Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems

  • Minghe Pei1 and

    Affiliated with

    • SungKag Chang2Email author

      Affiliated with

      Boundary Value Problems20092009:362983

      DOI: 10.1155/2009/362983

      Received: 5 February 2009

      Accepted: 14 July 2009

      Published: 19 August 2009

      Abstract

      We are concerned with the higher-order nonlinear three-point boundary value problems: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq1_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq2_HTML.gif with the three point boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq3_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq4_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq5_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq6_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq7_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq8_HTML.gif are continuous, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq9_HTML.gif are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.

      1. Introduction

      Higher-order boundary value problems were discussed in many papers in recent years; for instance, see [122] and references therein. However, most of all the boundary conditions in the above-mentioned references are for two-point boundary conditions [211, 14, 1722], and three-point boundary conditions are rarely seen [1, 12, 13, 16, 18]. Furthermore works for nonlinear three point boundary conditions are quite rare in literatures.

      The purpose of this article is to study the existence and uniqueness of solutions for higher order nonlinear three point boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ1_HTML.gif
      (1.1)
      with nonlinear three point boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq11_HTML.gif is a continuous function, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq12_HTML.gif are continuous functions, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq13_HTML.gif are arbitrary given constants. The tools we mainly used are the method of upper and lower solutions and Leray-Schauder degree theory.

      Note that for the cases of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq14_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq15_HTML.gif in the boundary conditions (1.2), our theorems hold also true. However, for brevity we exclude such cases in this paper.

      2. Preliminary

      In this section, we present some definitions and lemmas that are needed to our main results.

      Definition 2.1.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq16_HTML.gif are called lower and upper solutions of BVP (1.1), (1.2), respectively, if
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ3_HTML.gif
      (2.1)

      Definition 2.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq17_HTML.gif be a subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq18_HTML.gif . We say that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq19_HTML.gif satisfies the Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq20_HTML.gif if there exists a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq21_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ4_HTML.gif
      (2.2)

      Lemma 2.3 (see [10]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq22_HTML.gif be a continuous function satisfying the Nagumo condition on
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ5_HTML.gif
      (2.3)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq23_HTML.gif are continuous functions such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ6_HTML.gif
      (2.4)
      Then there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq24_HTML.gif (depending only on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq26_HTML.gif such that every solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq27_HTML.gif of (1.1) with
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ7_HTML.gif
      (2.5)

      satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq28_HTML.gif

      Lemma 2.4.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq29_HTML.gif be a continuous function. Then boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ8_HTML.gif
      (2.6)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ9_HTML.gif
      (2.7)

      has only the trivial solution.

      Proof.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq30_HTML.gif is a nontrivial solution of BVP (2.6), (2.7). Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq31_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq32_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq33_HTML.gif . We may assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq34_HTML.gif . There exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq35_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ10_HTML.gif
      (2.8)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq37_HTML.gif . From (2.6) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ11_HTML.gif
      (2.9)

      which is a contradiction. Hence BVP (2.6), (2.7) has only the trivial solution.

      3. Main Results

      We may now formulate and prove our main results on the existence and uniqueness of solutions for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq38_HTML.gif -order three point boundary value problem (1.1), (1.2).

      Theorem 3.1.

      Assume that

      (i)there exist lower and upper solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq39_HTML.gif of BVP (1.1), (1.2), respectively, such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ12_HTML.gif
      (3.1)
      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq40_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq41_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq42_HTML.gif is nonincreasing in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq43_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq44_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq45_HTML.gif is nonincreasing in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq46_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq47_HTML.gif and satisfies the Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq48_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ13_HTML.gif
      (3.2)

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq49_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq50_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq51_HTML.gif is nonincreasing in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq52_HTML.gif and nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq53_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq54_HTML.gif ;

      (iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq55_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq56_HTML.gif , and nonincreasing in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq57_HTML.gif and nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq58_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq59_HTML.gif

      Then BVP (1.1), (1.2) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq60_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq61_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ14_HTML.gif
      (3.3)

      Proof.

      For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq62_HTML.gif define
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ15_HTML.gif
      (3.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq63_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq64_HTML.gif .

      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq65_HTML.gif , we consider the auxiliary equation
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ16_HTML.gif
      (3.5)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq66_HTML.gif is given by the Nagumo condition, with the boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ17_HTML.gif
      (3.6)
      Then we can choose a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq67_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ18_HTML.gif
      (3.7)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ19_HTML.gif
      (3.8)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ20_HTML.gif
      (3.9)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ21_HTML.gif
      (3.10)

      In the following, we will complete the proof in four steps.

      Step 1.

