- Research Article
- Open Access
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.
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.
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
Now, for convenience, we make the following assumptions.
From [18, Lemma 5.4.1], we have the following lemma.
The following lemma can be easily proved.
The equality (2.13) now follows from (2.14) and (2.16).
which is a contradiction.
which is contradiction.
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:
which is a contradiction. Thus, (2.32) holds.
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.
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.
3. Further Discussions
In this section, we will use the following assumptions.
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.
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).
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).
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