- Open Access
Existence of positive solutions for singular fourth-order three-point boundary value problems
© Sun and Zhu; licensee Springer 2013
Received: 24 July 2012
Accepted: 20 February 2013
Published: 8 March 2013
In this article, we consider the boundary value problem , , subject to the boundary conditions and . In this setting, and are constants and is a parameter. By imposing a sufficient structure on the nonlinearity , we deduce the existence of at least one positive solution to the problem. The novelty in our setting lies in the fact that may be singular at and . Our results here are achieved by making use of the Krasnosel’skii fixed point theorem. We conclude with examples illustrating our results and the improvements that they present.
MSC:34B15, 34B25, 34B18.
where , are constants, is a parameter, may be singular at and/or . Here, by a positive solution we mean a function which is positive on and satisfies problem (1.1).
where is a closed set with measure zero and the nonlinear term may be singular for . The author showed the existence of n positive solutions by constructing a suitable integral equation and applying fixed point theorems on a cone.
The author obtained the existence and nonexistence of positive solutions by applying the Guo-Krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.
where the nonlinear term may be singular at , and . The author presented the existence of a positive solution by using the fixed point index theorem and the properties of Green’s function.
where is a constant.
Inspired and motivated by the works mentioned above, we deal with the existence and nonexistence of positive solutions to problem (1.1) by making use of the fixed point theorem together with the properties of Green’s function. The main features of the paper are as follows. Firstly, we apply the Taylor expansion formula to prove a lemma, and then we give a comparison lemma and construct a special cone. Secondly, we present the existence of positive solutions for problem (1.1). To our best knowledge, no paper has considered problem (1.1). The arguments are based upon the fixed point theorem for the special cone.
The paper is organized as follows. In Section 2, we give some properties of Green’s function associated with problem (1.1) and construct a suitable cone and transform problem (1.1) into an integral equation. In Section 3, we discuss the existence of at least one positive solution for problem (1.1).
2 Preliminary lemmas
Let be a Banach space of all continuous functions with the norm , .
Throughout the paper, we assume that
(H1) is continuous.
The proof is complete. □
It is well known that problem (1.1) has a positive solution if and only if u is a fixed point of A.
Lemma 2.3 Suppose that (H1)∼(H3) hold. Then .
The proof is complete. □
Lemma 2.4 Suppose that (H1)∼(H3) hold. Then is completely continuous.
Proof For any , we have .
By the Arzela-Ascoli theorem, we know that is completely continuous.
It shows that a completely continuous operator converges to an operator A uniformly on . Hence A is continuous.
Hence A is uniformly bounded.
Therefore A is equicontinuous. Consequently, A is completely continuous. □
, and , , or
, and , .
Then T has a fixed point in .
3 Main results
Theorem 3.1 Suppose that (H1), (H2) and (H3) hold. In addition, assume that the following conditions hold:
Then problem (1.1) has at least one positive solution for λ small enough, and problem (1.1) has no positive solution for λ large enough.
From (H4), there exists a constant such that for .
By Lemma 2.5 we know that problem (1.1) has at least one positive solution.
Hence as .
which is a contradiction. The proof is complete. □
Remark 3.1 The conclusion of Theorem 3.1 also holds if .
Theorem 3.2 Suppose that (H1), (H2) and (H3) hold. In addition, assume that
Then problem (1.1) has at least one positive solution for any .
It follows from Lemma 2.5 that problem (1.1) has at least one positive solution. □
Now, we give examples to illustrate the main results in the paper.
Then problem (4.1) has at least one positive solution if and .
Take . Notice, for any fixed , that and for and .
Obviously, conditions (H1)∼(H3) are satisfied.
Thus, the existence of a positive solution follows from Theorem 3.1 if and .
Then problem (4.3) has at least one positive solution.
Take . Notice, for any fixed , that and .
Obviously, conditions (H1)∼(H3) are satisfied.
Thus, the existence of a positive solution follows from Theorem 3.2.
The authors would like to express their thanks to the editor of the journal and the anonymous referees for their careful reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper. The authors were supported financially by the Foundation of Shanghai Municipal Education Commission (Grant No. DYL201105).
