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

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## 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## 1. Introduction

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).

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.

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 .

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 is increasing on Open image in new window .

Definition 2.1.

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 .

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.

Definition 2.3.

Definition 2.4.

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.

Proof. .

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.

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

Theorem 2.8.

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.

Step 2.

- (1)Open image in new window , then, we have(2.34)

- (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(2.35)

- (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 .

which is a contradiction.

- (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 ;

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

which is contradiction.

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 ;

- (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 ;

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

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 .

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

Step 4.

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.

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

Proof .

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.

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.

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 .

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

and Open image in new window .

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.

## 3. Further Discussions

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 .

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.

Then, (3.2) has at least eight solutions.

Proof .

which contradicts (3.9).

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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