Abstract: In this paper, by use of the fixed point
theorem, combining spectral theory of operator, the authors
establish the theorem on existence of positive solutions for
fourth-order boundary value problem with variable coefficient as
follows
\[\left\{ {\begin{array}{l} u^{(4)} + B(t){u}'' - A(t)u = f(t,u),0 < t < 1 ,\\ u(0) = u(1) = {u}''(0) = {u}''(1) = 0 \end{array}} \right.\]
\noindent where $A(t),B(t) \in C[0,1]$ and $f(t,u):[0,1]\times [0,\infty ) \to[0,\infty )$ is continuous.

Key words: Positive solutions, Fixed point theorem, Operator spectra