The paper contains several somewhat independent results.
One is related to nonlinear eigenvalue problems for the Hammerstein operator \[ \lambda x(t)=\int_\Omega k(t,s)f(s,x(s))\,ds. \] Under some nonnegativity condition for \(k\) on a subset \(\Omega_0\) it had previously been shown by the first and last author that a positive eigenvalue exists with a continuous eigenfunction which is nonnegative on \(\Omega_0\) and which has a given \(L_p\)-norm. This result is extended: In hypothesis and conclusion the nonnegativitity is relaxed to a mean-nonnegativity on \(\Omega_0\), i.e., on the nonnegativity of a corresponding integral over \(\Omega_0\).
In the second part, the existence of a solution of the problem \[ x(t)=\alpha[x]v(t)+\beta[x]w(t)+\lambda\int_0^1k(t,s)f(s,x(s))\,ds \] with given functions \(v,w\) and bounded linear functionals \(\alpha,\beta\) are considered in the space of continuous functions of bounded variations on \([0,1]\). To this end, a variant of Krasnoselskiĭ’s fixed point theorem and recent compactness result in the space of bounded variations by the second author are applied.
Finally, the results are applied to the second-order equation \(x''(t)=-\lambda f(t,x(t))\) with nonlocal boundary conditions of Riemann-Stieltjes type \[ x(0)=\int_0^1A(s)\,dx(s),\quad x(1)=\int_0^1B(s)\,dx(s). \]