Open Access

Positive Solutions for Fourth-Order Singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq1_HTML.gif -Laplacian Differential Equations with Integral Boundary Conditions

Boundary Value Problems20102010:862079

DOI: 10.1155/2010/862079

Received: 7 April 2010

Accepted: 12 August 2010

Published: 19 August 2010

Abstract

By employing upper and lower solutions method together with maximal principle, we establish a necessary and sufficient condition for the existence of pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq2_HTML.gif as well as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq3_HTML.gif positive solutions for fourth-order singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq4_HTML.gif -Laplacian differential equations with integral boundary conditions. Our nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq5_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq7_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq8_HTML.gif . The dual results for the other integral boundary condition are also given.

1. Introduction

In this paper, we consider the existence of positive solutions for the following nonlinear fourth-order singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq9_HTML.gif -Laplacian differential equations with integral boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq17_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq18_HTML.gif is nonnegative. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq20_HTML.gif . Throughout this paper, we always assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq22_HTML.gif and nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq23_HTML.gif satisfies the following hypothesis:

(H) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq24_HTML.gif is continuous, nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq25_HTML.gif and nonincreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq26_HTML.gif for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq27_HTML.gif , and there exists a real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq28_HTML.gif such that, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq29_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ2_HTML.gif
(1.2)
there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq30_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq32_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq33_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ3_HTML.gif
(1.3)

Remark 1.1.

Condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq34_HTML.gif is used to discuss the existence and uniqueness of smooth positive solutions in [1].

(i)Inequality (1.2) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ4_HTML.gif
(1.4)

Conversely, (1.4) implies (1.2).

(ii)Inequality (1.3) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ5_HTML.gif
(1.5)

Conversely, (1.5) implies (1.3).

Remark 1.2.

Typical functions that satisfy condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq35_HTML.gif are those taking the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq36_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq37_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq40_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq41_HTML.gif .

Remark 1.3.

It follows from (1.2) and (1.3) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ6_HTML.gif
(1.6)

Boundary value problems with integral boundary conditions arise in variety of different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. They include two point, three point, and nonlocal boundary value problems (see [25]) as special cases and have attracted much attention of many researchers, such as Gallardo, Karakostas, Tsamatos, Lomtatidze, Malaguti, Yang, Zhang, and Feng (see [613], e.g.). For more information about the general theory of integral equations and their relation to boundary value problems, the reader is referred to the book by Corduneanu [14] and Agarwal and O'Regan [15].

Recently, Zhang et al. [13] studied the existence and nonexistence of symmetric positive solutions for the following nonlinear fourth-order boundary value problems:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq43_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq46_HTML.gif is nonnegative, symmetric on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq48_HTML.gif is continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq49_HTML.gif are nonnegative, symmetric on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq50_HTML.gif .

To seek necessary and sufficient conditions for the existence of solutions to the ordinary differential equations is important and interesting, but difficult. Professors Wei [16, 17], Du and Zhao [18], Graef and Kong [19], Zhang and Liu [20], and others have done much excellent work under some suitable conditions in this direction. To the author's knowledge, there are no necessary and sufficient conditions available in the literature for the existence of solutions for integral boundary value problem (1.1). Motivated by above papers, the purpose of this paper is to fill this gap. It is worth pointing out that the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq51_HTML.gif permits singularity not only at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq52_HTML.gif but also at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq53_HTML.gif . By singularity, we mean that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq54_HTML.gif is allowed to be unbounded at the points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq56_HTML.gif .

2. Preliminaries and Several Lemmas

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq58_HTML.gif is called a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq59_HTML.gif (positive) solution of BVP (1.1) if it satisfies (1.1) ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq60_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq61_HTML.gif ). A https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq62_HTML.gif (positive) solution of (1.1) is called a psuedo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq63_HTML.gif (positive) solution if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq65_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq66_HTML.gif . Denote that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ8_HTML.gif
(2.1)

Definition 2.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq67_HTML.gif is called a lower solution of BVP (1.1) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq68_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ9_HTML.gif
(2.2)

Definition 2.2.

