- Research Article
- Open Access
© J.-P. Sun and F.-X. Jin. 2010
- Received: 5 November 2010
- Accepted: 14 December 2010
- Published: 19 December 2010
In this paper, we are concerned with the following nonlinear third-order -point boundary value problem: , , , , . Some existence criteria of solution and positive solution are established by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.
- Convex Subset
- Fixed Point Theorem
- Point Equation
- Nonnegative Solution
- Curve Beam
Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves, or gravity-driven flows and so on .
The main tool used was Mawhin's continuation theorem.
Throughout, we always assume that and . The purpose of this paper is to consider the local properties of on some bounded sets and establish some existence criteria of solution and positive solution for the BVP (1.3) by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.
The following theorem guarantees the existence of solution for the BVP (1.3).
On the basis of Theorem 2.2, we now give some existence results of nonnegative solution and positive solution for the BVP (1.3).
Assume that all the conditions of Theorem 2.3 are fulfilled. Then the BVP (1.3) has one positive solution if one of the following conditions is satisfied:
A simple calculation shows that and . Thus, if we choose and , then all the conditions of Theorem 2.3 and (i) of Corollary 2.4 are fulfilled. It follows from Corollary 2.4 that the BVP (2.35) has a positive solution.
This paper was supported by the National Natural Science Foundation of China (10801068).
- Greguš M: Third Order Linear Differential Equations, Mathematics and its Applications (East European Series). Volume 22. Reidel, Dordrecht, The Netherlands; 1987:xvi+270.View ArticleGoogle Scholar
- Anderson DR: Green's function for a third-order generalized right focal problem. Journal of Mathematical Analysis and Applications 2003,288(1):1–14. 10.1016/S0022-247X(03)00132-XMathSciNetView ArticleMATHGoogle Scholar
- Bai Z: Existence of solutions for some third-order boundary-value problems. Electronic Journal of Differential Equations 2008, 25: 1–6.View ArticleMathSciNetGoogle Scholar
- Feng Y, Liu S: Solvability of a third-order two-point boundary value problem. Applied Mathematics Letters 2005,18(9):1034–1040. 10.1016/j.aml.2004.04.016MathSciNetView ArticleMATHGoogle Scholar
- Guo L-J, Sun J-P, Zhao Y-H: Existence of positive solutions for nonlinear third-order three-point boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2008,68(10):3151–3158. 10.1016/j.na.2007.03.008MathSciNetView ArticleMATHGoogle Scholar
- Hopkins B, Kosmatov N: Third-order boundary value problems with sign-changing solutions. Nonlinear Analysis. Theory, Methods & Applications 2007,67(1):126–137. 10.1016/j.na.2006.05.003MathSciNetView ArticleMATHGoogle Scholar
- Ma R: Multiplicity results for a third order boundary value problem at resonance. Nonlinear Analysis. Theory, Methods & Applications 1998,32(4):493–499. 10.1016/S0362-546X(97)00494-XMathSciNetView ArticleMATHGoogle Scholar
- Sun J-P, Ren Q-Y, Zhao Y-H: The upper and lower solution method for nonlinear third-order three-point boundary value problem. Electronic Journal of Qualitative Theory of Differential Equations 2010, 26: 1–8.MathSciNetMATHGoogle Scholar
- Sun Y: Positive solutions for third-order three-point nonhomogeneous boundary value problems. Applied Mathematics Letters 2009,22(1):45–51. 10.1016/j.aml.2008.02.002MathSciNetView ArticleMATHGoogle Scholar
- Yao Q: Solution and positive solution for a semilinear third-order two-point boundary value problem. Applied Mathematics Letters 2004,17(10):1171–1175. 10.1016/j.aml.2003.09.011MathSciNetView ArticleMATHGoogle Scholar
- Du Z, Lin X, Ge W: On a third-order multi-point boundary value problem at resonance. Journal of Mathematical Analysis and Applications 2005,302(1):217–229. 10.1016/j.jmaa.2004.08.012MathSciNetView ArticleMATHGoogle Scholar
- Jin S, Lu S: Existence of solutions for a third-order multipoint boundary value problem with -Laplacian. Journal of the Franklin Institute 2010,347(3):599–606. 10.1016/j.jfranklin.2009.12.005MathSciNetView ArticleMATHGoogle Scholar
- Sun J-P, Zhang H-E: Existence of solutions to third-order -point boundary-value problems. Electronic Journal of Differential Equations 2008, 125: 1–9.MathSciNetGoogle Scholar
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