Multiplicity Results via Topological Degree for Impulsive Boundary Value Problems under Non-Well-Ordered Upper and Lower Solution Conditions
© Xu Xian et al. 2008
Received: 14 April 2008
Accepted: 26 August 2008
Published: 31 August 2008
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.
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).
where , and are two linear operators, , are constants. In  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 , , . In , we made the following assumption.
for some . Then the three-point boundary value problem (1.4) has at least six solutions .
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  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 ).
This paper is a continuation of the paper . 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 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 .
whenever for each and , for each .
and whenever for each and , for each .
From [18, Lemma 5.4.1], we have the following lemma.
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.
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.
From Lemma 2.5, is a completely continuous operator.
We only prove the case when or for some and some . The conclusion is achieved in four steps.
- (1), then, we have(2.34)
- (2); assume without loss of generality that and for some , then, we have(2.35)
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 .
which is a contradiction.
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 ;
while for each .
which is contradiction.
while for each . For case (4B), we have two subcases:
(4Ba) there exists small enough and such that for ;
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 ;
while for each .
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.
Now, we show (2.32). Suppose not, then we have the following two subcases:
(I)there exists such that ;
(II)there exists such that .
which is a contradiction. Thus, (2.32) holds.
The proof is complete.
From the proof of Theorem 2.8, we see that has no fixed point on
and for some .
Thus, has at least one fixed point . Since , then there exist such that and . The proof is complete.
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 .
The relationship of is different from that of [12, Theorems 9 and 10].
Similarly, we have the following result.
and for some .
From Theorems 2.10 and 2.14, we have the following 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.
and for some .
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.
3. Further Discussions
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 , 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.
for some . Then, (3.2) has at least eight solutions.
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.
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 .
Then, (3.2) has at least eight solutions.
which contradicts (3.9).
From (3.13)–(3.18), we see that are eight solutions of (3.2). The proof is complete.
We also can replace (3.3) by other bounding conditions, see .
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.
This paper is supported by Natural Science Foundation of Jiangsu Education Committee (04KJB110138) and China Postdoctoral Science Foundation (2005037712).
- Guo D: Periodic boundary value problems for second order impulsive integro-differential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 1997,28(6):983-997. 10.1016/S0362-546X(97)82855-6MathSciNetView ArticleMATHGoogle Scholar
- 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.View ArticleGoogle Scholar
- Guo D:Multiple positive solutions of a boundary value problem for th-order impulsive integro-differential equations in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2004,56(7):985-1006. 10.1016/j.na.2003.10.023MathSciNetView ArticleMATHGoogle Scholar
- Erbe LH, Liu X: Quasi-solutions of nonlinear impulsive equations in abstract cones. Applicable Analysis 1989,34(3-4):231-250. 10.1080/00036818908839897MathSciNetView ArticleMATHGoogle Scholar
- Liu X: Nonlinear boundary value problems for first order impulsive integro-differential equations. Applicable Analysis 1990,36(1-2):119-130. 10.1080/00036819008839925MathSciNetView ArticleMATHGoogle Scholar
- 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: 217-271.MATHGoogle Scholar
- 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; 2001:3-22.View ArticleGoogle Scholar
- Rachunková I, Tvrdý M: Non-ordered lower and upper functions in second order impulsive periodic problems. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2005,12(3-4):397-415.MathSciNetMATHGoogle Scholar
- Amann H, Ambrosetti A, Mancini G: Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Mathematische Zeitschrift 1978,158(2):179-194. 10.1007/BF01320867MathSciNetView ArticleMATHGoogle Scholar
- 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):420-452. 10.1006/jdeq.1998.3423MathSciNetView ArticleMATHGoogle Scholar
- Rachunková I, Tvrdý M:Periodic problems with -Laplacian involving non-ordered lower and upper functions. Fixed Point Theory 2005,6(1):99-112.MathSciNetMATHGoogle Scholar
- Rachunková I: Upper and lower solutions and topological degree. Journal of Mathematical Analysis and Applications 1999,234(1):311-327. 10.1006/jmaa.1999.6375MathSciNetView ArticleGoogle Scholar
- Rachunková I: Upper and lower solutions and multiplicity results. Journal of Mathematical Analysis and Applications 2000,246(2):446-464. 10.1006/jmaa.2000.6798MathSciNetView ArticleMATHGoogle Scholar
- 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):25-56.MathSciNetMATHGoogle Scholar
- Xu X, O'Regan D, Sun J: Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition. Mathematical and Computer Modelling 2007,45(1-2):189-200. 10.1016/j.mcm.2006.05.003MathSciNetView ArticleMATHGoogle Scholar
- Khan RA, Webb JRL: Existence of at least three solutions of a second-order three-point boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2006,64(6):1356-1366. 10.1016/j.na.2005.06.040MathSciNetView ArticleMATHGoogle Scholar
- Xian X: Three solutions for three-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2005,62(6):1053-1066. 10.1016/j.na.2005.04.017MathSciNetView ArticleMATHGoogle Scholar
- Guo D, Sun J, Liu Z: The Funtional Method for Nonlinear Ordinary Differential Equations. Shandong Science and Technology Press, Jinan, China; 1995.Google Scholar
- Liu B: Positive solutions of second-order three-point boundary value problems with change of sign. Computers & Mathematics with Applications 2004,47(8-9):1351-1361. 10.1016/S0898-1221(04)90128-9MathSciNetView ArticleMATHGoogle Scholar
- Gupta CP, Trofimchuk SI: Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator. Nonlinear Analysis: Theory, Methods & Applications 1998,34(4):489-507. 10.1016/S0362-546X(97)00584-1MathSciNetView ArticleMATHGoogle Scholar
- Ma R, Castaneda N:Existence of solutions of nonlinear -point boundary-value problems. Journal of Mathematical Analysis and Applications 2001,256(2):556-567. 10.1006/jmaa.2000.7320MathSciNetView ArticleMATHGoogle Scholar
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.