Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 125, pp. 1-9.

Title: Existence of solutions to third-order m-point boundary-value problems
  
Authors: Jian-Ping Sun (Lanzhou Univ. of Technology, China)
         Hai-E Zhang (Tangshan College, Tangshan, Hebei, China)

Abstract:
 This paper concerns  the  third-order m-point boundary-value problem
 $$\displaylines{
 u'''(t)+f(t,u(t),u'(t),u''(t))=0 ,\quad \hbox{a.e. } t\in (0,1), \cr
 u(0)=u'(0)=0, \quad u''(1)=\sum _{i=1}^{m-2}k_{i}u''(\xi_{i}),
 }$$
 where $f:[0,1]\times \mathbb{R}^{3}\to \mathbb{R}$ is 
 $L_p$-Caratheodory,
 $1\leq p<+\infty$, $0=\xi_0<\xi _1<\dots <\xi _{m-2}<\xi_{m-1}=1$,
 $k_i\in \mathbb{R}$ ($i=1,2,\dots ,m-2$) and  $\sum_{i=1}^{m-2}k_i\neq 1$.
 Some criteria for the existence of at least one solution are
 established by using the well-known Leray-Schauder Continuation Principle.
 
Submitted March 25, 2008. Published September 04, 2008.
Math Subject Classifications: 34B10, 34B15.
Key Words: Third-order m-point boundary-value problem; Caratheodory; 
           Leray-Schauder continuation principle.