`Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 357697, 14 pageshttp://dx.doi.org/10.1155/2012/357697`
Research Article

Existence of Unbounded Solutions for a Third-Order Boundary Value Problem on Infinite Intervals

School of Sciences, China University of Geosciences, Beijing 100083, China

Received 20 June 2012; Accepted 31 July 2012

Copyright © 2012 Hairong Lian and Junfang Zhao. 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

We generalize the unbounded upper and lower solution method to a third-order ordinary differential equation on the half line subject to the Sturm-Liouville boundary conditions. By using such techniques and the Schäuder fixed point theorem, some criteria are presented for the existence of solutions and positive ones to the problem discussed.

1. Introduction

Boundary value problems on infinite intervals, arising from the study of radially symmetric solutions of nonlinear elliptic equation [1], have received much attention in recent years. Because the infinite interval is noncompact, the discussion about BVPs on the half-line is more complicated. There have been many existence results for some boundary value problems of differential equations on the half line. The main methods are the extension of continuous solutions on the corresponding finite intervals under a diagonalization process, fixed point theorems in special Banach space or in special Fréchet space; see [112] and the references therein.

The method of upper and lower solutions is a powerful technique to deal with the existence of boundary value problems (BVPs). In many cases, when given one pair of well-ordered lower and upper solution, nonlinear BVPs always have at least one solution in the closed interval. To obtain this kind of result, we can employ topological degree theory, the monotone iterative technique, or critical theory. For details, we refer the reader to see [14, 7, 9, 1214] and therein.

When the method of upper and lower solution is applied to the infinite interval problems, diagonalization process is always used; see [2, 3, 7]. For example, in [3], Agarwal and O’Regan discussed a Sturm-Liouville boundary value problem of second-order differential equation: where . General existence criteria were obtained to guarantee the existence of bounded solutions. The methods used therein were based on a diagonalization arguments and existence results of appropriate boundary value problems on finite intervals.

In [12], Yan et al. investigated the boundary value problem where . By using the upper and lower solutions method and the fixed point theorem, the authors presented sufficient conditions for the existence of unbounded positive solutions. In [9], Lian and the coauthors discussed further the existence of the unbounded solutions.

There are many results of third-order boundary value problems on finite interval; see [14, 15] and the references therein. However, there has been few papers concerned with the upper and lower solutions technique for the boundary value problems of third-order differential equation on infinite intervals. In this paper, we aim to investigate a general Sturm-Liouville boundary value problem for third-order differential equation on the half line where , are continuous, , . The methods mainly depend on the unbounded upper and lower solutions method and topological degree theory. The nonlinear is admitted to involve in the high-order derivatives under the considerations of the Nagumo condition. The solutions obtained can be unbounded in this paper. The results obtained in this paper generalize those in [4].

2. Preliminaries

We present here some definitions and lemmas which are essential in the proof of the main results.

Definition 2.1. A function is called a lower solution of BVP (1.3) if Similarly, a function is called an upper solution of BVP (1.3) if

Definition 2.2. Given a positive function and a pair of functions satisfying and ; a function is said to satisfy the Nagumo condition with respect to the pair of functions , if there exist positive functions satisfying such that holds for all , and .
Let and consider the space defined by with the norm , where . By the standard arguments, we can prove that is a Banach space.

Lemma 2.3. If , then the BVP of third-order linear differential equation has a unique solution in . Moreover this solution can be expressed as where

Proof. It is easy to verify that (2.6) satisfies BVP (2.5). Now we show the uniqueness. Suppose is a solution of (2.5). Let , then we have By a direct calculation, we obtain the general solution of the above equation: Substituting this to the boundary condition, we arrive at Therefore, (2.8) has a unique solution where Furthermore, , so The proof is complete.

Theorem 2.4 (see [1]). Let . Then is relatively compact if the following conditions hold:(a)all functions from are uniformly bounded;(b)all functions from are equicontinuous on any compact interval of ;(c)all functions from are equiconvergent at infinity; that is, for any given , there exists a such that , for all and .
From the above results, we can obtain the following general criteria for the relative compactness of subsets in .

