Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System

  • Zigen Ouyang1Email author,

    Affiliated with

    • Yuming Chen2 and

      Affiliated with

      • Shuliang Zou3

        Affiliated with

        Boundary Value Problems20112011:475126

        DOI: 10.1155/2011/475126

        Received: 14 November 2010

        Accepted: 24 February 2011

        Published: 14 March 2011

        Abstract

        Though boundary value problems for fractional differential equations have been extensively studied, most of the studies focus on scalar equations and the fractional order between 1 and 2. On the other hand, delay is natural in practical systems. However, not much has been done for fractional differential equations with delays. Therefore, in this paper, we consider a boundary value problem of a general delayed nonlinear fractional system. With the help of some fixed point theorems and the properties of the Green function, we establish several sets of sufficient conditions on the existence of positive solutions. The obtained results extend and include some existing ones and are illustrated with some examples for their feasibility.

        1. Introduction

        In the past decades, fractional differential equations have been intensively studied. This is due to the rapid development of the theory of fractional differential equations itself and the applications of such construction in various sciences such as physics, mechanics, chemistry, and engineering [1, 2]. For the basic theory of fractional differential equations, we refer the readers to [37].

        Recently, many researchers have devoted their attention to studying the existence of (positive) solutions of boundary value problems for differential equations with fractional order [823]. We mention that the fractional order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq1_HTML.gif involved is generally in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq2_HTML.gif with the exception that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq3_HTML.gif in [12, 23] and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq4_HTML.gif in [8, 17]. Though there have been extensive study on systems of fractional differential equations, not much has been done for boundary value problems for systems of fractional differential equations [1820].

        On the other hand, we know that delay arises naturally in practical systems due to the transmission of signal or the mechanical transmission. Though theory of ordinary differential equations with delays is mature, not much has been done for fractional differential equations with delays [2431].

        As a result, in this paper, we consider the following nonlinear system of fractional order differential equations with delays,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq5_HTML.gif is the standard Riemann-Liouville fractional derivative of order   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq6_HTML.gif for some integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq8_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq10_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq11_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq12_HTML.gif is a nonlinear function from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq13_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq14_HTML.gif . The purpose is to establish sufficient conditions on the existence of positive solutions to (1.1) by using some fixed point theorems and some properties of the Green function. By a positive solution to (1.1) we mean a mapping with positive components on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq15_HTML.gif such that (1.1) is satisfied. Obviously, (1.1) includes the usual system of fractional differential equations when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq16_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq17_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq18_HTML.gif . Therefore, the obtained results generalize and include some existing ones.

        The remaining part of this paper is organized as follows. In Section 2, we introduce some basics of fractional derivative and the fixed point theorems which will be used in Section 3 to establish the existence of positive solutions. To conclude the paper, the feasibility of some of the results is illustrated with concrete examples in Section 4.

        2. Preliminaries

        We first introduce some basic definitions of fractional derivative for the readers' convenience.

        Definition 2.1 (see [3, 32]).

        The fractional integral of order   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq19_HTML.gif of a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq20_HTML.gif is defined as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ2_HTML.gif
        (2.1)

        provided that the integral exists on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq21_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq22_HTML.gif is the Gamma function.

        Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq23_HTML.gif has the semigroup property, that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ3_HTML.gif
        (2.2)

        Definition 2.2 (see [3, 32]).

        The Riemann-Liouville derivative of order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq24_HTML.gif of a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq25_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ4_HTML.gif
        (2.3)

        provided that the right-hand side is pointwise defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq26_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq27_HTML.gif .

        It is well known that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq28_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq29_HTML.gif . Furthermore, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq31_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq32_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq33_HTML.gif .

        The following results on fractional integral and fractional derivative will be needed in establishing our main results.

        Lemma 2.3 (see [10]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq34_HTML.gif . Then solutions to the fractional equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq35_HTML.gif can be written as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ5_HTML.gif
        (2.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq37_HTML.gif .

        Lemma 2.4 (see [10]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq38_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ6_HTML.gif
        (2.5)

        for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq40_HTML.gif .

        Now, we cite the fixed point theorems to be used in Section 3.

        Lemma 2.5 (the Banach contraction mapping theorem [33]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq41_HTML.gif be a complete metric space and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq42_HTML.gif be a contraction mapping. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq43_HTML.gif has a unique fixed point.

        Lemma 2.6 (see [16, 34]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq44_HTML.gif be a closed and convex subset of a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq45_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq46_HTML.gif is a relatively open subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq47_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq49_HTML.gif is completely continuous. Then at least one of the following two properties holds:

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq50_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq51_HTML.gif ;

        (ii)there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq53_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq54_HTML.gif .

        Lemma 2.7 (the Krasnosel'skii fixed point theorem [33, 35]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq55_HTML.gif be a cone in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq56_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq58_HTML.gif are open subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq59_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq61_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq62_HTML.gif is a completely continuous operator such that either

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq63_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq65_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq66_HTML.gif

        or

        (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq67_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq69_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq70_HTML.gif .

        Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq71_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq72_HTML.gif .

