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

  • Meiqiang Feng1Email author,

    Affiliated with

    • Xuemei Zhang2, 3 and

      Affiliated with

      • WeiGao Ge3

        Affiliated with

        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.
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ2_HTML.gif
        (1.2)

        where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq2_HTML.gif , the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq3_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq4_HTML.gif , and function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq5_HTML.gif may be singular at http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq7_HTML.gif satisfies the following conditions:

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

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

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

        here http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq13_HTML.gif with http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq16_HTML.gif such that
        http://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.
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ6_HTML.gif
        (P)

        where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq19_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq20_HTML.gif or/and at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq22_HTML.gif is nonnegative, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq23_HTML.gif .

        For the case of http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq34_HTML.gif positive solutions, where http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ7_HTML.gif
        (1.6)

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq41_HTML.gif denotes the integer part of number http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq44_HTML.gif with a fractional derivative of order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq45_HTML.gif that belongs to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq46_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ9_HTML.gif
        (1.8)

        for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq47_HTML.gif , where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq51_HTML.gif Then for any http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ10_HTML.gif
        (2.1)
        is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ11_HTML.gif
        (2.2)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ12_HTML.gif
        (2.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ13_HTML.gif
        (2.4)
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ15_HTML.gif
        (2.6)
        By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq53_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq54_HTML.gif . Thus,
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ17_HTML.gif
        (2.8)
        By (2.8) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq55_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq56_HTML.gif Similarly, we can obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq57_HTML.gif Then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ18_HTML.gif
        (2.9)
        By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq58_HTML.gif , we have
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ20_HTML.gif
        (2.11)

        where http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ21_HTML.gif
        (2.12)
        It follows that
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ23_HTML.gif
        (2.14)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq60_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq62_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq64_HTML.gif defined by (2.4) satisfies

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

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

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ24_HTML.gif
        (2.15)
        where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq68_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq70_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq71_HTML.gif .

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

          Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq75_HTML.gif , it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq76_HTML.gif is increasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq77_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq79_HTML.gif , for given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq80_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ28_HTML.gif
        (2.19)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ29_HTML.gif
        (2.20)
        Then, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ30_HTML.gif
        (2.21)
        and so,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ31_HTML.gif
        (2.22)
        Noticing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq81_HTML.gif , from (2.22), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ32_HTML.gif
        (2.23)

        Then, for given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq82_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq83_HTML.gif arrives at maximum at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq84_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq85_HTML.gif . This together with the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq86_HTML.gif is increasing on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq88_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq89_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq90_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ33_HTML.gif
        (2.24)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig1_HTML.jpg
        Figure 1

        Graph of functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq91_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq92_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq94_HTML.gif is increasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq95_HTML.gif .
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig2_HTML.jpg
        Figure 2

        Graph of function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq96_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq98_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq99_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq100_HTML.gif .
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig3_HTML.jpg
        Figure 3

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

        Remark 2.6.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq104_HTML.gif . From (2.15), for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq105_HTML.gif , we have
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq106_HTML.gif is decreasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq107_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ36_HTML.gif
        (2.27)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Fig4_HTML.jpg
        Figure 4

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

        Proposition 2.9.

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

        Proof.

        For http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq112_HTML.gif .

        Case 1.

        If http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ38_HTML.gif
        (2.29)
        It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq115_HTML.gif are bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq116_HTML.gif . So, there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq117_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ39_HTML.gif
        (2.30)

        Case 2.

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

        Case 3.

        If http://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
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq126_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ45_HTML.gif
        (2.36)

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

        Proposition 2.10.

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

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

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

        Proof.

        Using the properties of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq132_HTML.gif , definition of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq134_HTML.gif , the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq135_HTML.gif defined by (2.3) satisfies

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

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

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

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq140_HTML.gif is defined by (2.16), http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq142_HTML.gif is continuous for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq143_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq145_HTML.gif for each http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq147_HTML.gif , it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq148_HTML.gif .

        3. Preliminaries

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq149_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq151_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq152_HTML.gif Let
        http://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:

        () http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq156_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq157_HTML.gif is defined in Theorem 2.11;

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

        () http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq164_HTML.gif , where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq167_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq168_HTML.gif or/and at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq169_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq170_HTML.gif or/and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq171_HTML.gif .

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

        where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq176_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq177_HTML.gif is a fixed point of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq179_HTML.gif hold. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq181_HTML.gif is completely continuous.

        Proof.

        For any http://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 http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ53_HTML.gif
        (3.4)

        So, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq184_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq185_HTML.gif . Next by similar proof of Lemma http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq190_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq192_HTML.gif be two bounded open sets in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq193_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq194_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq195_HTML.gif . Let operator http://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) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq197_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq198_HTML.gif

        or

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

        is satisfied. Then http://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 http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ54_HTML.gif
        (4.1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq205_HTML.gif denotes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq206_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq207_HTML.gif and
        http://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 http://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:

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

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

        for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq219_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq220_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq221_HTML.gif for all http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq225_HTML.gif holds. Considering http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq226_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq227_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq228_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq229_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq230_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq231_HTML.gif . Then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq232_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ56_HTML.gif
        (4.3)
        that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq233_HTML.gif imply that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ57_HTML.gif
        (4.4)
        Next, turning to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq234_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq235_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ58_HTML.gif
        (4.5)

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

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

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

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

        Case 2.

        The Condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq242_HTML.gif satisfies. For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq243_HTML.gif , from (3.1) we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq244_HTML.gif . Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq245_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq246_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq247_HTML.gif , this together with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq248_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ62_HTML.gif
        (4.9)
        that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq249_HTML.gif imply that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq250_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq251_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq252_HTML.gif , this together with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq253_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ64_HTML.gif
        (4.11)
        that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq254_HTML.gif imply that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq255_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq256_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq257_HTML.gif with http://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 http://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 http://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:

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq263_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq264_HTML.gif (particularly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq265_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq269_HTML.gif , and the following two conditions:

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

        there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq275_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq278_HTML.gif , which satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ66_HTML.gif
        (5.1)

        Proof.

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

        If http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ67_HTML.gif
        (5.2)
        If http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq284_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq285_HTML.gif we have
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq286_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq287_HTML.gif , and a fixed point http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq290_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq291_HTML.gif . Noticing (5.5), we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq292_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq294_HTML.gif , and the following two conditions:

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

        there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq299_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq302_HTML.gif , which satisfy
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq303_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq304_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq305_HTML.gif hold. If there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq306_HTML.gif positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq307_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq308_HTML.gif such that

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq310_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq311_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq312_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq315_HTML.gif positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq316_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq317_HTML.gif

        Theorem 5.4.

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

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

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq329_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq332_HTML.gif positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq333_HTML.gif satisfying http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq336_HTML.gif and http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq341_HTML.gif , and
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq342_HTML.gif and http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_Equ73_HTML.gif
        (7.1)
        where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F720702/MediaObjects/13661_2010_Article_54_IEq345_HTML.gif hold. By simple computation, we have
        http://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 http://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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