      Show that every solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq68_HTML.gif of BVP (3.5), (3.6) satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ22_HTML.gif
      (3.11)

      independently of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq69_HTML.gif .

      Suppose that the estimate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq70_HTML.gif is not true. Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq71_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq72_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq73_HTML.gif . We may assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq74_HTML.gif . There exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq75_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ23_HTML.gif
      (3.12)

      There are three cases to consider.

      Case 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq76_HTML.gif ).

      In this case, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq78_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq79_HTML.gif , by (3.8), we get the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ24_HTML.gif
      (3.13)
      and for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq80_HTML.gif , we have the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ25_HTML.gif
      (3.14)

      Case 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq81_HTML.gif ).

      In this case,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ26_HTML.gif
      (3.15)
      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq82_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq83_HTML.gif , by (3.6) we have the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ27_HTML.gif
      (3.16)
      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq84_HTML.gif , by (3.9) and condition (iii) we can get the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ28_HTML.gif
      (3.17)

      Case 3 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq85_HTML.gif ).

      In this case,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ29_HTML.gif
      (3.18)
      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq86_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq87_HTML.gif , by (3.6) we have the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ30_HTML.gif
      (3.19)
      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq88_HTML.gif , by (3.10) and condition (iv) we can get the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ31_HTML.gif
      (3.20)
      By (3.6), the estimates
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ32_HTML.gif
      (3.21)

      are obtained by integration.

      Step 2.

      Show that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq89_HTML.gif such that every solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq90_HTML.gif of BVP (3.5), (3.6) satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ33_HTML.gif
      (3.22)

      independently of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq91_HTML.gif .

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ34_HTML.gif
      (3.23)
      and define the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq92_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ35_HTML.gif
      (3.24)
      In the following, we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq93_HTML.gif satisfies the Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq94_HTML.gif , independently of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq95_HTML.gif . In fact, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq96_HTML.gif satisfies the Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq97_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ36_HTML.gif
      (3.25)
      Furthermore, we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ37_HTML.gif
      (3.26)
      Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq98_HTML.gif satisfies the Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq99_HTML.gif , independently of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq100_HTML.gif . Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ38_HTML.gif
      (3.27)

      By Step 1 and Lemma 2.3, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq101_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq102_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq103_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq105_HTML.gif do not depend on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq106_HTML.gif , the estimate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq107_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq108_HTML.gif is also independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq109_HTML.gif .

      Step 3.

      Show that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq110_HTML.gif , BVP (3.5), (3.6) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq111_HTML.gif .

      Define the operators as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ39_HTML.gif
      (3.28)
      by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ40_HTML.gif
      (3.29)
      by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ41_HTML.gif
      (3.30)
      with
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ42_HTML.gif
      (3.31)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq112_HTML.gif is compact, we have the following compact operator:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ43_HTML.gif
      (3.32)
      defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ44_HTML.gif
      (3.33)

      Consider the set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq113_HTML.gif

      By Steps 1 and 2, the degree http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq114_HTML.gif is well defined for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq115_HTML.gif and by homotopy invariance, we get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ45_HTML.gif
      (3.34)
      Since the equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq116_HTML.gif has only the trivial solution from Lemma 2.4, by the degree theory we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ46_HTML.gif
      (3.35)
      Hence, the equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq117_HTML.gif has at least one solution. That is, the boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ47_HTML.gif
      (3.36)
      with the boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ48_HTML.gif
      (3.37)

      has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq118_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq119_HTML.gif .

      Step 4.

      Show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq120_HTML.gif is a solution of BVP (1.1), (1.2).

      In fact, the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq121_HTML.gif of BVP (3.36), (3.37) will be a solution of BVP (1.1), (1.2), if it satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ49_HTML.gif
      (3.38)
      By contradiction, suppose that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq122_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq123_HTML.gif . There exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq124_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ50_HTML.gif
      (3.39)

      Now there are three cases to consider.

      Case 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq125_HTML.gif ).

      In this case, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq126_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq127_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq129_HTML.gif . By conditions (i) and (ii), we get the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ51_HTML.gif
      (3.40)

      Case 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq130_HTML.gif ).

      In this case, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ52_HTML.gif
      (3.41)
      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq131_HTML.gif . By (3.37) and conditions (i) and (iii) we can get the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ53_HTML.gif
      (3.42)

      Case 3 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq132_HTML.gif ).

      In this case, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ54_HTML.gif
      (3.43)
      and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq133_HTML.gif . By (3.37) and conditions (i) and (iv) we can get the following contradiction:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ55_HTML.gif
      (3.44)
      Similarly, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq134_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq135_HTML.gif . Hence
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ56_HTML.gif
      (3.45)
      Also, by boundary condition (3.37) and condition (i), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ57_HTML.gif
      (3.46)
      Therefore by integration we have for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq136_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ58_HTML.gif
      (3.47)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ59_HTML.gif
      (3.48)

      Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq137_HTML.gif is a solution of BVP (1.1), (1.2) and satisfies (3.3).