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq69_HTML.gif is called an upper solution of BVP (1.1) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq70_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ10_HTML.gif
(2.3)
Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq71_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ11_HTML.gif
(2.4)

To prove the main results, we need the following maximum principle.

Lemma 2.3 (Maximum principle).

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq72_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq73_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq74_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq77_HTML.gif

Proof.

Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ12_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ13_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ14_HTML.gif
(2.7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ15_HTML.gif
(2.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ16_HTML.gif
(2.9)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq78_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq79_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq80_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq81_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ17_HTML.gif
(2.10)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ18_HTML.gif
(2.11)
then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ19_HTML.gif
(2.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ20_HTML.gif
(2.13)
By integration of (2.12), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ21_HTML.gif
(2.14)
Integrating again, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ22_HTML.gif
(2.15)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq82_HTML.gif in (2.15), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ23_HTML.gif
(2.16)
Substituting (2.13) and (2.16) into (2.15), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ24_HTML.gif
(2.17)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ25_HTML.gif
(2.18)
Notice that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ26_HTML.gif
(2.19)
therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ27_HTML.gif
(2.20)
Substituting (2.20) into (2.17), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ28_HTML.gif
(2.21)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ29_HTML.gif
(2.22)
Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq83_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq85_HTML.gif . From (2.21), it is easily seen that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq86_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq87_HTML.gif By (2.11), we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq88_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq89_HTML.gif Thus, we have proved that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq91_HTML.gif . Similarly, the solution of (2.5) and (2.7) can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ30_HTML.gif
(2.23)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ31_HTML.gif
(2.24)

By (2.23), we can get that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq92_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq93_HTML.gif

Lemma 2.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq94_HTML.gif holds. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq95_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq96_HTML.gif positive solution of BVP (1.1). Then there exist two constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq97_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ32_HTML.gif
(2.25)

Proof.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq98_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq99_HTML.gif positive solution of BVP (1.1). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq100_HTML.gif can be stated as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ33_HTML.gif
(2.26)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ34_HTML.gif
(2.27)
It is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ35_HTML.gif
(2.28)
By (2.26), for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq101_HTML.gif , we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ36_HTML.gif
(2.29)
From (2.26) and (2.27), we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ37_HTML.gif
(2.30)
Setting
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ38_HTML.gif
(2.31)

then from (2.29) and (2.30), we have (2.25).

Lemma 2.5.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq102_HTML.gif holds. And assume that there exist lower and upper solutions of BVP (1.1), respectively, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq104_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq106_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq107_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq108_HTML.gif . Then BVP (1.1) has at least one https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq109_HTML.gif positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq110_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq111_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq112_HTML.gif . If, in addition, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq113_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ39_HTML.gif
(2.32)

then the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq114_HTML.gif of BVP (1.1) is a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq115_HTML.gif positive solution.

Proof.

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq116_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq117_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq118_HTML.gif , we defined an auxiliary function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ40_HTML.gif
(2.33)

By condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq119_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq120_HTML.gif is continuous.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq121_HTML.gif be sequences satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq123_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq124_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq125_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq127_HTML.gif , be sequences satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ41_HTML.gif
(2.34)
For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq128_HTML.gif , consider the following nonsingular problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ42_HTML.gif
(2.35)
For convenience, we define linear operators as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ43_HTML.gif
(2.36)
By the proof of Lemma 2.3, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq129_HTML.gif is a solution of problem (2.35) if and only if it is the fixed point of the following operator equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ44_HTML.gif
(2.37)

By (2.33), it is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq130_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq131_HTML.gif is a bounded set. Moreover, by the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq132_HTML.gif , we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq133_HTML.gif is a compact operator and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq134_HTML.gif is a relatively compact set. So, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq135_HTML.gif is a completely continuous operator. In addition, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq136_HTML.gif is a solution of (2.35) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq137_HTML.gif is a fixed point of operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq138_HTML.gif . Using the Shauder's fixed point theorem, we assert that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq139_HTML.gif has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq140_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq141_HTML.gif , we can get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq142_HTML.gif

