Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq1_HTML.gif -Laplacian

Boundary Value Problems20102011:654871

DOI: 10.1155/2011/654871

Received: 2 February 2010

Accepted: 5 June 2010

Published: 29 June 2010

Abstract

By using the fixed point theory for completely continuous operator, this paper investigates the existence of positive solutions for a class of fourth-order impulsive boundary value problems with integral boundary conditions and one-dimensional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq2_HTML.gif -Laplacian. Moreover, we offer some interesting discussion of the associated boundary value problems. Upper and lower bounds for these positive solutions also are given, so our work is new.

1. Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. [1]; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al. [2]. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see, e.g., [318] and references cited therein).

Moreover, the theory of boundary-value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For boundary-value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [19], Karakostas and Tsamatos [20], Lomtatidze and Malaguti [21], and the references therein. For more information about the general theory of integral equations and their relation with boundary-value problems, we refer to the book of Corduneanu [22] and Agarwal and O'Regan [23].

On the other hand, boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary-value problems as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention in the literature. To identify a few, we refer the reader to [2446] and references therein. In particular, we would like to mention some results of Zhang et al. [34], Kang et al. [44], and Webb et al. [45]. In [34], Zhang et al. studied the following fourth-order boundary value problem with integral boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq3_HTML.gif is a positive parameter, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq5_HTML.gif is the zero element of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq6_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq7_HTML.gif . The authors investigated the multiplicity of positive solutions to problem (1.1) by using the fixed point index theory in cone for strict set contraction operator.

In [44], Kang et al. have improved and generalized the work of [34] by applying the fixed point theory in cone for a strict set contraction operator; they proved that there exist various results on the existence of positive solutions to a class of fourth-order singular boundary value problems with integral boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq8_HTML.gif and may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq9_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq10_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq11_HTML.gif are continuous and may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq12_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq13_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq14_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq15_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq16_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq17_HTML.gif are nonnegative, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq18_HTML.gif .

More recently, by using a unified approach, Webb et al. [45] considered the widely studied boundary conditions corresponding to clamped and hinged ends and many nonlocal boundary conditions and established excellent existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ3_HTML.gif
(1.3)
subject to various boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ4_HTML.gif
(1.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq19_HTML.gif denotes a linear functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq20_HTML.gif given by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq21_HTML.gif involving a Stieltjes integral, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq22_HTML.gif is a function of bounded variation.

At the same time, we notice that there has been a considerable attention on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq23_HTML.gif -Laplacian BVPs [18, 32, 35, 36, 38, 42] as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq24_HTML.gif -Laplacian appears in the study of flow through porous media ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq25_HTML.gif ), nonlinear elasticity ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq26_HTML.gif ), glaciology ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq27_HTML.gif ), and so forth. Here, it is worth mentioning that Liu et al. [43] considered the following fourth-order four-point boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ5_HTML.gif
(1.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq29_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq30_HTML.gif . By using upper and lower solution method, fixed-point theorems, and the properties of Green's function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq32_HTML.gif , the authors give sufficient conditions for the existence of one positive solution.

Motivated by works mentioned above, in this paper, we consider the existence of positive solutions for a class of boundary value problems with integral boundary conditions of fourth-order impulsive differential equations:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ6_HTML.gif
(1.6)

Here http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq33_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq34_HTML.gif -Laplace operator, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq37_HTML.gif (where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq38_HTML.gif is fixed positive integer) are fixed points with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq40_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq42_HTML.gif represent the right-hand limit and left-hand limit of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq43_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq44_HTML.gif , respectively, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq45_HTML.gif is nonnegative.

For the case of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq46_HTML.gif , problem (1.6) reduces to the problem studied by Zhang et al. in [33]. By using the fixed point theorem in cone, the authors obtained some sufficient conditions for the existence and multiplicity of symmetric positive solutions for a class of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq47_HTML.gif -Laplacian fourth-order differential equations with integral boundary conditions.

