Open Access

Monotone Positive Solution of Nonlinear Third-Order BVP with Integral Boundary Conditions

Boundary Value Problems20102010:874959

DOI: 10.1155/2010/874959

Received: 7 September 2010

Accepted: 31 October 2010

Published: 2 November 2010

Abstract

This paper is concerned with the following third-order boundary value problem with integral boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq1_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq2_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq4_HTML.gif . By using the Guo-Krasnoselskii fixed-point theorem, some sufficient conditions are obtained for the existence and nonexistence of monotone positive solution to the above problem.

1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [1].

Recently, third-order two-point or multipoint boundary value problems (BVPs for short) have attracted a lot of attention [217]. It is known that BVPs with integral boundary conditions cover multipoint BVPs as special cases. Although there are many excellent works on third-order two-point or multipoint BVPs, a little work has been done for third-order BVPs with integral boundary conditions. It is worth mentioning that, in 2007, Anderson and Tisdell [18] developed an interval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq5_HTML.gif values whereby a positive solution exists for the following third-order BVP with integral boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ1_HTML.gif
(1.1)
by using the Guo-Krasnoselskii fixed-point theorem. In 2008, Graef and Yang [19] studied the third-order BVP with integral boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ2_HTML.gif
(1.2)

For second-order or fourth-order BVPs with integral boundary conditions, one can refer to [2024].

In this paper, we are concerned with the following third-order BVP with integral boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ3_HTML.gif
(1.3)

Throughout this paper, we always assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq7_HTML.gif . Some sufficient conditions are established for the existence and nonexistence of monotone positive solution to the BVP (1.3). Here, a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq8_HTML.gif of the BVP (1.3) is said to be monotone and positive if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq11_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq12_HTML.gif . Our main tool is the following Guo-Krasnoselskii fixed-point theorem [25].

Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq13_HTML.gif be a Banach space and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq14_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq15_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq17_HTML.gif are bounded open subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq18_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq19_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq20_HTML.gif be a completely continuous operator such that either

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq21_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq23_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq24_HTML.gif , or

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq25_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq27_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq28_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq29_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq30_HTML.gif .

2. Preliminaries

For convenience, we denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq31_HTML.gif .

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq32_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq33_HTML.gif , the BVP
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ4_HTML.gif
(2.1)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ5_HTML.gif
(2.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ6_HTML.gif
(2.3)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq34_HTML.gif be a solution of the BVP (2.1). Then, we may suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ7_HTML.gif
(2.4)
By the boundary conditions in (2.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ8_HTML.gif
(2.5)
Therefore, the BVP (2.1) has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ9_HTML.gif
(2.6)

Lemma 2.2 (see [12]).

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq35_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ10_HTML.gif
(2.7)

Lemma 2.3 (see [26]).

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq36_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ11_HTML.gif
(2.8)

In the remainder of this paper, we always assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq39_HTML.gif .

Lemma 2.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq40_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq41_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq42_HTML.gif , then the unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq43_HTML.gif of the BVP (2.1) satisfies

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq45_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq46_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq48_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq49_HTML.gif .

Proof.

Since (1) is obvious, we only need to prove (2). By (2.2), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ12_HTML.gif
(2.9)

which indicates that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq50_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq51_HTML.gif .

On the one hand, by (2.9) and Lemma 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ13_HTML.gif
(2.10)
On the other hand, in view of (2.2) and Lemma 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ14_HTML.gif
(2.11)
It follows from (2.10) and (2.11) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ15_HTML.gif
(2.12)
which together with Lemma 2.2 implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ16_HTML.gif
(2.13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq52_HTML.gif be equipped with the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq53_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq54_HTML.gif is a Banach space. If we denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ17_HTML.gif
(2.14)
then it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq55_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq56_HTML.gif . Now, we define an operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq57_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq58_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ18_HTML.gif
(2.15)

Obviously, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq59_HTML.gif is a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq60_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq61_HTML.gif is a monotone nonnegative solution of the BVP (1.3).

Lemma 2.5.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq62_HTML.gif is completely continuous.

Proof.

First, by Lemma 2.4, we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq63_HTML.gif .