        3. Existence of Positive Solutions

        Throughout this paper, we let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq73_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq74_HTML.gif is a Banach space, where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ7_HTML.gif
        (3.1)

        In this section, we always assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq75_HTML.gif .

        Lemma 3.1.

        System (1.1) is equivalent to the following system of integral equations:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ8_HTML.gif
        (3.2)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ9_HTML.gif
        (3.3)

        Proof.

        It is easy to see that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq76_HTML.gif satisfies (3.2) then it also satisfies (3.2). So, assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq77_HTML.gif is a solution to (1.1). Integrating both sides of the first equation of (1.1) of order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq78_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq79_HTML.gif gives us
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ10_HTML.gif
        (3.4)
        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq80_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq81_HTML.gif . It follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ11_HTML.gif
        (3.5)
        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq82_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq83_HTML.gif . This, combined with the boundary conditions in (1.1), yields
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ12_HTML.gif
        (3.6)
        Similarly, one can obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ13_HTML.gif
        (3.7)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ14_HTML.gif
        (3.8)
           http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq84_HTML.gif . Then it follows from (3.8) and the boundary condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq85_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ15_HTML.gif
        (3.9)
        Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq86_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ16_HTML.gif
        (3.10)

        This completes the proof.

        The following two results give some properties of the Green functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq87_HTML.gif .

        Lemma 3.2.

        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq88_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq90_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq91_HTML.gif .

        Proof.

        Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq92_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq93_HTML.gif . It remains to show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq94_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq95_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq96_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq97_HTML.gif . We only need to show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq98_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq99_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq100_HTML.gif , let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ17_HTML.gif
        (3.11)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ18_HTML.gif
        (3.12)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ19_HTML.gif
        (3.13)

        Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq102_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq103_HTML.gif . It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq104_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq105_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq106_HTML.gif .

        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq107_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq108_HTML.gif and the proof is complete.

        Lemma 3.3.
        1. (i)

          If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq109_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq110_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq111_HTML.gif .

           
        2. (ii)

          If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq112_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq113_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq114_HTML.gif .

           
        Proof.
        1. (i)
          Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq115_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq116_HTML.gif . Now, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq117_HTML.gif , we have
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ20_HTML.gif
          (3.14)
           
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq118_HTML.gif is the function defined by (3.11). It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq119_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq120_HTML.gif . In summary, we have proved (i).
        1. (ii)
          Again, one can easily see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq121_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq122_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq123_HTML.gif , we have in this case that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ21_HTML.gif
          (3.15)
           

        which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq124_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq125_HTML.gif . To summarize, we have proved (ii) and this completes the proof.

        Now, we are ready to present the main results.

        Theorem 3.4.

        Suppose that there exist functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq127_HTML.gif , 2, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq128_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ22_HTML.gif
        (3.16)
        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq131_HTML.gif . If
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ23_HTML.gif
        (3.17)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ24_HTML.gif
        (3.18)

        then (1.1) has a unique positive solution.

        Proof.

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ25_HTML.gif
        (3.19)
        It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq132_HTML.gif is a complete metric space. Define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq133_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq134_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ26_HTML.gif
        (3.20)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq135_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ27_HTML.gif
        (3.21)
        Because of the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq137_HTML.gif , it follows easily from Lemma 3.2 that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq138_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq139_HTML.gif into itself. To finish the proof, we only need to show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq140_HTML.gif is a contraction. Indeed, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq141_HTML.gif , by (3.16) we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ28_HTML.gif
        (3.22)

        This, combined with Lemma 3.3 and (3.17) and (3.18), immediately implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq142_HTML.gif is a contraction. Therefore, the proof is complete with the help of Lemmas 3.1 and 2.5.

        The following result can be proved in the same spirit as that for Theorem 3.4.

        Theorem 3.5.

        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq143_HTML.gif , suppose that there exist nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq144_HTML.gif and nonnegative constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq145_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq146_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ29_HTML.gif
        (3.23)
        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq148_HTML.gif . If
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ30_HTML.gif
        (3.24)

        then (1.1) has a unique positive solution.

        Theorem 3.6.

        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq149_HTML.gif , suppose that there exist nonnegative real-valued functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq150_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ31_HTML.gif
        (3.25)
        for almost every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq151_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq152_HTML.gif . If
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ32_HTML.gif
        (3.26)

        then (1.1) has at least one positive solution.

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq154_HTML.gif be defined by (3.19) and (3.20), respectively. We first show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq155_HTML.gif is completely continuous through the following three steps.

        Step 1.

        Show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq156_HTML.gif is continuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq157_HTML.gif be a sequence in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq158_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq159_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq160_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq161_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq162_HTML.gif is continuous, it is uniformly continuous on any compact set. In particular, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq163_HTML.gif , there exists a positive integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq164_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ33_HTML.gif
        (3.27)
        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq166_HTML.gif . Then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq167_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ34_HTML.gif
        (3.28)
        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq169_HTML.gif . Therefore,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ35_HTML.gif
        (3.29)

        which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq170_HTML.gif is continuous.

        Step 2.