For the case of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq48_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq49_HTML.gif , problem (1.6) is related to fourth-order two-points boundary value problem of ODE. Under this case, problem (1.6) has received considerable attention (see, e.g., [4042] and references cited therein). Aftabizadeh [40] showed the existence of a solution to problem (1.6) under the restriction that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq50_HTML.gif is a bounded function. Bai and Wang [41] have applied the fixed point theorem and degree theory to establish existence, uniqueness, and multiplicity of positive solutions to problem (1.6). Ma and Wang [42] have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq51_HTML.gif is either superlinear or sublinear on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq52_HTML.gif by employing the fixed point theorem of cone extension or compression.

Being directly inspired by [18, 34, 43], in the present paper, we consider some existence results for problem (1.6) in a specially constructed cone by using the fixed point theorem. The main features of this paper are as follows. Firstly, comparing with [3943], we discuss the impulsive boundary value problem with integral boundary conditions, that is, problem (1.6) includes fourth-order two-, three-, multipoint, and nonlocal boundary value problems as special cases. Secondly, the conditions are weaker than those of [33, 34, 46], and we consider the case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq53_HTML.gif . Finally, comparing with [33, 34, 3943, 46], upper and lower bounds for these positive solutions also are given. Hence, we improve and generalize the results of [33, 34, 3943, 46] to some degree, and so, it is interesting and important to study the existence of positive solutions of problem (1.6).

The organization of this paper is as follows. We shall introduce some lemmas in the rest of this section. In Section 2, we provide some necessary background. In particular, we state some properties of the Green's function associated with problem (1.6). In Section 3, the main results will be stated and proved. Finally, in Section 4, we offer some interesting discussion of the associated problem (1.6).

To obtain positive solutions of problem (1.6), the following fixed point theorem in cones is fundamental, which can be found in [47, page 94].

Lemma 1.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq55_HTML.gif be two bounded open sets in Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq56_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq58_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq59_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq60_HTML.gif and let operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq61_HTML.gif be completely continuous. Suppose that one of the following two conditions is satisfied:

(a) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq62_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq63_HTML.gif ;

(b) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq64_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq65_HTML.gif .

Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq66_HTML.gif has at least one fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq67_HTML.gif .

2. Preliminaries

In order to define the solution of problem (1.6), we shall consider the following space.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq68_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ7_HTML.gif
(2.1)
Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq69_HTML.gif is a real Banach space with norm
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ8_HTML.gif
(2.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq70_HTML.gif .

A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq71_HTML.gif is called a solution of problem (1.6) if it satisfies (1.6).

To establish the existence of multiple positive solutions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq72_HTML.gif of problem (1.6), let us list the following assumptions:

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq74_HTML.gif ;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq76_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ9_HTML.gif
(2.3)
Write
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ10_HTML.gif
(2.4)

From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq77_HTML.gif , it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq78_HTML.gif .

We shall reduce problem (1.6) to an integral equation. To this goal, firstly by means of the transformation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ11_HTML.gif
(2.5)
we convert problem (1.6) into
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ12_HTML.gif
(2.6)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ13_HTML.gif
(2.7)

Lemma 2.1.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq80_HTML.gif hold. Then problem (2.6) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq81_HTML.gif given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ14_HTML.gif
(2.8)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ15_HTML.gif
(2.9)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ16_HTML.gif
(2.10)

Proof.

The proof follows by routine calculations.

Write http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq82_HTML.gif . Then from (2.9) and (2.10), we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq83_HTML.gif have the following properties.

Proposition 2.2.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq84_HTML.gif holds, then we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ17_HTML.gif
(2.11)

Proposition 2.3.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq85_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ18_HTML.gif
(2.12)

Proposition 2.4.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq86_HTML.gif holds, then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq87_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ19_HTML.gif
(2.13)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ20_HTML.gif
(2.14)

Proof.

By (2.6) and (2.12), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ21_HTML.gif
(2.15)
On the other hand, noticing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq88_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ22_HTML.gif
(2.16)

The proof of Proposition 2.4 is complete.

Remark 2.5.

From (2.9) and (2.13), we can obtain that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ23_HTML.gif
(2.17)

Lemma 2.6.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq90_HTML.gif hold, then problem (2.7) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq91_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq92_HTML.gif can be expressed in the following form:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ24_HTML.gif
(2.18)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ25_HTML.gif
(2.19)

and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq93_HTML.gif is defined in (2.10).

