Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales

  • Chenghua Gao1Email author and

    Affiliated with

    • Hua Luo2

      Affiliated with

      Boundary Value Problems20112011:198598

      DOI: 10.1155/2011/198598

      Received: 4 May 2010

      Accepted: 3 June 2010

      Published: 27 June 2011

      Abstract

      We discuss the existence of solutions for the first-order multipoint BVPs on time scale http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq1_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq4_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq5_HTML.gif is a parameter, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq6_HTML.gif is a fixed number, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq8_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq9_HTML.gif is regressive and rd-continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq13_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq14_HTML.gif . For suitable http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq15_HTML.gif , some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.

      1. Introduction

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq16_HTML.gif be a time scale (a nonempty closed subset of the real line http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq17_HTML.gif ). We discuss the existence of positive solutions to the first-order multipoint BVPs on time scale http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq18_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq19_HTML.gif is a fixed number, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq21_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq22_HTML.gif is regressive and rd-continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq23_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq25_HTML.gif is defined in its standard form; see [1, page 59] for details.

      The multipoint boundary value problems arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq26_HTML.gif parts of different densities can be set up as a multipoint boundary value problem [2]; also many problems in the theory of elastic stability can be handled by a multipoint problem [3]. So, the existence of solutions to multipoint boundary value problems have been studied by many authors; see [413] and the reference therein. Especially, in recent years the existence of positive solutions to multipoint boundary value problems on time scales has caught considerable attention; see [1014]. For other background on dynamic equations on time scales, one can see [1, 1518].

      Our ideas arise from [13, 16]. In [13], Tian and Ge discussed the existence of positive solutions to nonlinear first-order three-point boundary value problems on time scale http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq27_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq28_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq29_HTML.gif is regressive and rd-continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq31_HTML.gif . The existence results are based on Krasnoselskii's fixed-point theorem in cones and Leggett-Williams's theorem.

      As we can see, if we take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq34_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq35_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq36_HTML.gif , then (1.1) is reduced to (1.2). Because of the influence of the parameter http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq37_HTML.gif , it will be more difficult to solve (1.1) than to solve (1.2).

      In 2009, by using the fixed-point index theory, Sun and Li [16] discussed the existence of positive solutions to the first-order PBVPs on time scale http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq38_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ3_HTML.gif
      (1.3)

      For suitable http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq39_HTML.gif , they gave some existence, multiplicity, and nonexistence criteria of positive solutions.

      Motivated by the above results, by using the well-known fixed-point index theory [16, 19], we attempt to obtain some existence, multiplicity and nonexistence criteria of positive solutions to (1.1) for suitable http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq40_HTML.gif .

      The rest of this paper is arranged as follows. Some preliminary results including Green's function are given in Section 2. In Section 3, we obtain some useful lemmas for the proof of the main result. In Section 4, some existence and multiplicity results are established. At last, some nonexistence results are given in Section 5.

      2. Preliminaries

      Throughout the rest of this paper, we make the following assumptions:

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq42_HTML.gif is continuous and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq43_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq44_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq46_HTML.gif is rd-continuous, which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq47_HTML.gif (where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq48_HTML.gif is defined in [16, 18, 20]).

      Moreover, let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ4_HTML.gif
      (2.1)

      Our main tool is the well-known results from the fixed-point index, which we state here for the convenience of the reader.

      Theorem 2.1 (see [19]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq49_HTML.gif be a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq50_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq51_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq52_HTML.gif , we define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq53_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq54_HTML.gif is completely continuous such http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq55_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq56_HTML.gif .

      (i)If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq57_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq58_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ5_HTML.gif
      (2.2)
      (ii)If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq59_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq60_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ6_HTML.gif
      (2.3)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq61_HTML.gif be equipped with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq62_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq63_HTML.gif is a Banach space.

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq64_HTML.gif , we consider the following linear BVP:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ7_HTML.gif
      (2.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ8_HTML.gif
      (2.5)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq65_HTML.gif , define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ9_HTML.gif
      (2.6)

      Lemma 2.2.

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq66_HTML.gif , the linear BVP (2.4)-(2.5) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq67_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq68_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ10_HTML.gif
      (2.7)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ11_HTML.gif
      (2.8)

      Proof.

      By (2.4), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ12_HTML.gif
      (2.9)
      So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ13_HTML.gif
      (2.10)
      And so,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ14_HTML.gif
      (2.11)
      Combining this with (2.5), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ15_HTML.gif
      (2.12)

      Lemma 2.3.

      If the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq69_HTML.gif is defined in (2.7), then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq70_HTML.gif may be expressed by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ16_HTML.gif
      (2.13)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ17_HTML.gif
      (2.14)

      Proof.

