Open Access

New Existence Results for Higher-Order Nonlinear Fractional Differential Equation with Integral Boundary Conditions

Boundary Value Problems20102011:720702

DOI: 10.1155/2011/720702

Received: 16 March 2010

Accepted: 5 July 2010

Published: 20 July 2010

Abstract

This paper investigates the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions. The results are established by converting the problem into an equivalent integral equation and applying Krasnoselskii's fixed-point theorem in cones. The nonexistence of positive solutions is also studied.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode's analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. An excellent account in the study of fractional differential equations can be found in [15]. For the basic theory and recent development of the subject, we refer a text by Lakshmikantham [6]. For more details and examples, see [723] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored.

In [23], Zhang used a fixed-point theorem for the mixed monotone operator to show the existence of positive solutions to the following singular fractional differential equation.
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ1_HTML.gif
(1.1)
subject to the boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq1_HTML.gif is the standard Rimann-Liouville fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq2_HTML.gif , the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq3_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq4_HTML.gif , and function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq5_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq6_HTML.gif . The author derived the corresponding Green's function named by fractional Green's function and obtained some properties as follows.

Proposition 1.1.

Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq7_HTML.gif satisfies the following conditions:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq8_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq9_HTML.gif ;

(ii)there exists a positive function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq10_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ3_HTML.gif
(1.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq11_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ4_HTML.gif
(1.4)

here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq12_HTML.gif .

It is well known that the cone theoretic techniques play a very important role in applying Green's function in the study of solutions to boundary value problems. In [23], the author cannot acquire a positive constant taking instead of the role of positive function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq13_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq14_HTML.gif in (1.3). At the same time, we notice that many authors obtained the similar properties to that of (1.3), for example, see Bai [12], Bai and L https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq15_HTML.gif [13], Jiang and Yuan [14], Li et al, [15], Kaufmann and Mboumi [19], and references therein. Naturally, one wishes to find whether there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq16_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ5_HTML.gif
(1.5)

for the fractional order cases. In Section 2, we will deduce some new properties of Green's function.

Motivated by the above mentioned work, we study the following higher-order singular boundary value problem of fractional differential equation.
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ6_HTML.gif
(P)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq17_HTML.gif is the standard Rimann-Liouville fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq19_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq20_HTML.gif or/and at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq22_HTML.gif is nonnegative, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq23_HTML.gif .

For the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq24_HTML.gif , the boundary value problems () reduces to the problem studied by Eloe and Ahmad in [24]. In [24], the authors used the Krasnosel'skii and Guo [25] fixed-point theorem to show the existence of at least one positive solution if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq26_HTML.gif is either superlinear or sublinear to problem (). For the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq28_HTML.gif , the boundary value problems () is related to a m-point boundary value problems of integer-order differential equation. Under this case, a great deal of research has been devoted to the existence of solutions for problem (), for example, see Pang et al. [26], Yang and Wei [27], Feng and Ge [28], and references therein. All of these results are based upon the fixed-point index theory, the fixed-point theorems and the fixed-point theory in cone for strict set contraction operator.

The organization of this paper is as follows. We will introduce some lemmas and notations in the rest of this section. In Section 2, we present the expression and properties of Green's function associated with boundary value problem (). In Section 3, we discuss some characteristics of the integral operator associated with the problem () and state a fixed-point theorem in cones. In Section 4, we discuss the existence of at least one positive solution of boundary value problem (). In Section 5, we will prove the existence of two or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq34_HTML.gif positive solutions, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq35_HTML.gif is an arbitrary natural number. In Section 6, we study the nonexistence of positive solution of boundary value problem (). In Section 7, one example is also included to illustrate the main results. Finally, conclusions in Section 8 close the paper.

The fractional differential equations related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to fractional differential equations. The readers who are unfamiliar with this area can consult, for example, [16] for details.

Definition 1.2 (see [4]).

The integral
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ7_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq37_HTML.gif , is called Riemann-Liouville fractional integral of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq38_HTML.gif .

Definition 1.3 (see [4]).

For a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq39_HTML.gif given in the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq40_HTML.gif , the expression
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ8_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq41_HTML.gif denotes the integer part of number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq42_HTML.gif , is called the Riemann-Liouville fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq43_HTML.gif .

Lemma 1.4 (see [13]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq44_HTML.gif with a fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq45_HTML.gif that belongs to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq46_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ9_HTML.gif
(1.8)

for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq47_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq48_HTML.gif is the smallest integer greater than or equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq49_HTML.gif .

2. Expression and Properties of Green's Function

In this section, we present the expression and properties of Green's function associated with boundary value problem ().

Lemma 2.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq51_HTML.gif Then for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq52_HTML.gif , the unique solution of boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ10_HTML.gif
(2.1)
is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ11_HTML.gif
(2.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ12_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ13_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ14_HTML.gif
(2.5)

Proof.

