We investigate the following fourthorder fourpoint nonhomogeneous SturmLiouville boundary value problem: , , , where and are nonnegative parameters. Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters is also studied.
1. Introduction
Boundary value problems (BVPs for short) consisting of fourthorder differential equation and fourpoint homogeneous boundary conditions have received much attention due to their striking applications. For example, Chen et al. [1] studied the fourthorder nonlinear differential equation
with the fourpoint homogeneous boundary conditions
where . By means of the upper and lower solution method and Schauder fixed point theorem, some criteria on the existence of positive solutions to the BVP (1.1)–(1.3) were established. Bai et al. [2] obtained the existence of solutions for the BVP (1.1)–(1.3) by using a nonlinear alternative of LeraySchauder type. For other related results, one can refer to [3–5] and the references therein.
Recently, nonhomogeneous BVPs have attracted many authors' attention. For instance, Ma [6, 7] and L. Kong and Q. Kong [8–10] studied some secondorder multipoint nonhomogeneous BVPs. In particular, L. Kong and Q. Kong [10] considered the following secondorder BVP with multipoint nonhomogeneous boundary conditions
where and are nonnegative parameters. They derived some conditions for the above BVP to have a unique solution and then studied the dependence of this solution on the parameters and . Sun [11] discussed the existence and nonexistence of positive solutions to a class of thirdorder threepoint nonhomogeneous BVP. The authors in [12] studied the multiplicity of positive solutions for some fourthorder twopoint nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. For more recent results on higherorder BVPs with nonhomogeneous boundary conditions, one can see [13–16].
Inspired greatly by the abovementioned excellent works, in this paper we are concerned with the following SturmLiouville BVP consisting of the fourthorder differential equation:
and the fourpoint nonhomogeneous boundary conditions
where and are nonnegative parameters. Under the following assumptions:
(A1) and are nonnegative constants with , , , , and
(A2) is continuous and monotone increasing in for every ;
(A3)there exists such that
we prove the uniqueness of positive solution for the BVP (1.5)–(1.7) and study the dependence of this solution on the parameters .
2. Preliminary Lemmas
First, we recall some fundamental definitions.
Definition 2.1.
Let be a Banach space with norm . Then
(1)a nonempty closed convex set is said to be a cone if for all and , where is the zero element of
(2)every cone in defines a partial ordering in by
(3)a cone is said to be normal if there exists such that implies that
(4)a cone is said to be solid if the interior of is nonempty.
Definition 2.2.
Let be a solid cone in a real Banach space an operator, and Then is called a concave operator if
Next, we state a fixed point theorem, which is our main tool.
Lemma 2.3 (see [17]).
Assume that is a normal solid cone in a real Banach space and is a concave increasing operator. Then has a unique fixed point in
The following two lemmas are crucial to our main results.
Lemma 2.4.
Assume that and are defined as in (A1) and . Then for any the BVP consisting of the equation
and the boundary conditions (1.6) and (1.7) has a unique solution
where
Proof.
Let
Then
By (2.5) and (1.6), we know that
On the other hand, in view of (2.5) and (1.7), we have
So, it follows from (2.6) and (2.8) that
which together with (2.7) implies that
Lemma 2.5.
Assume that (A1) holds. Then
(1) for
(2) for
(3) for
3. Main Result
For convenience, we denote and . In the remainder of this paper, the following notations will be used:
(1) if at least one of approaches ;
(2) if for ;
(3) if for and at least one of them is strict.
Let . Then is a Banach space, where is defined as usual by the sup norm.
Our main result is the following theorem.
Theorem 3.1.
Assume that (A1)–(A3) hold. Then the BVP (1.5)–(1.7) has a unique positive solution for any , where . Furthermore, such a solution satisfies the following properties:
(P1)
(P2) is strictly increasing in , that is,
(P3) is continuous in , that is, for any given
Proof.
Let . Then is a normal solid cone in with For any , if we define an operator as follows:
then it is not difficult to verify that is a positive solution of the BVP (1.5)–(1.7) if and only if is a fixed point of .
Now, we will prove that has a unique fixed point by using Lemma 2.3.
First, in view of Lemma 2.5, we know that
Next, we claim that is a concave operator.
In fact, for any and it follows from (3.3) and (A3) that
which shows that is concave.
Finally, we assert that is an increasing operator.
Suppose that and By (3.3) and (A2), we have
which indicates that is increasing.
Therefore, it follows from Lemma 2.3 that has a unique fixed point which is the unique positive solution of the BVP (1.5)–(1.7). The first part of the theorem is proved.
In the rest of the proof, we will prove that such a positive solution satisfies properties (P1), (P2), and (P3).
First,
which together with for implies (P1).
Next, we show (P2). Assume that Let
Then for We assert that Suppose on the contrary that Since is a concave increasing operator and for given , is strictly increasing in , we have
which contradicts the definition of Thus, we get for And so,
which indicates that is strictly increasing in .
Finally, we prove (P3). For any given we first suppose that with From (P2), we know that
Let
Then and for If we define
then and
which together with the definition of implies that
So,
Therefore,
In view of (3.10) and (3.16), we obtain that
which together with the fact that as shows that
Similarly, we can also prove that
Hence, (P3) holds.
Acknowledgment
Supported by the National Natural Science Foundation of China (10801068).
References