Proof.

First suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq94_HTML.gif is a solution of problem (2.7).

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq95_HTML.gif it is easy to see by integration of problem (2.7) that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ26_HTML.gif
(2.20)
If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq96_HTML.gif then integrate from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq97_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq98_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ27_HTML.gif
(2.21)
Similarly, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq99_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ28_HTML.gif
(2.22)
Integrating again, we can get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ29_HTML.gif
(2.23)
Letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq100_HTML.gif in (2.23), we find
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ30_HTML.gif
(2.24)
Substituting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq101_HTML.gif and (2.24) into (2.23), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ31_HTML.gif
(2.25)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ32_HTML.gif
(2.26)
Therefore, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ33_HTML.gif
(2.27)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ34_HTML.gif
(2.28)
Then,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ35_HTML.gif
(2.29)

and the proof of sufficient is complete.

Conversely, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq102_HTML.gif is a solution of (2.18).

Direct differentiation of (2.18) implies, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq103_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ36_HTML.gif
(2.30)
Evidently,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ37_HTML.gif
(2.31)

The Lemma is proved.

Remark 2.7.

From (2.19), we can prove that the properties of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq104_HTML.gif are similar to that of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq105_HTML.gif .

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq106_HTML.gif is a solution of problem (1.6). Then from Lemmas 2.6 and 2.1, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ38_HTML.gif
(2.32)
For the sake of applying Lemma 1.1, we construct a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq107_HTML.gif via
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ39_HTML.gif
(2.33)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ40_HTML.gif
(2.34)

It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq108_HTML.gif is a closed convex cone of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq109_HTML.gif .

Define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq110_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ41_HTML.gif
(2.35)

From (2.35), we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq111_HTML.gif is a solution of problem (1.6) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq112_HTML.gif is a fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq113_HTML.gif .

Definition 2.8 (see [1]).

The set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq114_HTML.gif is said to be quasi-equicontinuous in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq115_HTML.gif if for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq116_HTML.gif there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq117_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq118_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ42_HTML.gif
(2.36)

We present the following result about relatively compact sets in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq119_HTML.gif which is a consequence of the Arzela-Ascoli Theorem. The reader can find its proof partially in [1].

Lemma 2.9.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq120_HTML.gif is relatively compact if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq121_HTML.gif is bounded and quasi-equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq122_HTML.gif .

Write
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ43_HTML.gif
(2.37)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq123_HTML.gif .

Lemma 2.10.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq124_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq125_HTML.gif hold. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq127_HTML.gif is completely continuous.

Proof.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq128_HTML.gif , it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq129_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ44_HTML.gif
(2.38)

From (2.35) and Remark 2.5, we obtain the following cases.

Case 1.

if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq130_HTML.gif , noticing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq131_HTML.gif , then we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ45_HTML.gif
(2.39)

Case 2.

if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq132_HTML.gif , noticing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq134_HTML.gif , then we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ46_HTML.gif
(2.40)

Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq135_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq136_HTML.gif . Also, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq137_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq138_HTML.gif . Hence we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq139_HTML.gif .

Next, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq140_HTML.gif is completely continuous.

It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq141_HTML.gif is continuous. Now we prove http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq142_HTML.gif is relatively compact.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq143_HTML.gif be a bounded set. Then, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq144_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ47_HTML.gif
(2.41)

Therefore http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq145_HTML.gif is uniformly bounded.

On the other hand, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq146_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq147_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ48_HTML.gif
(2.42)
and by the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq148_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ49_HTML.gif
(2.43)

and then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq149_HTML.gif is quasi-equicontinuous. It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq150_HTML.gif is relatively compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq151_HTML.gif by Lemma 2.9. So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq152_HTML.gif is completely continuous.

3. Main Results

In this section, we apply Lemma 1.1 to establish the existence of positive solutions of problem (1.6). We begin by introducing the following conditions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq154_HTML.gif .

