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

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.

1. Introduction

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

(1.1)

where , , , , , , , .

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 [1–5] 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 :

(1.2)

where , and are two linear operators, , 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], Rachnková and Tvrdý studied the existence of solutions of the nonlinear impulsive periodic boundary value problem

(1.3)

where , . 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, 9–14]. 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

(1.4)

where , , . In [15], we made the following assumption.

There exists such that

(1.5)

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

Theorem 1.1.

Suppose that holds, and are two strict lower solutions of (1.4), and are two strict upper solutions of (1.4), and , , , . Moreover, assume

(1.6)

for some . Then the three-point boundary value problem (1.4) has at least six solutions .

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 and six solutions in Theorem 1.1 can be illustrated roughly by Figure 1.

Figure 1

The positions of and six solutions 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 , 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

Let , is a map from into such that is continuous at , left continuous at and its right-hand limit at exits, and is a map from into such that and are continuous at , left continuous at and their right-hand limits and at exits. For each , let

(2.1)

where and . Then, is a real Banach space with the norm . The function 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.

.

is increasing on .

Let . Now, we define the ordering by

(2.2)

Definition 2.1.

The function is called a strict lower solution of (1.1) if

(2.3)

whenever or for some and some

(2.4)

whenever for each and , for each .

The function is called a strict upper solution of (1.1) if

(2.5)

whenever or for some and some

(2.6)

and whenever for each and , for each .

Definition 2.2.

Let , for all . We say that satisfies Nagumo condition with respect to if there exists function such that

(2.7)

Definition 2.3.

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

(2.8)

Definition 2.4.

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

(2.9)

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

Lemma 2.5.

is a relative compact set if and only if for all , and are uniformly bounded on and equicontinuous on each , where .

The following lemma can be easily proved.

Lemma 2.6.

Suppose that satisfies

(2.10)

Then

(2.11)

Lemma 2.7.

Let and . Then, is a solution of

(2.12)

if and only if satisfies

(2.13)

Proof. .

Let be a solution of (2.12). From Lemma 2.6, we have

(2.14)

Thus,

(2.15)

Using the boundary value condition , we have

(2.16)

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

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

Let us define the operator by

(2.17)

From Lemma 2.5, is a completely continuous operator.

Theorem 2.8.

Suppose that and hold. Let be pairs of strict lower and upper solution, and

(2.18)

Suppose that , , satisfies Nagumo condition with respect to . Moreover, the strict lower solutions and the strict upper solutions are well ordered whenever or for some and some . Then, there exist and sufficiently large such that for each and

(2.19)

where

(2.20)

Proof .

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

Step 1.

Since satisfies Nagumo condition with respect to , then there exists such that

(2.21)

Let . Take such that

(2.22)

and such that

(2.23)

Let . Define the functions by

(2.24)

For each , let us define the functions by

(2.25)

It is easy to see that there exists such that

(2.26)

Let us define the operator by

(2.27)

By (2.26), we have

(2.28)

From (2.28), we have for each . Let . Then, . By the properties of the Leray-Schauder degree, we have

(2.29)

Thus, has at least one fixed point . From Lemma 2.7, satisfies

(2.30)

Step 2.

Next, we will show that

(2.31)

(2.32)

We first show that

(2.33)

To begin, we show that for all . Suppose not, then there exists such that . Set for . There are a number of cases to consider.

• (1) , then, we have

  

  (2.34)

which is a contradiction.

• (2) ; assume without loss of generality that and for some , then, we have

  

  (2.35)

which is a contradiction.

• (3) There exist and such that . Assume without loss of generality that for some . We have the following two cases:

(3A) for each and ;

(3B) there exists such that .

For case (3A), there exists small enough such that and

(2.36)

Then, , is the maximum of on . Thus, . By (2.30), we have

(2.37)

which is a contradiction.

For case (3B), set for . For any , we have

(2.38)

This implies that is a local maximum. Since , then , . Therefore,

(2.39)

which is a contradiction.

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

(4A) there exists such that for and ;

(4B) there exists a subset such that

(2.40)

while for each .

First, we consider case (4A). Since is increasing on , then

(2.41)

Then, there exists small enough such that for and so for . Since is a strict upper solution, we have

(2.42)

Since for each , then we have . Similarly, we have . Therefore,

(2.43)

which is contradiction.