        Show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq171_HTML.gif maps bounded sets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq172_HTML.gif into bounded sets. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq173_HTML.gif be a bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq174_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq175_HTML.gif is bounded. Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq176_HTML.gif is continuous, there exists an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq177_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ36_HTML.gif
        (3.30)
        It follows that, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq179_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ37_HTML.gif
        (3.31)

        Immediately, we can easily see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq180_HTML.gif is a bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq181_HTML.gif .

        Step 3.

        Show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq182_HTML.gif maps bounded sets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq183_HTML.gif into equicontinuous sets. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq184_HTML.gif be a bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq185_HTML.gif . Similarly as in Step 2, there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq186_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ38_HTML.gif
        (3.32)
        Then, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq187_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq188_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq189_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ39_HTML.gif
        (3.33)

        Now the equicontituity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq190_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq191_HTML.gif follows easily from the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq192_HTML.gif is continuous and hence uniformly continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq193_HTML.gif .

        Now we have shown that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq194_HTML.gif is completely continuous. To apply Lemma 2.6, let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ40_HTML.gif
        (3.34)
        Fix http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq195_HTML.gif and define
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ41_HTML.gif
        (3.35)
        We claim that there is no http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq196_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq197_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq198_HTML.gif . Otherwise, assume that there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq200_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq201_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ42_HTML.gif
        (3.36)
        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq202_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ43_HTML.gif
        (3.37)

        Similarly, we can have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq203_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq204_HTML.gif . To summarize, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq205_HTML.gif , a contradiction to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq206_HTML.gif . This proves the claim. Applying Lemma 2.6, we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq207_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq208_HTML.gif , which is a positive solution to (1.1) by Lemma 3.1. Therefore, the proof is complete.

        As a consequence of Theorem 3.6, we have the following.

        Corollary 3.7.

        If all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq209_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq210_HTML.gif , are bounded, then (1.1) has at least one positive solution.

        To state the last result of this section, we introduce
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ44_HTML.gif
        (3.38)

        Theorem 3.8.

        Suppose that there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq211_HTML.gif and positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq212_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq213_HTML.gif such that

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq214_HTML.gif   for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq215_HTML.gif

        and

        (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq216_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq217_HTML.gif ,

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq218_HTML.gif . Then (1.1) has at least a positive solution.

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq219_HTML.gif be defined by (3.19) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq220_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq221_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq222_HTML.gif . From the proof of Theorem 3.6, we know that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq223_HTML.gif defined by (3.20) is completely continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq224_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq225_HTML.gif , it follows from Lemma 3.3 and condition (ii) that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ45_HTML.gif
        (3.39)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ46_HTML.gif
        (3.40)
        On the other hand, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq226_HTML.gif , it follows from Lemma 3.3 and condition (i) that, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq227_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ47_HTML.gif
        (3.41)
        if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq228_HTML.gif , whereas
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ48_HTML.gif
        (3.42)
        if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq229_HTML.gif . In summary,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ49_HTML.gif
        (3.43)

        Therefore, we have verified condition (ii) of Lemma 2.7. It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq230_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq231_HTML.gif , which is a positive solution to (1.1). This completes the proof.

        4. Examples

        In this section, we demonstrate the feasibility of some of the results obtained in Section 3.

        Example 4.1.

        Consider
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ50_HTML.gif
        (4.1)
        Here
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ51_HTML.gif
        (4.2)
        One can easily see that (3.16) is satisfied with
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ52_HTML.gif
        (4.3)
        Moreover,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ53_HTML.gif
        (4.4)
        and hence
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ54_HTML.gif
        (4.5)

        It follows from Theorem 3.4 that (4.1) has a unique positive solution on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq232_HTML.gif .

        Example 4.2.

        Consider
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ55_HTML.gif
        (4.6)
        Here
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ56_HTML.gif
        (4.7)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ57_HTML.gif
        (4.8)
        Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq233_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq234_HTML.gif satisfy (3.25). Moreover, simple calculations give us
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ58_HTML.gif
        (4.9)
        Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq235_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ59_HTML.gif
        (4.10)
        Choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq236_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq237_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ60_HTML.gif
        (4.11)
        Then, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq238_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq239_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ61_HTML.gif
        (4.12)
        for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq240_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq241_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ62_HTML.gif
        (4.13)

        By now we have verified all the assumptions of Theorem 3.8. Therefore, (4.6) has at least one positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq242_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq243_HTML.gif .

        Declarations

        Acknowledgment

        Supported partially by the Doctor Foundation of University of South China under Grant no. 5-XQD-2006-9, the Foundation of Science and Technology Department of Hunan Province under Grant no. 2009RS3019 and the Subject Lead Foundation of University of South China no. 2007XQD13. Research was partially supported by the Natural Science and Engineering Re-search Council of Canada (NSERC) and the Early Researcher Award (ERA) Pro-gram of Ontario.

        Authors’ Affiliations

        (1)
        School of Mathematics and Physics, School of Nuclear Science and Technology, University of South China
        (2)
        Department of Mathematics, Wilfrid Laurier University
        (3)
        School of Nuclear Science and Technology, University of South China

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        © Zigen Ouyang et al. 2011

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