Open Access

Positive Solutions of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq1_HTML.gif th-Order Nonlinear Impulsive Differential Equation with Nonlocal Boundary Conditions

Boundary Value Problems20102011:456426

DOI: 10.1155/2011/456426

Received: 25 March 2010

Accepted: 9 May 2010

Published: 10 June 2010

Abstract

This paper is devoted to study the existence, nonexistence, and multiplicity of positive solutions for the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq2_HTML.gif th-order nonlocal boundary value problem with impulse effects. The arguments are based upon fixed point theorems in a cone. An example is worked out to demonstrate the main results.

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 the recent years and is much richer than the corresponding theory of differential equations; see, for instance, [314] and their references.

At the same time, a class of boundary value problems with integral boundary conditions arise naturally in thermal conduction problems [15], semiconductor problems [16], hydrodynamic problems [17]. Such problems include two, three, and multipoint boundary value problems as special cases and attract much attention; see, for instance, [7, 8, 11, 1844] and references cited therein. In particular, we would like to mention some results of Eloe and Ahmad [19] and Pang et al. [22]. In [19], by applying the fixed point theorem in cones due to the work of Krasnosel'kii and Guo, Eloe and Ahmad established the existence of positive solutions of the following https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq3_HTML.gif th boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ1_HTML.gif
(1.1)
In [22], Pang et al. considered the expression and properties of Green's function for the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq4_HTML.gif th-order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq5_HTML.gif -point boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq6_HTML.gif . Furthermore, they obtained the existence of positive solutions by means of fixed point index theory.

Recently, Yang and Wei [23] and the author of [24] improved and generalized the results of Pang et al. [22] by using different methods, respectively.

On the other hand, it is well known that fixed point theorem of cone expansion and compression of norm type has been applied to various boundary value problems to show the existence of positive solutions; for example, see [7, 8, 11, 19, 23, 24]. However, there are few papers investigating the existence of positive solutions of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq7_HTML.gif th impulsive differential equations by using the fixed point theorem of cone expansion and compression. The objective of the present paper is to fill this gap. Being directly inspired by [19, 22], using of the fixed point theorem of cone expansion and compression, this paper is devoted to study a class of nonlocal BVPs for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq8_HTML.gif th-order impulsive differential equations with fixed moments.

Consider the following https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq9_HTML.gif th-order impulsive differential equations with integral boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ3_HTML.gif
(1.3)

Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq10_HTML.gif (where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq11_HTML.gif is fixed positive integer) are fixed points with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq12_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq13_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq14_HTML.gif represent the right-hand limit and left-hand limit of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq15_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq16_HTML.gif , respectively, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq17_HTML.gif is nonnegative.

For the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq18_HTML.gif , problem (1.3) reduces to the problem studied by Samoĭlenko and Perestyuk in [4]. By using the fixed point index theory in cones, the authors obtained some sufficient conditions for the existence of at least one or two positive solutions to the two-point BVPs.

Motivated by the work above, in this paper we will extend the results of [4, 19, 2224] to problem (1.3). On the other hand, it is also interesting and important to discuss the existence of positive solutions for problem (1.3) when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq19_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq20_HTML.gif . Many difficulties occur when we deal with them; for example, the construction of cone and operator. So we need to introduce some new tools and methods to investigate the existence of positive solutions for problem (1.3). Our argument is based on fixed point theory in cones [45].

To obtain positive solutions of (1.3), the following fixed point theorem in cones is fundamental which can be found in [45, page 93].

Lemma 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq22_HTML.gif be two bounded open sets in Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq23_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq25_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq26_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq27_HTML.gif and let operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq28_HTML.gif be completely continuous. Suppose that one of the following two conditions is satisfied:
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq29_HTML.gif ;

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq30_HTML.gif .

     

Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq31_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq32_HTML.gif .

2. Preliminaries

In order to define the solution of problem (1.3), we will consider the following space.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq33_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ4_HTML.gif
(2.1)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq34_HTML.gif is a real Banach space with norm
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ5_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq35_HTML.gif

A function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq36_HTML.gif is called a solution of problem (1.3) if it satisfies (1.3).

To establish the existence of multiple positive solutions in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq37_HTML.gif of problem (1.3), let us list the following assumptions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq39_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq41_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq42_HTML.gif .

Lemma 2.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq44_HTML.gif hold. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq45_HTML.gif is a solution of problem (1.3) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq46_HTML.gif is a solution of the following impulsive integral equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ6_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ7_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ8_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ9_HTML.gif
(2.6)

Proof.

First suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq47_HTML.gif is a solution of problem (1.3). It is easy to see by integration of (1.3) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ10_HTML.gif
(2.7)
Integrating again and by boundary conditions, we can get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ11_HTML.gif
(2.8)
Similarly, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ12_HTML.gif
(2.9)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq48_HTML.gif in (2.9), we find
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ13_HTML.gif
(2.10)
Substituting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq49_HTML.gif and (2.10) into (2.9), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ14_HTML.gif
(2.11)
Multiplying (2.11) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq50_HTML.gif and integrating it, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ15_HTML.gif
(2.12)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ16_HTML.gif
(2.13)
Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ17_HTML.gif
(2.14)

Then, the proof of sufficient is complete.

Conversely, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq51_HTML.gif is a solution of (2.3), direct differentiation of (2.3) implies that, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq52_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ18_HTML.gif
(2.15)
Evidently,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ19_HTML.gif
(2.16)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ20_HTML.gif
(2.17)

So https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq54_HTML.gif , and it is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq55_HTML.gif , and the lemma is proved.

Similar to the proof of that from [22], we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq56_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq57_HTML.gif have the following properties.

Proposition 2.2.

The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq58_HTML.gif defined by (2.5) satisfyong https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq59_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq60_HTML.gif .

Proposition 2.3.

There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq61_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ21_HTML.gif
(2.18)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq62_HTML.gif is defined in (2.20).

Proposition 2.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq63_HTML.gif , then one has

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq64_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq65_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq66_HTML.gif .

Proof.

From the properties of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq67_HTML.gif and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq68_HTML.gif , we can prove that the results of Proposition 2.4 hold.

Proposition 2.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq69_HTML.gif , the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq70_HTML.gif defined by (2.4) satisfies

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq71_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq72_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq73_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq74_HTML.gif , and

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ22_HTML.gif
(2.19)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq75_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ23_HTML.gif
(2.20)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq76_HTML.gif is defined in Proposition 2.3.

Proof.
  1. (i)

    From Propositions 2.2 and 2.4, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq77_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq78_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq79_HTML.gif .

     
  2. (ii)

    From (ii) of Proposition 2.2 and (ii) of Proposition 2.4, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq80_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq81_HTML.gif .

     

Now, we show that (2.19) holds.

In fact, from Proposition 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ24_HTML.gif
(2.21)

Then the proof of Proposition 2.5 is completed.

Remark 2.6.

From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq82_HTML.gif , it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq83_HTML.gif .

Lemma 2.7.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq85_HTML.gif hold. Then, the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq86_HTML.gif of problem (1.3) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq87_HTML.gif

Proof.

It is an immediate subsequence of the facts that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq88_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq89_HTML.gif .

Remark 2.8.

From (ii) of Proposition 2.5, one can find that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ25_HTML.gif
(2.22)
For the sake of applying Lemma 1.1, we construct a cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq90_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq91_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ26_HTML.gif
(2.23)
Define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq92_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ27_HTML.gif
(2.24)

Lemma 2.9.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq94_HTML.gif hold. Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq95_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq96_HTML.gif is completely continuous.

Proof.

From Proposition 2.5 and (2.24), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ28_HTML.gif
(2.25)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq97_HTML.gif .

Next, by similar arguments to those in [8] one can prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq98_HTML.gif is completely continuous. So it is omitted, and the lemma is proved.

3. Main Results

Write
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ29_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq99_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq100_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq101_HTML.gif

In this section, we apply Lemma 1.1 to establish the existence of positive solutions for BVP (1.3).

Theorem 3.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq103_HTML.gif hold. In addition, letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq105_HTML.gif satisfy the following conditions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq108_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq110_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq111_HTML.gif ,

BVP (1.3) has at least one positive solution.

Proof.

Considering https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq112_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq113_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ30_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq114_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ31_HTML.gif
(3.3)
here
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ32_HTML.gif
(3.4)
Now, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq115_HTML.gif , we prove that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ33_HTML.gif
(3.5)
In fact, if there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq116_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq117_HTML.gif . Noticing (3.2), then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ34_HTML.gif
(3.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ35_HTML.gif
(3.7)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq118_HTML.gif , which is a contraction. Hence, (3.2) holds.

Next, turning to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq119_HTML.gif . Case ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq120_HTML.gif ). https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq121_HTML.gif . There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq122_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ36_HTML.gif
(3.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq123_HTML.gif . Choose
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ37_HTML.gif
(3.9)
We show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ38_HTML.gif
(3.10)
In fact, if there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq124_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq125_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ39_HTML.gif
(3.11)
This and (3.9) imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ40_HTML.gif
(3.12)
So, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ41_HTML.gif
(3.13)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ42_HTML.gif
(3.14)
It is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ43_HTML.gif
(3.15)

In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq126_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq127_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq128_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq129_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq130_HTML.gif , which contracts https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq131_HTML.gif So, (3.15) holds. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq132_HTML.gif , this is also a contraction. Hence, (3.10) holds.

