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Solutions to a ThreePoint Boundary Value Problem
Advances in Difference Equations volume 2011, Article number: 894135 (2011)
Abstract
By using the fixedpoint index theory and LeggettWilliams fixedpoint theorem, we study the existence of multiple solutions to the threepoint 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 threepoint boundary value problem for a class of thirdorder differential equations with inhomogeneous threepoint boundary values,
where , , , and , are given functions. To the authors' knowledge, few results on thirdorder differential equations with inhomogeneous threepoint 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
Assume that

(H)
(1.3)
Define
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
where
It is not hard to see Lemmas 2.1 and 2.2.
Lemma 2.1.
Let be the unique solution of (1.1). Then
where
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
Since ,
So, . Therefore, , which means
The proof is completed.
Lemma 2.4.
Let be the unique solution of (1.1). Then
Proof.
From (2.3), it follows that
Hence,
By Lemmas 2.2 and 2.3, we get, for any ,
Thus,
Define a cone by
Set
Define an operator by
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 AscoliArzela 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
3. Main Results
In this section, we give new existence theorem about two positive solutions or three positive solutions for (1.1).
Write
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
for small enough.
Proof.
Since
there is such that
Let
Then, for any , it follows from Lemmas 2.2 and 2.3 and (3.4) that
Hence,
So
By Theorem 2.6, we have
On the other hand, since
there exist , such that
Let . Then, by a argument similar to that above, we obtain
By Theorem 2.6,
Finally, let , and let satisfy for any . Then, implies
which means that . Thus, , for all .
Using Theorem 2.6, we get
From (3.9)–(3.15) and , it follows that
Therefore, has fixed point and fixed point . Clearly, , are both positive solutions of the problem (1.1) and
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
for small enough.
Proof.
By
we see that there exists such that
Put
and let satisfy
Then Lemmas 2.2 and 2.3 and (3.20) implies that for any ,
So . Hence, , .
Applying Theorem 2.6, we have
Next, by
we know that there exists such that
Case 1.
is unbounded.
Define a function by
Clearly, is nondecreasing and , and
Taking , it follows from (3.26)–(3.28) that
By Lemmas 2.2 and 2.3 and (3.28), we have
So , and then .
Case 2.
is bounded.
In this case, there exists an such that
Choosing , we see by Lemmas 2.2 and 2.3 and (3.31) that
which implies , and then .
Therefore, in both cases, taking
we get
By Theorem 2.6, we have
Finally, put . Then implies that
that is, , and then , for all . By virtue of Theorem 2.6, we have
From (3.24), (3.35), (3.37), and , it follows that
Hence, has fixed point and fixed point . Obviously, , are both positive solutions of the problem (1.1) and
The proof of Theorem 3.2 is completed.
Theorem 3.3.
Let there exist , , , and with
such that
Then problem (1.1) has at least three positive solutions , , satisfying
for .
Proof.
Let
Then, is a nonnegative continuous concave functional on and for each . Let be in . Equation (3.43) implies that
Hence, . This means that .
Take
Then,
By (3.42), we have, for any ,
Therefore, in Theorem 2.7 holds.
By (3.41), we see that for any
So, . This means that of Theorem 2.7 holds.
Moreover, for any with , we have
which implies
So, in Theorem 2.7 holds. Thus, by Theorem 2.7, we know that the operator has at least three positive fixed points satisfying
4. Examples
In this section, we give three examples to illustrate our results.
Example 4.1.
Consider the problem
where , . Set
Then,
So, the condition is satisfied. Observe
Taking
we have
Thus, condition is satisfied.
Therefore, by Theorem 3.1, the problem (4.1) has at least two positive solutions and such that
for
Example 4.2.
Consider the problem
where , . Set
Then,
that is, the condition is satisfied. Moreover,
Taking
we get
Thus, condition is satisfied.
Consequently, by Theorem 3.2, we see that for
the problem (4.9) has at least two positive solutions and such that
Example 4.3.
For the problem (1.1), take , , and . Then,
Let
Then,
which implies
That is, the conditions of Theorem 3.3 are satisfied. Consequently, the problem (1.1) has at least three positive solutions for
satisfying
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Acknowledgments
This paper was supported partially by the NSF of China (10771202) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).
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Keywords
 Boundary Condition
 Differential Equation
 Banach Space
 Partial Differential Equation
 Unique Solution