Positive Solutions for Impulsive Equations of Third Order in Banach Space

Abstract

Using the fixed-point theorem, this paper is devoted to study the multiple and single positive solutions of third-order 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 third-order three-point boundary value problems (BVP, for short) in Banach space

(1.1)

where , , , . , , . is the zero element of .

Recently, third-order 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 three-point boundary value problem:

(1.2)

where , . The authors obtained at least one positive solutions of BVP (1.2) by using fixed-point 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 third-order two-point boundary value problem

(1.3)

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 fixed-point 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:

(2.1)

has a unique solution

(2.2)

where

(2.3)

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

(2.4)

In the paper, we define cone as follows:

(2.5)

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 fixed-point 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:

(2.6)

Define operators , and as follows:

(2.7)

As we know, BVP (1.1) has a positive solution if and only if is the fixed-point of .

Lemma 2.6.

is completely continuous.

Proof.

By condition (A) we get , , for all . For any , we have

(2.8)

Similarly

(2.9)

So .

Next, we prove is completely continuous. We first prove that is continuous. Let and such that . Let , then

(2.10)

By (A), we obtain

(2.11)

Hence

(2.12)

Since

(2.13)

By (2.11)–(2.13) and Lebesgue-dominated convergence theorem

(2.14)

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

(2.15)

which implies that . So, , it follows that is compact. The lemma is proved.

In this paper, denote

(2.16)

where or , and . is a dual cone of .

We list the assumptions:

(H₁) , , where ;

(H₂) , , , where and ;

(H₃) , , , , where and .

For convenience, denote

(2.17)

3. Main Results

Theorem 3.1.

Assume that (A), (H₁) 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

(3.1)

that is,

(3.2)

Similarly

(3.3)

So

(3.4)

Hence

(3.5)

Since , there exist and such that for and . Let . Then for any , by (H₁) and the definition of , we obtain

(3.6)

By (3.6) and (H)^'

(3.7)

Similarly, by , there exist and such that for and with . Let . Then for any ,

(3.8)

So we have by (3.8) and (H)^'

(3.9)

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.

(3.10)

Theorem 3.3.

Assume that (A) and (H₂) 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 (H₂), there exists , , such that for ,

(3.11)

for any with , where , such that

(3.12)

Let . Then for any , we obtain

(3.13)

Similarly

(3.14)

It follows that

(3.15)

which implies

(3.16)

On the other hand, by , there exists , such that for and . Let , . For any , we have

(3.17)

By the definition of we get

(3.18)

So

(3.19)

Hence

(3.20)

Therefore

(3.21)

By (3.16), (3.21) and Lemma 2.4, it is easily seen that has a fixed-point .

Corollary 3.4.

Let (A) and the following conditions hold, then BVP (1.1) has at least one positive solution while and .

(3.22)

Theorem 3.5.

Let (A) and (H₃) 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 ,

(3.23)

So

(3.24)

which implies

(3.25)

On the other hand, by and , there exist , , such that and

(3.26)

where satisfies

(3.27)

Let and . Then for any , we have

(3.28)

Similarly

(3.29)

Hence

(3.30)

So

(3.31)

By (3.25), (3.31), and Lemma 2.4, has a fixed-point .

Corollary 3.6.

Assume that (A) and the following conditions hold, then BVP (1.1) has at least one positive solution while and .

(3.32)

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 third-order boundary value problem:

(4.1)

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,

(4.2)

that is, . Similarly, , it is easy to see that , where , . In this example, , , and

(4.3)

and .

Let . By computing, we get

(4.4)

Above all, the conditions of Theorem 3.3 are satisfied. Then for any and , BVP (4.1) has at least one positive solution.