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Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq3_HTML.gif is a positive parameter, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq5_HTML.gif is the zero element of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq6_HTML.gif , and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq8_HTML.gif and may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq9_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq10_HTML.gif ; https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq13_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq14_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq15_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq16_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq17_HTML.gif are nonnegative, https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq19_HTML.gif denotes a linear functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq20_HTML.gif given by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq21_HTML.gif involving a Stieltjes integral, and https://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 https://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 https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq25_HTML.gif ), nonlinear elasticity ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq26_HTML.gif ), glaciology ( https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq29_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq31_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ6_HTML.gif
(1.6)

Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq33_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq34_HTML.gif -Laplace operator, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq37_HTML.gif (where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq40_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq41_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq43_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq44_HTML.gif , respectively, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq45_HTML.gif is nonnegative.

For the case of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq48_HTML.gif , and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq51_HTML.gif is either superlinear or sublinear on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq54_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq56_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq58_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq59_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq60_HTML.gif and let operator https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq62_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq63_HTML.gif ;

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

Then, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq68_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ7_HTML.gif
(2.1)
Then https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ8_HTML.gif
(2.2)

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

A function https://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 https://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:

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

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

From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq77_HTML.gif , it is clear that https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ12_HTML.gif
(2.6)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq79_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq81_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ14_HTML.gif
(2.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ15_HTML.gif
(2.9)
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq84_HTML.gif holds, then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ17_HTML.gif
(2.11)

Proposition 2.3.

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

Proposition 2.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq86_HTML.gif holds, then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq87_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ19_HTML.gif
(2.13)
where
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq88_HTML.gif , we obtain
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ23_HTML.gif
(2.17)

Lemma 2.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq89_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq91_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ24_HTML.gif
(2.18)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ25_HTML.gif
(2.19)

and https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ26_HTML.gif
(2.20)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq96_HTML.gif then integrate from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq97_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq98_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ27_HTML.gif
(2.21)
Similarly, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq99_HTML.gif we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ29_HTML.gif
(2.23)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq100_HTML.gif in (2.23), we find
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ30_HTML.gif
(2.24)
Substituting https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ31_HTML.gif
(2.25)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ32_HTML.gif
(2.26)
Therefore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ33_HTML.gif
(2.27)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ34_HTML.gif
(2.28)
Then,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq103_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ36_HTML.gif
(2.30)
Evidently,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq104_HTML.gif are similar to that of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq105_HTML.gif .

Suppose that https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq107_HTML.gif via
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ39_HTML.gif
(2.33)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq108_HTML.gif is a closed convex cone of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq109_HTML.gif .

Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq110_HTML.gif by
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq112_HTML.gif is a fixed point of operator https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq115_HTML.gif if for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq116_HTML.gif there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq117_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq118_HTML.gif then
https://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 https://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.

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

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

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

Lemma 2.10.

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

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq128_HTML.gif , it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq129_HTML.gif , and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq130_HTML.gif , noticing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq131_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ45_HTML.gif
(2.39)

Case 2.

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

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

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

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq144_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ47_HTML.gif
(2.41)

Therefore https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq146_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq147_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq148_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ49_HTML.gif
(2.43)

and then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq149_HTML.gif is quasi-equicontinuous. It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq150_HTML.gif is relatively compact on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq151_HTML.gif by Lemma 2.9. So https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq154_HTML.gif .

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

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

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

Theorem 3.1.

Assume that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq163_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ53_HTML.gif
(3.4)

Proof.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq165_HTML.gif with https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ54_HTML.gif
(3.5)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ55_HTML.gif
(3.6)
Now if we let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq167_HTML.gif , then (3.5) shows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ56_HTML.gif
(3.7)
Further, let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ57_HTML.gif
(3.8)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq169_HTML.gif implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ58_HTML.gif
(3.9)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ59_HTML.gif
(3.10)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq170_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq171_HTML.gif . Therefore, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq172_HTML.gif , (3.2) implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ60_HTML.gif
(3.11)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq173_HTML.gif implies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq174_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq175_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq176_HTML.gif . Hence, since for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq177_HTML.gif we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq180_HTML.gif hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq181_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq182_HTML.gif , then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq183_HTML.gif being sufficiently small and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq187_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq188_HTML.gif .

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

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ62_HTML.gif
(3.13)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ63_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ64_HTML.gif
(3.15)
where https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq192_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq194_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ66_HTML.gif
(3.17)

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq195_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq196_HTML.gif , (3.13) implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ67_HTML.gif
(3.18)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq197_HTML.gif implies
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ69_HTML.gif
(3.20)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ70_HTML.gif
(3.21)
Thus, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq198_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ71_HTML.gif
(3.22)
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq199_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq200_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq201_HTML.gif . Hence, since for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq202_HTML.gif we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq204_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq205_HTML.gif hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq207_HTML.gif ; then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq208_HTML.gif being sufficiently small and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq212_HTML.gif (3.1) of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq214_HTML.gif hold. In addition, letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq215_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq218_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq219_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq220_HTML.gif implies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq222_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ75_HTML.gif
(3.26)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq223_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq224_HTML.gif satisfy
https://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 https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ78_HTML.gif
(3.29)
Finally, we show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ79_HTML.gif
(3.30)
In fact, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq227_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq228_HTML.gif then by (2.18), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ80_HTML.gif
(3.31)
and it follows from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq229_HTML.gif that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq230_HTML.gif has two fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq231_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq232_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq233_HTML.gif . Hence, since for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq234_HTML.gif we have https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ82_HTML.gif
(4.1)

Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq236_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq237_HTML.gif -Laplace operator, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq238_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq239_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq240_HTML.gif (where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq242_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq243_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq244_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq246_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq247_HTML.gif , respectively, and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ83_HTML.gif
(4.2)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq249_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq250_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq251_HTML.gif , respectively, then we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq252_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ85_HTML.gif
(4.4)

Lemma 4.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq253_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq255_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq256_HTML.gif can be expressed in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ86_HTML.gif
(4.5)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ87_HTML.gif
(4.6)

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq258_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq260_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq261_HTML.gif can be expressed in the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ88_HTML.gif
(4.7)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq262_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq264_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq265_HTML.gif . But for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq266_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq267_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq269_HTML.gif , then we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq270_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq272_HTML.gif holds, then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq273_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ90_HTML.gif
(4.9)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ91_HTML.gif
(4.10)
where
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq274_HTML.gif , we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq275_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq277_HTML.gif by
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_Equ96_HTML.gif
(5.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq278_HTML.gif , and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq279_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq280_HTML.gif with
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq281_HTML.gif . Select https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq282_HTML.gif then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq283_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F654871/MediaObjects/13661_2010_Article_51_IEq284_HTML.gif with https://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

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© Meiqiang Feng. 2011

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