Now we consider case (4B). Since is increasing, then we have

(2.44)

while for each . For case (4B), we have two subcases:

(4Ba) there exists small enough and such that for ;

(4Bb) there exists small enough and , such that

(2.45)

For case (4Ba) as in case (4A), we can easily obtain a contradiction. For case (4Bb), we have

(2.46)

In the same way as in the proof of case (4A), we see that , and we have . Note that , and we have

(2.47)

which is a contradiction.

• (5) There exists a such that . Without loss of generality, we may assume that for each and . We have two subcases:

(5A) there exists such that for each ;

(5B) there exists a subset such that

(2.48)

while for each .

Since is increasing, then for case (5A), we have

(2.49)

and for case (5B), we have and

(2.50)

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

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

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

Step 3.

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

(I)there exists such that ;

(II)there exists such that .

We only consider case (II). A similar argument works for case (I). We may assume without loss of generality that . By the mean-value theorem, there exists such that

(2.51)

Let be such that , then, there exist?? such that , , , and for . Therefore,

(2.52)

Consequently,

(2.53)

On the other hand,

(2.54)

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

Step 4.

From the excision property of Leray-Schauder degree and (2.29), we have

(2.55)

From (2.31) and (2.32), we see that for each , and so

(2.56)

The proof is complete.

Remark 2.9.

From the proof of Theorem 2.8, we see that has no fixed point on

Theorem 2.10.

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

(2.57)

and for some .

Proof .

Set for , and for each . From Theorem 2.8, we see that there exist and large enough such that

(2.58)

where , , and . Then, has fixed points and , respectively. From the conditions of Theorem 2.10, we see that . Let be a continuous function on such that its graph passes the points and , and satisfies . By the well-known Weierstrass approximation theorem, there exists such that

(2.59)

It is easy to see that , and so is a nonempty open set. Note has no fixed point on , and . From (2.58), we have

(2.60)

Thus, has at least one fixed point . Since , then there exist such that and . 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 satisfies .

Remark 2.12.

The position of in Theorem 2.10 can be illustrated roughly by Figure 2.

Figure 2

The position of in Theorem 2. 10.

Remark 2.13.

The relationship of is different from that of [12, Theorems 9 and 10].

Similarly, we have the following result.

Theorem 2.14.

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

(2.61)

and for some .

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

Theorem 2.15.

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

Proof .

Now Theorem 2.10 guarantees that (1.1) has at least three solutions such that

(2.62)

and for some .

Also (1.1) has at least two solutions and such that

(2.63)

and .

Now Theorem 2.14 guarantees that (1.1) has at least two solutions such that

(2.64)

and .

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

Remark 2.16.

The position of in Theorem 2.15 can be illustrated roughly by Figure 3.

Figure 3

The position of in Theorem 2. 15.

3. Further Discussions

For simplicity, in this section, we will always assume that

(3.1)

In this case, (1.1) can be reduced to the following three-point boundary value problem

(3.2)

where and .

In this section, we will use the following assumptions.

Suppose that are two strict lower solutions, are two strict upper solutions of (1.1), , , and for some .

Recently, this multipoint boundary value problem has been studied by many authors, see [16, 17, 19–21] 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 , in this section we will use the space . First, we have the following theorem.

Theorem 3.1.

Suppose that holds, and

(3.3)

for some . Then, (3.2) has at least eight solutions.

Proof .

First, we show that there exist strict lower and upper solutions such that

(3.4)

Let . Now, we consider the following boundary value problem:

(3.5)

Let

(3.6)

By Lemma 2.7, we have

(3.7)

It is easy to see that and for each . Thus, for each , and therefore, , for . On the other hand, from (3.5), it is easy to see that is a strict upper solution of (1.1). Similarly, we can show the existence of . 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 holds, and there exists function such that

(3.8)

(3.9)

(3.10)

where , ,

(3.11)

Then, (3.2) has at least eight solutions.

Proof .

Let for each , and

(3.12)

Now, we consider the following boundary value problem:

(3.13)

From and (3.8), we see that are strict lower solutions of (3.13), and and are two strict upper solutions of (3.13). By Theorem 3.1, (3.13) has at least eight solutions . We need only to show that are solutions of (3.2). We claim that

(3.14)

We only show that for . If for some , then for some , where for . If , then , and so

(3.15)

which contradicts (3.9).

From Lemma 2.6, we have

(3.16)

and so

(3.17)

This implies that . Therefore, (3.14) holds. Integrating (3.14), we have

(3.18)

From (3.13)–(3.18), we see that 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.