Open Access

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

Boundary Value Problems20102010:236560

DOI: 10.1155/2010/236560

Received: 14 May 2010

Accepted: 11 October 2010

Published: 18 October 2010

Abstract

This paper investigates the second-order multipoint boundary value problem on the half-line https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq5_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq9_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq10_HTML.gif is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq11_HTML.gif is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

1. Introduction

In this paper, we consider the following second-order multipoint boundary value problem on the half-line
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq13_HTML.gif is continuous, in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq14_HTML.gif .

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 [510] 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 [1113] and have been also widely studied [1427]. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq15_HTML.gif , BVP (1.1) reduces to the following three-point BVP on the half-line:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ2_HTML.gif
(1.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq16_HTML.gif . Lian and Ge [16] only studied the solvability of BVP (1.2) by the Leray-Schauder continuation theorem. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq18_HTML.gif , and nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq19_HTML.gif is variable separable, BVP (1.1) reduces to the second order two-point BVP on the half-line
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ3_HTML.gif
(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.

2. Preliminaries and Lemmas

Denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq20_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq21_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ4_HTML.gif
(2.1)
For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq22_HTML.gif , define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ5_HTML.gif
(2.2)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq23_HTML.gif is a Banach space with the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq24_HTML.gif (see [17]).

The Arzela-Ascoli theorem fails to work in the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq25_HTML.gif due to the fact that the infinite interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq26_HTML.gif is noncompact. The following compactness criterion will help us to resolve this problem.

Lemma 2.1 (see [17]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq27_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq28_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq29_HTML.gif if the following conditions hold:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq30_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq31_HTML.gif ;

(b) the functions belonging to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq32_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq33_HTML.gif are locally equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq34_HTML.gif ;

(c) the functions from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq36_HTML.gif are equiconvergent, at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq37_HTML.gif .

Throughout the paper we assume the following.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq39_HTML.gif , and there exist nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq40_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq41_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ6_HTML.gif
(2.3)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq43_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq45_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ7_HTML.gif
(2.4)
Denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ8_HTML.gif
(2.5)

Lemma 2.2.

Supposing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq46_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq47_HTML.gif , then BVP
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ9_HTML.gif
(2.6)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ10_HTML.gif
(2.7)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ11_HTML.gif
(2.8)

in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq48_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq49_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq50_HTML.gif .

Proof.

Integrating the differential equation from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq51_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq52_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ12_HTML.gif
(2.9)
Then, integrating the above integral equation from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq53_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq54_HTML.gif , noticing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq56_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ13_HTML.gif
(2.10)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq57_HTML.gif , it holds that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ14_HTML.gif
(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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ15_HTML.gif
(2.12)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq58_HTML.gif , (2.12) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ16_HTML.gif
(2.13)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq59_HTML.gif , define operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq60_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ17_HTML.gif
(2.14)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ18_HTML.gif
(2.15)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ19_HTML.gif
(2.16)

Remark 2.3.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq61_HTML.gif is the Green function for the following associated homogeneous BVP on the half-line:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ20_HTML.gif
(2.17)
It is not difficult to testify that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ21_HTML.gif
(2.18)

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

Lemma 2.4.

Supposing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq62_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq63_HTML.gif hold, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq64_HTML.gif is completely continuous.

Proof.
  1. (1)

    First, we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq65_HTML.gif is well defined.

     
For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq66_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq67_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq68_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ22_HTML.gif
(2.19)
so
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ23_HTML.gif
(2.20)
Similarly,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ24_HTML.gif
(2.21)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ25_HTML.gif
(2.22)
Further,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ26_HTML.gif
(2.23)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ27_HTML.gif
(2.24)
On the other hand, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq70_HTML.gif , by Remark 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ28_HTML.gif
(2.25)
Hence, by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq71_HTML.gif , the Lebesgue dominated convergence theorem, and the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq72_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq73_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ29_HTML.gif
(2.26)

So, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq74_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq75_HTML.gif .

We can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq76_HTML.gif . In fact, by (2.23) and (2.24), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ30_HTML.gif
(2.27)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq77_HTML.gif is well defined.
  1. (2)

    We show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq78_HTML.gif is continuous.

     
Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq79_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq80_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq81_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq82_HTML.gif , and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq83_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq84_HTML.gif . The continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq85_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ31_HTML.gif
(2.28)
as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq86_HTML.gif . Moreover, since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ32_HTML.gif
(2.29)
we have from the Lebesgue dominated convergence theorem that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ33_HTML.gif
(2.30)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq87_HTML.gif is continuous.
  1. (3)

    We show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq88_HTML.gif is relatively compact.

