Existence of solutions to third-order m-point boundary-value problems

This paper concerns the third-order m-point boundary-value problem $$displaylines{ u'''(t)+f(t,u(t),u'(t),u''(t))=0 ,quad hbox{a.e. } tin (0,1), cr u(0)=u'(0)=0, quad u''(1)=sum _{i=1}^{m-2}k_{i}u''(xi_{i}), }$$ where $f:[0,1] imes mathbb{R}^{3} o mathbb{R}$ is $L_p$-Caratheodory, $1leq p<+infty$, $0=xi_0 Keywords Third-order m-point boundary-value problem --- Caratheodory --- Leray-Schauder continuation principle