, 4, 5] and the references therein).

The purpose of this paper is to study the multiplicity of solutions of the impulsive boundary value problems (1.1) by the method of upper and lower solutions. The method of lower and upper solutions has a very long history. Some of the ideas can be traced back to Picard [6]. This method deals mainly with existence results for various boundary value problems. For an overview of this method for ordinary differential equations, the reader is referred to [7]. Usually, when one uses the method of upper and lower solutions to study the existence and multiplicity of solutions of impulsive differential equations, one assumes that the upper solution is larger than the lower solution, that is, the condition that upper and lower solutions are well ordered. For example, Guo [1] studied the PBVP for second-order integrodifferential equations of mixed type in real Banach space Open image in new window :

where Open image in new window , Open image in new window and Open image in new window are two linear operators, Open image in new window , Open image in new window are constants. In [1] Guo first obtained a comparison result, and then, by establishing two increasing and decreasing sequences, he proved an existence result for maximal and minimal solutions of the PBVP (1.2) in the ordered interval defined by the lower and upper solutions.

However, to the best of our knowledge, only in the last few years, it was shown that existence and multiplicity for impulsive differential equation under the condition that the upper solution is not larger than the lower solution, that is, the condition of non-well-ordered upper and lower solutions. In [8], Rach Open image in new window nková and Tvrdý studied the existence of solutions of the nonlinear impulsive periodic boundary value problem
where Open image in new window , Open image in new window . Using Leray-Schauder degree, the authors of [8] showed some existence results for (1.3) under the non-well-ordered upper and lower solutions condition. For other results related to non-well-ordered upper and lower solutions, the reader is referred to [7, 9, 10, 11, 12, 13, 14]. Also, here we mention the main results of a very recent paper [15]. In that paper, we studied the second-order three-point boundary value problem

where Open image in new window , Open image in new window , Open image in new window . In [15], we made the following assumption.

Let the function Open image in new window be Open image in new window for Open image in new window . In [15], we proved the following theorem (see, [15, Theorem 3.4]).

Theorem 1.1.

for some Open image in new window . Then the three-point boundary value problem (1.4) has at least six solutions Open image in new window .

Theorem 1.1 establishes the existence of at least six solutions of the three-point boundary value problem (1.4) only under the condition of two pairs of strict lower and upper solutions. The positions of Open image in new window and six solutions Open image in new window in Theorem 1.1 can be illustrated roughly by Figure 1.
Figure 1

The positions of Open image in new window and six solutions Open image in new window in Theorem 1. 1.

In some sense, we can say that these two pairs of lower and upper solutions are parallel to each other. The position of these two pairs of lower and upper solutions is sharply different from that of the lower and upper solutions of the main results in [14, 16, 17]. The technique to prove our main results of [15] is to use the fixed-point index of some increasing operator with respect to some closed convex sets, which are translations of some special cones (see Open image in new window , Open image in new window of [15]).

This paper is a continuation of the paper [15]. The aim of this paper is to study the multiplicity of solutions of the impulsive boundary value problem (1.1) under the conditions of non-well-ordered upper and lower solutions. In this paper, we will permit the presence of impulses and the first derivative. The main ideas of this paper are to associate a Leray-Schauder degree with the lower or upper solution. We will give some multiplicity results for at least eight solutions. To obtain this multiplicity result, an additional pair of lower and upper solutions is needed, that is, we will employ a condition of three pairs of lower and upper solutions. The position of these three pairs of lower and upper solutions will be illustrated in Remark 2.16.

2. Results for at Least Eight Solutions

where Open image in new window and Open image in new window . Then, Open image in new window is a real Banach space with the norm Open image in new window . The function Open image in new window is called a solution of the boundary value problem (1.1) if it satisfies all the equalities of (1.1).

Now, for convenience, we make the following assumptions.

Open image in new window .

Open image in new window is increasing on Open image in new window .

Definition 2.1.

The function Open image in new window is called a strict lower solution of (1.1) if

whenever Open image in new window for each Open image in new window and Open image in new window , Open image in new window for each Open image in new window .

The function Open image in new window is called a strict upper solution of (1.1) if

and whenever Open image in new window for each Open image in new window and Open image in new window , Open image in new window for each Open image in new window .

Definition 2.2.

Let Open image in new window , Open image in new window for all Open image in new window . We say that Open image in new window satisfies Nagumo condition with respect to Open image in new window if there exists function Open image in new window such that

Definition 2.3.