We claim that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ45_HTML.gif
(2.38)
From this it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ46_HTML.gif
(2.39)
Indeed, suppose by contradiction that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq143_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq144_HTML.gif . By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq145_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ47_HTML.gif
(2.40)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ48_HTML.gif
(2.41)
On the other hand, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq146_HTML.gif is an upper solution of (1.1), we also have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ49_HTML.gif
(2.42)
Then setting
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ50_HTML.gif
(2.43)
By (2.41) and (2.42), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ51_HTML.gif
(2.44)
By Lemma 2.3, we can conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ52_HTML.gif
(2.45)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ53_HTML.gif
(2.46)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ54_HTML.gif
(2.47)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ55_HTML.gif
(2.48)
By Lemma 2.3, we can conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ56_HTML.gif
(2.49)

which contradicts the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq147_HTML.gif Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq148_HTML.gif is impossible.

Similarly, we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq149_HTML.gif So, we have shown that (2.38) holds.

Using the method of [21] and Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq150_HTML.gif .2 in [22], we can obtain that there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq151_HTML.gif positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq152_HTML.gif of (1.1) such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq153_HTML.gif and a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq154_HTML.gif converging to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq155_HTML.gif on any compact subintervals of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq156_HTML.gif .

In addition, if (2.32) holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq157_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq158_HTML.gif is absolutely integrable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq159_HTML.gif . This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq160_HTML.gif is a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq161_HTML.gif positive solution of (1.1).

3. The Main Results

Theorem 3.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq162_HTML.gif holds, then a necessary and sufficient condition for BVP (1.1) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq163_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ57_HTML.gif
(3.1)

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq164_HTML.gif is a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq165_HTML.gif positive solution of (1.1). Then both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq167_HTML.gif exist. By Lemma 2.4, there exist two constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq168_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ58_HTML.gif
(3.2)
Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq169_HTML.gif . This together with condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq170_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ59_HTML.gif
(3.3)
On the other hand, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq171_HTML.gif is a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq172_HTML.gif positive solution of (1.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ60_HTML.gif
(3.4)
Otherwise, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq173_HTML.gif . By the proof of Lemma 2.3, we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq175_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq176_HTML.gif which contradicts that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq177_HTML.gif is a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq178_HTML.gif positive solution. Therefore, there exists a positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq179_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq180_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq181_HTML.gif . By (1.6) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ61_HTML.gif
(3.5)
Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq182_HTML.gif , which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ62_HTML.gif
(3.6)
It follows from (3.3) and (3.6) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ63_HTML.gif
(3.7)

which is the desired inequality.

Sufficiency.

First, we prove the existence of a pair of upper and lower solutions. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq183_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq184_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ64_HTML.gif
(3.8)
Otherwise, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq185_HTML.gif , then there exists a real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq186_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq187_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq188_HTML.gif , which contradicts the condition that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq189_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq190_HTML.gif . In view of condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq191_HTML.gif and (3.8), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ65_HTML.gif
(3.9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ66_HTML.gif
(3.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq192_HTML.gif .

Suppose that (3.1) holds. Firstly, we define the linear operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq193_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq194_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ67_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ68_HTML.gif
(3.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq195_HTML.gif is given by (2.27). Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ69_HTML.gif
(3.13)
It is easy to know from (3.11) and (3.12) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq196_HTML.gif By Lemma 2.4, we know that there exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq197_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ70_HTML.gif
(3.14)
Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq198_HTML.gif sufficiently small, then by (3.10), we get that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq199_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ71_HTML.gif
(3.15)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ72_HTML.gif
(3.16)
Thus, from (3.14) and (3.16), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ73_HTML.gif
(3.17)
Considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq200_HTML.gif , it follows from (3.15), (3.17), and condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq201_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ74_HTML.gif
(3.18)
From (3.13) and (3.16), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ75_HTML.gif
(3.19)

Thus, we have shown that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq203_HTML.gif are lower and upper solutions of BVP (1.1), respectively.

Additionally, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq204_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq205_HTML.gif , by (3.17) and condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq206_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ76_HTML.gif
(3.20)

From (3.1), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq207_HTML.gif So, it follows from Lemma 2.5 that BVP (1.1) admits a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq208_HTML.gif positive solution such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq209_HTML.gif

Remark 3.2.