There exist numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq156_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ50_HTML.gif
(3.1)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ51_HTML.gif
(3.2)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq157_HTML.gif is defined in (2.4), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq158_HTML.gif are defined in (2.14), respectively, and write
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ52_HTML.gif
(3.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq159_HTML.gif denotes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq160_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq161_HTML.gif

Theorem 3.1.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq162_HTML.gif hold. Then problem (1.6) has at least one positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq163_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ53_HTML.gif
(3.4)

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq164_HTML.gif be the cone preserving, completely continuous operator that was defined by (2.35). For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq165_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq166_HTML.gif , (2.13), (2.19), and (3.1) imply
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ54_HTML.gif
(3.5)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ55_HTML.gif
(3.6)
Now if we let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq167_HTML.gif , then (3.5) shows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ56_HTML.gif
(3.7)
Further, let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ57_HTML.gif
(3.8)
Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq169_HTML.gif implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ58_HTML.gif
(3.9)
that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ59_HTML.gif
(3.10)
Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq170_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq171_HTML.gif . Therefore, for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq172_HTML.gif , (3.2) implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ60_HTML.gif
(3.11)
that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq173_HTML.gif implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ61_HTML.gif
(3.12)

Applying (b) of Lemma 1.1 to (3.7) and (3.12) yields that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq174_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq175_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq176_HTML.gif . Hence, since for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq177_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq178_HTML.gif , it follows that (3.4) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.1, we can prove the following results.

Corollary 3.2.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq180_HTML.gif hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq181_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq182_HTML.gif , then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq183_HTML.gif being sufficiently small and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq184_HTML.gif being sufficiently large, BVP (1.6) has at least one positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq185_HTML.gif with property (3.4).

Proof.

The proof is similar to that of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq186_HTML.gif of [6].

In Theorem 3.3, we assume the following condition on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq187_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq188_HTML.gif .

There exist numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq190_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ62_HTML.gif
(3.13)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ63_HTML.gif
(3.14)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ64_HTML.gif
(3.15)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq191_HTML.gif are defined in (2.14), and write
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ65_HTML.gif
(3.16)

Theorem 3.3.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq192_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq193_HTML.gif hold. Then problem (1.6) has at least one positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq194_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ66_HTML.gif
(3.17)

Proof.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq195_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq196_HTML.gif , (3.13) implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ67_HTML.gif
(3.18)
that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq197_HTML.gif implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ68_HTML.gif
(3.19)
Next, we turn to (3.14) and (3.15). From (3.14), (3.15), and (3.16), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ69_HTML.gif
(3.20)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ70_HTML.gif
(3.21)
Thus, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq198_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ71_HTML.gif
(3.22)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ72_HTML.gif
(3.23)
Then, (3.22) and (3.23) imply
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ73_HTML.gif
(3.24)

Applying (a) of Lemma 1.1 to (3.19) and (3.24) yields that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq199_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq200_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq201_HTML.gif . Hence, since for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq202_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq203_HTML.gif , it follows that (3.17) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.3, we can prove the following results.

Corollary 3.4.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq204_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq205_HTML.gif hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq207_HTML.gif ; then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq208_HTML.gif being sufficiently small and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq209_HTML.gif being sufficiently large, BVP (1.6) has at least one positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq210_HTML.gif with property (3.17).

Proof.

The proof is similar to that of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq211_HTML.gif of [6].

Theorem 3.5.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq212_HTML.gif (3.1) of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq213_HTML.gif and (3.14) and (3.15) of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq214_HTML.gif hold. In addition, letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq216_HTML.gif satisfy the following condition:

There is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq218_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq219_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq220_HTML.gif implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ74_HTML.gif
(3.25)
Then, problem (1.6) has at least two positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq222_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ75_HTML.gif
(3.26)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq224_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ76_HTML.gif
(3.27)

Proof.

If (3.1) of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq225_HTML.gif holds, similar to the proof of (3.7), we can prove that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ77_HTML.gif
(3.28)
If (3.14) and (3.15) of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq226_HTML.gif hold, similar to the proof of (3.23), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ78_HTML.gif
(3.29)
Finally, we show that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ79_HTML.gif
(3.30)
In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq227_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq228_HTML.gif then by (2.18), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ80_HTML.gif
(3.31)
and it follows from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq229_HTML.gif that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ81_HTML.gif
(3.32)

which implies that (3.30) holds.

