The upper and lower solution method for nonlinear third-order three-point boundary value problem

Jian-Ping Sun, Lanzhou University of Technology, Lanzhou, Gansu, P. R. China
Qiu-Yan Ren, Lanzhou University of Technology, Lanzhou, Gansu, P. R. China
Ya-Hong Zhao, Lanzhou University of Technology, Lanzhou, Gansu, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 26. (2010), pp. 1-8.

Abstract: This paper is concerned with the following nonlinear third-order three-point boundary value problem

\[\left\{
\begin{array}{l}
u^{\prime \prime \prime }(t)+f\left( t,u\left( t\right) ,u^{\prime}\left(t\right) \right) =0,\, t\in \left[ 0,1\right], \\
u\left( 0\right) =u^{\prime }\left( 0\right) =0,\, u^{\prime}\left( 1\right) =\alpha u^{\prime }\left( \eta \right),\label{1.1}
\end{array}
\right.\]

where $0<\eta <1$ and $0\leq \alpha <1.$ A new maximum principle is established and some existence criteria are obtained for the above problem by using the upper and lower solution method.

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