Chen, S, Ni, W, Wang, C: Positive solution of fourth order ordinary differential equation with fourpoint boundary conditions. Applied Mathematics Letters. 19(2), 161–168 (2006). Publisher Full Text

Bai, C, Yang, D, Zhu, H: Existence of solutions for fourth order differential equation with fourpoint boundary conditions. Applied Mathematics Letters. 20(11), 1131–1136 (2007). Publisher Full Text

Graef, JR, Yang, B: Positive solutions of a nonlinear fourth order boundary value problem. Communications on Applied Nonlinear Analysis. 14(1), 61–73 (2007)

Wu, H, Zhang, J: Positive solutions of higherorder fourpoint boundary value problem with Laplacian operator. Journal of Computational and Applied Mathematics. 233(11), 2757–2766 (2010). Publisher Full Text

Zhao, J, Ge, W: Positive solutions for a higherorder fourpoint boundary value problem with a Laplacian. Computers & Mathematics with Applications. 58(6), 1103–1112 (2009). PubMed Abstract  Publisher Full Text

Ma, R: Positive solutions for secondorder threepoint boundary value problems. Applied Mathematics Letters. 14(1), 1–5 (2001). Publisher Full Text

Ma, R: Positive solutions for nonhomogeneous point boundary value problems. Computers & Mathematics with Applications. 47(45), 689–698 (2004). PubMed Abstract  Publisher Full Text

Kong, L, Kong, Q: Secondorder boundary value problems with nonhomogeneous boundary conditions. I. Mathematische Nachrichten. 278(12), 173–193 (2005). Publisher Full Text

Kong, L, Kong, Q: Secondorder boundary value problems with nonhomogeneous boundary conditions. II. Journal of Mathematical Analysis and Applications. 330(2), 1393–1411 (2007). Publisher Full Text

Kong, L, Kong, Q: Uniqueness and parameter dependence of solutions of secondorder boundary value problems. Applied Mathematics Letters. 22(11), 1633–1638 (2009). Publisher Full Text

Sun, Y: Positive solutions for thirdorder threepoint nonhomogeneous boundary value problems. Applied Mathematics Letters. 22(1), 45–51 (2009). Publisher Full Text

do Ó, JM, Lorca, S, Ubilla, P: Multiplicity of solutions for a class of nonhomogeneous fourthorder boundary value problems. Applied Mathematics Letters. 21(3), 279–286 (2008). Publisher Full Text

Graef, JR, Kong, L, Kong, Q, Wong, JSW: Higher order multipoint boundary value problems with signchanging nonlinearities and nonhomogeneous boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations. 2010(28), 1–40 (2010)

Kong, L, Kong, Q: Higher order boundary value problems with nonhomogeneous boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 72(1), 240–261 (2010). PubMed Abstract  Publisher Full Text

Kong, L, Piao, D, Wang, L: Positive solutions for third order boundary value problems with Laplacian. Results in Mathematics. 55(12), 111–128 (2009). Publisher Full Text

Kong, L, Wong, JSW: Positive solutions for higher order multipoint boundary value problems with nonhomogeneous boundary conditions. Journal of Mathematical Analysis and Applications. 367(2), 588–611 (2010). Publisher Full Text

Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering,p. viii+275. Academic Press, Boston, Mass, USA (1988)