Lin et al. [23, 24] considered the existence and uniqueness of solutions for some fourth-order and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq210_HTML.gif conjugate boundary value problems when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq211_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ77_HTML.gif
(3.21)

under the following condition:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq213_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq214_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq215_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq216_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ78_HTML.gif
(3.22)

Lei et al. [25] and Liu and Yu [26] investigated the existence and uniqueness of positive solutions to singular boundary value problems under the following condition:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq218_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq220_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq222_HTML.gif is nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq223_HTML.gif and nonincreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq224_HTML.gif .

Obviously, (3.21)-(3.22) imply condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq225_HTML.gif and condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq226_HTML.gif implies condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq227_HTML.gif . So, condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq228_HTML.gif is weaker than conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq229_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq230_HTML.gif . Thus, functions considered in this paper are wider than those in [2326].

In the following, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq231_HTML.gif admits the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq232_HTML.gif , that is, nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq233_HTML.gif is not mixed monotone on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq234_HTML.gif , but monotone with respect https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq235_HTML.gif , BVP (1.1) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ79_HTML.gif
(3.23)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq236_HTML.gif satisfies one of the following:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq238_HTML.gif is continuous, nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq239_HTML.gif , for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq240_HTML.gif , there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq242_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq243_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq244_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ80_HTML.gif
(3.24)

Theorem 3.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq245_HTML.gif holds, then a necessary and sufficient condition for BVP (3.23) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq246_HTML.gif positive solution is that the following integral condition holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ81_HTML.gif
(3.25)

Proof.

The proof is similar to that of Theorem 3.1; we omit the details.

Theorem 3.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq247_HTML.gif holds, then a necessary and sufficient condition for problem (3.23) to have a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq248_HTML.gif positive solution is that the following integral condition holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ82_HTML.gif
(3.26)

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq249_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq250_HTML.gif positive solution of BVP (3.23). By Lemma 2.4, there exist two constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq252_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq253_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ83_HTML.gif
(3.27)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq254_HTML.gif be a constant such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq255_HTML.gif . By condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq256_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ84_HTML.gif
(3.28)
By virtue of (3.28), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ85_HTML.gif
(3.29)
By boundary value condition, we know that there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq257_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ86_HTML.gif
(3.30)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq258_HTML.gif by integration of (3.29), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ87_HTML.gif
(3.31)
Integrating (3.31), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ88_HTML.gif
(3.32)
Exchanging the order of integration, we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ89_HTML.gif
(3.33)
Similarly, by integration of (3.29), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ90_HTML.gif
(3.34)
Equations (3.33) and (3.34) imply that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ91_HTML.gif
(3.35)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq259_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq260_HTML.gif positive solution of BVP (1.1), there exists a positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq261_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq262_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq263_HTML.gif . On the other hand, choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq264_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq265_HTML.gif . By condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq266_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ92_HTML.gif
(3.36)
Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq267_HTML.gif , which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ93_HTML.gif
(3.37)
It follows from (3.35) and (3.37) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ94_HTML.gif
(3.38)

which is the desired inequality.

Sufficiency.

Suppose that (3.26) holds. Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ95_HTML.gif
(3.39)
It is easy to know, from (3.11) and (3.26), that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ96_HTML.gif
(3.40)
Thus, (3.12), (3.39), and (3.40) imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq268_HTML.gif By Lemma 2.4, we know that there exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq269_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ97_HTML.gif
(3.41)
Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq270_HTML.gif sufficiently small, then by (3.10), we get that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq271_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ98_HTML.gif
(3.42)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ99_HTML.gif
(3.43)
Thus, from (3.41) and (3.43), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ100_HTML.gif
(3.44)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq272_HTML.gif , it follows from (3.42)–(3.44) and condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq273_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ101_HTML.gif
(3.45)
From (3.39) and (3.43), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ102_HTML.gif
(3.46)

Thus, we have shown that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq275_HTML.gif are lower and upper solutions of BVP (1.1), respectively.

From the first conclusion of Lemma 2.5, we conclude that problem (1.1) has at least one https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq276_HTML.gif positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq277_HTML.gif .

4. Dual Results

Consider the fourth-order singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq278_HTML.gif -Laplacian differential equations with integral conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ103_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ104_HTML.gif
(4.2)
Firstly, we define the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq279_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ105_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq280_HTML.gif is given by (2.27).