Next, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq64_HTML.gif is a bounded set. Then there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq65_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq66_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq67_HTML.gif . Now, we will prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq68_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq69_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq70_HTML.gif . Then there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq71_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq72_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ19_HTML.gif
(2.16)
Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq73_HTML.gif , by Lemma 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ20_HTML.gif
(2.17)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq74_HTML.gif is uniformly bounded. At the same time, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq75_HTML.gif , in view of Lemma 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ21_HTML.gif
(2.18)
which shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq76_HTML.gif is also uniformly bounded. This indicates that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq77_HTML.gif is equicontinuous. It follows from Arzela-Ascoli theorem that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq78_HTML.gif has a convergent subsequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq79_HTML.gif . Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq80_HTML.gif converges in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq81_HTML.gif . On the other hand, by the uniform continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq82_HTML.gif , we know that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq83_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq84_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq85_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq86_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ22_HTML.gif
(2.19)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq87_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq88_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq89_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq90_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ23_HTML.gif
(2.20)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq91_HTML.gif is equicontinuous. Again, by Arzela-Ascoli theorem, we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq92_HTML.gif has a convergent subsequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq93_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq94_HTML.gif has a convergent subsequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq95_HTML.gif . Thus, we have shown that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq96_HTML.gif is a compact operator.

Finally, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq97_HTML.gif is continuous. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq98_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq99_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq100_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq101_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq102_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ24_HTML.gif
(2.21)
Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq104_HTML.gif , in view of Lemmas 2.2 and 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ25_HTML.gif
(2.22)
By applying Lebesgue Dominated Convergence theorem, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ26_HTML.gif
(2.23)

which indicates that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq105_HTML.gif is continuous. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq106_HTML.gif is completely continuous.

3. Main Results

For convenience, we define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ27_HTML.gif
(3.1)

Theorem 3.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq107_HTML.gif , then the BVP (1.3) has at least one monotone positive solution.

Proof.

In view of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq108_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq109_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ28_HTML.gif
(3.2)
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq110_HTML.gif , we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq111_HTML.gif so that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ29_HTML.gif
(3.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq112_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq113_HTML.gif , in view of (3.2) and (3.3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ30_HTML.gif
(3.4)
By integrating the above inequality on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq114_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ31_HTML.gif
(3.5)
which together with (3.4) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ32_HTML.gif
(3.6)
On the other hand, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq115_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq116_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ33_HTML.gif
(3.7)
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq117_HTML.gif , we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq118_HTML.gif , so that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ34_HTML.gif
(3.8)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq119_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq120_HTML.gif , in view of (3.7) and (3.8), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ35_HTML.gif
(3.9)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ36_HTML.gif
(3.10)

Therefore, it follows from (3.6), (3.10), and Theorem 1.1 that the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq121_HTML.gif has one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq122_HTML.gif , which is a monotone positive solution of the BVP (1.3).

Theorem 3.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq123_HTML.gif , then the BVP (1.3) has at least one monotone positive solution.

Proof.

The proof is similar to that of Theorem 3.1 and is therefore omitted.

Theorem 3.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq124_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq126_HTML.gif , then the BVP (1.3) has no monotone positive solution.

Proof.

Suppose on the contrary that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq127_HTML.gif is a monotone positive solution of the BVP (1.3). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq129_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq130_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ37_HTML.gif
(3.11)
By integrating the above inequality on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq131_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ38_HTML.gif
(3.12)
which together with (3.11) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ39_HTML.gif
(3.13)

This is a contradiction. Therefore, the BVP (1.3) has no monotone positive solution.

Similarly, we can prove the following theorem.

Theorem 3.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq132_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq134_HTML.gif , then the BVP (1.3) has no monotone positive solution.

Example 3.5.

Consider the following BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ40_HTML.gif
(3.14)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq136_HTML.gif , if we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq137_HTML.gif , then it is easy to compute that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ41_HTML.gif
(3.15)
which shows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ42_HTML.gif
(3.16)

So, it follows from Theorem 3.1 that the BVP (3.14) has at least one monotone positive solution.

Declarations

Acknowledgment

This work was supported by the National Natural Science Foundation of China (10801068).

Authors’ Affiliations

(1)
Department of Applied Mathematics, Lanzhou University of Technology

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Copyright

© The Author(s) Jian-Ping Sun and Hai-Bao Li. 2010

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