Electron. J. Diff. Eqns., Vol. 2008(2008), No. 125, pp. 1-9.

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

Jian-Ping Sun, Hai-E Zhang

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 4, 2008.
Math Subject Classifications: 34B10, 34B15.
Key Words: Third-order m-point boundary-value problem; Caratheodory; Leray-Schauder continuation principle.

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Jian-Ping Sun
Department of Applied Mathematics
Lanzhou University of Technology
Lanzhou, Gansu, 730050, China
email: jpsun@lut.cn
Hai-E Zhang
Department of Basic Teaching, Tangshan College
Tangshan, Hebei 063000, China
email: ninthsister@tom.com

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