Lan, K. Q. Multiple positive solutions of Hammerstein integral equations with singularities. (English) Zbl 0977.45001 Differ. Equ. Dyn. Syst. 8, No. 2, 175-192 (2000). Summary: We study the existence of multiple positive solutions of a Hammerstein integral equation of the form \[ z(t)=\int_G k(t,s)f \bigl(s, z(s)\bigr) dx\equiv Az(t), \] here \(G\) is a bounded closed set in \(R^n\). We allow \(k\) to have singularities in its second variable and \(f\) to have singularities in its first variable. We shall prove that \(A\) is compact under these weak conditions. We shall construct a relatively open set and prove that the fixed point index of \(A\) over the relatively open set is zero. We shall apply our results to the existence of multiple positive solutions for the equation \[ z''(t)+ f\bigl(t,z(t) \bigr)=0 \quad\text{a.e. on }[0,1], \] subject to the known separated boundary conditions. Our results generalize many known results. Cited in 1 ReviewCited in 26 Documents MSC: 45G05 Singular nonlinear integral equations 45M15 Periodic solutions of integral equations Keywords:multiple positive solutions; Hammerstein integral equation; singularities; fixed point index PDFBibTeX XMLCite \textit{K. Q. Lan}, Differ. Equ. Dyn. Syst. 8, No. 2, 175--192 (2000; Zbl 0977.45001)