Electron. J. Diff. Eqns., Vol. 2003(2003), No. 120, pp. 1-19.

Solvability of a (p, n-p)-type multi-point boundary-value problem for higher-order differential equations

Yuji Liu & Weigao Ge

Abstract:
In this article, we study the differential equation
$$ (-1)^{n-p} x^{(n)}(t)=f(t,x(t),x'(t),\dots,x^{(n-1)}(t)),\;\;0<t<1, $$
subject to the multi-point boundary conditions
$$\displaylines{ x^{(i)}(0)=0 \quad \hbox{for }i=0,1,\dots,p-1,\cr x^{(i)}(1)=0 \quad \hbox{for }i=p+1,\dots,n-1,\cr \sum_{i=1}^m\alpha_ix^{(p)}(\xi_i)=0, }$$
where $1\le p\le n-1$. We establish sufficient conditions for the existence of at least one solution at resonance and another at non-resonance. The emphasis in this paper is that $f$ depends on all higher-order derivatives. Examples are given to illustrate the main results of this article.

Submitted August 19, 2003. Published December 1, 2003.
Math Subject Classifications: 34K20, 92D25.
Key Words: Solvability, resonance, non-resonance, multi-point boundary-value problem, higher order differential equation.

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Yuji Liu
Department of Mathematics
Beijing Institute of Technology
Beijing, 100081, China
Department of Applied Mathematics
Hunan Institute of Technology, Hunan, 414000, China
email: liuyuji888@sohu.com
  Weigao Ge
Department of Applied Mathematics
Beijing Institute of Technology
Beijing, 100081, China

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