      When http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq71_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ18_HTML.gif
      (2.15)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq73_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ19_HTML.gif
      (2.16)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq75_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ20_HTML.gif
      (2.17)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq77_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ21_HTML.gif
      (2.18)

      When http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq78_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq79_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ22_HTML.gif
      (2.19)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq81_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ23_HTML.gif
      (2.20)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq83_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ24_HTML.gif
      (2.21)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq85_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ25_HTML.gif
      (2.22)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq87_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ26_HTML.gif
      (2.23)

      When http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq88_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ27_HTML.gif
      (2.24)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq90_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ28_HTML.gif
      (2.25)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq92_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ29_HTML.gif
      (2.26)
      ()For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq94_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ30_HTML.gif
      (2.27)

      Lemma 2.4.

      Green's function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq95_HTML.gif has the following properties.

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq96_HTML.gif ,

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq97_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq98_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq99_HTML.gif

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq100_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq101_HTML.gif

      Proof.

      This proof is similar to [13, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq102_HTML.gif ], so we omit it.

      Now, we define a cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq103_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq104_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ31_HTML.gif
      (2.28)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq105_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq106_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq108_HTML.gif .

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq109_HTML.gif , define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq110_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ32_HTML.gif
      (2.29)

      Similar to the proof of [13, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq111_HTML.gif ], we can see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq112_HTML.gif is completely continuous. By the above discussions, its not difficult to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq113_HTML.gif being a solution of BVP (1.1) equals the solution that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq114_HTML.gif is a fixed point of the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq115_HTML.gif .

      3. Some Lemmas

      Lemma 3.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq116_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq119_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ33_HTML.gif
      (3.1)

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq122_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ34_HTML.gif
      (3.2)

      Lemma 3.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq123_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq124_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq126_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ35_HTML.gif
      (3.3)

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq129_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ36_HTML.gif
      (3.4)

      Lemma 3.3.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq130_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq131_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ37_HTML.gif
      (3.5)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq132_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq133_HTML.gif .

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq134_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq136_HTML.gif . So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ38_HTML.gif
      (3.6)

      4. Some Existence and Multiplicity Results

      Theorem 4.1.

      Assume that (H1) and (H2) hold and that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq137_HTML.gif . Then the BVP (1.1) has at least two positive solutions for
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ39_HTML.gif
      (4.1)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq138_HTML.gif . Then it follows from (4.1) and Lemma 3.3 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ40_HTML.gif
      (4.2)
      In view of Theorem 2.1, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ41_HTML.gif
      (4.3)
      Now, combined with the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq139_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq140_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq141_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq143_HTML.gif uniformly, where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq144_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ42_HTML.gif
      (4.4)
      So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ43_HTML.gif
      (4.5)
      In view of (4.1), (4.4), (4.5), and Lemma 3.2, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ44_HTML.gif
      (4.6)
      It follows from Theorem 2.1 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ45_HTML.gif
      (4.7)
      By (4.3) and (4.7), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ46_HTML.gif
      (4.8)

      This shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq145_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq146_HTML.gif , which is a positive solution of the BVP (1.1).

      Now, by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq147_HTML.gif , there exits an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq148_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq149_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq151_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq152_HTML.gif is chosen so that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ47_HTML.gif
      (4.9)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq153_HTML.gif . Then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq155_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq156_HTML.gif . So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ48_HTML.gif
      (4.10)
      In view of (4.1), (4.9), and Lemma 3.2, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ49_HTML.gif
      (4.11)
      It follows from Theorem 2.1 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ50_HTML.gif
      (4.12)
      By (4.3) and (4.12), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ51_HTML.gif
      (4.13)

      This shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq157_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq158_HTML.gif , which is another positive solution of the BVP (1.1).

      Similar to the proof of Theorem 4.1, we have the following results.

      Theorem 4.2.

      Suppose that (H1) and (H2) hold and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ52_HTML.gif
      (4.14)

      Then,

      (i)equation (1.1) has at least one positive solution if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq159_HTML.gif ,

      (ii)equation (1.1) has at least one positive solution if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq160_HTML.gif ,

      (iii)equation (1.1) has at least two positive solutions if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq161_HTML.gif .

      Theorem 4.3.

      Assume that (H1) and (H2) hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq162_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq163_HTML.gif , then the BVP (1.1) has at least two positive solutions for
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ53_HTML.gif
      (4.15)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq164_HTML.gif . Then it follows from (4.15) and Lemma 3.3 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ54_HTML.gif
      (4.16)
      In view of Theorem 2.1, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ55_HTML.gif
      (4.17)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq165_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq166_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq167_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq169_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq170_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq171_HTML.gif So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ56_HTML.gif
      (4.18)
      In view of (4.15), (4.18), and Lemma 3.1, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ57_HTML.gif
      (4.19)
      It follows from Theorem 2.1 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ58_HTML.gif
      (4.20)
      By (4.17) and (4.20), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ59_HTML.gif
      (4.21)

      This shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq172_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq173_HTML.gif , which is a positive solution of the BVP (1.1).

      Now, by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq174_HTML.gif , there exists an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq175_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq176_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq178_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq179_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ60_HTML.gif
      (4.22)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq180_HTML.gif . Then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq181_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq183_HTML.gif . So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ61_HTML.gif
      (4.23)
      Combined with (4.22) and Lemma 3.1, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ62_HTML.gif
      (4.24)
      It follows from Theorem 2.1 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ63_HTML.gif
      (4.25)
      By (4.17) and (4.25), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ64_HTML.gif
      (4.26)

      This shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq184_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq185_HTML.gif , which is another positive solution of the BVP (1.1).

