Solutions to a Three-Point Boundary Value Problem

Abstract

By using the fixed-point index theory and Leggett-Williams fixed-point theorem, we study the existence of multiple solutions to the three-point boundary value problem ,  ; ; , where , are constants, is a parameter, and , are given functions. New existence theorems are obtained, which extend and complement some existing results. Examples are also given to illustrate our results.

1. Introduction

It is known that when differential equations are required to satisfy boundary conditions at more than one value of the independent variable, the resulting problem is called a multipoint boundary value problem, and a typical distinction between initial value problems and multipoint boundary value problems is that in the former case one is able to obtain the solutions depend only on the initial values, while in the latter case, the boundary conditions at the starting point do not determine a unique solution to start with, and some random choices among the solutions that satisfy these starting boundary conditions are normally not to satisfy the boundary conditions at the other specified point(s). As it is noticed elsewhere (see, e.g., Agarwal [1], Bisplinghoff and Ashley [2], and Henderson [3]), multi point boundary value problem has deep physical and engineering background as well as realistic mathematical model. For the development of the research of multi point boundary value problems for differential equations in last decade, we refer the readers to, for example, [1, 4–9] and references therein.

In this paper, we study the existence of multiple solutions to the following three-point boundary value problem for a class of third-order differential equations with inhomogeneous three-point boundary values,

(1.1)

where , , , and , are given functions. To the authors' knowledge, few results on third-order differential equations with inhomogeneous three-point boundary values can be found in the literature. Our purpose is to establish new existence theorems for (1.1) which extend and complement some existing results.

Let be an Banach space, and let be a cone in . A mapping is said to be a nonnegative continuous concave functional on if is continuous and

(1.2)

Assume that

• (H)
  

  (1.3)

Define

(1.4)

This paper is organized in the following way. In Section 2, we present some lemmas, which will be used in Section 3. The main results and proofs are given in Section 3. Finally, in Section 4, we give some examples to illustrate our results.

2. Lemmas

Let be a Banach Space with norm

(2.1)

where

(2.2)

It is not hard to see Lemmas 2.1 and 2.2.

Lemma 2.1.

Let be the unique solution of (1.1). Then

(2.3)

where

(2.4)

Lemma 2.2.

One has the following.

• (i) .

• (ii) .

• (iii) .

Lemma 2.3.

Let be the unique solution of (1.1). Then is nonnegative and satisfies .

Proof.

Let be the unique solution of (1.1). Then it is obvious that is nonnegative. By Lemmas 2.1 and 2.2, we have the following.

• (i) For ,

  

  (2.5)

that is, .

• (ii) For ,

  

  (2.6)

On the other hand, for , we have

(2.7)

Since ,

(2.8)

So, . Therefore, , which means

(2.9)

The proof is completed.

Lemma 2.4.

Let be the unique solution of (1.1). Then

(2.10)

Proof.

From (2.3), it follows that

(2.11)

Hence,

(2.12)

By Lemmas 2.2 and 2.3, we get, for any ,

(2.13)

Thus,

(2.14)

Define a cone by

(2.15)

Set

(2.16)

Define an operator by

(2.17)

Lemma 2.1 implies that (1.1) has a solution if and only if is a fixed point of .

From Lemmas 2.1 and 2.2 and the Ascoli-Arzela theorem, the following follow.

Lemma 2.5.

The operator defined in (2.17) is completely continuous and satisfies .

Theorem 2.6 (see [10]).

Let be a real Banach Space, let be a cone, and . Let operator be completely continuous and satisfy , . Then

(i)if ,  for all  , then ,

(ii)if ,  for all  , then .

Theorem 2.7 (see [8]).

Let be a completely continuous operator and a nonnegative continuous concave functional on such that for all . Suppose that there exist such that

• (a) and for ,

• (b) for ,

• (c) for with .

Then, has at least three fixed points in satisfying

(2.18)

3. Main Results

In this section, we give new existence theorem about two positive solutions or three positive solutions for (1.1).

Write

(3.1)

Theorem 3.1.

Assume that

;

there exists a constant such that , for , and .

Then, the problem (1.1) has at least two positive solutions and such that

(3.2)

for small enough.

Proof.