Let Open image in new window be strict upper solutions of (1.1) and Open image in new window for each Open image in new window . Then, we say the upper solutions Open image in new window are well ordered if for each Open image in new window , there exist Open image in new window and Open image in new window small enough such that

Definition 2.4.

Let Open image in new window be strict lower solutions of (1.1) and Open image in new window for each Open image in new window . Then, we say the lower solutions Open image in new window are well ordered if for each Open image in new window , there exist Open image in new window and Open image in new window small enough such that

From [18, Lemma 5.4.1], we have the following lemma.

Lemma 2.5.

Open image in new window is a relative compact set if and only if for all Open image in new window , Open image in new window and Open image in new window are uniformly bounded on Open image in new window and equicontinuous on each Open image in new window , where Open image in new window .

The following lemma can be easily proved.

Lemma 2.6.

Lemma 2.7.

if and only if Open image in new window satisfies

Proof. .

Let Open image in new window be a solution of (2.12). From Lemma 2.6, we have
Using the boundary value condition Open image in new window , we have

The equality (2.13) now follows from (2.14) and (2.16).

On the other hand, if Open image in new window satisfies (2.13), by direct computation, we can easily show that Open image in new window satisfies (2.12). The proof is complete.

Let us define the operator Open image in new window by

From Lemma 2.5, Open image in new window is a completely continuous operator.

Theorem 2.8.

Suppose that Open image in new window and Open image in new window hold. Let Open image in new window be Open image in new window pairs of strict lower and upper solution, and
Suppose that Open image in new window , Open image in new window , Open image in new window satisfies Nagumo condition with respect to Open image in new window . Moreover, the strict lower solutions Open image in new window and the strict upper solutions Open image in new window are well ordered whenever Open image in new window or Open image in new window for some Open image in new window and some Open image in new window . Then, there exist Open image in new window and Open image in new window sufficiently large such that for each Open image in new window and Open image in new window

Proof .

We only prove the case when Open image in new window or Open image in new window for some Open image in new window and some Open image in new window . The conclusion is achieved in four steps.

Step 1.

Since Open image in new window satisfies Nagumo condition with respect to Open image in new window , then there exists Open image in new window such that
For each Open image in new window , let us define the functions Open image in new window by
It is easy to see that there exists Open image in new window such that
Let us define the operator Open image in new window by
By (2.26), we have
From (2.28), we have Open image in new window for each Open image in new window . Let Open image in new window . Then, Open image in new window . By the properties of the Leray-Schauder degree, we have
Thus, Open image in new window has at least one fixed point Open image in new window . From Lemma 2.7, Open image in new window satisfies

Step 2.

Next, we will show that
We first show that
To begin, we show that Open image in new window for all Open image in new window . Suppose not, then there exists Open image in new window such that Open image in new window . Set Open image in new window for Open image in new window . There are a number of cases to consider.
which is a contradiction.
  1. (2)
    Open image in new window ; assume without loss of generality that Open image in new window and Open image in new window for some Open image in new window , then, we have
     
which is a contradiction.
  1. (3)

    There exist Open image in new window and Open image in new window such that Open image in new window . Assume without loss of generality that Open image in new window for some Open image in new window . We have the following two cases:

     

(3A) Open image in new window for each Open image in new window and Open image in new window ;

(3B) there exists Open image in new window such that Open image in new window .

For case (3A), there exists Open image in new window small enough such that Open image in new window and

which is a contradiction.

which is a contradiction.
  1. (4)

    There exists Open image in new window such that Open image in new window . Without loss of generality, we may assume Open image in new window for each Open image in new window and Open image in new window . (Otherwise, if there exists Open image in new window for some Open image in new window such that Open image in new window , then we can get a contradiction as in case (3)). In this case, we have the following two subcases:

     

(4A) there exists Open image in new window such that Open image in new window for Open image in new window and Open image in new window ;

(4B) there exists a subset Open image in new window such that

while Open image in new window for each Open image in new window .

First, we consider case (4A). Since Open image in new window is increasing on Open image in new window , then
Then, there exists Open image in new window small enough such that Open image in new window for Open image in new window and so Open image in new window for Open image in new window . Since Open image in new window is a strict upper solution, we have

which is contradiction.

