Electron. J. Diff. Eqns., Vol. 2007(2007), No. 45, pp. 1-10.

Positive solutions of a nonlinear higher order boundary-value problem

John R. Graef, Johnny Henderson, Bo Yang

Abstract:
The authors consider the higher order boundary-value problem
$$\displaylines{ u^{(n)}(t)= q(t)f(u(t)), \quad 0 \leq t \leq 1, \cr u^{(i-1)}(0) = u^{(n-2)}(p) = u^{(n-1)}(1)=0, \quad 1 \leq i \leq n-2, }$$
where $n\ge 4$ is an integer, and $p\in(1/2,1)$ is a constant. Sufficient conditions for the existence and nonexistence of positive solutions of this problem are obtained. The main results are illustrated with an example.

Submitted November 16, 2006. Published March 15, 2007.
Math Subject Classifications: 34B18.
Key Words: Existence and nonexistence of positive solutions; Guo-Krasnosel'skii fixed point theorem; higher order boundary value problem.

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John R. Graef
Department of Mathematics
University of Tennessee at Chattanooga
Chattanooga, TN 37403, USA
email: John-Graef@utc.edu
Johnny Henderson
Department of Mathematics
Baylor University
Waco, TX 76798-7328, USA
email: Johnny_Henderson@baylor.edu
Bo Yang
Department of Mathematics
Kennesaw State University
Kennesaw, GA 30144, USA
email: byang@kennesaw.edu

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