Since

(3.3)

there is such that

(3.4)

Let

(3.5)

Then, for any , it follows from Lemmas 2.2 and 2.3 and (3.4) that

(3.6)

Hence,

(3.7)

So

(3.8)

By Theorem 2.6, we have

(3.9)

On the other hand, since

(3.10)

there exist , such that

(3.11)

Let . Then, by a argument similar to that above, we obtain

(3.12)

By Theorem 2.6,

(3.13)

Finally, let , and let satisfy for any . Then, implies

(3.14)

which means that . Thus, , for all .

Using Theorem 2.6, we get

(3.15)

From (3.9)–(3.15) and , it follows that

(3.16)

Therefore, has fixed point and fixed point . Clearly, , are both positive solutions of the problem (1.1) and

(3.17)

The proof of Theorem 3.1 is completed.

Theorem 3.2.

Assume that

;

there exists a constant such that , for and .

Then, the problem (1.1) has at least two positive solutions and such that

(3.18)

for small enough.

Proof.

By

(3.19)

we see that there exists such that

(3.20)

Put

(3.21)

and let satisfy

(3.22)

Then Lemmas 2.2 and 2.3 and (3.20) implies that for any ,

(3.23)

So . Hence, , .

Applying Theorem 2.6, we have

(3.24)

Next, by

(3.25)

we know that there exists such that

(3.26)

Case 1.

is unbounded.

Define a function by

(3.27)

Clearly, is nondecreasing and , and

(3.28)

Taking , it follows from (3.26)–(3.28) that

(3.29)

By Lemmas 2.2 and 2.3 and (3.28), we have

(3.30)

So , and then .

Case 2.

is bounded.

In this case, there exists an such that

(3.31)

Choosing , we see by Lemmas 2.2 and 2.3 and (3.31) that

(3.32)

which implies , and then .

Therefore, in both cases, taking

(3.33)

we get

(3.34)

By Theorem 2.6, we have

(3.35)

Finally, put . Then implies that

(3.36)

that is, , and then , for all . By virtue of Theorem 2.6, we have

(3.37)

From (3.24), (3.35), (3.37), and , it follows that

(3.38)

Hence, has fixed point and fixed point . Obviously, , are both positive solutions of the problem (1.1) and

(3.39)

The proof of Theorem 3.2 is completed.

Theorem 3.3.

Let there exist , , , and with

(3.40)

such that

(3.41)

(3.42)

(3.43)

Then problem (1.1) has at least three positive solutions , , satisfying

(3.44)

for .

Proof.

Let

(3.45)

Then, is a nonnegative continuous concave functional on and for each . Let be in . Equation (3.43) implies that

(3.46)

Hence, . This means that .

Take

(3.47)

Then,

(3.48)

By (3.42), we have, for any ,

(3.49)

Therefore, in Theorem 2.7 holds.

By (3.41), we see that for any

(3.50)

So, . This means that of Theorem 2.7 holds.

Moreover, for any with , we have

(3.51)

which implies

(3.52)

So, in Theorem 2.7 holds. Thus, by Theorem 2.7, we know that the operator has at least three positive fixed points satisfying

(3.53)

4. Examples

In this section, we give three examples to illustrate our results.

Example 4.1.

Consider the problem

(4.1)

where , . Set

(4.2)

Then,

(4.3)

So, the condition is satisfied. Observe

(4.4)

Taking

(4.5)

we have

(4.6)

Thus, condition is satisfied.

Therefore, by Theorem 3.1, the problem (4.1) has at least two positive solutions and such that

(4.7)

for

(4.8)

Example 4.2.

Consider the problem

(4.9)

where , . Set

(4.10)

Then,

(4.11)

that is, the condition is satisfied. Moreover,

(4.12)

Taking

(4.13)

we get

(4.14)

Thus, condition is satisfied.

Consequently, by Theorem 3.2, we see that for

(4.15)

the problem (4.9) has at least two positive solutions and such that

(4.16)

Example 4.3.

For the problem (1.1), take , , and . Then,

(4.17)

Let

(4.18)

Then,

(4.19)

which implies

(4.20)

That is, the conditions of Theorem 3.3 are satisfied. Consequently, the problem (1.1) has at least three positive solutions for

(4.21)

satisfying

(4.22)