We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation \[ y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )\] \[\qquad \qquad \qquad \quad +\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\, ds,\] where certain asymptotic growth properties are imposed on the functions $f$, $H_1$ and $H_2$. Moreover, the functionals $\varphi _1$ and $\varphi _2$ are realizable as Stieltjes integrals with signed measures, which means that the nonlocal elements in the Hammerstein equation are possibly of a very general, sign-changing form. We focus here on the case where the kernel $(t,s)\mapsto G(t,s)$ is allowed to change sign and demonstrate the existence of at least one positive solution to the integral equation. As applications, we demonstrate that, by choosing $\gamma _1$ and $\gamma _2$ in particular ways, we obtain positive solutions to boundary value problems, both in the ODEs and elliptic PDEs setting, even when the Green's function is sign-changing, and, moreover, we are able to localize the range of admissible values of the parameter~$\lambda $. Finally, we also provide a result that for each $\lambda >0$ yields the existence of at least one positive solution.
Christopher S. Goodrich. "Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs." J. Integral Equations Applications 28 (4) 509 - 549, 2016. https://doi.org/10.1216/JIE-2016-28-4-509