- Research Article
- Open Access
- Published:
Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems
Boundary Value Problems volume 2009, Article number: 362983 (2009)
Abstract
We are concerned with the higher-order nonlinear three-point boundary value problems: with the three point boundary conditions
;
where
is continuous,
are continuous, and
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 [1–22] and references therein. However, most of all the boundary conditions in the above-mentioned references are for two-point boundary conditions [2–11, 14, 17–22], 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

with nonlinear three point boundary conditions

where ,
is a continuous function,
are continuous functions, and
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 or
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.
are called lower and upper solutions of BVP (1.1), (1.2), respectively, if

Definition 2.2.
Let be a subset of
. We say that
satisfies the Nagumo condition on
if there exists a continuous function
such that

Lemma 2.3 (see [10]).
Let be a continuous function satisfying the Nagumo condition on

where are continuous functions such that

Then there exists a constant (depending only on
and
such that every solution
of (1.1) with

satisfies
Lemma 2.4.
Let be a continuous function. Then boundary value problem


has only the trivial solution.
Proof.
Suppose that is a nontrivial solution of BVP (2.6), (2.7). Then there exists
such that
or
. We may assume
. There exists
such that

Then ,
. From (2.6) we have

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 -order three point boundary value problem (1.1), (1.2).
Theorem 3.1.
Assume that
(i)there exist lower and upper solutions of BVP (1.1), (1.2), respectively, such that

(ii) is continuous on
,
is nonincreasing in
on
, and
is nonincreasing in
on
and satisfies the Nagumo condition on
, where

(iii) is continuous on
, and
is nonincreasing in
and nondecreasing in
on
;
(iv) is continuous on
, and nonincreasing in
and nondecreasing in
on
Then BVP (1.1), (1.2) has at least one solution such that for each
,

Proof.
For each define

where ,
.
For , we consider the auxiliary equation

where is given by the Nagumo condition, with the boundary conditions

Then we can choose a constant such that




In the following, we will complete the proof in four steps.
Step 1.
Show that every solution of BVP (3.5), (3.6) satisfies

independently of .
Suppose that the estimate is not true. Then there exists
such that
or
. We may assume
. There exists
such that

There are three cases to consider.
Case 1 ().
In this case, and
. For
, by (3.8), we get the following contradiction:

and for , we have the following contradiction:

Case 2 ().
In this case,

and . For
, by (3.6) we have the following contradiction:

For , by (3.9) and condition (iii) we can get the following contradiction:

Case 3 ().
In this case,

and . For
, by (3.6) we have the following contradiction:

For , by (3.10) and condition (iv) we can get the following contradiction:

By (3.6), the estimates

are obtained by integration.
Step 2.
Show that there exists such that every solution
of BVP (3.5), (3.6) satisfies

independently of .
Let

and define the function as follows:

In the following, we show that satisfies the Nagumo condition on
, independently of
. In fact, since
satisfies the Nagumo condition on
, we have

Furthermore, we obtain

Thus, satisfies the Nagumo condition on
, independently of
. Let

By Step 1 and Lemma 2.3, there exists such that
for
. Since
and
do not depend on
, the estimate
on
is also independent of
.
Step 3.
Show that for , BVP (3.5), (3.6) has at least one solution
.
Define the operators as follows:

by

by

with

Since is compact, we have the following compact operator:

defined by

Consider the set
By Steps 1 and 2, the degree is well defined for every
and by homotopy invariance, we get

Since the equation has only the trivial solution from Lemma 2.4, by the degree theory we have

Hence, the equation has at least one solution. That is, the boundary value problem

with the boundary conditions

has at least one solution in
.
Step 4.
Show that is a solution of BVP (1.1), (1.2).
In fact, the solution of BVP (3.36), (3.37) will be a solution of BVP (1.1), (1.2), if it satisfies

By contradiction, suppose that there exists such that
. There exists
such that

Now there are three cases to consider.
Case 1 ().
In this case, since on
, we have
and
. By conditions (i) and (ii), we get the following contradiction:

Case 2 ().
In this case, we have

and . By (3.37) and conditions (i) and (iii) we can get the following contradiction:

Case 3 ().
In this case, we have

and . By (3.37) and conditions (i) and (iv) we can get the following contradiction:

Similarly, we can show that on
. Hence

Also, by boundary condition (3.37) and condition (i), we have

Therefore by integration we have for each ,

that is,

Hence 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 ,
and
, and a kind of estimating condition in Theorem 3.1.
Theorem 3.2.
Assume that
(i)there exist lower and upper solutions of BVP (1.1), (1.2), respectively, such that


(ii) and its first-order partial derivatives in
are continuous on
,
on
,  
on
and satisfy the Nagumo condition on
(iii) is continuous on
and continuously partially differentiable on
, and

(iv) is continuous on
and continuously partially differentiable on
, and

(v)there exists a function such that
on
and

Then BVP (1.1), (1.2) has a unique solution 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 and
are any two solutions of BVP (1.1), (1.2) satisfying (3.3). Let
. It is easy to show that
is a solution of the following boundary value problem



where for each ,

By conditions (ii), (iii), and (iv), we have that and

Now suppose that there exists such that
. Without loss of generality assume
, and let

It is easy to see that by condition (v), hence
. Let
. We have that
,
on
, and there exists a point
such that
. Furthermore
. In fact, if
, then
. By condition (v) and (3.55) we can easily show that

In particular

Hence

which contradicts to (3.54). Thus . Similarly we can show that
. Consequently
.
Now, there are two cases to consider, that is

If , then by (3.59) we have

Thus, by (3.53) and condition (v) we have

Consequently, by Taylor's theorem there exists such that

which is a contradiction.
A similar contradiction can be obtained if . Hence
on
. By (3.55), we obtain
on
. 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:


Let

Choose and
. It is easy to check that
, and
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
satisfying

Example 3.4.
Consider the following fourth-order three point BVP:


Let

Choose and
. It is easy to check that
, and
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
satisfying

References
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. 10.1137/0520049
Agarwal RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Teaneck, NJ, USA; 1986:xii+307.
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)00211-2
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-X
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.1994.1250
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.2005.04.023
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.2004.02.001
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.2004.04.016
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)00364-9
Grossinho MR, Minhós FM: Upper and lower solutions for higher order boundary value problems. Nonlinear Studies 2005, 12(2):165–176.
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.2005.04.029
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-M
Henderson J, Taunton RD: Solutions of boundary value problems by matching methods. Applicable Analysis 1993, 49(3–4):235–246. 10.1080/00036819308840175
Lin Y, Pei M: Positive solutions for two-point semipositone right focal eigenvalue problem. Boundary Value Problems 2007, 2007:-12.
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)90120-1
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)90232-3
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.2006.03.057
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. 10.1080/00036810601066061
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.
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/S0002-9939-98-04709-1
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)00122-7
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/1178227249
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Pei, M., Chang, S. Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems. Bound Value Probl 2009, 362983 (2009). https://doi.org/10.1155/2009/362983
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/362983
Keywords
- Boundary Condition
- Continuous Function
- Unique Solution
- Ordinary Differential Equation
- Functional Equation