By analogous methods, we have the following results.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq281_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq282_HTML.gif positive solution of problem (4.1). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq283_HTML.gif can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ106_HTML.gif
(4.4)

Theorem 4.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq284_HTML.gif holds, then a necessary and sufficient condition for (4.1) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq285_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ107_HTML.gif
(4.5)

Theorem 4.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq286_HTML.gif holds, then a necessary and sufficient condition for problem (4.2) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq287_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ108_HTML.gif
(4.6)

Theorem 4.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq288_HTML.gif holds, then a necessary and sufficient condition for problem (4.2) to have a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq289_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ109_HTML.gif
(4.7)
Consider the fourth-order singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq290_HTML.gif -Laplacian differential equations with integral conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ110_HTML.gif
(4.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ111_HTML.gif
(4.9)
Define the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq291_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ112_HTML.gif
(4.10)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq292_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq293_HTML.gif positive solution of problem (4.8). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq294_HTML.gif can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ113_HTML.gif
(4.11)

Theorem 4.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq295_HTML.gif holds, then a necessary and sufficient condition for problem (4.8) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq296_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ114_HTML.gif
(4.12)

Theorem 4.5.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq297_HTML.gif holds, then a necessary and sufficient condition for problem (4.9) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq298_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ115_HTML.gif
(4.13)

Theorem 4.6.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq299_HTML.gif holds, then a necessary and sufficient condition for problem (4.9) to have a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq300_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ116_HTML.gif
(4.14)
Consider the fourth-order singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq301_HTML.gif -Laplacian differential equations with integral conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ117_HTML.gif
(4.15)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ118_HTML.gif
(4.16)
Define the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq302_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ119_HTML.gif
(4.17)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq303_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq304_HTML.gif positive solution of problem (4.15). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq305_HTML.gif can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ120_HTML.gif
(4.18)

Theorem 4.7.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq306_HTML.gif holds, then a necessary and sufficient condition for problem (4.15) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq307_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ121_HTML.gif
(4.19)

Theorem 4.8.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq308_HTML.gif holds, then a necessary and sufficient condition for problem (4.16) to have a pseudo- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq309_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ122_HTML.gif
(4.20)

Theorem 4.9.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq310_HTML.gif holds, then a necessary and sufficient condition for problem (4.16) to have a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq311_HTML.gif positive solution is that the following integral condition holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ123_HTML.gif
(4.21)

Declarations

Acknowledgments

The project is supported financially by a Project of Shandong Province Higher Educational Science and Technology Program (no. J10LA53) and the National Natural Science Foundation of China (no. 10971179).

Authors’ Affiliations

(1)
Department of Mathematics, Huazhong University of Science and Technology
(2)
Department of Mathematics, Liaocheng University
(3)
Department of Applied Mathematics, Shandong University of Science and Technology

