The author considers the following Hammerstein integral equation:
\[ y(t) = \int_0^1 k(t,s) g(s) f(s,y(s))\,ds, \quad t\in[0,1],\tag{1} \]
where \( f: [0,1]\times\mathbb{R}_+ \to\mathbb{R}\) is continuous, and such that \(f>-\eta \) for some \(\eta >0; \) the kernel \(k: [0,1]\times[0, 1] \to \mathbb{R}_+ \) is continuous; \(g: [0,1]\to\mathbb{R}_+ \) is measurable. The author uses the fixed-point index theory to prove the existence of one or several positive solutions for problem (1). Furthermore, he applies the obtained results to some \(n\)th order ordinary differential equations with boundary conditions.