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

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

(1.1)

with nonlinear three point boundary conditions

(1.2)

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

(2.1)

Definition 2.2.

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

(2.2)

Lemma 2.3 (see [10]).

Let be a continuous function satisfying the Nagumo condition on

(2.3)

where are continuous functions such that

(2.4)

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

(2.5)

satisfies

Lemma 2.4.

Let be a continuous function. Then boundary value problem

(2.6)

(2.7)

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

(2.8)

Then , . From (2.6) we have

(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 -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

(3.1)

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

(3.2)

(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 ,

(3.3)

Proof.

For each define

(3.4)

where , .

For , we consider the auxiliary equation

(3.5)

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

(3.6)

Then we can choose a constant such that

(3.7)

(3.8)

(3.9)

(3.10)

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

Step 1.

Show that every solution of BVP (3.5), (3.6) satisfies

(3.11)

independently of .

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

(3.12)

There are three cases to consider.

Case 1 ().

In this case, and . For , by (3.8), we get the following contradiction:

(3.13)

and for , we have the following contradiction:

(3.14)

Case 2 ().

In this case,

(3.15)

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

(3.16)

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

(3.17)

Case 3 ().

In this case,

(3.18)

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

(3.19)

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

(3.20)

By (3.6), the estimates

(3.21)

are obtained by integration.

Step 2.

Show that there exists such that every solution of BVP (3.5), (3.6) satisfies

(3.22)

independently of .

Let

(3.23)

and define the function as follows:

(3.24)

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

(3.25)

Furthermore, we obtain

(3.26)

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

(3.27)

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:

(3.28)

by

(3.29)

by

(3.30)

with

(3.31)

Since is compact, we have the following compact operator:

(3.32)

defined by

(3.33)

Consider the set

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

(3.34)

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

(3.35)

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

(3.36)

with the boundary conditions

(3.37)

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

(3.38)

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

(3.39)

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:

(3.40)

Case 2 ().

In this case, we have

(3.41)

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

(3.42)

Case 3 ().

In this case, we have

(3.43)

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

(3.44)

Similarly, we can show that on . Hence

(3.45)

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

(3.46)

Therefore by integration we have for each ,

(3.47)

that is,

(3.48)

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

(3.49)

(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

(3.50)

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

(3.51)

(v)there exists a function such that on and

(3.52)

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

(3.53)

(3.54)

(3.55)

where for each ,

(3.56)

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

(3.57)

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

(3.58)

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

(3.59)

In particular

(3.60)

Hence

(3.61)

which contradicts to (3.54). Thus . Similarly we can show that . Consequently .

Now, there are two cases to consider, that is

(3.62)

If , then by (3.59) we have

(3.63)

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

(3.64)

Consequently, by Taylor's theorem there exists such that

(3.65)

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:

(3.66)

(3.67)

Let

(3.68)

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

(3.69)

Example 3.4.

Consider the following fourth-order three point BVP:

(3.70)

(3.71)

Let

(3.72)

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

(3.73)