- Research Article
- Open Access
Existence of Positive Solutions for Nonlocal Fourth-Order Boundary Value Problem with Variable Parameter
© Xiaoling Han et al. 2011
- Received: 26 November 2010
- Accepted: 14 January 2011
- Published: 23 January 2011
- Variable Parameter
- Point Theorem
- Fixed Point Theorem
- Lower Solution
- Fixed Point Theory
The existence of positive solutions for nonlinear fourth-order multipoint boundary value problems has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point theory, and the method of upper and lower solutions (see, e.g., [1–15] and references therein). The multipoint boundary value problem is in fact a special case of the boundary value problem with integral boundary conditions.
under the assumption:
Obviously, BVP(1.1) can be regarded as the special case of BVP(1.2) with . Since the parameters is variable, we cannot expect to transform directly BVP(1.2) into an integral equation as in . We will apply the cone fixed point theory, combining with the operator spectra theorem to establish the existence of positive solutions of BVP(1.2). Our results generalize the main result in .
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Assume that (A1), (A2) hold. Then,
Computations yield the following results.
Lemma 2.4 (see ).
Lemma 2.5 (see ).
It is similar to Lemma 6 of , so we omit it.
Lemma 2.7 (see ).
The proof is similar to (i), so we omit it.
By the proof of Theorem 3.1, we know that (1) from the condition , there exists , such that , , (2) from the condition , there exists , , such that , , (3) from the condition (ii), there exists , , such that , . By the use of Krasnoselskii's fixed point theorem, it is easy to know that (1.2) has at least two positive solutions.
The proof is similar to Corollary 3.2, so we omit it.
This work is sponsored by the NSFC (no. 11061030), NSFC (no. 11026060), and nwnu-kjcxgc-03-69, 03-61.
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