Applying Lemma 1.1 to (3.28), (3.29), and (3.30) yields that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq230_HTML.gif has two fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq231_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq232_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq233_HTML.gif . Hence, since for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq234_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq235_HTML.gif , it follows that (3.26) holds. This and Lemma 2.9 complete the proof.

Remark 3.6.

Similar to the proof of that of [5], we can prove that problem (1.6) can be generalized to obtain many positive solutions.

4. Discussion

In this section, we offer some interesting discussions associated with problem (1.6).

Discussion.

Generally, it is difficult to obtain the upper and lower bounds of positive solutions for nonlinear higher-order boundary value problems (see, e.g., [33, 34, 3943, 46, 48, 49] and their references).

For example, we consider the following problems:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ82_HTML.gif
(4.1)

Here http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq236_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq237_HTML.gif -Laplace operator, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq238_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq239_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq240_HTML.gif (where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq241_HTML.gif is fixed positive integer) are fixed points with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq242_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq243_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq244_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq245_HTML.gif represent the right-hand limit and left-hand limit of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq246_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq247_HTML.gif , respectively, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq248_HTML.gif is nonnegative.

By means of the transformation (2.5), we can convert problem (4.1) into
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ83_HTML.gif
(4.2)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ84_HTML.gif
(4.3)
Using the similar proof of that of Lemmas 2.1 and 2.6, we can obtain the following results. In addition, if we replace http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq249_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq250_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq251_HTML.gif , respectively, then we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq252_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ85_HTML.gif
(4.4)

Lemma 4.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq253_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq254_HTML.gif hold, then BVP (4.2) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq255_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq256_HTML.gif can be expressed in the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ86_HTML.gif
(4.5)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ87_HTML.gif
(4.6)

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq257_HTML.gif is defined in (2.10).

Lemma 4.2.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq259_HTML.gif hold, then BVP (4.3) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq260_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq261_HTML.gif can be expressed in the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ88_HTML.gif
(4.7)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ89_HTML.gif
(4.8)

It is not difficult to prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq262_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq263_HTML.gif have the similar properties to that of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq264_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq265_HTML.gif . But for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq266_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq267_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq268_HTML.gif have no property (2.13). In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq269_HTML.gif , then we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq270_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq271_HTML.gif have the following properties.

Proposition 4.3.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq272_HTML.gif holds, then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq273_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ90_HTML.gif
(4.9)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ91_HTML.gif
(4.10)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ92_HTML.gif
(4.11)

Proof.

We only consider (4.9). By (4.3), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ93_HTML.gif
(4.12)
On the other hand, noticing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq274_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ94_HTML.gif
(4.13)

Similarly, we can prove that (4.10) holds, too.

Remark 4.4.

Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq275_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq276_HTML.gif .

From (4.9) and (4.10), we can only define a cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq277_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ95_HTML.gif
(4.14)

which implies that we cannot obtain the lower bounds for the positive solutions of problem (4.1).

5. Example

To illustrate how our main results can be used in practice, we present an example.

Example 5.1.

Consider the following boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ96_HTML.gif
(5.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq278_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ97_HTML.gif
(5.2)

Conclusion.

Equation(5.1) has at least one positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq279_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq280_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ98_HTML.gif
(5.3)

Proof.

By simple computation, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq281_HTML.gif . Select http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq282_HTML.gif then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq283_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ99_HTML.gif
(5.4)

By Theorem 3.1, (5.1) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq284_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq285_HTML.gif .

Declarations

Acknowledgments

The authors thank the referee for his/her careful reading of the manuscript and useful suggestions. This work is sponsored by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201010772018), and Beijing Municipal Education Commission (71D0911003).