Now we consider case (4B). Since Open image in new window is increasing, then we have

while Open image in new window for each Open image in new window . For case (4B), we have two subcases:

(4Ba) there exists Open image in new window small enough and Open image in new window such that Open image in new window for Open image in new window ;

For case (4Ba) as in case (4A), we can easily obtain a contradiction. For case (4Bb), we have
In the same way as in the proof of case (4A), we see that Open image in new window , Open image in new window and we have Open image in new window . Note that Open image in new window , and we have
which is a contradiction.
  1. (5)

    There exists a Open image in new window such that Open image in new window . Without loss of generality, we may assume that Open image in new window for each Open image in new window and Open image in new window . We have two subcases:

     

(5A) there exists Open image in new window such that Open image in new window for each Open image in new window ;

(5B) there exists a subset Open image in new window such that

while Open image in new window for each Open image in new window .

Since Open image in new window is increasing, then for case (5A), we have
and for case (5B), we have Open image in new window and

while Open image in new window for each Open image in new window . Therefore, we can use the same method as in case (4) to obtain a contradiction.

From the discussions of (1)–(5), we see that Open image in new window for Open image in new window . Similarly, we can prove that Open image in new window for Open image in new window . Thus, (2.33) holds.

Next, we prove that Open image in new window . If the inequality Open image in new window does not hold, then either there exists Open image in new window such that Open image in new window or there exists Open image in new window such that Open image in new window . Set Open image in new window for Open image in new window . Then, we have either Open image in new window or Open image in new window for some Open image in new window . Essentially the same reasoning as in (1)–(5) above yields a contradiction. Thus, Open image in new window . Similarly, Open image in new window . Consequently, (2.31) holds.

Step 3.

Now, we show (2.32). Suppose not, then we have the following two subcases:

(I)there exists Open image in new window such that Open image in new window ;

(II)there exists Open image in new window such that Open image in new window .

We only consider case (II). A similar argument works for case (I). We may assume without loss of generality that Open image in new window . By the mean-value theorem, there exists Open image in new window such that
Consequently,
On the other hand,

which is a contradiction. Thus, (2.32) holds.

Step 4.

From the excision property of Leray-Schauder degree and (2.29), we have
From (2.31) and (2.32), we see that Open image in new window for each Open image in new window , and so

The proof is complete.

Remark 2.9.

From the proof of Theorem 2.8, we see that Open image in new window has no fixed point on Open image in new window

Theorem 2.10.

Suppose that Open image in new window , Open image in new window hold, Open image in new window are strict lower solutions, Open image in new window are strict upper solutions, Open image in new window , Open image in new window , Open image in new window for some Open image in new window , and?? Open image in new window satisfies Nagumo condition with respect to Open image in new window . Moreover, the strict lower solutions Open image in new window are well ordered whenever Open image in new window or Open image in new window for some Open image in new window and some Open image in new window . Then, (1.1) has at least three solutions Open image in new window , and Open image in new window , such that

and Open image in new window for some Open image in new window .

Proof .

where Open image in new window , Open image in new window , and Open image in new window . Then, Open image in new window has fixed points Open image in new window and Open image in new window , respectively. From the conditions of Theorem 2.10, we see that Open image in new window . Let Open image in new window be a continuous function on Open image in new window such that its graph passes the points Open image in new window and Open image in new window , and satisfies Open image in new window . By the well-known Weierstrass approximation theorem, there exists Open image in new window such that
It is easy to see that Open image in new window , and so Open image in new window is a nonempty open set. Note Open image in new window has no fixed point on Open image in new window , and Open image in new window . From (2.58), we have

Thus, Open image in new window has at least one fixed point Open image in new window . Since Open image in new window , then there exist Open image in new window such that Open image in new window and Open image in new window . The proof is complete.

Remark 2.11.

Theorem 2.10 is a partial generalization of the main results of [16, Theorem 2.2]. Here, we do not need to assume that Open image in new window satisfies Open image in new window .

Remark 2.12.

The position of Open image in new window in Theorem 2.10 can be illustrated roughly by Figure 2.
Figure 2

The position of Open image in new window in Theorem 2. 10.

Remark 2.13.

The relationship of Open image in new window is different from that of [12, Theorems 9 and 10].

Similarly, we have the following result.

Theorem 2.14.

Suppose that Open image in new window , Open image in new window hold, Open image in new window are strict lower solutions of (1.1), Open image in new window and Open image in new window are strict upper solutions of (1.1), Open image in new window , Open image in new window , Open image in new window for some Open image in new window , and?? Open image in new window satisfies Nagumo condition with respect to Open image in new window . Moreover, the strict upper solutions Open image in new window are well ordered whenever Open image in new window or Open image in new window for some Open image in new window and some Open image in new window . Then, (1.1) has at least three solutions Open image in new window such that

and Open image in new window for some Open image in new window .

From Theorems 2.10 and 2.14, we have the following Theorem 2.15.

