Uniqueness and Parameter Dependence of Positive Solution of Fourth-Order Nonhomogeneous BVPs
© Jian-Ping Sun and Xiao-Yun Wang. 2010
Received: 23 February 2010
Accepted: 11 July 2010
Published: 28 July 2010
We investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville 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.
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.  obtained the existence of solutions for the BVP (1.1)–(1.3) by using a nonlinear alternative of Leray-Schauder type. For other related results, one can refer to [3–5] and the references therein.
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  discussed the existence and nonexistence of positive solutions to a class of third-order three-point nonhomogeneous BVP. The authors in  studied the multiplicity of positive solutions for some fourth-order two-point nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. For more recent results on higher-order BVPs with nonhomogeneous boundary conditions, one can see [13–16].
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.
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.
Next, we state a fixed point theorem, which is our main tool.
Lemma 2.3 (see ).
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.
Assume that (A1) holds. Then
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.
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:
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.
which shows that is -concave.
Finally, we assert that is an increasing operator.
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).
which together with for implies (P1).
which indicates that is strictly increasing in .
Hence, (P3) holds.
Supported by the National Natural Science Foundation of China (10801068).
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