      Similar to the proof of Theorem 4.3, we have the following results.

      Theorem 4.4.

      Suppose that (H1) and (H2) hold and that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ65_HTML.gif
      (4.27)

      Then,

      (i)equation (1.1) has at least one positive solution if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq186_HTML.gif ,

      (ii)equation (1.1) has at least one positive solution if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq187_HTML.gif ,
      1. (iii)

        equation (1.1) has at least two positive solutions if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq188_HTML.gif .

         

      Theorem 4.5.

      Suppose that (H1) and (H2) hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq189_HTML.gif , then the BVP (1.1) has at least one positive solution for
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ66_HTML.gif
      (4.28)

      Proof.

      We only deal with the case that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq190_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq191_HTML.gif . The other three cases can be discussed similarly.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq192_HTML.gif satisfy (4.28) and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq193_HTML.gif be chosen such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ67_HTML.gif
      (4.29)
      From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq194_HTML.gif , we know that there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq195_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq196_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq197_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq198_HTML.gif . So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ68_HTML.gif
      (4.30)
      This combines with (4.29) and Lemma 3.2, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ69_HTML.gif
      (4.31)
      It follows from Theorem 2.1 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ70_HTML.gif
      (4.32)
      On the other hand, from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq199_HTML.gif , there exists an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq200_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq201_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq203_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq204_HTML.gif . Then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq205_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq207_HTML.gif . So,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ71_HTML.gif
      (4.33)
      Combined with (4.29) and Lemma 3.1, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ72_HTML.gif
      (4.34)
      It follows from Theorem 2.1 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ73_HTML.gif
      (4.35)
      By (4.32) and (4.35), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ74_HTML.gif
      (4.36)

      which implies that the BVP (1.1) has at least one positive solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq208_HTML.gif .

      Remark 4.6.

      By making some minor modifications to the proof of Theorem 4.5, we can obtain the existence of at least one positive solution, if one of the following conditions is satisfied:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq209_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq210_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq211_HTML.gif .

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq212_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq214_HTML.gif .

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq215_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq216_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq217_HTML.gif .

      (iv) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq218_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq219_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq220_HTML.gif .

      Remark 4.7.

      From Conditions (ii) and (iv) of Remark 4.6, we know that the conclusion in Theorem 4.5 holds for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq221_HTML.gif in these two cases. By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq222_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq223_HTML.gif , there exist two positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq224_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq225_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ75_HTML.gif
      (4.37)
      This is the condition of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq226_HTML.gif of [13]. By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq227_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq228_HTML.gif , there exist two positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq229_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq230_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ76_HTML.gif
      (4.38)

      This is the condition of Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq231_HTML.gif of [13]. So, our conclusions extend and improve the results of [13].

      5. Some Nonexistence Results

      Theorem 5.1.

      Assume that (H1) and (H2) hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq232_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq233_HTML.gif , then the BVP (1.1) has no positive solutions for sufficiently small http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq234_HTML.gif .

      Proof.

      In view of the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq235_HTML.gif , there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq236_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq237_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq238_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq239_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq240_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ77_HTML.gif
      (5.1)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ78_HTML.gif
      (5.2)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq241_HTML.gif and we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ79_HTML.gif
      (5.3)

      We assert that the BVP (1.1) has no positive solutions for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq242_HTML.gif .

      Suppose on the contrary that the BVP (1.1) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq243_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq244_HTML.gif . Then from (5.3) and Lemma 3.2, we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ80_HTML.gif
      (5.4)

      which is a contradiction.

      Theorem 5.2.

      Assume that (H1) and (H2) hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq245_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq246_HTML.gif , then the BVP (1.1) has no positive solutions for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq247_HTML.gif .

      Proof.

      By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq248_HTML.gif , there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq249_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq250_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq251_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq252_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq253_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq254_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq255_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ81_HTML.gif
      (5.5)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ82_HTML.gif
      (5.6)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq256_HTML.gif and we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ83_HTML.gif
      (5.7)

      We assert that the BVP (1.1) has no positive solutions for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq257_HTML.gif .

      Suppose on the contrary that the BVP (1.1) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq258_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq259_HTML.gif . Then from (5.7) and Lemma 3.1 we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ84_HTML.gif
      (5.8)

      which is a contradiction.

      Corollary 5.3.

      Assume that (H1) and (H2) hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq260_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq261_HTML.gif , then the BVP (1.1) has no positive solutions for sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq262_HTML.gif .

      Declarations

      Acknowledgments

      This work was supported by the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), the NSF of Liaoning Province (no. L09DJY065), and NWNU-LKQN-09-3

      Authors’ Affiliations

      (1)
      Department of Mathematics, Northwest Normal University
      (2)
      School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics

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      © C. Gao and H. Luo. 2011

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