# Existence Of Positive Solutions For Third-order Boundary Value Problems

Volume 4, Issue 1, pp 8 - 18 Publication Date: January 18, 2012
• 685 Views

### 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.

### 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, Bo Yang, Multiple positive solutions to a three point third order boundary value problem, Discrete Contin. Dyn. Syst. 2005 (Suppl.) 1-8.
[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, Netherlands, 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 dfferential equations, Commun Nonlinear Sci Numer Simulat 14 (2009) 3792-3797.
[13] Y. 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, H. , Y. Liu, Multiple positive solutions to third-order three-point singular semipositone boundary value problem, Proc. Indian Sci. 114 (2004) 409-422.