By Lemma 1.4, we can reduce the equation of problem (2.1) to an equivalent integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ15_HTML.gif
(2.6)
By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq53_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq54_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ16_HTML.gif
(2.7)
Differentiating (2.7), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ17_HTML.gif
(2.8)
By (2.8) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq55_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq56_HTML.gif Similarly, we can obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq57_HTML.gif Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ18_HTML.gif
(2.9)
By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq58_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ19_HTML.gif
(2.10)
Therefore, the unique solution of BVP (2.1) is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ20_HTML.gif
(2.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq59_HTML.gif is defined by (2.4).

From (2.11), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ21_HTML.gif
(2.12)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ22_HTML.gif
(2.13)
Substituting (2.13) into (2.11), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ23_HTML.gif
(2.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq61_HTML.gif are defined by (2.3), (2.4), and (2.5), respectively. The proof is complete.

From (2.3), (2.4), and (2.5), we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq62_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq63_HTML.gif have the following properties.

Proposition 2.2.

The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq64_HTML.gif defined by (2.4) satisfies

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq65_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq66_HTML.gif ;

(ii)for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq67_HTML.gif , one has

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ24_HTML.gif
(2.15)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ25_HTML.gif
(2.16)
Proof.
  1. (i)

    It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq68_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq70_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq71_HTML.gif .

     
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq72_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ26_HTML.gif
(2.17)
So, by (2.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ27_HTML.gif
(2.18)
Similarly, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq73_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq74_HTML.gif .
  1. (ii)

    Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq75_HTML.gif , it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq76_HTML.gif is increasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq77_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq78_HTML.gif .

     
On the other hand, from the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq79_HTML.gif , for given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq80_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ28_HTML.gif
(2.19)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ29_HTML.gif
(2.20)
Then, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ30_HTML.gif
(2.21)
and so,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ31_HTML.gif
(2.22)
Noticing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq81_HTML.gif , from (2.22), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ32_HTML.gif
(2.23)

Then, for given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq82_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq83_HTML.gif arrives at maximum at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq84_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq85_HTML.gif . This together with the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq86_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq87_HTML.gif , we obtain that (2.15) holds.

Remark 2.3.

From Figure 1, we can see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq88_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq89_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq90_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ33_HTML.gif
(2.24)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig1_HTML.jpg
Figure 1

Graph of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq91_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq92_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq93_HTML.gif .

Remark 2.4.

From Figure 2, we can see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq94_HTML.gif is increasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq95_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig2_HTML.jpg
Figure 2

Graph of function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq96_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq97_HTML.gif .

Remark 2.5.

From Figure 3, we can see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq98_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq99_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq100_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig3_HTML.jpg
Figure 3

Graph of function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq101_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq103_HTML.gif .

Remark 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq104_HTML.gif . From (2.15), for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq105_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ34_HTML.gif
(2.25)

Remark 2.7.

From (2.25), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ35_HTML.gif
(2.26)

Remark 2.8.

From Figure 4, it is easy to obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq106_HTML.gif is decreasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq107_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ36_HTML.gif
(2.27)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig4_HTML.jpg
Figure 4

Graph of function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq108_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq109_HTML.gif .

Proposition 2.9.

There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq110_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ37_HTML.gif
(2.28)

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq111_HTML.gif , we divide the proof into the following three cases for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq112_HTML.gif .

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq113_HTML.gif , then from (i) of Proposition 2.2 and Remark 2.5, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ38_HTML.gif
(2.29)
It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq115_HTML.gif are bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq116_HTML.gif . So, there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq117_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ39_HTML.gif
(2.30)

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq118_HTML.gif , then from (2.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ40_HTML.gif
(2.31)
On the other hand, from the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq119_HTML.gif , we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq120_HTML.gif takes its maximum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq121_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq122_HTML.gif . So
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ41_HTML.gif
(2.32)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq123_HTML.gif . Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq124_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ42_HTML.gif
(2.33)

Case 3.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq125_HTML.gif , from (i) of Proposition 2.2, it is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ43_HTML.gif
(2.34)
In view of Remarks 2.6–2.8, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ44_HTML.gif
(2.35)
From (2.35), there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq126_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ45_HTML.gif
(2.36)

Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq127_HTML.gif and using (2.30), (2.33), and (2.36), it follows that (2.28) holds. This completes the proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ46_HTML.gif
(2.37)

Proposition 2.10.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq128_HTML.gif , then one has

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq129_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq130_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq131_HTML.gif .

Proof.

Using the properties of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq132_HTML.gif , definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq133_HTML.gif , it can easily be shown that (i) and (ii) hold.