Authors’ Affiliations

(1)
School of Applied Science, Beijing Information Science & Technology University

References

  1. Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Singapore; 1989:xii+273.View Article
  2. Benchohra M, Henderson J, Ntouyas S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications. Volume 2. Hindawi Publishing Corporation, New York, NY, USA; 2006:xiv+366.View Article
  3. Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems 2010, 4(1):134-141. 10.1016/j.nahs.2009.09.002MathSciNetMATH
  4. Baĭnov DD, Simeonov PS: Systems with Impulse Effect. Ellis Horwood, Chichester, UK; 1989:255.MATH
  5. Samoĭlenko AM, Perestyuk NA: Impulsive Differential Equations. Volume 14. World Scientific, Singapore; 1995:x+462.MATH
  6. Yan J: Existence of positive periodic solutions of impulsive functional differential equations with two parameters. Journal of Mathematical Analysis and Applications 2007, 327(2):854-868. 10.1016/j.jmaa.2006.04.018View ArticleMathSciNetMATH
  7. Feng M, Xie D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. Journal of Computational and Applied Mathematics 2009, 223(1):438-448. 10.1016/j.cam.2008.01.024View ArticleMathSciNetMATH
  8. Nieto JJ: Basic theory for nonresonance impulsive periodic problems of first order. Journal of Mathematical Analysis and Applications 1997, 205(2):423-433. 10.1006/jmaa.1997.5207View ArticleMathSciNetMATH
  9. Nieto JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters 2002, 15(4):489-493. 10.1016/S0893-9659(01)00163-XView ArticleMathSciNetMATH
  10. Guo D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space. Applied Mathematics and Computation 2003, 143(2-3):233-249. 10.1016/S0096-3003(02)00356-9View ArticleMathSciNetMATH
  11. Liu X, Guo D: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. Journal of Mathematical Analysis and Applications 1997, 216(1):284-302. 10.1006/jmaa.1997.5688View ArticleMathSciNetMATH
  12. Agarwal RP, O'Regan D: Multiple nonnegative solutions for second order impulsive differential equations. Applied Mathematics and Computation 2000, 114(1):51-59. 10.1016/S0096-3003(99)00074-0View ArticleMathSciNetMATH
  13. Liu B, Yu J:Existence of solution of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq286_HTML.gif -point boundary value problems of second-order differential systems with impulses. Applied Mathematics and Computation 2002, 125(2-3):155-175. 10.1016/S0096-3003(00)00110-7View ArticleMathSciNetMATH
  14. Agarwal RP, Franco D, O'Regan D: Singular boundary value problems for first and second order impulsive differential equations. Aequationes Mathematicae 2005, 69(1-2):83-96. 10.1007/s00010-004-2735-9View ArticleMathSciNetMATH
  15. Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. Journal of Mathematical Analysis and Applications 2006, 321(2):501-514. 10.1016/j.jmaa.2005.07.076View ArticleMathSciNetMATH
  16. Jankowski T: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments. Applied Mathematics and Computation 2008, 197(1):179-189. 10.1016/j.amc.2007.07.081View ArticleMathSciNetMATH
  17. Jankowski T: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Applied Mathematics and Computation 2008, 202(2):550-561. 10.1016/j.amc.2008.02.040View ArticleMathSciNetMATH
  18. Feng M, Du B, Ge W:Impulsive boundary value problems with integral boundary conditions and one-dimensional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq287_HTML.gif -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(9):3119-3126. 10.1016/j.na.2008.04.015View ArticleMathSciNetMATH
  19. Gallardo JM: Second-order differential operators with integral boundary conditions and generation of analytic semigroups. Rocky Mountain Journal of Mathematics 2000, 30(4):1265-1291. 10.1216/rmjm/1021477351View ArticleMathSciNetMATH
  20. Karakostas GL, Tsamatos PCh: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electronic Journal of Differential Equations 2002, 2002(30):1-17.
  21. 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
  22. Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge, UK; 1991:x+366.View ArticleMATH
  23. Agarwal RP, O'Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+341.View ArticleMATH
  24. Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differential Equations and Applications 2008, 15(1-2):45-67. 10.1007/s00030-007-4067-7View ArticleMathSciNetMATH
  25. Webb JRL, Infante G: Non-local boundary value problems of arbitrary order. Journal of the London Mathematical Society 2009, 79(1):238-258.View ArticleMathSciNetMATH
