Open Access

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

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: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq1_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq2_HTML.gif with the three point boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq3_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq4_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq5_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq6_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq7_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq8_HTML.gif are continuous, and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq11_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq12_HTML.gif are continuous functions, and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq14_HTML.gif or https://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.

https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ3_HTML.gif
(2.1)

Definition 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq17_HTML.gif be a subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq18_HTML.gif . We say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq19_HTML.gif satisfies the Nagumo condition on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq20_HTML.gif if there exists a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq21_HTML.gif such that
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ5_HTML.gif
(2.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq23_HTML.gif are continuous functions such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq24_HTML.gif (depending only on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq26_HTML.gif such that every solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq27_HTML.gif of (1.1) with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ7_HTML.gif
(2.5)

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

Lemma 2.4.

Let https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ8_HTML.gif
(2.6)
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq31_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq32_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq33_HTML.gif . We may assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq34_HTML.gif . There exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq35_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ10_HTML.gif
(2.8)
Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq37_HTML.gif . From (2.6) we have
https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ12_HTML.gif
(3.1)
(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq40_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq41_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq42_HTML.gif is nonincreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq43_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq44_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq45_HTML.gif is nonincreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq46_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq47_HTML.gif and satisfies the Nagumo condition on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq48_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ13_HTML.gif
(3.2)

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

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq55_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq56_HTML.gif , and nonincreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq57_HTML.gif and nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq58_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq60_HTML.gif such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq61_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ14_HTML.gif
(3.3)

Proof.

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

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

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq65_HTML.gif , we consider the auxiliary equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ16_HTML.gif
(3.5)
where https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq67_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ18_HTML.gif
(3.7)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ19_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ20_HTML.gif
(3.9)
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ22_HTML.gif
(3.11)

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

Suppose that the estimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq70_HTML.gif is not true. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq71_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq72_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq73_HTML.gif . We may assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq74_HTML.gif . There exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq75_HTML.gif such that
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq76_HTML.gif ).

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

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

In this case,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ26_HTML.gif
(3.15)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq82_HTML.gif . For https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ27_HTML.gif
(3.16)
For https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ28_HTML.gif
(3.17)

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

In this case,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ29_HTML.gif
(3.18)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq86_HTML.gif . For https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ30_HTML.gif
(3.19)
For https://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:
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq89_HTML.gif such that every solution https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ33_HTML.gif
(3.22)

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

Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ34_HTML.gif
(3.23)
and define the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq92_HTML.gif as follows:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq93_HTML.gif satisfies the Nagumo condition on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq94_HTML.gif , independently of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq95_HTML.gif . In fact, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq96_HTML.gif satisfies the Nagumo condition on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq97_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ36_HTML.gif
(3.25)
Furthermore, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ37_HTML.gif
(3.26)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq98_HTML.gif satisfies the Nagumo condition on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq99_HTML.gif , independently of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq100_HTML.gif . Let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq101_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq102_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq103_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq105_HTML.gif do not depend on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq106_HTML.gif , the estimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq107_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq108_HTML.gif is also independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq109_HTML.gif .

Step 3.

Show that for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq111_HTML.gif .

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

Consider the set https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq114_HTML.gif is well defined for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq115_HTML.gif and by homotopy invariance, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ45_HTML.gif
(3.34)
Since the equation https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ46_HTML.gif
(3.35)
Hence, the equation https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ47_HTML.gif
(3.36)
with the boundary conditions
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq118_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq119_HTML.gif .

Step 4.

Show that https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq122_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq123_HTML.gif . There exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq124_HTML.gif such that
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq125_HTML.gif ).

In this case, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq126_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq127_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq128_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ51_HTML.gif
(3.40)

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

In this case, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ52_HTML.gif
(3.41)
and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ53_HTML.gif
(3.42)

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

In this case, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ54_HTML.gif
(3.43)
and https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq134_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq135_HTML.gif . Hence
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq136_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ58_HTML.gif
(3.47)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ59_HTML.gif
(3.48)

Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq139_HTML.gif and https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ60_HTML.gif
(3.49)

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

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

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq151_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq152_HTML.gif and continuously partially differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq153_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ61_HTML.gif
(3.50)
(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq154_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq155_HTML.gif and continuously partially differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq156_HTML.gif , and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq157_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq158_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq159_HTML.gif and

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq161_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq163_HTML.gif . It is easy to show that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ64_HTML.gif
(3.53)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ65_HTML.gif
(3.54)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ66_HTML.gif
(3.55)

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq166_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq167_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq168_HTML.gif . Without loss of generality assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq169_HTML.gif , and let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq170_HTML.gif by condition (v), hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq171_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq172_HTML.gif . We have that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq174_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq175_HTML.gif , and there exists a point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq176_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq177_HTML.gif . Furthermore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq178_HTML.gif . In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq179_HTML.gif , then https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ70_HTML.gif
(3.59)
In particular
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ71_HTML.gif
(3.60)
Hence
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq181_HTML.gif . Similarly we can show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq182_HTML.gif . Consequently https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ73_HTML.gif
(3.62)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq184_HTML.gif , then by (3.59) we have
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq185_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq186_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq187_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq188_HTML.gif . By (3.55), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq189_HTML.gif on https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ77_HTML.gif
(3.66)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ78_HTML.gif
(3.67)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ79_HTML.gif
(3.68)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq192_HTML.gif . It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq193_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq195_HTML.gif satisfying
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ81_HTML.gif
(3.70)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ82_HTML.gif
(3.71)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_Equ83_HTML.gif
(3.72)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq197_HTML.gif . It is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq198_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F362983/MediaObjects/13661_2009_Article_842_IEq200_HTML.gif satisfying
https://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

