 Research Article
 Open Access
 Published:
Multiplicity Results via Topological Degree for Impulsive Boundary Value Problems under NonWellOrdered Upper and Lower Solution Conditions
Boundary Value Problems volume 2008, Article number: 197205 (2008)
Abstract
Some multiplicity results for solutions of an impulsive boundary value problem are obtained under the condition of nonwellordered upper and lower solutions. The main ideas of this paper are to associate a LeraySchauder degree with the lower or upper solution.
1. Introduction
In this paper, we study multiplicity of solutions of the impulsive boundary value problem
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 secondorder integrodifferential equations of mixed type in real Banach space :
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 nonwellordered upper and lower solutions. In [8], Rachnková and Tvrdý studied the existence of solutions of the nonlinear impulsive periodic boundary value problem
where , . Using LeraySchauder degree, the authors of [8] showed some existence results for (1.3) under the nonwellordered upper and lower solutions condition. For other results related to nonwellordered 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 secondorder threepoint boundary value problem
where , , . In [15], we made the following assumption.
There exists such that
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
for some . Then the threepoint boundary value problem (1.4) has at least six solutions .
Theorem 1.1 establishes the existence of at least six solutions of the threepoint 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.
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 fixedpoint 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 nonwellordered 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 LeraySchauder 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 righthand limit at exits, and is a map from into such that and are continuous at , left continuous at and their righthand limits and at exits. For each , let
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
Definition 2.1.
The function is called a strict lower solution of (1.1) if
whenever or for some and some
whenever for each and , for each .
The function is called a strict upper solution of (1.1) if
whenever or for some and some
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
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
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
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
Then
Lemma 2.7.
Let and . Then, is a solution of
if and only if satisfies
Proof. .
Let be a solution of (2.12). From Lemma 2.6, we have
Thus,
Using the boundary value condition , we have
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
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
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
where
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
Let . Take such that
and such that
Let . Define the functions by
For each , let us define the functions by
It is easy to see that there exists such that
Let us define the operator by
By (2.26), we have
From (2.28), we have for each . Let . Then, . By the properties of the LeraySchauder degree, we have
Thus, has at least one fixed point . From Lemma 2.7, satisfies
Step 2.
Next, we will show that
We first show that
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
Then, , is the maximum of on . Thus, . By (2.30), we have
which is a contradiction.
For case (3B), set for . For any , we have
This implies that is a local maximum. Since , then , . Therefore,
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
while for each .
First, we consider case (4A). Since is increasing on , then
Then, there exists small enough such that for and so for . Since is a strict upper solution, we have
Since for each , then we have . Similarly, we have . Therefore,
which is contradiction.
Now we consider case (4B). Since is increasing, then we have
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
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 , and we have . Note that , and we have
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
while for each .
Since is increasing, then for case (5A), we have
and for case (5B), we have and
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 meanvalue theorem, there exists such that
Let be such that , then, there exist?? such that , , , and for . Therefore,
Consequently,
On the other hand,
which is a contradiction. Thus, (2.32) holds.
Step 4.
From the excision property of LeraySchauder degree and (2.29), we have
From (2.31) and (2.32), we see that for each , and so
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
and for some .
Proof .
Set for , and for each . From Theorem 2.8, we see that there exist and large enough such that
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 wellknown Weierstrass approximation theorem, there exists such that
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
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.
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
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
and for some .
Also (1.1) has at least two solutions and such that
and .
Now Theorem 2.14 guarantees that (1.1) has at least two solutions such that
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.
3. Further Discussions
For simplicity, in this section, we will always assume that
In this case, (1.1) can be reduced to the following threepoint boundary value problem
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
for some . Then, (3.2) has at least eight solutions.
Proof .
First, we show that there exist strict lower and upper solutions such that
Let . Now, we consider the following boundary value problem:
Let
By Lemma 2.7, we have
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 NagumoKnoblochSchmitt condition. For this kind of bounding condition, the reader is referred to [13].
Theorem 3.3.
Suppose holds, and there exists function such that
where , ,
Then, (3.2) has at least eight solutions.
Proof .
Let for each , and
Now, we consider the following boundary value problem:
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
We only show that for . If for some , then for some , where for . If , then , and so
which contradicts (3.9).
