Journal of Applied Mathematics and Stochastic Analysis

Volume 2008 (2008), Article ID 589480, 11 pages

http://dx.doi.org/10.1155/2008/589480

## Unbounded Solutions of a Boundary Value Problem for Abstract *n*th-Order Differential Equations on an Infinite Interval

^{1}School of Sciences, Qingdao Agricultural University, Shandong, Qingdao 266109, China^{2}School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China^{3}Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 29 January 2007; Revised 28 May 2007; Accepted 17 October 2007

Academic Editor: Aizicovici Sergiu

Copyright © 2008 Zhenbin Liu et al. 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.

#### Abstract

The existence of unbounded nonnegative solutions of a boundary value problem for *n*th-order differential equations defined on an infinite interval is obtained by means of the Mönch fixed-point theorem. An example is then presented to demonstrate the application of our results.

#### 1. Introduction

Over the last few years, have focused their research on the study of boundary value problems for nonlinear differential and integral equations defined on an infinite interval, and various theoretical results have been obtained [1–8]. In [4], the existence of multiple positive solutions of a boundary value problem (BVP) for th-order nonlinear impulsive integral-differential equations defined on an infinite interval in a Banach space is obtained by means of the fixed-point index theory of completely continuous operators. However, the result requires the use of the measures of noncompactness condition (where ) and the normal and solid cone in a real Banach space. In [6], by using the Mönch fixed-point theorem, a class of infinite boundary value problems for first-order impulsive differential equations in a Banach space is considered and the existence of positive solutions is obtained, but the solutions are limited to bounded solutions only.

To generalize and further develop the existing results in this field, in this paper we discuss the existence of unbounded solutions for a class of th-order nonlinear differential equations defined on an infinite interval in a Banach space by using the Mönch fixed-point theorem under certain conditions weaker than those in [4]. The boundary value problem in question is as follows: where , , , , , in which is a real Banach space.

Let be the space of all continuous functions , and let be the Banach space of all strongly measurable functions with , equipped with the norm . Let , then it is clear that is a Banach space with norm Let , then it is also easy to see that is a Banach space with norm Let be a cone of the Banach space , and , where denotes the zero element of . Then, it is obvious that is a cone in space , and is a cone in space .

*Deffinition 1.1. *A function is called a
nonnegative solution of BVP(1.1) if satisfies (1.1) for .

The rest of the paper is organized as follows. In Section 2, we give some lemmas which provide a theoretical basis for the proof of our main results. The main theorem is presented and proved in Section 3. In Section 4, an example is given to demonstrate the application of our results.

#### 2. Some Lemmas

Here we first list some assumptions to be used throughout the rest of the paper.

There exist and such that , and

There exists such that where denotes the Kuratowski measure of noncompactness in . For details on the definition and properties of the measure of noncompactness, the reader is referred to [9, 10].

In the following, we give various lemmas which are to be used for the proof of the main results to be presented in Section 3.

Lemma 2.1 ( See [4]). *Let () be satisfied. Then is a solution of BVP(1.1) if and only if is a solution of the following integral equation: *

*Proof. *If is a solution of BVP(1.1), then by condition we have the convergence of the infinite integral Integrating the first equation in (1.1) from to , we have By virtue of , let in (2.5), we get From the second equation in (1.1), we have Substituting (2.6) into (2.7) yields for , From (2.5) and (2.6), we have Integrating (2.9) from to and using (2.8) for , we have It is not difficult to show by mathematical induction that satisfies (2.3). Conversely, if is a solution
of (2.3), then direct differentiation of (2.3) gives
Consequently, , and by (2.11) and (2.12), it is easy to see that satisfies (1.1). The proof of Lemma 2.1 is completed.

We now consider an operator defined by By Lemma 2.1, is a solution of BVP(1.1) if and only if is a fixed point of the operator in .

Lemma 2.2 (See [10, 11,
13]). *Let be a Banach
space, let
be a finite
interval, and let
be a countable
set. Assume that there exists such that . Then and
*

Lemma 2.3. *Let be a Banach
space, and . If
is a countable
set and there exists such that , then
is integrable
on , and
*

*Proof. *By for any and , we get for all and . As , is integrable
in . For any , from , there exists such that . So for any and , we have
Let
By (2.16) for any and , we obtain
where denotes the
distance of a point to a set. Using the same method, we have
So
Hence, for any , we have
where denotes the
Hausdorff distance of the sets and , that is,
So, by (2.21), we obtain for any ,
Hence, we have
By Lemma 2.3 for any , we have
Hence, from (2.24) and (2.25), we have
Therefore, (2.15) is satisfied. The proof of Lemma 2.3
is completed.

*Remark 2.4. *Lemma 2.3 generalizes
Lemma 2.2 from a finite
interval to an infinite interval, and it plays an important role in studying
the differential equations defined on an infinite interval. It should be
emphasized that Lemma 2.3 has no counterpart in the existing literature.

Lemma 2.5 (See [2, Lemma 7]). *Let
be satisfied,
and let be a countable
bounded set. Then
**
where denotes the
Kuratowski measure of noncompactness in , .*

Lemma 2.6. *If condition
is satisfied,
then the operator
is continuous
and
*

*Proof. *For any , we have
So, by (2.30) and condition for , we have
where . Hence, . Therefore, (2.28) is satisfied.

Now we show that the operator is continuous.
Let , . Then and . By the integrability of and for any , there exists such that
On the other
hand, by the continuity of , it is easy to see that
Hence, for the above , there exists a natural number such that for
any ,
Thus, from
(2.13), (2.32), (2.34), and for any , we obtain
Therefore, the continuity of is proved. So,
the proof of Lemma 2.6 is completed.

Lemma 2.7 ([10, 12]
(Mönch)). *Let
be a closed and
convex subset of and . Assume that operator
has the
following property:
**
Then has a fixed
point in .*

#### 3. Main Result

In this section, we present the main results we obtained.

Theorem 3.1. *Suppose and
hold. Then
BVP(1.1) has at least one nonnegative solution.*

*Proof. *Choose
and let . Obviously, is a bounded
convex closed set. For any , by (2.28), and the
definition of the norm , we have
from which and the definition of the norm , we obtain
Therefore, by Lemma 2.6, we conclude that is a continuous
operator from to .

Next, we prove that is a relatively
compact set if is a countable
set satisfying for some . From this and the properties of the Kuratowski
measure of noncompactness in , we have
On the other
hand, by (2.28), Lemmas 2.2−2.5, and for any and , we have
Thus, by (3.4),
(3.5), and Lemma 2.5, we get
Hence, by , we have , that is, is a relatively
compact set in . Therefore, by Lemma 2.7, we conclude that has at least
one fixed point in , that is, BVP(1.1) has at least
one solution in .

#### 4. An Example

*Example 4.1. *Consider the following
boundary value problem for a second-order differential equation defined on an
infinite interval

*Conclusion*

BVP(4.1) has at least one nonnegative solution.

*Proof. *Let
with norm . is a cone in a
Banach space and BVP(4.1) is
in the form of BVP(1.1) in with . In this situation, , , and in which
It is clear that . For any , , we have
So we get
where , , . From (4.1) and (5.3), we have
So, we have
Hence, the condition is satisfied.
By (5.3) for any bounded sets , we get
where , . So
Hence, the condition is satisfied.
Therefore, our conclusion follows from Theorem 3.1.

#### Acknowledgments

The authors are grateful to the referees whose comments have led to a number of significant improvements of the paper. The first and second authors are supported financially by the National Natural Science Foundation of China (10771117) and the State Education Commission Doctoral Foundation of China (20060446001). The third author is supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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