- Research Article
- Open Access
Monotone Positive Solution of Nonlinear Third-Order BVP with Integral Boundary Conditions
© The Author(s) Jian-Ping Sun and Hai-Bao Li. 2010
Received: 7 September 2010
Accepted: 31 October 2010
Published: 2 November 2010
This paper is concerned with the following third-order boundary value problem with integral boundary conditions , where and . By using the Guo-Krasnoselskii fixed-point theorem, some sufficient conditions are obtained for the existence and nonexistence of monotone positive solution to the above problem.
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 .
Throughout this paper, we always assume that and . Some sufficient conditions are established for the existence and nonexistence of monotone positive solution to the BVP (1.3). Here, a solution of the BVP (1.3) is said to be monotone and positive if , and for . Our main tool is the following Guo-Krasnoselskii fixed-point theorem .
Lemma 2.2 (see ).
Lemma 2.3 (see ).
which implies that is equicontinuous. Again, by Arzela-Ascoli theorem, we know that has a convergent subsequence in . Therefore, has a convergent subsequence in . Thus, we have shown that is a compact operator.
3. Main Results
The proof is similar to that of Theorem 3.1 and is therefore omitted.
This is a contradiction. Therefore, the BVP (1.3) has no monotone positive solution.
Similarly, we can prove the following theorem.
So, it follows from Theorem 3.1 that the BVP (3.14) has at least one monotone positive solution.
This work was supported by the National Natural Science Foundation of China (10801068).
- Gregus M: Third Order Linear Differential Equations, Mathematics and Its Applications. Reidel, Dordrecht, the Netherlands; 1987.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-XMATHMathSciNetView ArticleGoogle Scholar
- Du Z, Ge W, Lin X: Existence of solutions for a class of third-order nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 2004,294(1):104-112. 10.1016/j.jmaa.2004.02.001MATHMathSciNetView ArticleGoogle Scholar
- Feng Y: Solution and positive solution of a semilinear third-order equation. Journal of Applied Mathematics and Computing 2009,29(1-2):153-161. 10.1007/s12190-008-0121-9MathSciNetView ArticleGoogle 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.2004.04.016MATHMathSciNetView ArticleGoogle 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.2007.03.008MATHMathSciNetView ArticleGoogle Scholar
- Henderson J, Tisdale CC: Five-point boundary value problems for third-order differential equations by solution matching. Mathematical and Computer Modelling 2005,42(1-2):133-137. 10.1016/j.mcm.2004.04.007MATHMathSciNetView ArticleGoogle 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.2006.05.003MATHMathSciNetView ArticleGoogle Scholar
- Liu Z, Debnath L, Kang SM: Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem. Computers & Mathematics with Applications 2008,55(3):356-367. 10.1016/j.camwa.2007.03.021MATHMathSciNetView ArticleGoogle Scholar
- Liu Z, Ume JS, Kang SM: Positive solutions of a singular nonlinear third order two-point boundary value problem. Journal of Mathematical Analysis and Applications 2007,326(1):589-601. 10.1016/j.jmaa.2006.03.030MATHMathSciNetView ArticleGoogle 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-XMATHMathSciNetView ArticleGoogle 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.2008.02.002MATHMathSciNetView ArticleGoogle Scholar
- Sun Y: Positive solutions of singular third-order three-point boundary value problem. Journal of Mathematical Analysis and Applications 2005,306(2):589-603. 10.1016/j.jmaa.2004.10.029MATHMathSciNetView ArticleGoogle Scholar
- Yang B: Positive solutions of a third-order three-point boundary-value problem. Electronic Journal of Differential Equations 2008,2008(99):1-10.Google Scholar
- Yao Q: Positive solutions of singular third-order three-point boundary value problems. Journal of Mathematical Analysis and Applications 2009,354(1):207-212. 10.1016/j.jmaa.2008.12.057MATHMathSciNetView ArticleGoogle Scholar
- Yao Q: Successive iteration of positive solution for a discontinuous third-order boundary value problem. Computers & Mathematics with Applications 2007,53(5):741-749. 10.1016/j.camwa.2006.12.007MATHMathSciNetView ArticleGoogle Scholar
- Yao Q, Feng Y: The existence of solution for a third-order two-point boundary value problem. Applied Mathematics Letters 2002,15(2):227-232. 10.1016/S0893-9659(01)00122-7MATHMathSciNetView ArticleGoogle Scholar
- Anderson DR, Tisdell CC: Third-order nonlocal problems with sign-changing nonlinearity on time scales. Electronic Journal of Differential Equations 2007,2007(19):1-12.MathSciNetView ArticleGoogle Scholar
- Graef JR, Yang B: Positive solutions of a third order nonlocal boundary value problem. Discrete and Continuous Dynamical Systems. Series S 2008,1(1):89-97.MATHMathSciNetGoogle Scholar
- Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(1):364-371. 10.1016/j.na.2007.12.007MATHMathSciNetView ArticleGoogle Scholar
- Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. Journal of Computational and Applied Mathematics 2008,222(2):351-363. 10.1016/j.cam.2007.11.003MATHMathSciNetView ArticleGoogle Scholar
- Kong L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2010,72(5):2628-2638. 10.1016/j.na.2009.11.010MATHMathSciNetView ArticleGoogle Scholar
- Zhang X, Feng M, Ge W: Existence result of second-order differential equations with integral boundary conditions at resonance. Journal of Mathematical Analysis and Applications 2009,353(1):311-319. 10.1016/j.jmaa.2008.11.082MATHMathSciNetView ArticleGoogle Scholar
- Zhang X, Ge W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Computers & Mathematics with Applications 2009,58(2):203-215. 10.1016/j.camwa.2009.04.002MATHMathSciNetView 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.Google Scholar
- Erbe LH, Wang H: On the existence of positive solutions of ordinary differential equations. Proceedings of the American Mathematical Society 1994,120(3):743-748. 10.1090/S0002-9939-1994-1204373-9MATHMathSciNetView ArticleGoogle Scholar
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