     
(a) Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq89_HTML.gif be a bounded subset. Then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq90_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq91_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq92_HTML.gif . By the similar proof of (2.20) and (2.22), if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq93_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ34_HTML.gif
(2.31)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq94_HTML.gif is uniformly bounded.

(b) For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq95_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq96_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ35_HTML.gif
(2.32)
Thus, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq97_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq98_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq99_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ36_HTML.gif
(2.33)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq100_HTML.gif is arbitrary, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq102_HTML.gif are locally equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq103_HTML.gif .

(c) For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq104_HTML.gif , from (2.27), we have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ37_HTML.gif
(2.34)

which means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq105_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq106_HTML.gif are equiconvergent at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq107_HTML.gif . By Lemma 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq108_HTML.gif is relatively compact.

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq109_HTML.gif is completely continuous. The proof is complete.

Lemma 2.5 (see [28, 29]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq110_HTML.gif be Banach space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq111_HTML.gif be a bounded open subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq112_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq113_HTML.gif be a completely continuous operator. Then either there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq114_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq115_HTML.gif , or there exists a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq116_HTML.gif .

Lemma 2.6 (see [28, 29]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq117_HTML.gif be a bounded open set in real Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq118_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq119_HTML.gif be a cone of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq120_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq121_HTML.gif be completely continuous. Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ38_HTML.gif
(2.35)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ39_HTML.gif
(2.36)

3. Existence Result

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq122_HTML.gif hold. Then BVP (1.1) has at least one unbounded solution.

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq123_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq124_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq125_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq126_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq127_HTML.gif . Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ40_HTML.gif
(3.1)

From Lemmas 2.2 and 2.4, BVP (1.1) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq128_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq129_HTML.gif is a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq130_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq131_HTML.gif . So, we only need to seek a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq132_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq133_HTML.gif .

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq134_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq135_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ41_HTML.gif
(3.2)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ42_HTML.gif
(3.3)

which contradicts https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq136_HTML.gif . By Lemma 2.5, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq137_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq138_HTML.gif . Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq139_HTML.gif , boundary conditions imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq140_HTML.gif is an unbounded solution of BVP (1.1).

4. Existence and Uniqueness of Positive Solution

In this section, we restrict the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq141_HTML.gif and discuss the existence and uniqueness of positive solution for BVP (1.1).

Define the cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq142_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ43_HTML.gif
(4.1)

Lemma 4.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq144_HTML.gif hold. Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq145_HTML.gif is completely continuous.

Proof.

Lemma 2.4 shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq146_HTML.gif is completely continuous, so we only need to prove https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq147_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq148_HTML.gif , and from Remark 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ44_HTML.gif
(4.2)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ45_HTML.gif
(4.3)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq149_HTML.gif .

Theorem 4.2.

Suppose that conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq151_HTML.gif hold and the following condition holds:

suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq153_HTML.gif and there exist nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq154_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq155_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ46_HTML.gif
(4.4)

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

Proof.

We first show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq156_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq157_HTML.gif . By (4.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ47_HTML.gif
(4.5)
By Lemma 4.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq158_HTML.gif is completely continuous. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq159_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq160_HTML.gif . Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ48_HTML.gif
(4.6)
For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq161_HTML.gif , by (4.5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ49_HTML.gif
(4.7)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq162_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq163_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq164_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq165_HTML.gif . Then, Lemma 2.6 yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq166_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq167_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq168_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq169_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq170_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq171_HTML.gif is a contraction. In fact, by (4.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ50_HTML.gif
(4.8)

So, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq172_HTML.gif is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP (1.1).

5. Examples

Example 5.1.

Consider the following BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ51_HTML.gif
(5.1)
We have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ52_HTML.gif
(5.2)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ53_HTML.gif
(5.3)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq173_HTML.gif , and it is easy to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq174_HTML.gif is satisfied. By direct calculations, we can obtain that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq175_HTML.gif . By Theorem 3.1, BVP (5.1) has an unbounded solution.

Example 5.2.

Consider the following BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ54_HTML.gif
(5.4)
In this case, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ55_HTML.gif
(5.5)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ56_HTML.gif
(5.6)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq176_HTML.gif . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.

Declarations

Acknowledgments

The authors are grateful to the referees for valuable suggestions and comments. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 10771117) and the Natural Science Foundation of Shandong Province of China (Y2007A23, Y2008A24). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Qufu Normal University
(2)
Department of Mathematics and Statistics, Curtin University of Technology

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© Lishan Liu et al. 2010

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