Existence of Positive Solutions for Third-order Boundary Value Problems

Volume 4, Issue 1, pp 8--18 Publication Date: January 18, 2012
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Authors

N. Nyamoradi - Department of Mathematics, Faculty of Sciences Razi University, 67149 Kermanshah, Iran

Abstract

In this work, by employing the Guo-Krasnoselskii fixed point theorem, we study the existence of positive solutions to the third-order two-point non-homogeneous boundary value problem $\begin{cases} -u'''(t)=a(t)f(t,v(t)),\\ -v'''(t)=b(t)h(t,u(t)),\\ u(0)=u'(0)=0, \alpha u'(1)+\beta u''(1)=0,\\ v(0)=v'(0)=0, \alpha v'(1)+\beta v''(1)=0, \end{cases}$ where $\alpha\geq 0$ and $\beta\geq 0$ with $\alpha+\beta> 0$ are constant.

Keywords

• Positive solution
• Two-point boundary value problem
• Fixed point theorem.

•  34B18

References

• [1] D. R. Anderson, Green's function for a third- order generalized right focal problem, Math. Anal. Appl., 288 (2003), 1--14

• [2] D. R. Anderson, J. M. Davis, Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. Math. Anal. Appl., 267 (2002), 135--157

• [3] Z. Bai, X. Fei, Existence of triple positive solutions for a third order generalized right focal problem, Math. Inequal. Appl., 9 (2006), 437--444

• [4] A. Boucherif, N. Al-Malki, Nonlinear three-point third order boundary value problems, Appl. Math. Comput., 190 (2007), 1168--1177

• [5] J. R. Graef, B. Yang, Multiple positive solutions to a three point third order boundary value problem, Proceedings of the Fifth International Conference on Dynamical Systems and Differential Equations, 2005 (2005), 337--344

• [6] M. R. Grossinho, F. M. Minhos, Existence result for some third order separated boundary value problems, Nonlinear. Anal., 47 (2001), 2407--2418

• [7] D. Guo, V. Lakshmikantham, Nonlinear problem in Abstract Cones, Academic Press, New York (1988)

• [8] L. J. Guo, J. P. Sun, Ya H. Zhao, Existence of positive solutions for nonlinear third-order three-point boundary value problems, Nonlinear Anal., 68 (2008), 3151--3158

• [9] L. Hu, L. L. Wang, Multiple positive solutions of boundary value problems for systems of non-linear second-order differential equations, J. Math. Anal. Appl., 335 (2007), 1052--1060

• [10] M. A. Krasnoselskii, Positive solutions of operator equations, Noordhoff, Groningen (1964)

• [11] R. W. Leggett, L. R. Williams, Multiple positive fixed point of nonlinear operators on orderd Banach space, Indiana Univ. Math. J., 28 (1979), 673--688

• [12] Y. Li, Y. Guo, G. Li, Existence of positive solutions for systems of nonlinear third-order differential equations, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 3792--3797

• [13] Y. P. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl., 306 (2005), 589--603

• [14] Q. Yao, The existence and multiplicity of positive solutions of a third-order three-point boundary value problem, Acta Math. Appl. Sin., 19 (2003), 117--122

• [15] H. Yu, L. Haiyan, Y. Liu, Multiple positive solutions to third-order three-point singular semipositone boundary value problem, Proceedings Mathematical Sciences, 114 (2004), 409--422