- Ma R, Zhang J, Fu S: The method of upper and lower solutions for fourth-order two point boundary value problems. J. Math. Anal. Appl. 1997, 215: 415-422. 10.1006/jmaa.1997.5639MathSciNetView ArticleGoogle Scholar
- Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 1986, 116: 415-426. 10.1016/S0022-247X(86)80006-3MathSciNetView ArticleGoogle Scholar
- Yang Y: Fourth-order two-point boundary value problems. Proc. Am. Math. Soc. 1988, 104: 175-180. 10.1090/S0002-9939-1988-0958062-3View ArticleGoogle Scholar
- Gupta CP: Existence and uniqueness results for the bending of an elastic beam equation at resonance. J. Math. Anal. Appl. 1988, 135: 208-225. 10.1016/0022-247X(88)90149-7MathSciNetView ArticleGoogle Scholar
- Wei Z: Positive solutions of singular boundary value problems of fourth-order differential equations. Acta Math. Sin. 1999, 42: 715-722.Google Scholar
- Ma R, Wang H: On the existence of positive solutions of fourth order ordinary differential equation. Appl. Anal. 1995, 59: 225-231. 10.1080/00036819508840401MathSciNetView ArticleGoogle Scholar
- Harjani J, López B, Sadarangani K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal. 2011, 74: 1749-1760. 10.1016/j.na.2010.10.047MathSciNetView ArticleGoogle Scholar
- Yao Q: Existence and multiplicity of positive solutions to a class of nonlinear cantilever beam equations. J. Syst. Sci. Math. Sci. 2009, 1: 63-69.Google Scholar
- Sun Y: Positive solutions for third-order three-point nonhomogeneous boundary value problems. Appl. Math. Lett. 2009, 22: 45-51. 10.1016/j.aml.2008.02.002MathSciNetView ArticleGoogle Scholar
- Zhang K, Wang C: The existence of positive solutions of a class of fourth-order singular boundary value problems. Acta Math. Sci., Ser. A 2009, 29: 127-135.Google Scholar
- O’Regan D: Solvability of some fourth (and higher) order singular boundary value problems. J. Math. Anal. Appl. 1991, 161: 78-116. 10.1016/0022-247X(91)90363-5MathSciNetView ArticleGoogle Scholar
- Schroder J: Fourth order two-point boundary value problems; estimates by two-sided bounds. Nonlinear Anal. 1984, 8: 107-114. 10.1016/0362-546X(84)90063-4MathSciNetView ArticleGoogle Scholar
- Guo L, Sun J, Zhao Y: Existence of positive solution for nonlinear third-order three-point boundary value problem. Nonlinear Anal. 2008, 68: 3151-3158. 10.1016/j.na.2007.03.008MathSciNetView ArticleGoogle Scholar
- Bai Z, Wang H: On the positive solutions of some nonlinear fourth-order beam equations. J. Math. Anal. Appl. 2002, 270: 357-368. 10.1016/S0022-247X(02)00071-9MathSciNetView ArticleGoogle Scholar
- Graef JR, Qian C, Yang B: A three point boundary value problem for nonlinear fourth-order differential equations. J. Math. Anal. Appl. 2003, 287: 217-233. 10.1016/S0022-247X(03)00545-6MathSciNetView ArticleGoogle Scholar
- Yao Q: Existence and iteration of n symmetric positive solutions for a singular two-point boundary value problem. Comput. Math. Appl. 2004, 47: 1195-1200. 10.1016/S0898-1221(04)90113-7MathSciNetView ArticleGoogle Scholar
- Yao Q: Local existence of multiple positive solutions to a singular cantilever beam equation. J. Math. Anal. Appl. 2010, 363: 138-154. 10.1016/j.jmaa.2009.07.043MathSciNetView ArticleGoogle Scholar
- Agarwal RP: On fourth order boundary value problems arising in beam analysis. Differ. Integral Equ. 1989, 2: 91-110.Google Scholar
- Ji C, O’Regan D, Yan B, Agarwal RP: Nonexistence and existence of positive solutions for second order singular three-point boundary value problems with derivative dependent and sign-changing nonlinearities. J. Appl. Math. Comput. 2011, 36: 61-87. 10.1007/s12190-010-0388-5MathSciNetView ArticleGoogle Scholar
- Sun Y, Liu L, Zhang J, Agarwal RP: Positive solutions of singular three-point boundary value problems for second order differential equations. J. Comput. Appl. Math. 2009, 230: 738-750. 10.1016/j.cam.2009.01.003MathSciNetView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.Google Scholar
- Deimling K: Nonlinear Functional Analysis. Springer, New York; 1985.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.