Unbounded Solutions of a Boundary Value Problem for Abstract nth-Order Differential Equations on an Infinite Interval

Abstract

The existence of unbounded nonnegative solutions of a boundary value problem for nth-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.