Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

In this paper, we consider the following second-order multipoint boundary value problem on the half-line

(1.1)

where , and is continuous, in which .

The study of multipoint boundary value problems (BVPs) for second-order differential equations was initiated by Bicadze and Samarskĭ [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. Since then, great efforts have been devoted to nonlinear multi-point BVPs due to their theoretical challenge and great application potential. Many results on the existence of (positive) solutions for multi-point BVPs have been obtained, and for more details the reader is referred to [5–10] and the references therein. The BVPs on the half-line arise naturally in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium [11–13] and have been also widely studied [14–27]. When , BVP (1.1) reduces to the following three-point BVP on the half-line:

(1.2)

where . Lian and Ge [16] only studied the solvability of BVP (1.2) by the Leray-Schauder continuation theorem. When , , and nonlinearity is variable separable, BVP (1.1) reduces to the second order two-point BVP on the half-line

(1.3)

Yan et al. [17] established the results of existence and multiplicity of positive solutions to the BVP (1.3) by using lower and upper solutions technique.

Motivated by the above works, we will study the existence results of unbounded (positive) solution for second order multi-point BVP (1.1). Our main features are as follows. Firstly, BVP (1.1) depends on derivative, and the boundary conditions are more general. Secondly, we will study multi-point BVP on infinite intervals. Thirdly, we will obtain the unbounded (positive) solution to BVP (1.1). Obviously, with the boundary condition in (1.1), if the solution exists, it is unbounded. Hence, we extend and generalize the results of [16, 17] to some degree. The main tools used in this paper are Leray-Schauder nonlinear alternative and the fixed point index theory.

The rest of the paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, the existence of unbounded solution is established. In Section 4, the existence and uniqueness of positive solution are obtained. Finally, we formulate two examples to illustrate the main results.

Denote , where . Let

(2.1)

For any , define

(2.2)

then is a Banach space with the norm (see [17]).

The Arzela-Ascoli theorem fails to work in the Banach space due to the fact that the infinite interval is noncompact. The following compactness criterion will help us to resolve this problem.

Lemma 2.1 (see [17]).

Let . Then, is relatively compact in if the following conditions hold:

(a) is bounded in ;

(b) the functions belonging to and are locally equicontinuous on ;

(c) the functions from and are equiconvergent, at .

Throughout the paper we assume the following.

Suppose that , and there exist nonnegative functions with such that

(2.3)

, where

(2.4)

Denote

(2.5)

Lemma 2.2.

Supposing that with , then BVP

(2.6)

has a unique solution

(2.7)

where

(2.8)

in which , and for .

Proof.

Integrating the differential equation from to , one has

(2.9)

Then, integrating the above integral equation from to , noticing that and , we have

(2.10)

Since , it holds that

(2.11)

By using arguments similar to those used to prove Lemma 2.2 in [9], we conclude that (2.7) holds. This completes the proof.

Now, BVP (1.1) is equivalent to

(2.12)

Letting , (2.12) becomes

(2.13)

For , define operator by

(2.14)

Then,

(2.15)

Set

(2.16)

Remark 2.3.

is the Green function for the following associated homogeneous BVP on the half-line:

(2.17)

It is not difficult to testify that

(2.18)

Let us first give the following result of completely continuous operator.

Lemma 2.4.

Supposing that and hold, then is completely continuous.

For any , there exists such that . Then,

(2.19)

so

(2.20)

Similarly,

(2.21)

(2.22)

Further,

(2.23)

(2.24)

On the other hand, for any and , by Remark 2.3, we have

(2.25)

Hence, by , the Lebesgue dominated convergence theorem, and the continuity of , for any , we have

(2.26)

So, for any .

We can show that . In fact, by (2.23) and (2.24), we obtain

(2.27)

Hence, is well defined.

1. (2)

   We show that is continuous.

    

Suppose , and . Then, as , and there exists such that . The continuity of implies that

(2.28)

as . Moreover, since

(2.29)

we have from the Lebesgue dominated convergence theorem that

(2.30)

Thus, is continuous.

1. (3)

   We show that is relatively compact.

    

(a) Let be a bounded subset. Then, there exists such that for all . By the similar proof of (2.20) and (2.22), if , one has

(2.31)

which implies that is uniformly bounded.

(b) For any , if , we have

(2.32)

Thus, for any there exists such that if , then

(2.33)

Since is arbitrary, then and are locally equicontinuous on .

(c) For , from (2.27), we have

(2.34)

which means that and are equiconvergent at . By Lemma 2.1, is relatively compact.

Therefore, is completely continuous. The proof is complete.

Lemma 2.5 (see [28, 29]).

Let be Banach space, be a bounded open subset of , and be a completely continuous operator. Then either there exist such that , or there exists a fixed point .

Lemma 2.6 (see [28, 29]).

Let be a bounded open set in real Banach space , let be a cone of , and let be completely continuous. Suppose that

(2.35)

Then,

(2.36)

In this section, we present the existence of an unbounded solution for BVP (1.1) by using the Leray-Schauder nonlinear alternative.

Theorem 3.1.

Suppose that conditions hold. Then BVP (1.1) has at least one unbounded solution.

Proof.

Since , by , we have , a.e. , which implies that . Set

(3.1)

From Lemmas 2.2 and 2.4, BVP (1.1) has a solution if and only if is a fixed point of in . So, we only need to seek a fixed point of in .

Suppose such that . Then

(3.2)

Therefore,

(3.3)

which contradicts . By Lemma 2.5, has a fixed point . Letting , boundary conditions imply that is an unbounded solution of BVP (1.1).

In this section, we restrict the nonlinearity and discuss the existence and uniqueness of positive solution for BVP (1.1).

Define the cone as follows:

(4.1)

Lemma 4.1.

Suppose that and hold. Then, is completely continuous.

Proof.

Lemma 2.4 shows that is completely continuous, so we only need to prove . Since , and from Remark 2.3, we have

(4.2)

Then,

(4.3)

Therefore, .

Theorem 4.2.

Suppose that conditions and hold and the following condition holds:

suppose that and there exist nonnegative functions with such that

(4.4)

Then, BVP (1.1) has a unique unbounded positive solution.

Proof.

We first show that implies . By (4.4), we have

(4.5)

By Lemma 4.1, is completely continuous. Let . Then, . Set

(4.6)

For any , by (4.5), we have

(4.7)

Therefore, for all , that is, for any . Then, Lemma 2.6 yields , which implies that has a fixed point . Let . Then, is an unbounded positive solution of BVP (1.1).

Next, we show the uniqueness of positive solution for BVP (1.1). We will show that is a contraction. In fact, by (4.4), we have

(4.8)

So, is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP (1.1).

Example 5.1.

Consider the following BVP:

(5.1)

We have

(5.2)

Let

(5.3)

Then, , and it is easy to prove that is satisfied. By direct calculations, we can obtain that . By Theorem 3.1, BVP (5.1) has an unbounded solution.

Example 5.2.

Consider the following BVP:

(5.4)

In this case, we have

(5.5)

Let

(5.6)

Then, . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.