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Positive Solutions for Impulsive Equations of Third Order in Banach Space
Advances in Difference Equations volume 2010, Article number: 185701 (2010)
Abstract
Using the fixedpoint theorem, this paper is devoted to study the multiple and single positive solutions of thirdorder boundary value problems for impulsive differential equations in ordered Banach spaces. The arguments are based on a specially constructed cone. At last, an example is given to illustrate the main results.
1. Introduction
The purpose of this paper is to establish the existence of positive solutions for the following thirdorder threepoint boundary value problems (BVP, for short) in Banach space
where , , , . , , . is the zero element of .
Recently, thirdorder boundary value problems (cf. [1–9]) have attracted many authors attention due to their wide range of applications in applied mathematics, physics, and engineering, especially in the bridge issue. To our knowledge, most papers in literature concern mainly about the existence of positive solutions for the cases in which the spaces are real and the equations have no parameters. And many authors consider nonlinear term have same linearity. In this paper, we consider the existence of solutions when the nonlinear terms have different properties, the space is abstract and the equations have two different parameters.
In [3], Guo et al. studied the following nonlinear threepoint boundary value problem:
where , . The authors obtained at least one positive solutions of BVP (1.2) by using fixedpoint theorem when is sublinear or suplinear.
In [8], Yao and Feng used the upper and lower solutions method proved some existence results for the following thirdorder twopoint boundary value problem
Inspired by the above work, the aim of this paper is to establish some simple criteria for the existence of nontrivial solutions for BVP (1.1) under some weaker conditions. The new features of this paper mainly include the following aspects. Firstly, we consider the system (1.1) in abstract space while [3, 8] talk about equations in real space (). Secondly, we obtained the positive solutions when the two parameters have different ranges. Thirdly, and in system (1.1) may have different properties. Fourthly, in system (1.1) not only contains but also , which is much more complicated. Finally, the main technique used here is the fixedpoint theory and a special cone is constructed to study the existence of nontrivial solutions.
We recall some basic facts about ordered Banach spaces . The cone in induces a partial order on , that is, if and only if , is said to be normal if there exists a positive constant such that implies , without loss of generality, suppose, in present paper, the normal constant . denotes the measure of noncompactness (cf. [10]).
Some preliminaries and a number of lemmas to the derivation of the main results are given in Section 2, then the proofs of the theorems are given in Section 3, followed by an example, in Section 4, to demonstrate the validity of our main results.
2. Preliminaries and Lemmas
In this paper we will consider the Banach space , denote and , is continuous at and is left continuous at , the right limit exists, . For any we define and for .
For convenience, let us list the following assumption.
(A), , , . For any and , is relatively compact in , where .
Lemma 2.1.
Assume that , then for any , the following boundary value problem:
has a unique solution
where
Proof.
The proof is similar to Lemma 2.2 in [3], we omit it.
Lemma 2.2 (see [3]).
Assume that and . Then for any , where , .
Lemma 2.3 (see [3]).
Let and , then for any , , where
In the paper, we define cone as follows:
Lemma 2.4 (see [10]).
Let be a Banach space and be a cone. Suppose and are bounded open sets, , , is completely continuous such that either

(i)
for any and for any or

(ii)
for any and for any .
Then has a fixedpoint in .
Lemma 2.5.
The vector is a solution of differential systems (1.1) if and only if is the solution of the following integral systems:
Define operators , and as follows:
As we know, BVP (1.1) has a positive solution if and only if is the fixedpoint of .
Lemma 2.6.
is completely continuous.
Proof.
By condition (A) we get , , for all . For any , we have
Similarly
So .
Next, we prove is completely continuous. We first prove that is continuous. Let and such that . Let , then
By (A), we obtain
Hence
Since
By (2.11)–(2.13) and Lebesguedominated convergence theorem
So is continuous. Similarly, is continuous. It follows that is continuous.
Next we prove is compact. Let be bounded, and . Let for some , then , . It is easy to see that is equicontinuous. By condition (A) we have
which implies that . So, , it follows that is compact. The lemma is proved.
In this paper, denote
where or , and . is a dual cone of .
We list the assumptions:
(H_{1}) , , where ;
(H_{2}) , , , where and ;
(H_{3}) , , , , where and .
For convenience, denote
3. Main Results
Theorem 3.1.
Assume that (A), (H_{1}) and the following condition (H)^{'} hold, then BVP (1.1) has at least two positive solution while and .
(H)^{'}:; ; , where , .
Proof.
Let , then for , we have
that is,
Similarly
So
Hence
Since , there exist and such that for and . Let . Then for any , by (H_{1}) and the definition of , we obtain
By (3.6) and (H)^{'}
Similarly, by , there exist and such that for and with . Let . Then for any ,
So we have by (3.8) and (H)^{'}
By (3.5), (3.7), (3.9) and Lemma 2.4 we get that BVP (1.1) has at least two positive solutions with .
Corollary 3.2.
Assume that (A) and the following condition hold, then the conclusion of Theorem 3.1 also holds.
Theorem 3.3.
Assume that (A) and (H_{2}) hold, then BVP (1.1) has at least one positive solution when and .
Proof.
By Lemma 2.6, we see that is completely continuous. By (H_{2}), there exists , , such that for ,
for any with , where , such that
Let . Then for any , we obtain
Similarly
It follows that
which implies
On the other hand, by , there exists , such that for and . Let , . For any , we have
By the definition of we get
So
Hence
Therefore
By (3.16), (3.21) and Lemma 2.4, it is easily seen that has a fixedpoint .
Corollary 3.4.
Let (A) and the following conditions hold, then BVP (1.1) has at least one positive solution while and .
Theorem 3.5.
Let (A) and (H_{3}) hold, then BVP (1.1) has at least one positive solution while and .
Proof.
Since , we choose , such that for and . Let . Then for any ,
So
which implies
On the other hand, by and , there exist , , such that and
where satisfies
Let and . Then for any , we have
Similarly
Hence
So
By (3.25), (3.31), and Lemma 2.4, has a fixedpoint .
Corollary 3.6.
Assume that (A) and the following conditions hold, then BVP (1.1) has at least one positive solution while and .
4. An Example
In this section, we construct an example to demonstrate the application of our main results obtained in Section 3. Consider the following thirdorder boundary value problem:
Conclusion.
BVP(4.1) has at least one positive solution.
Proof.
, , . Define . , . , . , we know that , let , then for any , . It is easy to see that (A) is satisfied. On the other hand,
that is, . Similarly, , it is easy to see that , where , . In this example, , , and
and .
Let . By computing, we get
Above all, the conditions of Theorem 3.3 are satisfied. Then for any and , BVP (4.1) has at least one positive solution.
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Acknowledgments
The author thanks Professor Liu and Professor Lou for many useful discussions and helpful suggestions. The work was partially supported by NSFC (10971155) and Innovation program of Shanghai Municipal Education Commission (09ZZ33).
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Cai, J. Positive Solutions for Impulsive Equations of Third Order in Banach Space. Adv Differ Equ 2010, 185701 (2010). https://doi.org/10.1155/2010/185701
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DOI: https://doi.org/10.1155/2010/185701
Keywords
 Banach Space
 Partial Order
 Nonlinear Term
 Differential System
 Normal Constant