We study the existence of positive solutions of the integral equation $$x\left(t\right)=\mu {\int }_0^{1}k(t,s)f(s,x\left(s\right),x^{\text{'}}\left(s\right),...,x^{(n-1)}\left(s\right))ds,n\ge 2$$ in both $C^{n-1}[0,1]$ and $W^{n-1,p}[0,1]$ spaces, where $p\ge 1$ and $\mu >0$. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.