From Lemma 2.6, we have
and so
This implies that . Therefore, (3.14) holds. Integrating (3.14), we have
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.
References
 1.
Guo D: Periodic boundary value problems for second order impulsive integrodifferential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 1997,28(6):983997. 10.1016/S0362546X(97)828556
 2.
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.
 3.
Guo D:Multiple positive solutions of a boundary value problem for thorder impulsive integrodifferential equations in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2004,56(7):9851006. 10.1016/j.na.2003.10.023
 4.
Erbe LH, Liu X: Quasisolutions of nonlinear impulsive equations in abstract cones. Applicable Analysis 1989,34(34):231250. 10.1080/00036818908839897
 5.
Liu X: Nonlinear boundary value problems for first order impulsive integrodifferential equations. Applicable Analysis 1990,36(12):119130. 10.1080/00036819008839925
 6.
Picard E: Sur l'application des méthodes d'approximations successives à l' étude de certaines équations différentielles ordinaries. Journal de Mathématiques Pures et Appliquées 1893, 9: 217271.
 7.
De Coster C, Habets P: An overview of the method of lower and upper solutions for ODEs. In Nonlinear Analysis and Its Applications to Differential Equations. Volume 43. Birkhauser, Boston, Mass, USA; 2001:322.
 8.
Rachunková I, Tvrdý M: Nonordered lower and upper functions in second order impulsive periodic problems. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2005,12(34):397415.
 9.
Amann H, Ambrosetti A, Mancini G: Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Mathematische Zeitschrift 1978,158(2):179194. 10.1007/BF01320867
 10.
De Coster C, Henrard M: Existence and localization of solution for second order elliptic BVP in presence of lower and upper solutions without any order. Journal of Differential Equations 1998,145(2):420452. 10.1006/jdeq.1998.3423
 11.
Rachunková I, Tvrdý M:Periodic problems with Laplacian involving nonordered lower and upper functions. Fixed Point Theory 2005,6(1):99112.
 12.
Rachunková I: Upper and lower solutions and topological degree. Journal of Mathematical Analysis and Applications 1999,234(1):311327. 10.1006/jmaa.1999.6375
 13.
Rachunková I: Upper and lower solutions and multiplicity results. Journal of Mathematical Analysis and Applications 2000,246(2):446464. 10.1006/jmaa.2000.6798
 14.
Habets P, Omari P: Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order. Topological Methods in Nonlinear Analysis 1996,8(1):2556.
 15.
Xu X, O'Regan D, Sun J: Multiplicity results for threepoint boundary value problems with a nonwellordered upper and lower solution condition. Mathematical and Computer Modelling 2007,45(12):189200. 10.1016/j.mcm.2006.05.003
 16.
Khan RA, Webb JRL: Existence of at least three solutions of a secondorder threepoint boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2006,64(6):13561366. 10.1016/j.na.2005.06.040
 17.
Xian X: Three solutions for threepoint boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2005,62(6):10531066. 10.1016/j.na.2005.04.017
 18.
Guo D, Sun J, Liu Z: The Funtional Method for Nonlinear Ordinary Differential Equations. Shandong Science and Technology Press, Jinan, China; 1995.
 19.
Liu B: Positive solutions of secondorder threepoint boundary value problems with change of sign. Computers & Mathematics with Applications 2004,47(89):13511361. 10.1016/S08981221(04)901289
 20.
Gupta CP, Trofimchuk SI: Existence of a solution of a threepoint boundary value problem and the spectral radius of a related linear operator. Nonlinear Analysis: Theory, Methods & Applications 1998,34(4):489507. 10.1016/S0362546X(97)005841
 21.
Ma R, Castaneda N:Existence of solutions of nonlinear point boundaryvalue problems. Journal of Mathematical Analysis and Applications 2001,256(2):556567. 10.1006/jmaa.2000.7320
Acknowledgments
This paper is supported by Natural Science Foundation of Jiangsu Education Committee (04KJB110138) and China Postdoctoral Science Foundation (2005037712).
Author information
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 Existence Result
 Lower Solution
 Integrodifferential Equation
 Impulsive Differential Equation
 Multiplicity Result