References

  1. Aftabizadeh AR, Gupta CP, Xu J-M: Existence and uniqueness theorems for three-point boundary value problems. SIAM Journal on Mathematical Analysis 1989, 20(3):716–726. 10.1137/0520049MATHMathSciNetView ArticleGoogle Scholar
  2. Agarwal RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Teaneck, NJ, USA; 1986:xii+307.MATHView ArticleGoogle Scholar
  3. Agarwal RP, Wong F-H: Existence of positive solutions for non-positive higher-order BVPs. Journal of Computational and Applied Mathematics 1998, 88(1):3–14. 10.1016/S0377-0427(97)00211-2MATHMathSciNetView ArticleGoogle Scholar
  4. Agarwal RP, Grace SR, O'Regan D: Semipositone higher-order differential equations. Applied Mathematics Letters 2004, 17(2):201–207. 10.1016/S0893-9659(04)90033-XMATHMathSciNetView ArticleGoogle Scholar
  5. Cabada A: The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. Journal of Mathematical Analysis and Applications 1994, 185(2):302–320. 10.1006/jmaa.1994.1250MATHMathSciNetView ArticleGoogle Scholar
  6. Cabada A, Grossinho MR, Minhós F: Extremal solutions for third-order nonlinear problems with upper and lower solutions in reversed order. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(6):1109–1121. 10.1016/j.na.2005.04.023MATHMathSciNetView ArticleGoogle Scholar
  7. Du Z, Ge W, Lin X: Existence of solutions for a class of third-order nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 2004, 294(1):104–112. 10.1016/j.jmaa.2004.02.001MATHMathSciNetView ArticleGoogle Scholar
  8. Feng Y, Liu S: Solvability of a third-order two-point boundary value problem. Applied Mathematics Letters 2005, 18(9):1034–1040. 10.1016/j.aml.2004.04.016MATHMathSciNetView ArticleGoogle Scholar
  9. Grossinho MR, Minhós FM: Existence result for some third order separated boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2001, 47(4):2407–2418. 10.1016/S0362-546X(01)00364-9MATHMathSciNetView ArticleGoogle Scholar
  10. Grossinho MR, Minhós FM: Upper and lower solutions for higher order boundary value problems. Nonlinear Studies 2005, 12(2):165–176.MATHMathSciNetGoogle Scholar
  11. Grossinho MR, Minhós FM, Santos AI: Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(7):1235–1250. 10.1016/j.na.2005.04.029MATHMathSciNetView ArticleGoogle Scholar
  12. Gupta CP, Lakshmikantham V: Existence and uniqueness theorems for a third-order three-point boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 1991, 16(11):949–957. 10.1016/0362-546X(91)90099-MMATHMathSciNetView ArticleGoogle Scholar
  13. Henderson J, Taunton RD: Solutions of boundary value problems by matching methods. Applicable Analysis 1993, 49(3–4):235–246. 10.1080/00036819308840175MATHMathSciNetView ArticleGoogle Scholar
  14. Lin Y, Pei M: Positive solutions for two-point semipositone right focal eigenvalue problem. Boundary Value Problems 2007, 2007:-12.Google Scholar
  15. Moorti VRG, Garner JB: Existence-uniqueness theorems for three-point boundary value problems for nth-order nonlinear differential equations. Journal of Differential Equations 1978, 29(2):205–213. 10.1016/0022-0396(78)90120-1MATHMathSciNetView ArticleGoogle Scholar
  16. Murty KN, Rao YS: A theory for existence and uniqueness of solutions to three-point boundary value problems. Journal of Mathematical Analysis and Applications 1992, 167(1):43–48. 10.1016/0022-247X(92)90232-3MATHMathSciNetView ArticleGoogle Scholar
  17. Pei M, Chang SK: Existence and uniqueness of solutions for third-order nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 2007, 327(1):23–35. 10.1016/j.jmaa.2006.03.057MATHMathSciNetView ArticleGoogle Scholar
  18. Shi Y, Pei M: Two-point and three-point boundary value problems for nth-order nonlinear differential equations. Applicable Analysis 2006, 85(12):1421–1432. 10.1080/00036810601066061MATHMathSciNetView ArticleGoogle Scholar
  19. Wang L, Pei M: Existence and uniqueness for nonlinear third-order two-point boundary value problems. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2007, 14(3):321–332.MathSciNetGoogle Scholar
  20. Wong F-H: An application of Schauder's fixed point theorem with respect to higher order BVPs. Proceedings of the American Mathematical Society 1998, 126(8):2389–2397. 10.1090/S0002-9939-98-04709-1MATHMathSciNetView ArticleGoogle Scholar
  21. Yao Q, Feng Y: The existence of solution for a third-order two-point boundary value problem. Applied Mathematics Letters 2002, 15(2):227–232. 10.1016/S0893-9659(01)00122-7MATHMathSciNetView ArticleGoogle Scholar
  22. Zhao WL: Existence and uniqueness of solutions for third order nonlinear boundary value problems. The Tohoku Mathematical Journal 1992, 44(4):545–555. 10.2748/tmj/1178227249MATHMathSciNetView ArticleGoogle Scholar

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.