Case ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq133_HTML.gif ). https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq134_HTML.gif . There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq135_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ44_HTML.gif
(3.16)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq136_HTML.gif . If we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq137_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq138_HTML.gif . Choose
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ45_HTML.gif
(3.17)

We prove that (3.10) holds.

In fact, if there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq139_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq140_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ46_HTML.gif
(3.18)
This and (3.17) imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ47_HTML.gif
(3.19)
So, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ48_HTML.gif
(3.20)
From (3.20), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ49_HTML.gif
(3.21)
So, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ50_HTML.gif
(3.22)
From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq141_HTML.gif , we can find that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ51_HTML.gif
(3.23)

Similar to the proof in case ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq142_HTML.gif ), we can show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq143_HTML.gif . Then, from (3.23), we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq144_HTML.gif , which is a contraction. Hence, (3.10) holds.

Applying (i) of Lemma 1.1 to (3.2) and (3.10) yields that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq145_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq146_HTML.gif . Thus, it follows that BVP (1.3) has at least one positive solution, and the theorem is proved.

Theorem 3.2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq148_HTML.gif hold. In addition, letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq150_HTML.gif satisfy the following conditions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq153_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq155_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq156_HTML.gif ,

BVP (1.3) has at least one positive solution.

Proof.

Considering https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq157_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq158_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq159_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq160_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq161_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq162_HTML.gif .

Similar to the proof of (3.2), we can show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ52_HTML.gif
(3.24)
Next, turning to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq163_HTML.gif . Under condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq164_HTML.gif , similar to the proof of (3.10), we can also show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ53_HTML.gif
(3.25)

Applying (i) of Lemma 1.1 to (3.24) and (3.25) yields that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq165_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq166_HTML.gif . Thus, it follows that BVP (1.3) has one positive solution, and the theorem is proved.

Theorem 3.3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq167_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq168_HTML.gif hold. In addition, letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq169_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq170_HTML.gif satisfy the following condition:

there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq172_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq174_HTML.gif implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ54_HTML.gif
(3.26)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq175_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq176_HTML.gif , BVP (1.3) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq178_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq179_HTML.gif .

Proof.

We choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq180_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq181_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq182_HTML.gif holds, similar to the proof of (3.2), we can prove that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ55_HTML.gif
(3.27)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq183_HTML.gif holds, similar to the proof of (3.24), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ56_HTML.gif
(3.28)
Finally, we show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ57_HTML.gif
(3.29)
In fact, if there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq184_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq185_HTML.gif then by (2.23), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ58_HTML.gif
(3.30)
and it follows from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq186_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ59_HTML.gif
(3.31)

that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq187_HTML.gif , which is a contraction. Hence, (3.29) holds.

Applying Lemma 1.1 to (3.27), (3.28), and (3.29) yields that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq188_HTML.gif has two fixed points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq189_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq190_HTML.gif . Thus it follows that BVP (1.3) has two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq191_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq192_HTML.gif . The proof is complete.

Our last results corresponds to the case when problem (1.3) has no positive solution. Write
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ60_HTML.gif
(3.32)

Theorem 3.4.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq193_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq194_HTML.gif , then problem (1.3) has no positive solution.

Proof.

Assume to the contrary that problem (1.3) has a positive solution, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq195_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq196_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq197_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq198_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ61_HTML.gif
(3.33)

which is a contradiction, and this completes the proof.

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

Example 3.5.

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ62_HTML.gif
(3.34)

Conclusion.

BVP (3.34) has at least one positive solution.

Proof.

BVP (3.34) can be regarded as a BVP of the form (1.3), where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ63_HTML.gif
(3.35)
It is not difficult to see that conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq200_HTML.gif hold. In addition,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ64_HTML.gif
(3.36)

Then, conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq201_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq202_HTML.gif of Theorem 3.1 hold. Hence, by Theorem 3.1, the conclusion follows, and the proof is complete.

Declarations

Acknowledgment

This work is supported by the National Natural Science Foundation of China (10771065), the Natural Sciences Foundation of Heibei Province (A2007001027), 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). The authors thank the referee for his/her careful reading of the paper and useful suggestions.

Authors’ Affiliations

(1)
School of Science, Beijing Information Science & Technology University
(2)
Department of Mathematics and Physics, North China Electric Power University

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© Meiqiang Feng et al. 2011

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