The author proves several fixed point results for completely continuous maps on cones in Banach spaces in the spirit of R. W. Leggett and L. R. Williams [Indiana Univ. Math. J. 28, 673–688 (1979; Zbl 0421.47033)] {the page numbers for this item are missing in the bibliography}. The details are (to put it mildly) somewhat technical. In addition, the author proves the existence of a positive solution to the boundary problem \(u''(t)+f(t,u(t))=0\) on \([0,1]\) with zero boundary conditions with a non-negative continuous right hand side \(f:[0,1]\times\mathbb{R}_+\to\mathbb{R}_+\).