Theorem 2.5. Given continuous functions satisfying with a positive constant. Let . Then is relatively compact if the following conditions hold:(a)all functions from are uniformly bounded;(b)the functions from are equicontinuous on any compact interval of , ;(c)the functions from are equiconvergent at infinity, .

Proof. Set , then . From conditions (a)–(c), we have is relatively compact in . Therefore, for any sequence , it has a convergent subsequence. Without loss of generality, we denote it this sequence. Then there exists such that
Set , then . Noticing that all functions from are uniformly continuous, we can obtain that . So is relatively compact.

3. Main Results

In this section, we present the existence criteria for the existence of solutions and positive solutions of BVP (1.3). We first cite conditions (H1) and (H2) here.(H1):(1) BVP (1.3) has a pair of upper and lower solutions in with ;(2) satisfies the Nagumo condition with respect to and .(H2): For any and , it holds

Lemma 3.1. Suppose condition (H1) holds. And suppose further that the following condition holds:(H3) there exists a constant such that .
If is a solution of (1.3) satisfying then there exists a constant (without relations to ) such that .

Proof. Let and , such that where is the nonhomogeneous boundary value, , then . If it is untrue, we have the following three cases.

Case 1. Consider Without loss of generality, we suppose . While for any , which is a contraction. So there must exist such that .

Case 2. Consider Just take and we can complete the proof.

Case 3. There exists such that or .
Suppose that . Obviously, concludes that . For and are arbitrary, we have .
Similarly if , we can also obtain that .
Thus there exists , just related with , and , such that .

Remark 3.2. Condition (H3) is necessary for an a priori estimation of in Lemma 3.1. Because the upper and lower solutions are in , and are at most linearly increasing, especially at infinity. Otherwise, and may be equal to infinity.

Theorem 3.3. Suppose and the conditions (H1)–(H3) hold. Then BVP (1.3) has at least one solution such that

Proof. Let be the same definition in Lemma 3.1 and consider the boundary value problem where Firstly we prove that BVP (3.10) has at least one solution . To this end, define the operator by By Lemma 2.3, we can see that the fixed points of coincide with the solutions of BVP (3.10). So it is enough to prove that has a fixed point.
We claim that is completely continuous.
is well defined. For any and it holds where . By the Lebesgue-dominated convergence theorem, we have so .
is continuous. For any convergent sequence in , there exists such that . Similarly, we have so is continuous.
is compact. Let be any bounded subset of , then there exists such that . For any , one has where , so is uniformly bounded. Meanwhile, for any , if , we have that is, is equicontinuous. From Theorem 2.5, we can see that if is equiconvergent at infinity, then is relatively compact. In fact, Then we can obtain that is completely continuous.
By the Schäuder fixed point theorem, has at least one fixed point . Next we will prove satisfying . If does not hold, then, Because , so there are two cases.

Case 1. Consider Easily, . While by the boundary condition, we have which is a contraction.

Case 2. There exists such that So, . Unfortunately,
Consequently, holds for all . Similarly, we can show that for all . Noticing that , from the inequality , we can obtain that . Lemma 3.1 guarantee that . So, that is, is a solution of BVP (1.3).

Remark 3.4. For finite interval problem, it is sharp to define the lower and upper solutions satisfying and ; see [15].
If , we can establish a criteria for the existence of positive solutions.

Theorem 3.5. Let be continuous and . Suppose the condition (H2) holds and the following conditions hold.(P1) BVP (1.3) has a pair of positive upper and lower solutions satisfying (P2) For any , there exists satisfying such that holds for all , and .
Then BVP (1.3) with has at least one solution such that

Proof. Choose and consider the boundary value problem (3.10) except substituting by Similarly, we can obtain that (3.10) has at least one solution satisfying and . Because and , we have Consequently, the solution is a positive solution of (1.3).

Funding

This research is supported by the National Natural Science Foundation of China (nos. 11101385 and 60974145) and by the Fundamental Research Funds for the Central Universities.

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