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Multiple positive solutions of Hammerstein integral equations with singularities. (English) Zbl 0977.45001

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.

MSC:

45G05 Singular nonlinear integral equations
45M15 Periodic solutions of integral equations
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