Theorem 2.11.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq134_HTML.gif , the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq135_HTML.gif defined by (2.3) satisfies

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq136_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq137_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq138_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq139_HTML.gif , and

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ47_HTML.gif
(2.38)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ48_HTML.gif
(2.39)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq140_HTML.gif is defined by (2.16), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq141_HTML.gif is defined in Proposition 2.9.

Proof.
  1. (i)

    From Propositions 2.2 and 2.10, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq142_HTML.gif is continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq143_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq144_HTML.gif .

     
  2. (ii)

    From (ii) of Proposition 2.2 and (ii) of Proposition 2.10, we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq145_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq146_HTML.gif .

     

Now, we show that (2.38) holds.

In fact, from Proposition 2.9, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ49_HTML.gif
(2.40)

Then the proof of Theorem 2.11 is completed.

Remark 2.12.

From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq147_HTML.gif , it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq148_HTML.gif .

3. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq150_HTML.gif denote a real Banach space with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq151_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq152_HTML.gif Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ50_HTML.gif
(3.1)

To prove the existence of positive solutions for the boundary value problem (), we need the following assumptions:

() https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq155_HTML.gif on any subinterval of (0,1) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq156_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq157_HTML.gif is defined in Theorem 2.11;

() https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq160_HTML.gif uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq161_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq162_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq164_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq165_HTML.gif is defined by (2.37).

From condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq166_HTML.gif , it is not difficult to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq167_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq168_HTML.gif or/and at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq169_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq170_HTML.gif or/and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq171_HTML.gif .

Define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq172_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ51_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq173_HTML.gif is defined by (2.3).

Lemma 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq174_HTML.gif hold. Then boundary value problems () has a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq176_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq177_HTML.gif is a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq178_HTML.gif .

Proof.

From Lemma 2.1, we can prove the result of this lemma.

Lemma 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq179_HTML.gif hold. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq180_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq181_HTML.gif is completely continuous.

Proof.

For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq182_HTML.gif , by (3.2), we can obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq183_HTML.gif . On the other hand, by (ii) of Theorem 2.11, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ52_HTML.gif
(3.3)
Similarly, by (2.38), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ53_HTML.gif
(3.4)

So, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq184_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq185_HTML.gif . Next by similar proof of Lemma https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq186_HTML.gif in [13] and Ascoli-Arzela theorem one can prove https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq187_HTML.gif is completely continuous. So it is omitted.

To obtain positive solutions of boundary value problem (), the following fixed-point theorem in cones is fundamental which can be found in [25, page 94].

Lemma 3.3 (Fixed-point theorem of cone expansion and compression of norm type).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq189_HTML.gif be a cone of real Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq190_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq192_HTML.gif be two bounded open sets in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq193_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq195_HTML.gif . Let operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq196_HTML.gif be completely continuous. Suppose that one of the two conditions

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq197_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq198_HTML.gif

or

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq200_HTML.gif

is satisfied. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq201_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq202_HTML.gif .

4. Existence of Positive Solution

In this section, we impose growth conditions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq203_HTML.gif which allow us to apply Lemma 3.3 to establish the existence of one positive solution of boundary value problem (), and we begin by introducing some notations:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ54_HTML.gif
(4.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq205_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq206_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq207_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ55_HTML.gif
(4.2)

Theorem 4.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq208_HTML.gif hold. In addition, one supposes that one of the following conditions is satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq210_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq211_HTML.gif (particularly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq213_HTML.gif ).

there exist two constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq215_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq216_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq217_HTML.gif is nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq218_HTML.gif

for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq219_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq220_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq221_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq222_HTML.gif . Then boundary value problem () has at least one positive solution.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq224_HTML.gif be cone preserving completely continuous that is defined by (3.2).

Case 1.

The condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq225_HTML.gif holds. Considering https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq226_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq227_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq228_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq229_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq230_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq231_HTML.gif . Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq232_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ56_HTML.gif
(4.3)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq233_HTML.gif imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ57_HTML.gif
(4.4)
Next, turning to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq234_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq235_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ58_HTML.gif
(4.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq236_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq237_HTML.gif .

Set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ59_HTML.gif
(4.6)

then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq238_HTML.gif .

Chose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq239_HTML.gif . Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq240_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ60_HTML.gif
(4.7)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq241_HTML.gif imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ61_HTML.gif
(4.8)

Case 2.

The Condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq242_HTML.gif satisfies. For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq243_HTML.gif , from (3.1) we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq244_HTML.gif . Therefore, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq245_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq246_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq247_HTML.gif , this together with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq248_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ62_HTML.gif
(4.9)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq249_HTML.gif imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ63_HTML.gif
(4.10)
On the other hand, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq250_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq251_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq252_HTML.gif , this together with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq253_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ64_HTML.gif
(4.11)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq254_HTML.gif imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ65_HTML.gif
(4.12)

Applying Lemma 3.3 to (4.4) and (4.8), or (4.10) and (4.12), yields that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq255_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq256_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq257_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq258_HTML.gif . Thus it follows that boundary value problems () has a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq260_HTML.gif , and the theorem is proved.

