Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations

Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of integral equations of the form $$u(t)=\gamma(t){\alpha}[u]+\int_{G}k(t,s)f(s,u(s))\,ds$$ where $G$ is a compact set in $\R^{n}$, ${\alpha}[u]$ is a positive functional, $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We apply our results to some nonlocal BVPs that arise in some heat flow problems to prove the existence of multiple nonzero solutions under suitable conditions. For one of the BCs we study solutions lose positivity as a parameter decreases. For a certain parameter range, not all solutions can be positive but we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.