References

  1. Du X, Zhao Z:Existence and uniqueness of smooth positive solutions to a class of singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq312_HTML.gif -point boundary value problems. Boundary Value Problems 2009, 2009:-13.
  2. Graef JR, Webb JRL: Third order boundary value problems with nonlocal boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,71(5-6):1542-1551. 10.1016/j.na.2008.12.047MathSciNetView ArticleMATH
  3. Graef JR, Yang B: Positive solutions of a third order nonlocal boundary value problem. Discrete and Continuous Dynamical Systems. Series S 2008,1(1):89-97.MathSciNetMATH
  4. Infante G, Webb JRL: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proceedings of the Edinburgh Mathematical Society. Series II 2006,49(3):637-656. 10.1017/S0013091505000532MathSciNetView ArticleMATH
  5. Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society. Second Series 2006,74(3):673-693. 10.1112/S0024610706023179MathSciNetView ArticleMATH
  6. Gallardo JM: Second-order differential operators with integral boundary conditions and generation of analytic semigroups. The Rocky Mountain Journal of Mathematics 2000,30(4):1265-1292. 10.1216/rmjm/1021477351MathSciNetView ArticleMATH
  7. Karakostas GL, Tsamatos PCh: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electronic Journal of Differential Equations 2002, (30):1-17.
  8. Lomtatidze A, Malaguti L: On a nonlocal boundary value problem for second order nonlinear singular differential equations. Georgian Mathematical Journal 2000,7(1):133-154.MathSciNetMATH
  9. Yang Z: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2005,62(7):1251-1265. 10.1016/j.na.2005.04.030MathSciNetView ArticleMATH
  10. Yang Z: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2006,65(8):1489-1511. 10.1016/j.na.2005.10.025MathSciNetView ArticleMATH
  11. Yang Z: Positive solutions of a second-order integral boundary value problem. Journal of Mathematical Analysis and Applications 2006,321(2):751-765. 10.1016/j.jmaa.2005.09.002MathSciNetView ArticleMATH
  12. Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. Journal of Computational and Applied Mathematics 2008,222(2):351-363. 10.1016/j.cam.2007.11.003MathSciNetView ArticleMATH
  13. Zhang X, Feng M, Ge W:Symmetric positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq313_HTML.gif -Laplacian fourth-order differential equations with integral boundary conditions. Journal of Computational and Applied Mathematics 2008,222(2):561-573. 10.1016/j.cam.2007.12.002MathSciNetView ArticleMATH
  14. Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge, UK; 1991:x+366.View ArticleMATH
  15. Agarwal RP, O'Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001.View ArticleMATH
  16. Wei Z:Positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq314_HTML.gif th-order singular sub-linear https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq315_HTML.gif -point boundary value problems. Applied Mathematics and Computation 2006,182(2):1280-1295. 10.1016/j.amc.2006.05.014MathSciNetView ArticleMATH
  17. Wei Z:A necessary and sufficient condition for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq316_HTML.gif th-order singular super-linear https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq317_HTML.gif -point boundary value problems. Journal of Mathematical Analysis and Applications 2007,327(2):930-947. 10.1016/j.jmaa.2006.04.056MathSciNetView ArticleMATH
  18. Du X, Zhao Z: A necessary and sufficient condition of the existence of positive solutions to singular sublinear three-point boundary value problems. Applied Mathematics and Computation 2007,186(1):404-413. 10.1016/j.amc.2006.07.120MathSciNetView ArticleMATH
  19. Graef JR, Kong L: Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems. Journal of Mathematical Analysis and Applications 2007,331(2):1467-1484. 10.1016/j.jmaa.2006.09.046MathSciNetView ArticleMATH
  20. Zhang X, Liu L:A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq318_HTML.gif -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2008,68(10):3127-3137. 10.1016/j.na.2007.03.006MathSciNetView ArticleMATH
  21. Zhang Y: Positive solutions of singular sublinear Emden-Fowler boundary value problems. Journal of Mathematical Analysis and Applications 1994,185(1):215-222. 10.1006/jmaa.1994.1243MathSciNetView ArticleMATH
  22. Hartman P: Ordinary Differential Equations. 2nd edition. Birkhäuser, Boston, Mass, USA; 1982:xv+612.MATH
  23. Lin X, Jiang D, Li X: Existence and uniqueness of solutions for singular fourth-order boundary value problems. Journal of Computational and Applied Mathematics 2006,196(1):155-161. 10.1016/j.cam.2005.08.016MathSciNetView ArticleMATH
  24. Lin X, Jiang D, Li X:Existence and uniqueness of solutions for singular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq319_HTML.gif conjugate boundary value problems. Computers & Mathematics with Applications 2006,52(3-4):375-382. 10.1016/j.camwa.2006.03.019MathSciNetView ArticleMATH
  25. Lei P, Lin X, Jiang D: Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):2773-2779. 10.1016/j.na.2007.08.049MathSciNetView ArticleMATH
  26. Liu Y, Yu H: Existence and uniqueness of positive solution for singular boundary value problem. Computers & Mathematics with Applications 2005,50(1-2):133-143. 10.1016/j.camwa.2005.01.022MathSciNetView ArticleMATH

Copyright

© The Author(s) Xingqiu Zhang and Yujun Cui. 2010

This article is published under license to BioMed Central Ltd. 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.