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
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 .
2. The Preliminary Lemmas
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 ).
3. The Main Results
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.
- Bai Z: The method of lower and upper solutions for a bending of an elastic beam equation. Journal of Mathematical Analysis and Applications 2000,248(1):195–202. 10.1006/jmaa.2000.6887MATHMathSciNetView ArticleGoogle Scholar
- Bai Z: The upper and lower solution method for some fourth-order boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2007,67(6):1704–1709. 10.1016/j.na.2006.08.009MATHMathSciNetView ArticleGoogle Scholar
- Chai G: Existence of positive solutions for fourth-order boundary value problem with variable parameters. Nonlinear Analysis. Theory, Methods & Applications 2007,66(4):870–880. 10.1016/j.na.2005.12.028MATHMathSciNetView ArticleGoogle Scholar
- Feng H, Ji D, Ge W: Existence and uniqueness of solutions for a fourth-order boundary value problem. Nonlinear Analysis. Theory, Methods & Applications 2009,70(10):3561–3566. 10.1016/j.na.2008.07.013MATHMathSciNetView ArticleGoogle Scholar
- Li Y: Positive solutions of fourth-order boundary value problems with two parameters. Journal of Mathematical Analysis and Applications 2003,281(2):477–484. 10.1016/S0022-247X(03)00131-8MATHMathSciNetView ArticleGoogle Scholar
- Li Y: Positive solutions of fourth-order periodic boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2003,54(6):1069–1078. 10.1016/S0362-546X(03)00127-5MATHMathSciNetView ArticleGoogle Scholar
- Liu X-L, Li W-T: Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters. Mathematical and Computer Modelling 2007,46(3–4):525–534. 10.1016/j.mcm.2006.11.018MATHMathSciNetView ArticleGoogle Scholar
- Ma H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Analysis. Theory, Methods & Applications 2008,68(3):645–651. 10.1016/j.na.2006.11.026MATHMathSciNetView ArticleGoogle Scholar
- Ma R: Existence of positive solutions of a fourth-order boundary value problem. Applied Mathematics and Computation 2005,168(2):1219–1231. 10.1016/j.amc.2004.10.014MATHMathSciNetView ArticleGoogle Scholar
- Pang C, Dong W, Wei Z: Multiple solutions for fourth-order boundary value problem. Journal of Mathematical Analysis and Applications 2006,314(2):464–476. 10.1016/j.jmaa.2005.04.008MATHMathSciNetView ArticleGoogle Scholar
- Wei Z, Pang C: Positive solutions and multiplicity of fourth-order m-point boundary value problems with two parameters. Nonlinear Analysis. Theory, Methods & Applications 2007,67(5):1586–1598. 10.1016/j.na.2006.08.001MATHMathSciNetView ArticleGoogle Scholar
- Yang Y, Zhang J: Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Analysis. Theory, Methods & Applications 2008,69(4):1364–1375. 10.1016/j.na.2007.06.035MATHMathSciNetView ArticleGoogle Scholar
- Yao QL, Bai ZB: Existence of positive solutions of a BVP for . Chinese Annals of Mathematics. Series A 1999,20(5):575–578.MATHMathSciNetGoogle Scholar
- Yao Q: Local existence of multiple positive solutions to a singular cantilever beam equation. Journal of Mathematical Analysis and Applications 2010,363(1):138–154. 10.1016/j.jmaa.2009.07.043MATHMathSciNetView ArticleGoogle Scholar
- Zhao J, Ge W: Positive solutions for a higher-order four-point boundary value problem with a p-Laplacian. Computers & Mathematics with Applications 2009,58(6):1103–1112. 10.1016/j.camwa.2009.04.022MATHMathSciNetView ArticleGoogle Scholar
- Bai Z: Positive solutions of some nonlocal fourth-order boundary value problem. Applied Mathematics and Computation 2010,215(12):4191–4197. 10.1016/j.amc.2009.12.040MATHMathSciNetView ArticleGoogle Scholar
- Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.MATHGoogle Scholar
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