Electron. J. Diff. Eqns., Vol. 2008(2008), No. 25, pp. 1-6.

Existence of solutions for some third-order boundary-value problems

Zhanbing Bai

Abstract:
In this paper concerns the third-order boundary-value problem
$$\displaylines{ u'''(t)+ f(t, u(t),u'(t), u''(t))=0, \quad 0 < t < 1, \cr r_1 u(0) - r_2 u' (0)= r_3 u(1) + r_4 u'(1)= u''(0)=0. }$$
By placing certain restrictions on the nonlinear term f, we prove the existence of at least one solution to the boundary-value problem with the use of lower and upper solution method and of Schauder fixed-point theorem. The construction of lower or upper solutions is also presented.

Submitted September 21, 2007. Published February 22, 2008.
Math Subject Classifications: 34B15.
Key Words: Third-order boundary-value problem; lower and upper solutions; fixed-point theorem.

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Zhanbing Bai
Institute of Mathematics
Shandong University of Science and Technology
Qingdao 266510, China
email: zhanbingbai@163.com

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