      Now we give a uniqueness theorem by assuming additionally the differentiability for functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq140_HTML.gif , and a kind of estimating condition in Theorem 3.1.

      Theorem 3.2.

      Assume that

      (i)there exist lower and upper solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq141_HTML.gif of BVP (1.1), (1.2), respectively, such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ60_HTML.gif
      (3.49)

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq142_HTML.gif

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq143_HTML.gif and its first-order partial derivatives in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq144_HTML.gif are continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq146_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq147_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq148_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq149_HTML.gif and satisfy the Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq150_HTML.gif

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq151_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq152_HTML.gif and continuously partially differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq153_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ61_HTML.gif
      (3.50)
      (iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq154_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq155_HTML.gif and continuously partially differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq156_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ62_HTML.gif
      (3.51)

      (v)there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq157_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq158_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq159_HTML.gif and

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ63_HTML.gif
      (3.52)

      Then BVP (1.1), (1.2) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq160_HTML.gif satisfying (3.3).

      Proof.

      The existence of a solution for BVP (1.1), (1.2) satisfying (3.3) follows from Theorem 3.1.

      Now, we prove the uniqueness of solution for BVP (1.1), (1.2). To do this, we let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq162_HTML.gif are any two solutions of BVP (1.1), (1.2) satisfying (3.3). Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq163_HTML.gif . It is easy to show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq164_HTML.gif is a solution of the following boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ64_HTML.gif
      (3.53)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ65_HTML.gif
      (3.54)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ66_HTML.gif
      (3.55)

      where for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq165_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ67_HTML.gif
      (3.56)
      By conditions (ii), (iii), and (iv), we have that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq166_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ68_HTML.gif
      (3.57)
      Now suppose that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq167_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq168_HTML.gif . Without loss of generality assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq169_HTML.gif , and let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ69_HTML.gif
      (3.58)
      It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq170_HTML.gif by condition (v), hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq171_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq172_HTML.gif . We have that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq173_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq174_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq175_HTML.gif , and there exists a point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq176_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq177_HTML.gif . Furthermore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq178_HTML.gif . In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq179_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq180_HTML.gif . By condition (v) and (3.55) we can easily show that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ70_HTML.gif
      (3.59)
      In particular
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ71_HTML.gif
      (3.60)
      Hence
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ72_HTML.gif
      (3.61)

      which contradicts to (3.54). Thus http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq181_HTML.gif . Similarly we can show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq182_HTML.gif . Consequently http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq183_HTML.gif .

      Now, there are two cases to consider, that is
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ73_HTML.gif
      (3.62)
      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq184_HTML.gif , then by (3.59) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ74_HTML.gif
      (3.63)
      Thus, by (3.53) and condition (v) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ75_HTML.gif
      (3.64)
      Consequently, by Taylor's theorem there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq185_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ76_HTML.gif
      (3.65)

      which is a contradiction.

      A similar contradiction can be obtained if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq186_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq187_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq188_HTML.gif . By (3.55), we obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq189_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq190_HTML.gif . This completes the proof of the theorem.

      Next we give two examples to demonstrate the application of Theorem 3.2.

      Example 3.3.

      Consider the following third-order three point BVP:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ77_HTML.gif
      (3.66)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ78_HTML.gif
      (3.67)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ79_HTML.gif
      (3.68)
      Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq192_HTML.gif . It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq193_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq194_HTML.gif are lower and upper solutions of BVP (3.66), (3.67) respectively, and all the assumptions in Theorem 3.2 are satisfied. Therefore by Theorem 3.2 BVP (3.66), (3.67) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq195_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ80_HTML.gif
      (3.69)

      Example 3.4.

      Consider the following fourth-order three point BVP:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ81_HTML.gif
      (3.70)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ82_HTML.gif
      (3.71)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ83_HTML.gif
      (3.72)
      Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq196_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq197_HTML.gif . It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq198_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq199_HTML.gif are lower and upper solutions of BVP (3.70), (3.71), respectively, and all the assumptions in Theorem 3.2 are satisfied. Therefore by Theorem 3.2 BVP (3.70), (3.71) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq200_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ84_HTML.gif
      (3.73)

      Authors’ Affiliations

      (1)
      Department of Mathematics, Bei Hua University
      (2)
      Department of Mathematics, Yeungnam University

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      Copyright

      © M. Pei and S. K. Chang. 2009

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.