Theorem 2.15.

Suppose that Open image in new window , Open image in new window hold, Open image in new window are three strict lower solutions of (1.1), Open image in new window are three strict upper solutions of (1.1), Open image in new window , Open image in new window , Open image in new window for some Open image in new window , and?? Open image in new window satisfies Nagumo conditions with respect to Open image in new window . Moreover, the strict lower solutions Open image in new window and the strict upper solutions Open image in new window are well ordered whenever Open image in new window or Open image in new window for some Open image in new window and some Open image in new window . Then, (1.1) has at least eight solutions.

Proof .

Now Theorem 2.10 guarantees that (1.1) has at least three solutions Open image in new window such that

and Open image in new window for some Open image in new window .

Also (1.1) has at least two solutions Open image in new window and Open image in new window such that

and Open image in new window .

Now Theorem 2.14 guarantees that (1.1) has at least two solutions Open image in new window such that

and Open image in new window .

Also (1.1) has at least one solution Open image in new window such that Open image in new window and Open image in new window for some Open image in new window . It is easy to see that Open image in new window are distinct eight solutions of (1.1). The proof is complete.

Remark 2.16.

The position of Open image in new window in Theorem 2.15 can be illustrated roughly by Figure 3.
Figure 3

The position of Open image in new window in Theorem 2. 15.

3. Further Discussions

For simplicity, in this section, we will always assume that
In this case, (1.1) can be reduced to the following three-point boundary value problem

where Open image in new window and Open image in new window .

In this section, we will use the following assumptions.

Open image in new window Suppose that Open image in new window are two strict lower solutions, Open image in new window are two strict upper solutions of (1.1), Open image in new window , Open image in new window , and Open image in new window for some Open image in new window .

Recently, this multipoint boundary value problem has been studied by many authors, see [16, 17, 19, 20, 21] and the references therein. The goal of this section is to prove some multiplicity results for (3.2) using the condition of two pairs of strict upper and lower solutions. As we can see from [13], some bounding condition on the nonlinear term is needed. Instead of the space Open image in new window , in this section we will use the space Open image in new window . First, we have the following theorem.

Theorem 3.1.

for some Open image in new window . Then, (3.2) has at least eight solutions.

Proof .

First, we show that there exist strict lower and upper solutions Open image in new window such that
Let Open image in new window . Now, we consider the following boundary value problem:
By Lemma 2.7, we have

It is easy to see that Open image in new window and Open image in new window for each Open image in new window . Thus, Open image in new window for each Open image in new window , and therefore, Open image in new window , Open image in new window for Open image in new window . On the other hand, from (3.5), it is easy to see that Open image in new window is a strict upper solution of (1.1). Similarly, we can show the existence of Open image in new window . Then, by Theorem 2.15, the conclusion holds.

Remark 3.2.

Obviously, the condition (3.3) is restrictive. In the following, we will make use of a weaker condition. We study the multiplicity of solutions of (3.2) under a Nagumo-Knobloch-Schmitt condition. For this kind of bounding condition, the reader is referred to [13].

Theorem 3.3.

Suppose Open image in new window holds, and there exists function Open image in new window such that

Then, (3.2) has at least eight solutions.

Proof .

Now, we consider the following boundary value problem:
From Open image in new window and (3.8), we see that Open image in new window are strict lower solutions of (3.13), and Open image in new window and Open image in new window are two strict upper solutions of (3.13). By Theorem 3.1, (3.13) has at least eight solutions Open image in new window . We need only to show that Open image in new window are solutions of (3.2). We claim that

which contradicts (3.9).

From Lemma 2.6, we have
This implies that Open image in new window . Therefore, (3.14) holds. Integrating (3.14), we have

From (3.13)–(3.18), we see that Open image in new window are eight solutions of (3.2). The proof is complete.

Remark 3.4.

We also can replace (3.3) by other bounding conditions, see [13].

Remark 3.5.

To end this paper, we point out that the results of this paper can be applied to study the multiplicity of radial solutions of elliptic differential equation in an annulus with impulses at some radii.

Notes

Acknowledgments

This paper is supported by Natural Science Foundation of Jiangsu Education Committee (04KJB110138) and China Postdoctoral Science Foundation (2005037712).

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Copyright information

© Xu Xian et al. 2008

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsXuzhou Normal UniversityXuzhouChina
  2. 2.Department of MathematicsNational University of IrelandGalwayIreland
  3. 3.Department of Mathematical ScienceFlorida Institute of TechnologyMelbourneUSA

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