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].
2. Preliminary Lemmas
First, we recall some fundamental definitions.
Next, we state a fixed point theorem, which is our main tool.
Lemma 2.3 (see ).
The following two lemmas are crucial to our main results.
Assume that (A1) holds. Then
3. Main Result
Our main result is the following theorem.
Hence, (P3) holds.
Supported by the National Natural Science Foundation of China (10801068).
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