  26. Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society 2006, 74(3):673-693. 10.1112/S0024610706023179View ArticleMathSciNetMATH
  27. Ahmad B, Nieto JJ:The monotone iterative technique for three-point second-order integrodifferential boundary value problems with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq288_HTML.gif -Laplacian. Boundary Value Problems 2007, 2007:-9.
  28. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.
  29. Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 9(4):1727-1740.View ArticleMathSciNetMATH
  30. 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.2007.11.003View ArticleMathSciNetMATH
  31. Zhang X, Feng M, Ge W:Multiple positive solutions for a class of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq289_HTML.gif -point boundary value problems. Applied Mathematics Letters 2009, 22(1):12-18. 10.1016/j.aml.2007.10.019View ArticleMathSciNetMATH
  32. Feng M, Ge W:Positive solutions for a class of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq290_HTML.gif -point singular boundary value problems. Mathematical and Computer Modelling 2007, 46(3-4):375-383. 10.1016/j.mcm.2006.11.009View ArticleMathSciNetMATH
  33. Zhang X, Feng M, Ge W:Symmetric positive solutions for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq291_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.2007.12.002View ArticleMathSciNetMATH
  34. Zhang X, Feng M, Ge W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3310-3321. 10.1016/j.na.2007.09.020View ArticleMathSciNetMATH
  35. 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.2005.09.002View ArticleMathSciNetMATH
  36. Ma R:Positive solutions for multipoint boundary value problem with a one-dimensional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq292_HTML.gif -Laplacian. Computational & Applied Mathematics 2001, 42: 755-765.View ArticleMATH
  37. Bai Z, Huang B, Ge W:The iterative solutions for some fourth-order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq293_HTML.gif -Laplace equation boundary value problems. Applied Mathematics Letters 2006, 19(1):8-14. 10.1016/j.aml.2004.10.010View ArticleMathSciNetMATH
  38. Liu B, Liu L, Wu Y: Positive solutions for singular second order three-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(12):2756-2766. 10.1016/j.na.2006.04.005View ArticleMathSciNetMATH
  39. Zhang X, Liu L:A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq294_HTML.gif -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(10):3127-3137. 10.1016/j.na.2007.03.006View ArticleMathSciNetMATH
  40. Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems. Journal of Mathematical Analysis and Applications 1986, 116(2):415-426. 10.1016/S0022-247X(86)80006-3View ArticleMathSciNetMATH
  41. Bai Z, Wang H: On positive solutions of some nonlinear fourth-order beam equations. Journal of Mathematical Analysis and Applications 2002, 270(2):357-368. 10.1016/S0022-247X(02)00071-9View ArticleMathSciNetMATH
  42. Ma R, Wang H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis 1995, 59(1–4):225-231.MathSciNetMATH
  43. Liu L, Zhang X, Wu Y:Positive solutions of fourth order four-point boundary value problems with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq295_HTML.gif -Laplacian operator. Journal of Mathematical Analysis and Applications 2007, 326(2):1212-1224. 10.1016/j.jmaa.2006.03.029View ArticleMathSciNetMATH
  44. Kang P, Wei Z, Xu J: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Applied Mathematics and Computation 2008, 206(1):245-256. 10.1016/j.amc.2008.09.010View ArticleMathSciNetMATH
  45. Webb JRL, Infante G, Franco D: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions. Proceedings of the Royal Society of Edinburgh 2008, 138(2):427-446.MathSciNetMATH
  46. Ma H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(3):645-651. 10.1016/j.na.2006.11.026View ArticleMathSciNetMATH
  47. Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.MATH
  48. Eloe PW, Ahmad B:Positive solutions of a nonlinear http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq296_HTML.gif th order boundary value problem with nonlocal conditions. Applied Mathematics Letters 2005, 18(5):521-527. 10.1016/j.aml.2004.05.009View ArticleMathSciNetMATH
  49. Hao X, Liu L, Wu Y:Positive solutions for nonlinear http://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq297_HTML.gif th-order singular nonlocal boundary value problems. Boundary Value Problems 2007, 2007:-10.

Copyright

© Meiqiang Feng. 2011

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.