Theorem 4.2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq261_HTML.gif hold. In addition, one supposes that the following condition is satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq263_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq264_HTML.gif (particularly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq265_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq266_HTML.gif ).

Then boundary value problem () has at least one positive solution.

5. The Existence of Multiple Positive Solutions

Now we discuss the multiplicity of positive solutions for boundary value problem (). We obtain the following existence results.

Theorem 5.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq269_HTML.gif , and the following two conditions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq271_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq272_HTML.gif (particularly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq273_HTML.gif );

there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq275_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq276_HTML.gif

Then boundary value problem () has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq278_HTML.gif , which satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ66_HTML.gif
(5.1)

Proof.

We consider condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq279_HTML.gif . Choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq280_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq281_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq282_HTML.gif , then by the proof of (4.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ67_HTML.gif
(5.2)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq283_HTML.gif , then similar to the proof of (4.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ68_HTML.gif
(5.3)
On the other hand, by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq284_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq285_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ69_HTML.gif
(5.4)
By (5.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ70_HTML.gif
(5.5)

Applying Lemma 3.3 to (5.2), (5.3), and (5.5) yields that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq286_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq287_HTML.gif , and a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq288_HTML.gif Thus it follows that boundary value problem () has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq290_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq291_HTML.gif . Noticing (5.5), we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq293_HTML.gif . Therefore (5.1) holds, and the proof is complete.

Theorem 5.2.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq294_HTML.gif , and the following two conditions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq297_HTML.gif ;

there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq299_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq300_HTML.gif

Then boundary value problem () has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq302_HTML.gif , which satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ71_HTML.gif
(5.6)

Theorem 5.3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq303_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq304_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq305_HTML.gif hold. If there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq306_HTML.gif positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq307_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq308_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq310_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq311_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq312_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq313_HTML.gif

Then boundary value problem () has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq315_HTML.gif positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq316_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq317_HTML.gif

Theorem 5.4.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq318_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq319_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq320_HTML.gif hold. If there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq321_HTML.gif positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq322_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq323_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq325_HTML.gif is nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq326_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq327_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq329_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq330_HTML.gif

Then boundary value problem () has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq332_HTML.gif positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq333_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq334_HTML.gif

6. The Nonexistence of Positive Solution

Our last results corresponds to the case when boundary value problem () has no positive solution.

Theorem 6.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq336_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq337_HTML.gif , then boundary value problem () has no positive solution.

Proof.

Assume to the contrary that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq339_HTML.gif is a positive solution of the boundary value problem (). Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq341_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ72_HTML.gif
(6.1)

which is a contradiction, and complete the proof.

Similarly, we have the following results.

Theorem 6.2.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq342_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq343_HTML.gif , then boundary value problem () has no positive solution.

7. Example

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

Example 7.1.

Consider the following boundary value problem of nonlinear fractional differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ73_HTML.gif
(7.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ74_HTML.gif
(7.2)
It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq345_HTML.gif hold. By simple computation, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ75_HTML.gif
(7.3)

thus it follows that problem (7.1) has a positive solution by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq346_HTML.gif .

8. Conclusions

In this paper, by using the famous Guo-Krasnoselskii fixed-point theorem, we have investigated the existence and multiplicity of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and obtained some easily verifiable sufficient criteria. The interesting point is that we obtain some new positive properties of Green's function, which significantly extend and improve many known results for fractional order cases, for example, see [1215, 19]. The methodology which we employed in studying the boundary value problems of integer-order differential equation in [28] can be modified to establish similar sufficient criteria for higher-order nonlinear fractional differential equations. It is worth mentioning that there are still many problems that remain open in this vital field except for the results obtained in this paper: for example, whether or not we can obtain the similar results of fractional differential equations with p-Laplace operator by employing the same technique of this paper, and whether or not our concise criteria can guarantee the existence of positive solutions for higher-order nonlinear fractional differential equations with impulses. More efforts are still needed in the future.

Declarations

Acknowledgments

The authors thank the referee for his/her careful reading of the manuscript and useful suggestions. These have greatly improved this paper. 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), the 2010 level of scientific research of improving project (5028123900), the Graduate Technology Innovation Project (5028211000) and Beijing Municipal Education Commission (71D0911003).

Authors’ Affiliations

(1)
School of Applied Science, Beijing Information Science & Technology University
(2)
Department of Mathematics and Physics, North China Electric Power University
(3)
Department of Mathematics, Beijing Institute of Technology

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© Meiqiang Feng et al. 2011

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