Electron. J. Diff. Eqns., Vol. 2003(2003), No. 89, pp. 1-12.

Positive solutions of boundary-value problems for 2m-order differential equations

Yuji Liu & Weigao Ge

This article concerns the existence of positive solutions to the differential equation
 (-1)^m x^{(2m)}(t)=f(t,x(t),x'(t),\dots,x^{(m)}(t)),
 \quad  0 less than t less than \pi,
subject to boundary condition
$$  x^{(2i)}(0)=x^{(2i)}(\pi)=0, $$
or to the boundary condition
$$  x^{(2i)}(0)=x^{(2i+1)}(\pi)=0,  $$
for $i=0,1,\dots,m-1$. Sufficient conditions for the existence of at least one positive solution of each boundary-value problem are established. Motivated by references [7,17,21], the emphasis in this paper is that $f$ depends on all higher-order derivatives.

Submitted June 23, 2003. Published September 4, 2003.
Math Subject Classifications: 34B18, 34B15, 34B27
Key Words: Higher-order differential equation, boundary-value problem, positive solution, fixed point theorem

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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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